SAMPLED-DATA REPETITIVE CONTROL · First, a methodology is developed for designing optimal...

SAMPLED-DATA REPETITIVE CONTROL SYSTEMS

Ali Langari

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Electrical and Cornputer Engineering University of Toronto

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@ Copyright by Ali Langari, 1997

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To Tadj Khadje-Masoudi and Akbar Langari,

mg parents

Sampled-Data Repetitive Control Svstems

A thesis submitted in conformity with the requirements for the Degree of Doctor of Philosophy

Department of Electrical and Cornputer Engineering University of Toronto

@ Copyright by Ali Langari, 1997

Abstract

Repetitive control is ernployed in numerous industrial app

systems to track or reject unknown periodic signals of a known period. This thesis

takes a novel approach to the design and analysis of such systerns, by introducing a

useful performance measure, referred to as the induced power-norm. This measure

represents the maximum power-nom of the steady-state error vector in the system,

for al1 periodic inputs of unit power-norm. The approach taken here is also new in

that it is a sampled-data formulation. Hence, the intersample beha-vior is directly

taken into account.

First, a methodology is developed for designing optimal sampled-data repetitive

controllers, based on minimizing the power-norm of the steady-state error vector for

a given periodic input. It is shown that such an optimal controller always esists.

This methodology is then generalized to the case of an unknown periodic input by

minimizing the induced poiver-norm. Fast discretization is verified to be a usefuI

computat ional tool for obtaining suboptimal controllers in both met hodologies. To

demonstrate these methodologies, active suppression of fan noise present in an acous-

tic duct is discussed with promising results.

Also formulated and analyzed in this thesis is a robust tracking problem for

sampled-data repetitive control systems in the presence of structured linear peri-

odically time-varying perturbations. Specifically, we investigate whether the induced

power-norm of the closed-loop system remains below a given bound for a class of such

perturbations. The result is stated in terms of a necessary and sufficient condition

that involves Dullerud's generalized notion of structu~ed singular values for operators.

Computational aspects are addressed wi th a numerical example.

Acknowledgement s

First and foremost, 1 would like to thank my supervisor, Professor Bruce Francis.

1 am indebted to him not only for the support and guidance that he generously

offered me throughout this research, but also for the invaluable impact that he left

on my approach to engineering and scientific problems. What 1 learned frorn him is

a treasure that 1 wil1 always carry with me.

1 also wish to thank Chris Derventzis, who has never been short of a perfect friend.

1 cannot thank him enough for the wise advice and support he has given me over the

years, in particular, for standing by me through the rough times.

-4s weI1, rny marmest thanks to Benoit Boulet, Mark Lanrford and Ryan Leduc for their

pure friendship and encouragement throughout. As network system administrators,

it is thanks to them that I always had my computer system up and running.

And lastly, there are no words that will suffice to thank my parents, my brothers

Reza and Abdol, and my only sister Ladan. I would have never come so far without

their rock solid support and unconditional love.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . 1.1 Active Noise Control in an Acoustic Duct 3

. . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivation for this Thesis 4

. . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions of the Thesis 9

1.4 The Remaining Chapters . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Fundamental Building Blocks 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Periodic Signals 13

2.2 Input-Output Relationship for LTI Systems . . . . . . . . . . . . . . 16

2.3 Funct ion-Valued Periodic Signals . . . . . . . . . . . . . . . . . . . . 20

. . . . . . . . . . . . . . . . . . . . . . 2.4 Lifting and Periodic Systerns 33

3 Repetitive Control: An Overview 35

3.1 Repetitive Control and The Interna1 Mode1 Principle . . . . . . . . . 35

3.2 The Induced Power-Norm Approach . . . . . . . . . . . . . . . . . . . 40

4 Sampled-Data Repetitive Control: A Known Periodic Input 44

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The Optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . 48

. . . . . . . . . . . . . 4.3 Problem Reformulation via Fast Discretization 53

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Suboptimal Controllers 57

4.5 Convergence Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Design Example 73

4.7 Controller Optimality for Other Periodic Inputs . . . . . . . . . . . . 76

5 Sampled-Data Repetitive Control: Unknown Periodic Inputs 84

5.1 Induced Power-Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Summary of Design Procedure . . . . . . . . . . . . . . . . . . . . . . 94

5.5 Convergence Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6 Fan Noise Suppression in an .4 coustic Duct . . . . . . . . . . . . . . . 100

6 Robustness Analysis of Sampled-Data Repetitive Control Systems 114

6.1 Stability and Tracking Robustness . . . . . . . . . . . . . . . . . . . . 114

6.2 Robustness Analysis Setup . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4 Generalized Structured Singular Values . . . . . . . . . . . . . . . . . 122

6.5 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.6 Cornputational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7 Conclusions 136

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

List of Figures

1.1 Periodic signal generator . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 A continuous-time repetitive control system . . . . . . . . . . . . . . . 7

1.3 Sampled-data repetitive control setup . . . . . . . . . . . . . . . . . . 8

2.1 Maximum singular value plot of f i . . . . . . . . . . . . . . . . . . . . . 21

2.2 Continuous lifting illustrated . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Block diagram for lifting . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Sampled-data repetitive control system . . . . . . . . . . . . . . . . . . 27

2.5 Continuous lifting to get a LTI system . . . . . . . . . . . . . . . . . . 27

2.6 LTI lifted system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Fast discretized system . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Two-rate discrete-time system . . . . . . . . . . . . . . . . . . . . . . . 33

2.9 Lifting gives a single-rate system . . . . . . . . . . . . . . . . . . . . . 33

2.10 Single-rate lifted system . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 A repetitive control system; w is periodic of period T . . . . . . . . . . 36

3.2 Generator K, and stabilizing controller Ks bring tracking to the system . 36

3.3 Periodic signal generator . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Plant P and generator Kg are now grouped into G . . . . . . . . . . . 40

3.5 High-frequency tones are attenuated by the fictitious filter F . . . . . . 43

4.1 SampIed-data repetitive control system . . . . . . . . . . . . . . . . . . 45

4.2 Discretization of the sampled-data system . . . . . . . . . . . . . . . . 46

4.3 Continuous lifting to get an LTI system . . . . . . . . . . . . . . . . . 46

4.4 LTI lifted system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 S tabiiizing controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Approximating signais by fast.discretization . . . . . . . . . . . . . . . 54

4.7 Approximation by way of fast discretization . . . . . . . . . . . . . . . 54

4.8 Fast discretized system . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.10 Lifting gives a single-rate systern . . . . . . . . . . . . . . . . . . . . . 37

4.11 Single-rate lifted system . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.12 Stabilizing controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.13 Fast-discretization technique is used to analyze the sampled-data setup . 62

4.14 Lifted setup for convergence analysis . . . . . . . . . . . . . . . . . . . 71

4.15 Simplified diagram for convergence analysis . . . . . . . . . . . . . . . 72

4.16 The setup for the design example . . . . . . . . . . . . . . . . . . . . . 74

4.17 Stead-state tracking error for different number of intersample points . 75

. . . . . . . . . . . 4.18 Reference signal and the tracking error for N = 10 76

5.1 Sampled-data repetitive control system . . . . . . . . . . . . . . . . . . 85

5.2 Fictitious filter F shapes the spectrum of the exogenous input w . . . 86

5.3 Fast discretization to emulate Figure 4.1. . . . . . . . . . . . . . . . . 87

5.4 Two-rate discrete-time system . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 (a) Lifting . (b) Single-rate LTI system . . . . . . . . . . . . . . . . . . 89

5.6 S tabilizing controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.7 Computing the induced power-norm of by fast discretization . . . . 96

5.8 Acoustic duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.9 Block diagram for active noise control in duct . . . . . . . . . . . . . . 103

3.10 Sampled-data setup for the duct control system . . . . . . . . . . . . . 106

5.11 Bode plot for the system from w to z for 8 modes . . . . . . . . . . . . 108

5.12 Bode plot for the system from ui to z for 80 modes . . . . . . . . . . . 108

5.13 Input to the duct a t x, . . . . . . . . . . . . . . . . . . . . . . . . . . 110

. . . . . . . . . 5.14 One period of the steady-state noise at x, (Method 2) 112

vii

5.15 One period of the steady-state noise at xo. with controllers designed

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by Method 1 113

. . . . . . . . . . . . . . . . 6.1 A sarnpled-data repetitive control system 115

6.2 Sampled-data repetitive control system mith uncertainty . . . . 116

6.3 Continuous-time lifting of the perturbed repetitive control system . . . 119

. . . . . . . . . . . . . . . 6.4 The lifted system is single-rate discretetime 119

. . . . . . . . . . . . . 6.5 Upper LFT of Al equals the input-output map 123

. . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Lifted uncertain system 126

6.7 Plot of upper bound for p versus w , predicted by the Small

. . . Gain Theorem (dashed) and weighted norm minimization (solid) 134

viii

List of Symbols

Symbol Page Description

R 13 Real number set

@ 15 Cornplex number set

Rn 13 n-dimensional vector space over real numbers

@L 15 n-dimensional vector space over complex numbers

E 28 Rn for any n

W, 13 Set of non-negative real numbers

Z+ 13 Set of non-negative integers

IV 25 Set of natural numbers

L(&, P) 13 Linear space of al1 functions frorn I& to RP

(continuous- t ime signals)

Lz 23 Linear space of al1 square integrable functions from l& to C

P(Z+, W) 13 Linear space of al1 functions frorn Z+ to IRP

(discrete-time signals)

G 7 Generalized finite-dimensional linear time-invariant (FDLTI)

continuous-time plant

g(s) 26 Transfer matrix of G

F 7 Low-pass filter

Kc 7 Continuous-time controller

K 7 FDLTI discrete-time controller

&A) 26 Transfer matriu of K (A = z- ' )

S 7 Periodic sampler

H 7 Periodic zero-order hold, synchronized with S

h 21 Sarnpling period of S

SN 32 Periodic sampler of sampling period h / N (fast sampler)

Periodic zerworder hold synchronized with SN (fast hold)

Ratio of the sarnpling rates of SN and S

SN restricted to [O, h)

HN restricted to {O, 1,. . . , N - 1)

Usually, the Planck's constant! In this thesis, h / N

Set of all sequences from {O, 1, . . . , N - 1) to 0

Period of reference commands/disturbance signals

Continuous-time periodic input of period T

Error vector in the system, controlled output

Periodic component of z

Transient component of z

Control input

Measured output

power-nom of periodic signal w of period T' T 1/2

defined by [$ w(t)'w(t)dt]

Set of al1 continuous-time periodic signals of period T

of finite power-norm

Discrete-time periodic input of period m

power-norm of discrete-tirne periodic inputs of period m,

defined by - w (n)'w(n) [m , 1 '" Set of al1 discrete-time periodic signals of period m

A - transform of discrete-time signal w E Q,

Discrete Fourier Transform (DFT) of w in R, 2n

e- j ; in the DFT formulas, (2.6) and (2.7)

Linear space of al1 functions from [O, h) to é'

Linear space of al1 squareintegrable functions in K:

Linear space of al1 functions in K of finite p - norm

Linear space of al1 functions in K of finite oo - norm

Linear space of al1 functions from %+ to IC

( function-valued discrete-time signals)

Linear space of al1 functions from Z+ to &

Linear space of a11 square-integrable functions in e(Z+, K2)

The mapping from a signal u E 12(lC2) to its X - transform û

Set of al1 function-valued discrete-time periodic signals

of period m

Number of sampling periods h in T (Tlh is assumed to be

an integer.)

Continuous-t ime lifting

Discrete-t ime lifting

L restricted to {O, 1,. . . , N - 1)

Operator from w to z in Figure 5.1

Induced power-norm from w to z

Operator from g := L,w to g := L,z

Operator from w to C- in Figure 5.4

Xnduced power-norm from w to C-

Operator from g to m c in Figure 5.5

Induced power-norrn from g to - c Delay-by-h operator on L2

Unilateral shift on & (K2)

Open unit disc in @

Closed unit disc in @

Unit circle in @

Linear space of al1 hnctions Û from a to &, analytic on ]ID

Set of al1 bounded linear transformations on

a normed space X

Set of ail bounded linear transformations on Kz

Linear space of al1 functions g from IÜ) to L(K2), analytic on

Linear space of al1 hnctions in N, (Q L(R2 )),

Set of al1 h-periodic operators with the transfer function of

their lifted associated continous on BD

Perturbation to the plant

Continuously-lifted A, L;'A L,

Operator-valued transfer function of

Operator value of &A) at a given A

Robust stability uncertainty set

Robust tracking uncertainty set

Set of al1 structured perturbations

Set of al1 periodically tirne-varying structured perturbations

Open unit bal1 in XpTV

Structured singular value

Upper linear fractioanl transformation

Chapter 1

Introduction

Repetitive control is applied to a systern to rnake it track or reject unknown peri-

odic reference commands or disturbance signals of which only the period is known.

Thus, repetitive control systems are servomechanisms with periodic exogenous sig-

nals. However, there are a few reasons that make these systems stand out.

One is the large number and wide range of applications that they have found ever

since they were introduced to the control community by houe et al. [I'JK+81]. Among

those applications are the following: rejection of power supply interferences [PiH86]'

control of robotic manipulators which perform a repetitive task, such as painting or

picking and placing [OHN8T, TAT88, SHKTSO], accurate placement of the readlnrrite

head on a selected track of a disk-drive system, where the eccentricity of the track

causes major run-outs [CT90], restoration of periodic signals distorted by nonlin-

ear measurement devices [HHS92], control of peristaltic purnps intended for periodic

pumping of blood in dialysis machines [HS93], noncircular machining [TT88, TT941,

attitude stabilization of satellites [BM92, LHP+94], precision control of compact disc

mechanisms [DSVS95], rejection of unknown periodic load disturbances in continuous

steel casting processes [MTBR96], active suppression of vibrations (Hi1961, and active

noise attenuation in finite-length ducts [Hu95, Hu96j. In al1 these applications, there

are periodic signals to be tracked or rejected, and depending on the application, there

are certain practical issues arising from the periodicity of the signals that need to be

considered.

Chapter 1. Introduction 2

In addition, repetitive control systems have posed t heoretical challenges to re-

searchers st udying t hem, especially in continuous- time formulations [HON85, HY85,

HYON88, YH88, Yarn931. The most fundamental theoretical issue is that an arbi-

trary peï-ïodic signal may have an infinite number of harrnonics. This makes the

space of exogenous signals infinite-dimensional. Hence intuitively speaking. to have

steady-state tracking of al1 those signals, or simply put, perfect traclnng, an infinite-

dimensional controller would be needed. Dealing with an infinite-dimensional system

involves more effort and requires that many questions be re-investigated, the moût

important one being stabilizability of the system. In fact, we will see in Chapter 3

that there is an important set of plants, namely the set of al1 strictly proper plants,

that cannot be stabilized if the requirement of perfect tracking is imposed on them.

Repetitive control can also be regarded as a learning scheme. This is because, in

reaching the steady state, the error signal goes down in amplitude frorn one period to

the other, which is interpreted as if the plant is learning, as periods go by. With this

interpret ation. repetitive control resembles what is called iterative learning control,

where the error in tracking a finite-duration reference signal is reduced by means of

correcting the input in each iteration, based on the error observed in the preceding

iterations. For example, see [..\KM83, AKMT85: hIK85, BCGB]. However, there is

a subtle difference between iterative learning and what ive cal1 repetitive control. In

iterative learning, the plant is always reset to the same initial conditions after every

iteration, whereas, in repetitive control, the control process continues frorn where i t

has finished at the end of the preceding period. This also means that, in iterative

learning, the control action need not be causal. Keeping in mind the differences

between the two techniques, we see that they can be applied interchangably, thus

enabling a wider variety of applications.

In the next section, we give a brief description of the application of repetitive

control to cancelling fan noise in acoustic ducts. This is then followed by a brief

historical review of the previous work on repetitive control with a focus on the ideas

motivating this thesis. Finally, we surnmarize the main contributions of the thesis,

followed by a glance at the remaining chapters.

Cha~ter 1. Introduction 3

1.1 Active Noise Control in an Acoustic Duct

_i\coustic noise suppression in the ducts or pipework used in Heating, Ventilation and

Air Conditioning (HVAC) systems, or the exhaust systems in numerous industrial

applications is of significant importance. For instance, concert halls and hospitals

naturally require a low level of noise. Another example is industrial work environ-

ments. In these areas, it is very critical to reduce the noise from fans in air handlers

and ventilation ducts, as the prolonged exposure to such noise has been shown to re-

sult in fatigue and loss of concentration in people. -4 minimum level of acoustic noise

is also desirable in office buildings, classrooms, meeting rooms, and living rooms.

Traditionally. passive silencers are used for noise attenuation in ducts. These

silencers, which are constructed from some energy absorbent materials, line the ducts,

especially around the corners, to reduce the turbulence caused by the rapid movernent

of air. Passive silencers are used because they attenuate noise over a Mde range of

frequencies. However, they carry some disadvantages. They are ineffective in low

frequencies, t hey are bulky, t hey cause flow restrictions and, more important ly, tliey

are costly.

An alternative approach that is of current and increasing interest is active control

[NE92, HRS93, Hu95. Hu96: KN+96]. This approach is based on the concept that

reducing the noise along the duct and a t its opening to a roorn reduces the overall level

of the noise in that room. In this respect. certain locations are picked in the duct and

a secondary sound wave is injected into i t through an array of secondary sources, with

the goal of counterbalancing or minimizing the noise of the original source a t those

points. Because of their small size, active noise controllers reduce the noise without

much physical modification to the duct. At the same time, low-frequency noise can

be attenuated efficiently by means of a properly designed controller. Also, in contrast

to passive silencers, active noise controllers can cost less. For a comprehensive review

of the work on active noise control systems, see [KM96].

Now, since the fan noise, which is a main contributing factor to the overall duct

noise, is a periodic disturbance, repetitive control makes a potentially useful candi-


date for its active cancellation. In fact, Hu shows in [Hu95, Hu961 that, by using

repetitive control, periodic noise can be attenuated extensively. In this thesis, we will

also demonstrate the suitability of repetitive control as a useful technique in mini-

mizing periodic noise, but also take into account the intersample behavior that rnight

arise from the A/D and D/A operations in the digital irnplementation of such a con-

troller. In [Hu951 and [Hu961 a digital implementation of a continuous-time repetitive

controller, originally proposed in [HYON88], and a discrete-time internal mode1 are

used, respectively, to cancel out periodic disturbances of a given period. The disad-

vantage of discretizing continuous-time controllers is that high sampling rates, of a t

least a few times the bandwidth of both the plant and the disturbance, are required

to stay close to the original continuous-tirne performance. Moreover, discretctime

controllers require high sampling rates, because the intersample behavior is ignored

in their design. As we will see, the frequency response of a duct can have quite a

broadband nature. This means that, since the plant bandwidth is high to begin with,

very high sampling rates would be required for control. By way of simulation, we will

show that at a fkxed: low sarnpling rate, the sampled-data technique yields a much

better performance, thus relaxhg the requirement for high sampling rates.

1.2 Motivation for this Thesis

The work on repetitive control was pioneered by houe, Nakano, and their CO-workers

[INI81, INKf81], in application to the highly accurate control of a proton syn-

chrotron magnet power supply. This was followed by theoretical studies by Hara

et al. [HON85] and by Hara and Yamamoto [HY85]. X more detailed analysis a g

peared later in [HYON88, YH88]. These works, which incorporate the internal mode1

principle, imply that, in order to achieve steady-state tracking of periodic signals of

a given period T, a compensator which can generate al1 such signals should be in-

cluded in the closed loop. (Just as in the case of tracking steps, where an integrator

is introduced in the loop.) A rigorom derivation of such a generator is presented

in Chapter 3. However to understand some of the issues relating to these papers,


consider the delay system of Figure 1.1: where the delay equals 7'. As illustrated in

this figure, a desired periodic waveform of period T can be generated a t the output of

this system, by feeding a single cycle of that waveform into its input. Let us denote

this system by K, and let the tramfer function for it be k&). Then from the block

diagram we have that, 1

It can be seen that k,(s) has an infinite nurnber of poles on the imaginary a i s

at f j 2 k s / T , k = 0,1, . . .; hence, it is an infinite-dimensional system. Yamamoto

proved [Yam93], that, in order to have perfect tracking, it is in fact necessary to

include (1.1) in the closed loop.

Figure 1.1: Periodic signal generator.

However, it was shown in [HY85, HO.1851 that the class of systems which can

be stabilized with k,(s) in the loop is very limited, namely the class of systems

whose transfer functions have a relative degree of zero. In other words, simultaneous

tracking and stability with the proposed infinite-dimensional controller is not possible

for strict ly proper plants, which represent rnost practical systems. One may attribute

this to the unreasonable requirement that tracking should occur for arbitrary periodic

signals, including the high-frequency components that are not in the passband of the

plant.

To overcorne this limitation, houe et al. in [INKC81] and Hara et al. in [HOPI851

weighted the delay term in IÊ,(s) by a stable low-pas filter Q(s) of infinity-norm Iess


than or equal to 1. That is, &(s) is replaced by

with

In this technique, called modified repetitive cont~ol, the stabilizability constraint on

strictly proper plants is removed; however, high-frequency tones cannot be tracked

anyrnore. -4 controller synthesis algorithm based on this modification was introduced

in [HOP4851 for single-input, single output (SISO) minimum phase plants and it was

extended in [HY85] to nonminimum phase plants. The generalization of the syn-

thesis algorithm to the rnultivariable case appeared in [HYON88]. Later, using the

same modified compensator ( 1.2), Peery and 0zbaY presented a design met hodology

in [P093, PO961 with the added feature of robust performance in a standard %,

framework [DFTSP] .

Now let us see if we can explain why this low-pass filtering is so helpful. By

examining kgm0,(s), we realize that weighting the delay term by a low-pass filter

removes the unstable high-frequency poles of the controller. Therefore' the controller

adds only a finite number of unstable poles to the system, which now becomes easier

to stabilize than when there is an infinite number of unstable poles present. At the

same tirne: removing the high-frequency poles of the controller violates the interna1

model principle since the resulting model doesn7t account for the high order harrnonics

of the periodic signais. Thus, the modified-repetitive-control remedy trades tracking

ability for stability by considering a finite number of unstable poles.

We can now raise two points. First, since perfect tracking is not possible, we need

some sort of a tracking mesure, that is, a tool that can be used to compare the

performance of different controllers. Secondly, by low-pass filtering the delay term,

one is basically Iimited to finite-dimensional controllers. Noting these two points,

this thesis is motivated by the following question: why not seek from the beginning

a finite-dimensional controller and require that it tracks only periodic inputs which


are restncted to have negligi ble hi&-frequency components? To demonst rate t his, we

consider the setup of Figure 1.2, where G is a finite-dimensional linear time-invariant

(FDLTI) generalized plant, F is a low-pass filter, Kc is a continuous-time controller

to be designed, w is a periodic input of period T, and z is the tracking error that

we want to be small. We propose the design of a finite-dimensional Kc to minimize

an appropriate tracking measure. Our criterion is that the power of the steady-state

tracking error should be minimized for the worst periodic input w of unit power.

Note that, since there are high-frequency components of w that can leak through the

low-pass filter F (since the filter is not ideal), it is not be possible to have tracking

for e v e l input W.

Figure 1.2: h continuous-time repetitive control system.

Lin and Ho [LHSL] have proposed a finite-dimensional controller as well. However?

their design does not consider a tracking measure. It simply includes an interna1

mode1 of a finite number of single tones and then solves a sensitivity minimization

problem.

In addition, it is well-known that the flexibility and convenience of digital imple-

mentation of controllers are accompanied by sarnpling rate limitations and repetitive

controllers are no exceptions. Thus, one mandate of a design methodology for repeti-

tive controllers should be to take possible intersample behavior into account. For this

reason, we consider the sampled-data setup of Figure 1.3, where solid and dotted lines

represent continuous-time and discrete-time signals, respectively. The filter F is now

absorbed into the generalized plant G and S and H denote, respectively, the periodic

sampler and zero-order hold. Our goal here is to design a finite-dimensional discrete-


time controller K to make the continuous-time plant G tracklreject continuous- time

periodic signals w of a given period T. This is done by defining a performance mea-

sure, called the induced power-nom, which also considers the intersample behavior

of the steady-state tracking error. In contrast, previous works are based either on

an analog design with a digital implementation [NH86], which may require very high

sampling rates to stay close to the analog performance, or a discretization of the

analog system followed by a discrete-time design, which doesn't take the intersample

behavior into account [TTC89]. For a comprehensive review of discrete-time repeti-

tive control systems, see [Hi196].

Figure 1.3: Sampled-data repetitive control setup.

F i n a l l ~ the work in this thesis is rnotivated by the fact that, a repetitive controller

which offers a certain level of performance for the plant mode1 might not perform as

well when i t is implernented. Hence, developing robustness analysis tools is great ly

desired. This work aims at formulating a robustness analysis problem for repetitive

control systems in a sampled-data framework. While robustness issues of repetitive

control systerns with respect to plant variations have been the subject of study of

previous works [TT94, Hi1941, the major effort in those has been in the context of

digital control, using a discretized plant. Since in practice the discrete-tirne con-

troller is connected to an analog plant: a sampled-data approach to this issue makes

more sense.

C h a ~ t e r 1. Introduction 9

1.3 Contributions of the Thesis

The contributions of this thesis can be summarized as follows:

The introduction of a tracking measure for repetitive control systems [LF94a].

This measure, referred to as the induced power-norm, represents the power of

the steady-state tracking error for the worst-case periodic input of unit power

and is well-defined for periodically tirne-varying sampled-data systems, as well

as LTI continuous- and discrete-time systems. An important feature of the

induced power-nom of a sampled-data system is that it takes the intersample

error into account.

The formulation of a design problem for sampled-data repetitive control systems

wit h known periodic inputs, where minimization of the steady-st ate tracking er-

ror is used to design the controller [LF95]. The main feature of this formulation

is that it places a major emphasis on the intersample error or ripple, as is re-

ferred to in some of the previous works. In [FE86], Franklin et al. proposed a

technique for ripple-free sampled-data servomechanisms. But t hey require t hat

the continuous-time part of the system include an internal mode1 for the peri-

odic inputs. This is a strong condition and in the case that the continuous-time

plant does not have the required internal model, some analog pre-compensation

is necessary. Hara et al. [HTKSO] proposed a different technique for ripple atten-

uation in repetitive control systems, which involves generalized hold functions

and asynchronous sampling. Compared to the standard sampled-data design,

where the hold is zero-order and is synchronous with the sampler, this tech-

nique is far more cornplex. For a general treatment of ripple-free sampled-data

systems, see [Yam94].

The development of a design methodology for sampled-data repetitive con-

trol systems with unknown periodic inputs, where minimization of the induced

power-norm is used to design the controller [LF94a]. Fast discretization is veri-

fied to be a useful computational tool in both obtaining suboptimal controllers

Cha~ter 1. Introduction 10

and evaluating their tracking performance.

The formulation and analysis of a robust tracking problem for sampled-data

repetitive control systerns [LF96]. Specifically, we give a necessary and sufficient

condition that can be used to investigate whether the induced power-nom of

a sampled-data repetitive control system remains below a given bound for a

class of s tructured linear periodically time-varying perturbations. Our result

is based on Dullerud's frarnework for treatment of robust stability and robust

performance of sampled-data systems with respect to such perturbations.

The demonstration of the suitability of repetitive control as a useful technique

in minimizing periodic noise in acoustic ducts. In this example, we show that,

for a h e d sampling rate, the methodology developed in this thesis, yields a

much better performance than a pure discrete-time design.

1.4 The Remaining Chapters

The rest of this thesis is organized as follows.

Chapter 2 contains the fundamental building blocks that we will use for the anal-

ysis and design of sampled-data repetitive control systems. We begin with the main

ingredient for repetitive control, continuous- and discrete-time periodic signals. Then

we introduce the power-norm as a tool for measuring such signals. This is followed

by studying the behavior of multi-input, multi-output (MIMO) discrete-time LTI

systems which are excited by discrete-time periodic signals. Then, we define a new

measure for these systerns, called the induced power-norm, which helps identi- how

big the steady-state component of the output can be, provided we know how big the

periodic input is. We close the chapter with the notion of lifting which allows us to

convert Our sarnp fed-data system to a discre te-time LTI system.

In Chapter 3, ive present a concise overview of continuous-time repetitive control

systems. Specifically, we demonstrate in a rigorous manner that simultaneous stability

and perfect tracking of arbitrary periodic signals is not possible for strictly proper

Chapter 1- Introduction Il

plants. This then leads us to a problem formulation that sets the base for the approach

of this work to the design and analysis of repetitive control systems.

In Chapter 4 we formulate a design problem for sampled-data repetitive control

systems when the periodic input is known. Since the input is fixed, it can not be

arbitrarily high-frequency; thus, we will not lowpass the input in this formulation.

Our approach is to minimize the power-norm of the steady-state tracking error; this

wa- the intersample behavior is taken into account. We prove that an optimal

controller always exists but, due to the sampled-data nature of the problem, it is not

easily computable. Therefore, an approximation technique called fast Çiscretization

is introduced, which gives us a suboptimal controller that can be easily computed.

However, the introduction of this approximation brings up some convergence issues.

Finally, we show that the performance of a given sampled-data repetitive controller

can be computed to any degree of accuracy with fast discretization. In preparation

for the next chapter, we show that under certain mild conditions in the SIS0 case,

the resulting controller is also optimal for al1 periodic inputs that involve the same

harrnonics as the input on which the design is based.

In Chapter 5 , the aim is a t generalizing the methodology of Chapter 4 to the

case where the periodic input is unknown. To this end, the definition of the induced

power-norm from Chapters 2 and 3 is extended to sampled-data systems, however,

now taking the intersample behavior into consideration. Minimization of this measure

is then used as a criterion for designing sampled-data repetitive controllers. As in

Chapter 4, we exploit fast discretization to find a suboptimal controller and verify

that fast discretization can be used to compute the induced power-norm of a given

sampled-data repetitive control system to any desired degree of accuracy. Finally, we

apply the methodologies developed in this chapter and Chapter 4 to the suppression

of fan noise in an acoustic duct.

In Chapter 6, we formulate and analyze a robust tracking problem for sampled-

data repetitive control systems in the presence of st ructured linear periodically time-

varying perturbations. We show that necessary and sufficient conditions can be ob-

tained that guarantee tracking robustness of the systern under such perturbations.


In preparation for this result, we will see an overview of Dullerud's framework for

stability robustness of sampled-data systems.

Finally, in Chapter 7 we state our concluding remarks and some directions for

future research.

Chapter 2

Fundament al Building Blocks

This chapter contains notation, preliminaries and some of the building blocks common

to most of the coming chapters. FVe begin with the definitions of periodic signals and

systems, which constitute the main ingredient of sampled-data repetitive control.

2.1 Periodic Signals

Repetitive control is concerned wit h the steady-state response of systems to periodic

inputs. -4 well-known fact is that a periodic input to a stable LTI system produces a

periodic output (same period) when the time interval is (-m. CG). If the time interval

is [O: co), the output is the sum of a transient response and a periodic signal. However,

a sampled-data repetitive control system is tirne-varying and hence its response to

periodic inputs is not so simple. To investigate the situation in this case, this thesis

assumes that the tirne origin for signals is at zero and rigorously analyzes the response

of a sarnpled-data system to periodic inputs.

Let W, Et+, and Z+ denote the spaces of real, nonnegative real, and nonnegative

integer numbers. Denote by C(&, WP) the linear space of al1 functions from Ik& to RP,

that is, continuous-tinie signals, and by e(Z+, RP) the space of al1 functions from

Z+ to RP , discrete-tirne signals (sequences). -4 continuous-time signal w in L(& , B )

is periodic of period T (T > O) if

Chapter 2. Fundamental Building Blocks 14

-4 familiar example is cos crt in which case S is equal to 2n/a. Let

M)T = {W : w is of perioci T, hlT w(tI1w(t)dt < m

where prime denotes transpose (conjugate transpose is denoted by *). For w in WT

define its power-nom by

T 1/2

llwll,=[$~ w(t)lw(t)dt] -

Analogously, a discrete-time signal w in t(Z+,lRP) satisfying

is periodic of period m (m > O ) . Define

R, = {W : w is of period m).

For LL: E R, define its power-nom by

For ease of notation, the norm subscripts are dropped hereafter.

For periodic discrete-time signals the following results are useful. First , recall the

standard discrete Fourier transform (DFT) equations [0S89]:


where W := e-j% Let @ denote the set of cornplex numbers. Hence, the DFT

coefficients of a discrete-time periodic signal are in Q in general. Syrnbolically, ive

have

Notice that G ( k ) is of period m too.

For a general (not necessarily penodic) discrete-time signal #(n), its A-transform

(where X = 2-') is defined by

It follows frorn (2.7) and (2.8) that the A-transform of a periodic signal w ( n ) can be

written in the form

Conversely, one can easily show that any discrete-time signal w ( n ) whose A-transform

is or can be expressed in the form

for some W ( k ) E C, k = 0 , 1 , . . . ! rn - 1: is periodic of period m and

That is, W ( k ) is the DFT of w ( n ) .

The following lemma shows that the power-norm of a periodic discrete-time signal

equals the power-norm of its DFT, divided by m.


Lemma 2.1 For w(n ) periodic of penod rn

Proof Noting that

we get

m - l m - l m- 1

2.2 Input- Out put Relationship for LTI Systems

Xow, we will look at the behavior of stable multi-input, multi-output LTI discrete-

time systems when they are excited by periodic inputs. We will veri% that the

response of such systems to periodic inputs converges to a steady-state component .

Shen, having the power-norm as a tool for measuring the size of periodic signals, we

define a new measure called the induced power-nom for such systems which helps

identify how big this steady-state component can be, if we know how big the periodic

input is. But first a general definition.


Definition 2.1 Let F be a linear transformation of a normed linear space X into a

normed linear space Y . The induced n o m of F is

If 1 1 Fil < cq then F is called a bounded linear transformation.

Now, consider an LTI discrete-time system P with input v, output -$ and transfer

matriv p ( X ) . It is assumed that @(A) is defined and continuous on the closed unit disc

and is analytic and bounded in its interior. Denote the set of al1 square-summable

functions in e(Z+, P) by t,. The assumption on P then implies its stability (bound-

edness) on 112. Examples of such systems are LTI systems with rational transfer

functions with al1 the poles outside the closed unit d ix . From (2.9), for u periodic of

period m we have

From this, one can decompose 4 into steady-state and transient components, &,, $tr:

The advantage of such a decomposition is the following. Since p is continuous, each

term in (2.13) is analytic and bounded in IXI < 1. Therefore &,(A) is analytic

and bounded in IXI < 1. This implies that $,(k) + O. On the other liand, by


the discussion around (2.10), &,(A) represents a penodic signal of period m. So

eventually, @ approaches its periodic steady-state component, .Sr,,. -41~0, note that

v +-+ @,, is a Iinear transformation on 0,.

Definition 2.2 Suppose P is an &stable LTI discrete-time systern. Its induced

pomer-norm is

Remark 2.1 The induced power-norm depends on the input period, m.

The next result gives an explicit formula for J*,. Let a denote the maximum

singular value for a rnatrix [ZGD95].

Lemma 2.2 For an 12-stable LTI discrete-time system P,

m-t

From (3.9) and (2.14)


This proves that

it remains to show that J*v achieves this upper bound. In this respect, we construct

a signal u E Wr rvhich turns the above inequalities into equalities. In doing so, we

note that v is restricted to be real by the definition of R, (page 14). Let km, denote

the index at which a [fi (wk)] takes its maximum value and let 6 be a vector where

I l @ (IVk-,) 611 achieves its maximum (Euclidean) nom. Now, Wkmaz is either real

or complex. If it is real, which will happen if km, = O or km, = m/2 for even rn:

then p (Wkma-) and, thus, ü are real. In this case, signal v with DFT coefficients

will be real and it can be easily verified that v clears al1 the above inequalities to

equalities. For complex PVkrnuz : 6 (IVkrnar) is complex and, thus, is complex. In t his

case, it suffices to set

ü k = k , , - 6 , k = m - kmaz

0 : else:

where bar denotes the conjugate, and note that p ( c V ~ - ~ ~ ~ ~ ) = p ( l V k m = ) and

Ilfi ( p p - k m a z ) 511 = 8 [fi (pprm-km.. 11 IlW rn

Example: Consider the multivariable system

This system is analytic for IXI 5 1 and hence is stable. So by the analysis performed

in this section, the response of this system to a periodic input approaches a steady

state that is periodic with the same period as that of the input. Let us compute

the induced power-norm of this system for periodic inputs of period m = 3. So

Chaoter 2. Fundamental Building Blocks 20

2 T W = ë j * . By Lemma 2.2, we only have to cornpute the maximum singular value of

p ( X ) for 5 values of A, that is, for Wo, W1, W2, W3 and W" These values are listed

in Table 2.1. Thus by (2.17), J ~ , = 4.5795.

Table 2.1: Maximum singular values of p.

We have also brought in Table 2.2 the induced power-norm of the given system

for different input periods. From this table, we observe that the induced power-norm

is a function of the input period, m, confirming Remark 2.1. Xlso, we see that for

large values of rn, .J*, converges to a final value. This can be explained from the

mavimum singular value plot of p in Figure 2.1, as follows. -4s we consider larger

periods, more points from the singular value plot participate in the induced power-

norm computation, as is suggested by Lemma 2.2. Hence, for large periods, the

induced power-norm of the system approaches the maximum of this plot, 3.4915 in

t his case. This maximum is in fact the infinity norrn of 1 lpllm.

Table 2.2: Maximum singular value of p.

2.3 Function-Valued Periodic Signals

Discrete-time periodic signals that were introduced in Section 2.1 are vector-valued,

that is, they take values in RP. In contrast to these finite-dimensional signals, we


Figure 2.1: Maximum singular value plot of P .

will have discrete-time function-valued periodic signals that are infinite-dimensional.

These signals are useful in sampled-data repetitive control.

Let h > O; it will be the sampling period in later sections. Denote by K: the space of

functions from [O; h) to Cl. Elements of this space are vector-valued continuous-time

functions defined over the interval [O, h). Define

for p < oo and

for p = m. Specifically, we are interested in I C 2 , that is, the space of al1 functions

in K that are square-integrable. Then denote by t(Z+, K) the space of al1 functions

from Z+ to I C , function-valued discrete-time signals and by QZ+, iCz) the space of al1

function-valued discrete-time signals that take values in K2. A signal w in t(Z+, &)

of penod rn is then defined exactly in the same way that a vector-valued periodic

signal is defined, that is, by (2.4). The difference is that now w takes values in K2.

Denote by n,(lC2) the space of al1 function-valued periodic signals of period rn and


define the power-norm for w in R,&) by

Observing that discrete Fourier transform equation pair (2.6, 2.7) do not depend

on the space where the periodic signals take their values in, they will still be valid for

function-valued periodic signals. However, the Fourier transform takes values in Kz

nom

The next lemma gives a frequenc-domain expression for the power-norm

of function-valued periodic signals. The proof is similar to that of Lemma 2.1 and

hence is omitted.

Lemma 2.3 For v(n) E R,(IC2)

The notion of A-transform too can be extended to function-valued signals [Hil48,

SNFiO]. Let ez(lC2) denote the space of functions from Z+ to & that are square-

Denote the open unit disk, the closed unit disk, and the unit circle by D, 6. and aD,

respectively. A function 6 : ID -+ K2 is analytic if for each Xo E D the limit

??(A) - G(X,,) lim

X+Xo X - Xo

exists in ICz . Define 'H2(D, I C z ) to be the space of functions mapping D to K2 that are

analytic on D and such that

.- II~II&,~-,> -- SUP '- j I I ~ ( T ~ - " ) II:, de < m . o<r<12n 0


For v E e2(lCp), define the A-transform

From [SNF70], 6

proposition holds,

Proposition 2.1

phism.

E X2 (4 K2). Moreover from the same reference, the following

where h denotes the mapping from v to 6.

The mapping A : e2 (&) + R2 (D, ICz ) defines an isometric isomor-

This simply means that this mapping is linear, one-to-one, onto and norm preserving.

Here too, as in (2.9), the A-transform of a periodic signal IJ E R,(K2) can be

written in the form

û(A) = - w 5 - ACv-k' m k=O

-4s well, a discrete-time function-valued signal u with a A-transform that can be

espressed as

for some ( k ) E XI2 , k = 0,1, . . . ! rn - 1, is a periodic signal of period m and its DFT.

C ( k ) , equals ü(k).

The next section d l tell us how function-valued signals appear in our analysis of

sampled-data systems.

2.4 Lifting and Periodic Systems

Lifting is a mathematical technique that is used to associate LTI systems to a certain

kind of time-wrying systems, namely periodic systems. We shall see the definition

of lifting and periodic systems momentarily, but the advantage of lifting is already

clear: By lifting we get LTI systems. The lifting technique will be a very important

deïelopmental tool in this thesis.


There are two types of lifting, continuous-time and discrete-time, which will be

discussed in the sequel.

Continuous-Time Lifting and Periodic Systems

Continuous lifting L, is a linear transformation that allows converting a vector-

valued continuous-time signal into a function-valued discrete-tirne, defined by

This is depicted in Figure 2.2. Continuous lifting was introduced

n f Z+.

by Yamamoto

[Yam90]. FVe see that the lifted signal is a discrete-time signal that takes

it is a function-valued signal. The block diagram of Figure 2.3 is used

lifting.

values in iC;

to represent

O h 2h 3h .th t O 1 2 3 n

Figure 2 2: Continuous lifting illustrated.

Figure 2.3: Block diagram for lifting.


The inverse of lifting, L;', exists and is given by

L;l : e(z+, n) -t r(&, a?), = L ; ~ ~

~ ( ~ + n h ) = [ ( ) ( ) , r ~ [ O , h ) , ~ E Z + .

We are particularly interested in the operation of lifting on &(& Q), abbrevi-

ated by L2 from now on, and on WT. This allows us to have some nice properties for

lifting including the following result [BPFTS 11.

Proposition 2.2 L, and L i L are isometric isomorphisms between L2 and 12(K2) .

-4 similar result exists for periodic signals. Let Pi denote the set of positive integers.

If T is an integer multiple of h, that is, T = Mh for some hf E Pl, then for w E WT,

w := L,w E RAI(&) and - l141wr = Ilillnu(icl)-

The following result gives a restatement of this fact.

Proposition 2.3 If T = M h for some LW E N, then L, is an isornetric isornorphism

from WT to Rnr(lC2).

For any normed space X , denote by L ( X ) the space of al1 bounded linear trans-

formations on X. Then it follo~vs from Proposition 2.2 that for P E L(&): the lzfted

system? P := L,PL;l , is in L ( e 2 ( K 2 ) ) and their induced norms are equal.

Now we will see how lifting helps to associate LTI systems to periodic systems.

Let Dh denote the delay-by-h operator on L2, that is, for f E L2<

and let us denote the unilateral shift on t 2 ( K 2 ) by Li, that is, for u E e2 (&), (UV) ( k ) =

v(k - 1). One can then see that UL, = LcDh and DhL;' = L F ~ L L

An operator P : L2 + L2 is called h-periodzc if it satisfies

Cha~ter 2. Fundamental Building Blocks 26

In other words, delaying the input to an h-periodic system by h delays the output by

the same amount. For h-periodic P, the lifted system, P = L,PL;', satisfies

up = U-L,PL;~

= LcDhPL;

= LcPDhL;

= L,PL;'u

= - PU.

This shows that the lifted system is LTI. In fact the following statement holds.

Proposition 2.4 The mapping P + P is an isometric isomorphism from the space

of bounded h-periodic operators on L2 to the space of bounded LTI operators on

MX2).

By the definition of h-periodic systems, al1 LTI operators on t2 are h-periodic

for al1 h > 0. Another familiar example is the standard sampled-data system drawn

in Figure 2.4. The generalized plant G is continuous-time FDLTI mith state-space

representation

and the controller is discrete-time LTI. The sampling operator S and the zero-order

hold operator H are of pen'od h and defined as follows:


Figure 2.4: Sampled-data repetitive control system.

It is straightforward to check that the system from w to z , in Figure 2.4 too

is h-periodic and hence by (2.23) the lifted system, 9, = L,<PKL;', displayed in

Figure 2.5, is LTI. First absorb the lifting operator together with the sample and

hold devices into the plant G to arrive at Figure 2.6. The lifted plant G is

Figure 2.5: Continuous lifting to get a LTI system.

Given the state space representation for G, one can obtain a state space representation

for G [BPFTSl]. We have


where

and

here E denotes Rn for any n. Assuming that the state space representation for the

FDLTI controller K is given by

that of 9, will be

where

The notion of transfer function can be generalized to operators on & (Kz) such as

Cha~ter 2. Fundarnental Building Blocks 29

Figure 2.6: LTI lifted system.

lifted periodic systems. The function g : b + L(&) is analytic at Xo E liD if

exists in L(IC2). Let %,(ID, L(K2)) be the normed space of functions <î from B to

L(K2) that are analytic on ID, along with the norm

For elernent ij E ;Hm (Do L(IC2)), define the multiplication operator, Ôg : '&(ID. K 2 ) +

712 (Q x2 1. b~

(@a) (A) = i j ( ~ ) f i ( ~ )

for û in R2(D, K2) and A E D. The folloMng connection holds between the linear

operators on lz (K2) and functions in %,(ID, L(K2 )), [FF90].

Proposition 2.5 (i) If G : e 2 ( l i z ) -t 12(1C2) is bounded, LTI and causal, then there

exists 9 in R, (D, L(K2)) so that G = l V L j A . (ii) Every multiplication operator Gg, defined from a function j in X,(IID, L(K2)), defines a bounded operator on 3L2(D, K2).

Moreover, the R2(D, K2) + X2 (D, K2) induced norm of the operator is equal to IlijIl oc.

With this result, one can see that H,(Q L(ICÎ)) is isomorphic with the space

of bounded LTI operators on t2(lC2), Le., it is formed of the transfer functions for

such operators. These transfer functions, which are operator-valued, could even have


discontinuity on the unit circle. It is technically more convenient to work with a

subspace of R,(Q L(XÎ)), denoted by A(Q L(&)), which consists of members of

?&,(ID, L(K2)) that are continuous on the unit circle. For g E A(JiD, L(Kz )), the n o m

can be written as

Now, define

Thus, members of LA(QL(r)) have the transfer function of their lifted

A(D, L(&)). From Propositions 2.1, 2.2 and 2.5? it is immediate that

associates in

&(D,L(K, 1) is

a subspace of L(L2). &O, it is straightfomard to check that members of LA(D,L(h2))

comrnute with Dh. That is, they are h-periodic. However, there are h-periodic oper-

ators that d o n t find themselves in Ld(nL(K, ) ) , those for which the transfer function

for the associated lifted system is not continuous on B.

Discrete-Time Lifting and Periodic Systems

Discrete-tirne lifting is a mathematical construction similar to continuous-time lifting.

This type of lifting is cornmonly used (under the name "blocking") in rnultirate signal

processing [Vai93] and is due to Friedland [FriGO]. By using lifting one can convert a

multirate periodic system to a single-rate system. Define the lzfiing operotor L by

So, if v is a signal which is referred ta

referred to period h. Note though that

that of v.

subperiod h/N, its lified associate 2 can be

the dimension of the lifted signal is N times


The inverse of lifting, L-l , is defined by

L-1 : e(z+, wN) + t(z+, RP), v = L - I ~

From these definitions, we see that L ( L - ' ) rnaps periodic signals of period MN (M)

to periodic signals of period ILI ( M N ) . Moreover the power-nom is scaled up (dom)

under L ( L - l ) by the constant factor fi, since for 2 = Lu: where o is periodic of

period MN, we have

11211 = ~ ~ I I V I I - (2.39)

Figure 2.7 displays a multirate system that will be used in later sections. Here,


the generalized plant G is continuoustime FDLTI with

and the controller K is discrete-time FDLTI. Samplers S and SN are periodic of

Figure 2.7: Fast discretized system.

periods h and h / N , respectively. and synchrooized with them correspondingly are

hold devices H and Hlv. This is an example of Wperiodic systems for mhich the

output shifts by !V samples if the input does. Similar to the pure sampled-data case:

discrete lifting can be used to associate an LTI system to this periodic system. First

absorb the sampiers and holds into the plant G to get the setup in Figure 2.8. Here

Then introduce the discrete-time lifting operator and its inverse in this setup to get

the setup in Figure 2.9. The system from g to - < is single rate. Take the lifting and

its inverse into P as in Figure 2.10 where


Figure 2.8: Two-rate discrete-time system.

Figure 2.9: Lifting gives a single-rate systern.

It is straightfonvard to show that P is LTI. A state space representation for P from [CF951 is as follows. Define


Figure 2.10: Single-rate lifted system.

We have now developed enough tools for Our treatment of sampled-data repetitive

control systems.

Chapter 3

Repetitive Control: An Overview

In this chapter, we will have a quick overview of repetitive control systems in contin-

uous time. Our purpose in doing so is to study a fundamental limitation of repetitive

control systems. This will then lead us to a design rnethodology which forms the

main idea of this thesis.

3.1 Repetitive Control and The Interna1 Mode1

Principle

Consider the unity-feedback setup shown in Figure 3.1 where the plant P is SIS0

FDLTI and the input ut is an arbitrary periodic input of period S. We intend to

design the controller I< to rnake the output y follow w, that is. our goal is to get the

tracking error z to go to zero as we proceed in time.

In tracking problems like this, the interna1 mode1 principle proposed by Francis and

Wonharn [FW75] plays a key role. According to this principle, for the tracking error

2 to go to zero in the steady state, it is necessary and sufficient that the generator for

the reference command be included in the stable closed loop. By the generator for a

reference cornmand, ive mean a linear systern Kg which for some initial conditions and

no input generates that reference command at its output. For instance, the generator

for step reference commands would be just a simple integrator. Inclusion of Kg in the

Chapter 3. Repetitive Control: An Overview 36

Figure 3.1: A repetitive control systern; w is periodic of period T.

stable closed loop would then mean a system such as the one in Figure 3.2: ivhere K, is

a stabilizing controller. In other words, controller K of Figure 3.1 has two components:

a generator of the reference cornmand, h;, and a stabilizing controller, Ks.

Figure 3.2: Generator I(, and stabilizing controller I i , bring tracking to the system.

Now, letk denote the transfer function of hg by k,(s). Then in the case of step

tracking, kg (s ) = ils, that is, the pole of kg@) is the same as the pole of the Laplace

transform of the reference command - in this case: a step signal. .4nalogously, if

tracking of a çinusoid, say, sin(2at/T), were desirable, k&) would need two poleç at

f jP*/T, i.e., the poles of the Laplace transform for sin(27rtlT). In other words.

would be a candidate to be taken in the loop.

-4n arbitrary periodic input w E WT, however, may have an infinite number of


harrnonics [WB82], with a Fourier series representation given by

Intuitively, the internal mode1 principle then suggests that the loop should include

an infinite product of the form

-j2r/T 1 j 2 ~ l T k,(s) = . . - x x - x X . - .

s + j2irlT s s - j2?r/T

Based on the identity

Yamamoto shows in (Yam931 that

kg ( s ) = ~ e - ~ ~ / ~ 1

1 - e-Ts'

Since ~ e - ~ ~ / ~ is just a delay term, we can simply take

to act as the generator of the reference command.

diagram for kg(s) in Figure 3.3, we observe that

In fact, by looking a t the block

kg(s) can produce any periodic

waveform, by just being exposed to one period of that waveform a t its input.

There is however a subtle point that needs to be emphasized here. The internal

model principle in its original form assumes that the generator for the reference

commands is finite-dimensional, which is certainly not the case with (3.3). So the

legitimacy of using the internal model principle needs to be verified in the infinite-

dimensional case. This was done by Yamamoto in [Yam93]. Specifically, he proved

the need for the infinite-dimensional compensator (3.3) for perfect tracking.

Despite the applicability of the internal model principle in the infinite-dimensional

C h a ~ t e r 3. Reoetitive Control: An Overview 38

Figure 3.3: Periodic signal generator.

case, perfect tracking of periodic signals is not possible for al1 plants. This was

shown by Hara and Yamamoto in [h'Y85] where they proved that with (3.3) in the

loop, stabilization cannot be achieved for strictly proper plants, which represent most

practical systems.

Denote by R, the space of al1 scalar cornplex-valued functions of a cornplex

variable that are analytic and bounded in the open right half-plane.

Definition 3.1 Controller & is admissible if its transfer function is proper and the

ratio of two functions in 31,.

The class of admissible controllers, hence, includes al1 the FDLTI controllers. which

have real rational transfer functions, as well as controllers t hat involve delays, such

as (3.3).

Theorem 3.1 In Figure 3.2, suppose that P is strictly proper and that k,(s) is given

by (3.3). Then there is no admissible controller K, that can stabilize the closed loop.

We present a proof here which compared to the original one in [HY85] is more

concise'. First we need a definition.

Definition 3.2 Functions a, b E R, are strongly coprime if there esist x, y E R,

such that ax + by = 1.

'1 am grateful to Abie Feintuch for suggesting this proof.


The celebrated Corona Theorem frorn complex function theory can be called to check

for the strong coprimeness of a pair of functions a, b E 31- Let @+ denote the closed

right half-plane.

Lemma 3.1 [Gar81] For a pair of functions a, b E 31, to be strongly coprimeo it is

necessary and sufficient that inf [la(s)l + Ib(s)l] > 0. 3€4&

Shen, let G be a given LTI system with transfer function g(s). Unlike P in Figure 3.1:

G is not required to be finite-dimensional.

Definition 3.3 Transfer function Q is said to have a strong coprime factorization

over 3L, if there are strongly coprime a, 6 E 31, such that Q = alb.

We need one more result for the proof of Theorem 3.1.

Theorem 3.2 [Smi89] In Figure 3.4, suppose that G is stabilizable, that is, there

is an admissible controller Ks that stabilizes the closed loop. Then g has a strong

coprime factorization over N,.

Proof of Theorem 3.1 Let

This choice for g makes the setup of Figure 3.1 to be the same as the setup of interest

in Figure 3.2 mith the repetitive generator (3.3) in the loop. Then, let

be a coprirne factorization of f i , where f i and d are in Yi, [DFT92]. Since P is strictly

proper, î1 will have an additional property that

lim fi(s) = O. S 4 0 0

Also 1 - e-Ts has arbitrarily large roots on the imaginary mis. Thus

inf [ l i r ( s ) ~ + ld(s) (I - ëTS) I] = O, s e ' ç


Figure 3.4: Plant P and generator hs are now grouped into G.

So by Lemma 3.1 the numerator and the denominator in

are not strongly coprime. By Theorem 3.2 then, G is not stabilizable, which implies

that P with (3.3) in the loop is not stabilizable.

This result in fact shows that simultaneous stabilization and tracking of unknown

periodic signals is not achievable when the plant is strictly proper. There is an

intuitive explanation for this fundamental limitation as well. A strictly proper plant

attenuates the high frequency harmonies to a great extent. As a result, the controller

will try to compensate for this by creating extremely large inputs to the plant which

will consequently result in the instability of the systern.

Now that perfect tracking of arbitrary periodic signals is not possible. a compro-

mise has to be made. We wiII discuss in the next section an outline of the approach

of this thesis in making this compromise, as well as the proposed design methodology

in the next section.

The Induced Power-Norm Approach

In this section, we intend to formulate a design problem for continuous-time repetitive

control systems. Our sole reason in doing this formulation is to motivate the approach

Chapter 3. Repetitive Control: An Ovewiew 41

of this thesis; hence Ive Rrill not solve this problem. It should be emphasized that this

formulation is not restricted to the SIS0 case.

Let G be a stable muiti-input, multi-output FDLTT continuous-time plant with

input w, output z and transfer matrk j ( s ) . -4 weil-known fact is that for w E Wr,

the output z is the sum of a transient response and a periodic signal of the same

period T. In fact if we denote this periodic signal by z,, and bnng in the Fourier

series representation (3.2) for w, ive get

It is also clear that the mapping from w to z,, is linear on WT.

Definition 3.4 For a stable FDLTI plant G, the induced power -nom is

The induced power-norm, hence, is a measure of how big the steady-state output can

be when the input ranges over al1 its possibilities in WT. Since z usually represents the

tracking error, the induced power-norm is in fact a tracking measure. The following

result is obtained.

Lemma 3.2

Proof First we note that for w E WT with Fourier series representation (3.2):

Then similar to the proof of Lemma 2.2, we can show that


To show that the upperbound is actually achieved, we note that given E > O, there

eiusts no > O such that

Because of symmetry, it is also true that

Let 5 be a unit-norm vector where 6 &?Y- IÜ achieves its mavirnum (Euclidean) ( - " " ) norm. In (3.2) set

w ? n=no - w , n=-no

0 : else.

Then noting that symmetry irnplies 114 (e-j") ~ 1 1 = 5 [i (e-j") ] IIzII, 1.e get

from above

which shows that llzss 1 1 can be made arbitrarily close to the upper bound. H

New; having the induced power-norm as a tracking measure and knowing that

stabilization of the system in Figure 3.1 with the infinite-dimensional generator in

the loop fails for strictly proper plants, we would like to find out what is the best

tracking that is achievable with an FDLTI controller.

Problem 3.1 (Optimal repetitive control for unknown periodic inputs)

With respect to Figure 3.1,

minirnize K: FDLTI and stabilizing

w€Wr,llwll=l

By recalling Lemma 3.2 and letting TK denote the system from w to i in Figure 3.1,


this problem boils down to

minimize supa pK (eiT)]. nEZ

K: FDLTI and stabilizing

FVe will not need an extensive analysis on Figure 3.1 before we realize that the result

of this minimization cannot be less than 1' regardless of what the strictly proper plant

is. The reason for this is of course the direct path that connects w to z.

Noting that WT is too large a class of periodic signals for tracking, it should be

restricted. One way of imposing this restriction is to introduce a fictitious lowpass

filter F at the input as in Figure 3.5 and let WT be the class of periodic inputs to

the filter. In this way. what used to appear a t r without the filter F, will now have

its high frequency harrnonics attenuated, certainly a more realistic setup.

Figure 3.5: High-frequency tones are attenuated by the fictitious filter F.

As was mentioned earlier, we will not try to solve (3.7) in this thesis. Instead. we

design a sampled-data controller. The next two chapters deal with this.

Chapter 4

Sampled-Data Repetitive Control:

A Known Periodic Input

In this chapter we formulate the sampled-data repetirive control problem for known

periodic inputs. While the interna1 mode1 principle makes it very simple to find a

solution to this problem in continuous time, the solution in a sampled-data framework

where intersample behavior can occur, is not straightfonvard. We then show that in

the SIS0 case, the resulting controller is also optimal for al1 periodic inputs that are

constituted by the same harrnonics that are present in the periodic signal for which

the controlier was originally designed.

4.1 Problem Formulation

The setup of interest is shown in Figure 4.1. The plant G is a continuous-time FDLTI

system with a minimal realization of the form

in which the matrices & and D21 are set to zero for ivell-posedness and because of

44

Chapter 4. Sampled-Data Repetitive Control: A Known Periodic Input 45

the presence of the sarnpler. Assume also that (A, B2) is stabilizable and that (C2, A)

is detectable. Exogenous input w contains dl reference commands or disturbance

signals, which is assurned to be a known periodic signal of penod T, that is, w E WT,

and the output z contains al1 the tracking errors to be minimized. Control input and

measured outputs are denoted by u and y, respectively.


The controller K is a sampled-data controller with two assumptions regarding its

sampling period h: (1) There is an integer nurnber of sampling periods within one

period of the periodic input, i-e., T = Mh for some integer M , and (2) the rnatrix

A in the realization of G satisfies the nonpathological sampling condition that for

each eigenvalue X of A with Re X 2 O, none of the points A + j y. k # O, is an

eigenvalue of A. This latter condition guarantees stabilizability of the system. Bring

in Figure 4.2, the discretization of the systern in Figure 4.1. The class of controllers

over whicb the optimization is performed is C, the class of FDLTI causal controllers

for which the A-matrix of the discretized system in Figure 4.2 is stable. As shown

in [CF91], under mild technical conditions these controllers achieve interna1 stability

for the sampled-data setup.

Due to the presence of the sampler, the system in Figure 4.1 from w to 2 , denoted

by Q K , is not an LTI system. However, since the sampler and hold are periodic of

period h and synchronized, for K E C, < P K defines an h-periodic operator on L2. This

allows us to have sorne of the nice properties that appear in LTI signals and systems

t heory as the following theorem, originally proved in [LF94b], indicates.

Chapter 4. Sarnpled-Data Repetitive Control: A Known Periodic Input 46

Figure 4.2: Discretization of the sampled-data system.

Theorem 4.1 Suppose K E C. Then in the sampled-data setup of Figure 4.1: for

each w E WT, there exist unique signals z,, in WT and rt, in C2 such t hat z = z,, + zt,.

Moreover, the mapping from w to z,, is linear on WT.

Figure 4.3: Continuous lifting to get an LTI system.

Proof The map aK is h-periodic. Therefore, the lifted map from g t o Z; gK, shown

in Figure 4.3, is LTI. Since the controller I( is stabilizing the system, Q K is bounded

on L2. By Proposition 2.2, <PK = Lc@&', defines a bounded map on e2(&). Thus,

by Proposition 2.5, there exists 4, in U,(D, L(ICî)) so that 9, = A. Also, - K

w E Wr and T = Mh imply that g E Hence from (2.20) its A-transform is

where &(k) is the DFT of ~ ( n ) . Therefore

Chapter 4. Sampled-Data Repetitive Control: A Known Periodic Inout 47

where

and

Figure 4.4: LTI lifted systern.

( v k ) + (A) - c Ul(k) -K -K -h-

4 (wk) )(k) 1 hl- 1

-K s - r

= -9s 2 (A) + &.(A),

Noting that &, is of the form in (2.21), it is a periodic signal of period M. Also in

the expression for &,,

is in Xm(D, L(IC2)) because it is analytic and bounded on ID. Therefore, &, E

R2 (D, &). By Proposition 2.1, zt,. E e2(1C2).

Now, define z,, := L;'&, and zh := L&, Thus, z = z,, + zt,, z,, E WT and by

Proposition 2.2, qr E L2. So the first statement of the theorem is proved. Also, from

above it is seen t h a t the mapping from w to z,, is linear. rn


Corollary 4.1 The DFT coefficients of ts are

2 (k) = 4 (wk) &(k). ss -K

In this decomposition, the steady-state component, zs,, is in WT and hence it is

periodic of period T. The transient component, z,,, is in L2 and therefore

which shows that z,, will have less and less energy over each period as penods go by.

So, in this sense the output z approaches a unique steady-state periodic component,

. Now, to have good tracking, emphasis has to be put on the power of z,,. So we

pose the following problem.

Problem 4.1 (Optimal sampled-data repetitive control for a known peri-

odic input) With respect to Figure 4.1,

Remark 4.1 Since z,, is a continuous-time signal, Problem 4.1 takes the intersample

error directly into the design.

We will solve this problem in the next section.

4.2 The Optimal Controller

To find the solution, we note that

from Proposition 2.3 and that


from Lemma 2.1. These along with Corollary 4.1 allow us to write Problem 4.

The advantage to this new form is that it tells us of the dependence of the functional

to be minimized on two elements: one that doesn7t depend on the controller K, iZ(k) ,

and one that does, 6 (IVk). The next step is to use a parametrization technique to -K

represent the dependence of 4 (IVk) on K in a more explicit rnanner. With respect -h'

to the state-space representation of d, given in (2.25), al1 the controllers in C can be

represented as the input-output system of the block diagram in Figure 4.5 [Doy84];

where Q is stable FDLTI and J has the realization

in which F and H are any matrices such that Ad + B2dF and .qd + HC2 are stable.

For a reference see [CF95]. By utilizing this characterization in Figure 4.4, the input-

output transfer matrix from - to 1 will be an affine function of i ( X ) , tha t is,

where

Chapter 4. Sampled-Data Repetitive Control: A Known Periodic l n ~ u t 50

Figure 4.5: Stabilizing

. - . . -

controllers.

1

Then (4.6) becomes

M- 1 2 rninirnize (wk) &(k) + 4 (Wk) 4 (wk) $ (IVk) y ( k ) 11 . (4.10)

-2 i ( A ) stable k=O

The optimization functional in (4.10) involves several elements. Let's pause here

to remind ourselves about the spaces that these elements live in or operate on: &(k)

is the DFT of ~ ( k ) , the (continuously) lifted w(t) , and hence it is in K2. Also

from (2.27), (2.28), (2.29) and (2.30) ive see that 4 (w') is in L(K2) and that -1

( F V ~ and $ ( W k ) are bounded linear transformations from IE to K2 and from K2 -2

to El respectively. The remaining element , 4 (Wk), is a complex-valued matrix.

The solution to (4.10) is then as follows. Note that the functional in (4.10) depends

on d(X) only a t the points Wk; hence the solution is not unique.

Lemma 4.1 -4 minirnizing Q-parameter for (4.10) is

Cha~te r 4. Sampled-Data Re~etitive Control: A Known Periodic l n ~ u t 51

where

[that is: r ( n ) - DFT-L i ( k ) ] , and T(k) solves

Proof The solution in (4.10) is in general greater than or equal to

M - 1

minimize -2

0 ( 1 ) . . . ( 1 - 1) k=O

(4.14)

Specifically. they are equal if there is a stable Q(X) that satisfies

On the other hand, since the optimization variables are independent. (4.14) is

equivalent to (4.13). Hence if one interpolates the points T(k) that solve the mini-

mization problem in (4 .13) with a polynomial, which of course is statle, an optimal

Q(X) is obtained. W

Polynomial (4.11) is the unique one of degree M - 1 that interpolates the points DFT i ( k ) . [Proof: From (4.1 l ) , q (Wk) - r(r(n Of course, there are other polyno-

mials of higher degree that interpolate, as well as, conceivably, lower order rational

functions t hat interpolate. The minimizing Q-parameter obtained from this theo- a rem can be plugged back in the general representation of al1 stabilizing controllers,

Figure 4.5, to get the controller itself.

So, one only has to find the solution to M minimization problerns in (4 .13) . The

remainder of this section outlines how (4.13) might be solved nurnerically. These

minirnization problerns al1 share the general form


minimize I l a + BXcll , X

where a is in K2, c is in E and B and X are bounded operators from E to IC2 and

from E to IE, respectively. Thus X is a rnatriu. The next step is to define

which takes (4.15) to

minimize 1 la + Bxll . 2

where x is in E again. (We can always backsolve for X such that x = Xc as long

as c # O.) To solve (4.17) we note from [Lue691 that K2 can be expressed as the

orthogonal direct sum of

C := Range@) (4.18)

and

Since E is finite dimensional, so is Cl hence it is closed. Denote by n the orthopro-

jector of K2 onto E. Then

a = a l +a2 (4.20)

where

al := na and a2 := (1- n ) a .

Now equation

is guaranteed to have at least one solution. Any solution will be a solution to (4.17)

as well and the corresponding minimum will be Ila2il.

The solution, as can be seen, involves a n operator equation and hence is not easy

to find. This can be attributed to the fact that the sampled-data systern of interest


experiences both continuous- and discrete-time dpamics which makes a simple solu-

tion not directly available. In the next section, Ive take a different approach where an

approximation of the input-output pair is used to find controllers that in performance

can be as dose as desired to the optimal performance.

4.3 Problem Reformulat ion via Fast Discret kat ion

Problern 4.1 involves both continuous-time and discrete-time signals and hence is not

arnenable to a simple andysis. Instead, a problem close to the original problem is

formulated here which, a s we shall see, has a state-space solution that can be easily

coded in MATLAB.

Before we start reformulating the problem, note that there are two time periods

in Figure 4.1: T is the period of the periodic input w; h is the sampling period. By

assumption' T = Mh, where !\f is an integer.

The new formulation follows Keller and Anderson (KA921 and is based on approxi-

mating two signals: the periodic signal w for which the design of optimal sampled-data

repetitive controller is desired and the output z, which we would like to minirnize.

The way this approximation is done is the following. First, w is sampled by a sampler

SN of sampling period h / N , iV being an integer. Since SN clocks iV-times faster than

S, it is called a fast sampler. Sarnples of w which are a t a rate of $ are then held by

a fast hold HN spchronized with SN. This process is illustrated in Figure 4.6 for the

input signal, W. Obviously, if w is periodic of period T, u will be periodic of period

M N and 6 will be periodic of period T, that is,

By applying fast-discretization to the sarnpled-data setup of Figure 4.1, we arrive

a t Figure 4.7. Here, 1V is a design parameter. In this way, the periodic input to the

plant may not be exactly the one intended, but by choosing N large enough it can be

made as close as desired. Also, by choosing iV large, we will still be able to control

Chapter 4. Sampled-Data Re~etitive Control: A Known Periodic l n ~ u t 54

Figure 4.6: Approximating signals by fast-discretization.

the intersample behavior of the output. So, it is expected that the setup of Figure 4.7

will emulate the sampled-data setup of Figure 4.1 for large N .

Figure 4.7: Approximation by way of fast discretization.

We need the following result before we can state a design problem for the new

setup.

Theorem 4.2 For K stabilizing in Figure 4.7, and for w E WT. output 2 can be

uniquely decomposed into a steady-state component Z,, E WT and a transient com-

ponent Zt, € &.

Proof The system in Figure 4.7 is h-periodic. Use continuous-time lifting in the

same way as in Theorem 4.1. rn

The new design problem can then be stated as

To see the advantage of fast discretization, let's move to Figure 4.8 where al1 the

signals are discrete-time. This setup has three time periods: T and h as before, and


h/N, the period of the fast sampler. Thus T = Mh = M N ( h / N ) . Then take the

samplers and hold devices into the plant G as shown in Figure 4.9 where

Figure 4.8: Fast discretized system.


Theorem 4.3 In Figure 4.5, suppose K E C. For each periodic si of period i V N .

there exist unique signals Gs in nwN and Gr in e2(Z+) such that C = Gs + C,, . Moreover the mapping from w to Cs is linear on Clnrhi

Being similar to that of Theorem 4.1, the proof is omitted.

NOW Ctr E &(Z+) implies that it goes to zero in the limit as n goes to infinity,

that is,


This means that C approaches a steady-state behavior, namely, Cs,. The following

result allows restating (4.23) in terms of Cs,.

Corollary 4.2 In Figure 4.7, suppose K E C. Then

The minimization problem (4.23) then takes the form

But

minimize II H N C s 11. KEC

So (4.25) reduces to

This problem relates to Figure 4.9 and involves discrete-time signals only. .L\lsoo note

that the class of controllers over which the minimization is performed is the same as

before.

Figure 4.9 is the two-rate discrete-time system of Section 2.4; some of the signals

are referred to the base period h on the real-time dock while the others are referred to


the subpenod h / N of the base period. As we saw in Chapter 2, discrete-time lifting

helps associate an LTI counterpart to this system. The lifted system is shown in

Figure 4.10 with P given by (2.40). For K E C, Theorem 4.3 allows a decomposition

of C into two cornponents for w E f i M N , its steady state component Cs, and its

transient component CL,. Since - C = LC and g = Lw, for a stabilizing controller,

there is a natural decomposition for - C when g E Rhf , namely, - < = < 4 s + Gr, wvhere Ls := LC, E Rn[ and Lr := LGr E &(2+). -41~0, from (2.39) we see that

Ili_// = 0 IIG,II. Therefore, (4.26) can be stated in terrns of the lifted output L_:

minimize IILs 11. K EC

Figure 4.10: Lifting gives a single-rate system.

Wow the system from g to - C , shown in Figure 4.11 with P given by (2.41), is

not only single-rate but also LTI. MTith a similar approach to what was taken in

Section 4.2, we can find a subotimal controller that minimizes the norm in (4.27) and

hence the norm in (4.26).

4.4 Suboptimal Controllers

In this section we present the solution to (4.27). Let's denote by the mapping

from w to C in Figure 4.9. Then the lifted map &!, = LQKL-' in Figure 4.11 is LTI

and thus by Lemma 2.1 and (2.15), (4.27) can be written as


Figure 4.11:

minimize

KEC

Single-rate lifted system.

TO find the solution to this minimization problem, ive follow the same path that we

took in Section 4.2 for the system with the continuous lifting, narnely, we parametrize

al1 controllers in C. With respect to the state space representation of P from Sec-

tion 2.4, al1 the stabilizing controllers can be represented as the input-output system

of the block diagram in Figure 4.12 [Doy84], where Q is stable FDLTI and J has the

realization

j ( ~ ) = I l ] in which F and H are matrices such that Ad + B2# and .Ad + HC2 are stable.

Figure 4.12: Stabilizing controllers.

We see here that characterization of the stabilizing controllers is the same as what


we had for the true sampled-data system of Figure 4.1. With this characterization,

the input-output transfer rnatrîu in Figure 4.11 Erom g to - C is an affine function of

@(A), that is,

4 0 ) = $, (A) + 4 (4 @(A) -K -2 (4.30)

where

<L (A) = 3

Shen (4.28) becornes

M- 1 2 minirnize C II$, (IVk) )(k) + -2 6 (IVk) 4 (IVk) -3 )G ( p l i k ) a(k) 1 1 - (4.32)

The main difference with the sampled-data case is that now the participating ele-

ments in the minimization are vector- and rnatrix-valued as opposed to function- and

operator-valued.

For the solution, ive have a finite-dimensional counterpart to Lemma 4.1.

Lemma 4.2 A minimizing Q-parameter for problem (4.32) is

where

Chaoter 4. Samded-Data Re~etitive Control: A Known Periodic l n ~ u t 60

[that is, r(n) = DFT-L T(k)], and i ( k ) solves

rninimize II$, (wk) &(k) + -2 4 (IVk) F(k ) & (wk) a ( k ) 11 k = O, 1,. . . L ~ I - 1.

f ( k )

(4.35)

The proof is omitted.

So, one only haç to find the solution to iCI minimization problems in (4 .35) . These

minimization problems share the general f o m

minimize Ils + BXcll , (4.36) X

where a and c are complex vectors and B and X are complex matrices. This rnini-

mization problem always has a solution. If c = O, then any X is a solution and the

mininium is Ilall. If c # O, then the solution can be obtained in two steps as follows.

First find a solution to

minimize Ils + Bxll . 2

Then find S by obtaining any solution to

Noting that (4.38) can be formulated as a minimization problem, two least squares

minimization problerns need to be solved to find the solution to (1.36). This can

be easily coded in Say MATLAB. Since (4.37) and (4.38) might have more than one

solution, depending on other possible performance specifications. a particular solution

pair may prove to be better than others. But as long as the power-norm minimization

is the sole performance criterion, al1 solution pairs to (4.37) and (4.38) are as good

since they al1 solve (4.32) .


4.5 Convergence Issues

In Section 4.1, Problem 4.1 we stated the optimal sampled-data repetitive control

problem for a known periodic input. There, we used the power-norm of the steady-

state component of the output, z,,, to be the tracking measure which means that

11zSs11 is what is looked at mhen it cornes to evaluating the performance of a given

repetitive controller in a sampled-data setup. However, we learned in Section 4.2 that

computing this norm might not be so easy due to the sampled-data nature of the

problem. S o deal with the computational complexity, a reformulation of Problem 4.1

was introduced in (4.23), as outlined in Figure 4.7, which relies on fast discretization.

This approach uses the power-norm of the steady-state component of the approximate

signal iss as the tracking mesure. The fact that i,, is an approximation to z,, brings

up two problems. The first problem, which we cal1 the analysis problem, is whether

for a given controller K , one can compute Ilzssll to any degree of accuracy by means of

fast discretization. That is, for a given controller, whether one can get i,, arbitrarily

close to z,,, or even Ilissll to llzssll for that matter, by choosing N large enough.

-4s we will show in this section, the answer here is affirmative, under a very mild

condition on the periodic input W. The other problem is the design problem. When

fast discret ization is used, since the input-output pair is approximated. the controller

may fa11 afar from optimality and we would then be interested to know whether

controllers can be designed that in performance converge to the optimal controller.

This remains as an open problem.

For the analysis problem, we concentrate on the setup of Figure 4.13. In this

figure, outputs of the sampled-data setup in Figure 4.1 and the setup in Figure 4.7

are compared for the same periodic input w and for the same controller. From

Theorem 4.2? i,, is in Wr. The following convergence result is obtained.

Theorem 4.4 In Figure 4.13, suppose that K E C and that w in WT is such that

lim ll(I - HNSN)wII = 0. N+oo

(4.39)


Figure 4.13: Fast-discretization technique is used to analyze the sampied-data setup.

lim llzss - Zs3 11 = O. N+oo

Moreover, the convergence rate in this lirnit is a t least Z/N.

Remark 4.2 Assurnption (4.39) is merely a statement of the fact that for the fast-

discretization technique to provide us with a good approximation of the controller

performance, it should first give a good approximation of the periodic input signal.

To understand this better: consider the following periodic input that is of period

T = 1,

O t rational

1 othenvise.

It is straightforward to see that

Chapter 4- Sampled-Data Repetitive Control: A Known Periodic Input 63

for al1 t regardless of what the value of iV is. Therefore Z(t ) = O for al1 t as well.

So there will be no convergence in this case. This example is of course an extreme

case and (4.39) holds for a fairly large class of signals. For example al1 piecewise

continuous signals satisfy (4 .39) , and hence (4.39) is no restriction in practice.

Theorem 4.4 can be proved relatively easily by continuous lifting and the help of

some convergence results. First we will see what is the effect of lifting on H N S N 7

which is a main block in Figure 4.13. Let EN denote the space of al1 finite sequences

from {O, 1,. . . , N - 1) to a?. Also let h := h/iV be the sampling period of the fast

sample and hold. So ti + O as 1V + m. Define

and

The operators SA. and H , would have been, respective15 restrictions of SN and Hlv

to [O: h) and {O, 1, . . . , N - 1) if they were acting on real-valued signals. Now for

signal g E e(Z+, K), - y = L,HlvSN L;'g is also a signal in e(Z+, I C ) and for al1 n E Z+

That is, for each n

Y (4 = H N S N d n ) -

Also, it is straightfonvard to verify that for z E Rh&), - y is also in ilhf (G) and


Next we will see a series of intermediate convergence results. For the first result,

we need a special case of M. Riesz's general Convexity Theorem [SW71].

Lemma 4.3 Suppose that G is a bounded linear transformation on ICI and I C , with

induced norms hll and hl,, respectively. Shen for every O 5 p 5 oo: G : K,, + IC, 1 1-1

is bounded, with induced norrn Mp < M1'Mm

Let f be an element in K i and let F be a linear transformation on K given by

Define

and

f h := sup I l f ( t ) l l - t E W 4

Theorem 4.5 If limt,o II = O and fh is finite, then (1 - ENSN) F converges to

zero in the induced norrn for every O 5 p 5 cm.

This theorem is essentially the same as Theorem 9.3.3 in [CF95]. The difference is

that there the time interval of interest is infinite whereas here it is of a finite duration.

Proof By Lemma 4.3, it suffices to show that lw and A&, for (1 - &S,v)F tend

to zero as 1V tends to infinity. Take u in ICI or Km and set y = Fu. For t E [O, h): let

k be such that kh 5 t < ( k + 1)h. Then

Therefore

Chapter 4. Sampled-Data Repetitive Control: A Known Periodic input 65

(kf 1)h

< JI1 fh(t - I IU(T) 11dT + f h / I u ( T ) 1 d ~ - (4-42) kh

Now for v E K,

Hence ( 1 1 fh 1 1 , + h fh) , which by assumption vanishes as N -+ oo, is an upper bound

for .Mm. Therefore Mm + O as N + m.

For u E /Cl; integrate 4.42 to get

+ J k h 1144 l l d ~ d t

( k f 1)h t

= [ / h(t - r ) I I u ( ~ ) l l d r d t

If we add al1 the inequalities that we get for k = 0,1, . . . : N - 1,

But


So ( I l f h l l + A fn) is an upper bound for ML and hence for AdP as well. Thus 1 4 goes

to zero as N goes to infinity. M

We need the following standard lemma for some more convergence results.

Lemma 4.4 Suppose that A is a square matrk . Then

Proof The result is immediate from

= I

the Taylor series for eA?

Lemma 4.5 In the state space representation for 9, given in (2.32): for every IC,

norm

lim ll(I - H,vSN) Dl1 II = 0. N - w

(4.45)

Proof We use Lemma 4.4 to get

From here it is immediate that llfn 1 1 goes to zero. -41~0 it is obvious that fh is finite.

The staternent of the lernma is then immediate from Theorem 4.5.

Cha~te r 4. Sarnded-Data Re~etit ive Control: A Known Periodic l n ~ u t 67

Corollary 4.3 The rate of convergence of ll(I - &SN)Ql1ll to zero as N + O is a t

least 1 / N .

Lemma 4.6 In the state space representation for gK given in (LX), the K2-to-E

induced norm of B, satisfies

Proof For w E K2

The resiilt is obtained by bringing out the constants and using the Cauchy-Schwarz

inequality.

Lemma 4.7 In the state space representation for 9, given in (2.32),

(Eto-ICz induced norm) .

Proof From (2.35) we note that

D, C,


So it suffices to show that

and

To see (4.:

(4.50)

49): for t E [O, h), Ive let k be such that kh 5 t 5 (k + 1)h. Then

Hence

By a simple integration

For the limit in (4.50), again we let for t E [O, h): k b e such that kh 5 t $ (k + 1)h.

Then


Hence

But

which by Lemma 4.4 and Cauchy-Schwarz inequdity gives

Corollary 4.4 The rate of convergence of ll(I - &,*SN)&LII to zero as iV -t O is at

least 1/N.

Lemma 4.8 For a stabilizing controller I< in Figure 4.1, the transfer function 4, ( A )

of the lifted system satisfies

Proof From (2.32)

4 (A) = QI, + ACd (XI - &) -' al. -

Since K is stabilizing, the inverse of XI - ilcl exists for A E b and

II(I-H~sN)&,#)(I 5 l l ( I - H ~ S N ) Q l l l l


Lemmas 4-5-47 then imply (4 .53) .

CoroUary 4.5 The rate of convergence of (1 - H~s~)&,(X)II to zero as iV + O is

at least 1/N for al1 X E B. Il

Corollary 4.6 For a stabilizing controller and iV in N, HN&4 ( A ) is bounded for -K

al1 X in ID> and

Proof of Theorem 4.4 The lifted setup for convergence analysis is shown in Fig-

ure 4.14 with a state space representation for eK given by (2.32). Since the steady-

state error signal z,, - Es, is in WT, Proposition 2.3 allows us to m i t e

Also from Lemma 2.3 ive can express the power-norrn of the lifted signal L, - 3, in

terms of the power-norm of its DFT, &(k) - &,(k): that is,

So it is enough to show that

Based on Corollary 4.1 and (4.41), the block diagram in Figure 4.15 shows how

i ( k ) and g , ( k ) are related to &(k) . From this diagram -9.9


Figure 4.14: Lifted setup for convergence analysis.

This implies that

and


Figure 4.15: Simplified diagram for convergence analysis.

We start by proving (4.57). Proposition 2.3 implies that

By calling Lemma 2.3 and (4.41) we get

The assumption made in (4.39) then implies that

From here (4.57) will be immediate if nre note that

To see (4.58) and (4.59), we just need to note that


and

and cal1 Lemma 4.8 and (4.61).

The statement about the convergence rate is a direct resuit of Corollary 4.5. W

Corollary 4.7 In Figure 4.13, suppose that K is in C and that w E WT is such that

4.6 Design Exarnple

In this section we present a simple but illustrative example to demonstrate the design

technique developed in Chapter 4. Consider the feedback setup of Figure 4.16. The

plant P is a second order system with transfer function

The reference signal w is the periodic signal

of period T = 10. The sarnpling period is chosen to be h = 0.5. This h is intentionally

comparable to the time constants of the plant, TI = 1 and T* = 0.25. From a practical

viewpoint, this h would be considered too large. Thus there are h1 = 20 sampling

intervals in each period of the periodic reference signal. FVe design the controller


for zero, one, and nine intersample points, that is, for N = 1,2, and 10. The case

1V = 1 represents the conventional discrete-time design, where the emphasis is put on

the error signal only a t the sampling instants, whereas for N = 2 and N = 10, the

power of the error signal is computed and minimized for one and nine extra points in

between the sampling instants, respectively. Thus one expects to see bet ter tracking

in the latter cases.

Figure 4.16: The setup for the design example.

To do the design, we first put the setup of Figure 4.16 in the form of the setup in

Figure 4.1 and incorporate the technique developed in Section 4.4. The stead-state

tracking error is shown in Figure 4.17 for fV = 1,2, and 10. As can be seen, the

steady-state tracking error is zero a t the sampling instants for 1V = 1, but not for

N = 2 or 10. However, we observe that by taking just one intersample point' i.e.,

1V = 2, the overall error reduces dramatically. The error is further decreased if we

take an even larger nurnber of intersample points, i.e., N = 10. -4s we take more

intersample points, not rnuch improvement in the performance is observed and the

error signais almost coincide with that for iV = 10. This convergence is not surprising

as by taking more intersample points, the system in Figure 4.8 is expected to give

a better emulation of the sampled-data setup in Figure 4.1. Vie have aiso Iisted the

power-norm of the steady-state tracking error for different values of N in Table 4.1

for a quantitative cornparison. This table shows that the power of the error reduces

almost 50% by taking one intersample point. Finally; in Figure 4.18 we have plotted

the reference signal w and the output y of the system with the controller designed for

N = 10 to show that the transient response of the system is acceptable.


Table 4.1: Power-nom of the steady-state error for different values of N.

295 Tlrne

Figure 4.17: Steady-state tracking error for different number of intersample points.


L L I 1 l O 5 1 0 15 2 O 25 3 0 3 5 4 0 4 5 5 0

Tirn e

Figure 4.18: Reference signal and the tracking error for N = 10.

4.7 Controller Optimality for Ot her Periodic In-

puts

Our effort in this chapter so far has been on designing an optimal sampled-data

repetitive controller for tracking a fixed known periodic input. In this regard, we

formulated Problem 4.1 and showed in Section 4.2 that there is alivays a solution

CO this problem. That is, for a fked known u E WT in Figure 4.1: there is always

a stabilizing controller K that would rninimize the power-norm of the steady-state

component of the tracking error 2. However, K is not necessarily optimal for other

periodic inputs in WT. Considering that the goal in repetitive control is likely not

just tracking a single periodic input, we would like to know how K will perform for

other periodic inputs in WT. In this section, ive will show that in fact in the case of

SIS0 systems, K retains its optimality for some other signals in WT. As we will see,

these signais form quite a large class of periodic signals under practical assumptions.

Suppose that in our main setup in Figure 4.1, the known input is a one-dimensional

periodic signal w E WT that has a finite Fourier series representation


where îù(i) E @, n = O 1 . . . 1 For this input, we know from Section 4.2 that

there is an optimal controller that rninimizes the power-norm of the steady-state

component z,, of the tracking error z. This minimization is embodied in Problem 4.1.

Now consider ano ther input, w l , whose Fourier series representat ion involves the

same sinusoids that w does, but its Fourier coefficients are not necessarily the same.

Let

be the corresponding Fourier series. A different design problem would be to find

a controller that minimizes the power-norm of the steady-state cornponent of the

tracking error for wl. Denote this tracking error by 21. This minimization problem

can then be stated as

Problem 4.2

minirnize II q ,, 1 1 . KEC

-4s \vas stated earlier in this section, a solution to Problem 4.1 need not in general

be a solution to Problem 4.2. However, as we will see shortly, this can indeed be the

case under some mild conditions. First me need some preliminaries. LVe start by a

standard result from algebra.

Lemma 4.9 [BM65] For given integers i E Z and m E N, there exist unique integers

q ( i , rn) and rj i , rn) such that

Next recall from Proposition 2.3 that since T = Mh for some Ad E N, for peri-

odic signal w E WT, the (continuously) lifted signal g, Le., L,w, is a discrete-time,

function-valued periodic signal of period M. The following lernma shows that when


w is a pure tone, the iCI Fourier coefficients &(k), k = O, 1, . . . , iCI - 1, of zu, given

from (2.6) by

can be found very simply. Recall that &(k) are functions on the time interval [O, h).

First define the function sequence 6 : [O, h) + @, i E Z by

Lemma 4.10 For a k e d i in Z, let w E WT be given by

Then for O 5 k 5 1bI - 1,

k = r ( i , M ) Y (i) = { -

O else.

Proof The lifted signal g is a discrete-time, function-valued signal. Its nth sarnple,

n = 0: 1,2 , . . ., is a function piece defined on the tirneinterval [O,h), described by

Noting that T = M h and W = ë j b , we get


By substituting (4.72) in (4.68), we obtain

Now the above sum equals hl whenever k - i is divisible by hl. By Lemma 4.9, there

is one and only one such k in [O, M - 11. The sum equals zero otherwise. rn

Now since Lemma 4.9 states that r( i , 1l.I) is a unique integer in [O, M - 11, we

can see that of the M Fourier coefficients of -, only one is nonzero. For example, in

the case that M = 5, that is, T = 5h, and i = 1, that is, w ( t ) = e'F, the Fourier

coefficients of g are

O k = O

k = 1

O 2 5 k < 4 ,

since ~ ( 1 ' 5 ) = 1. The correspondhg coefficients wheo i = -1 or w(t) = e - j y are

This is because -1 = -1 x 5 + 4, which implies that r(-l,5) = 4. For w( t ) = 1:

these coefficients are simply given by

It is also interesting to see a case wbere w E WT is not just a single harrnonic.

Since the DFT equation (4.68) is linear, the corresponding Fourier coefficients for

such w may be obtained by summing up those of the constituting terms, which are


in turn easily cornputable frorn Lemma 4.10. For instance, let 1 = 2 in (4.64), that is,

For iM = 5, the Fourier coefficients of the exponentials with i = -1,0,1 are given

by (4.73-4.75) and those of the ones with i = -2,2 can be computed similarly. Hence

A point to observe here is that, for each k in [0,4], &(k) involves only one of the

coefficients in the Fourier senes representation (4.76) for w, namely, zù(k). Clearly,

with M = 5 this observation holds for al1 1 5 2. However, this will not be the case

when 1 is greater than 2. For example, for

the Fourier coefficients of g are

In obtaining this relation, we note from Lemma 4.9 that ~ ( 2 , s ) = r ( - 3 , 5 ) = 2. From

Lernrna 4.10, this rneans that the nonzero Fourier coefficient for pure tones e'? and

ej-, t > O, both happen at k = 2. This in turn irnplies that G(2) and zE(-3)

should both appear in the expression for c (2) . The presence of G(3) and iu ( -2 ) in

the expression for i ( 3 ) can be explained similarly.

This example brings our attention to the important fact that when the number

of harrnonics in (4.64), i.e., 21 + 1, is more than M , alzaszng can occur. Whether

the periodic input for which the repetitive controller is designed has aliasing plays a


crucial role in the controller optimality for other periodic inputs. This will be shown

in the main result of this section. But let's first summarize Our observations in the

following lemma.

Lemma 4.11 Suppose that in (4.64)

Then the Fourier coefficients of g are given by

Proof From Lemma 4.9 we see that

Now constraint (4.83) on 1 irnplies that

This inequality together with Lemma 4.10 and linearity of the DFT formula (4.68)

gives (4.81). rn

Theorem 4.6 Suppose that

and that al1 the 21 + 1 coefficients present in (4.64) are nonzero. Shen any K E C

that solves Problem 4.1 is a solution to Problem 4.2 also.

Proof In Section 4.2, we learned that by using controller parametrization, Prob-


lem 4.1 boils down to

The assumption made on 1 allows us to bring in Lemma 4.11 to reduce the optimiza-

tion functional in (4.84) to

k=hf -1

Since al1 the 6 ( k )

for each k = 0,. . .

are nonzero by assumption, then Q is optimal iff it solves

4 , pvk) ck +$ pvk) a p k ) & (wk) t,l12

1 and

for each k = ~ b f - 1 , . .. , M - 1. II

From the proof, we note the importance of the assumptions made in the statement

of this theorem. Without constraint (4.83) and with G ( k ) nonzero, the input will have

aliasing as it was illustrated by an example earlier. With aliasing present, some of the

Fourier coefficients of will involve more than one of the coefficients G ( k ) in (4.64) .

This means that the optimization functional in (4.84) will not be linear in l~E(k ) l2 as in (4.85) and hence can not be broken down into independent pieces as in (4.86)

and (4.87). Thus the optimal Q will not in general be independent of the coefficients

of the exponentials in (4.64); it will be input-dependent.

Another point to note here is that constraint (4.83) need not necessarily limit 2 to

be small. Indeed, 1 could be quite large since M could be large due to the fact that


there are typically many sampling periods in one period of the periodic input. This

theorem, thus, suggests a strategy for designing a sampled-data repetitive controller

that is optimal for a large class of penodic inputs: Pick any periodic input described

by (4.64) in which 2 is the largest integer that is less than 9 and has al1 the

exponentials with nonzero coefficients, and cal1 the design algorithm developed in

this chapter. Homever, one realizes from the proof that such a strategy stays limited

to the SIS0 case. Our intention is to extend this strategy to EvIIEYIO systems by

introducing the notion of induced power-norm for sampled-data systems.

Chapter 5

Sampled-Data Repetitive Control:

Unknown Periodic Inputs

In the preceding chapter, we developed a methodology for designing a sampled-data

repetitive controller for k n o m periodic inputs. We also showed that, under certain

mild conditions in the SIS0 case, the resulting controller is not only optimal for the

input on which the design is based, but also for al1 ot her periodic inputs that share the

same harrnonics. This chapter aims at generalizing this methodology to the case where

the periodic input is unknown. In this respect, a useful measure called the induced

power-norm is introduced. This measure represents the power of the steady-state error

vector in the system for the worst periodic input of unit power. The induced power-

norm also takes the intersample bahavior into consideration. Minirnization of this

measure is then used as a criterion for designing sarnpled-data repetitive controllers.

5.1 Induced Power-Norm

Consider the setup shown in Figure 5.1. This setup is the same as that in Figure 4.1

except that now the input w is not known; it can be any signal in WT normalized to

llwll = 1. In the preceding chapter, one result of Theorem 4.1 was that the output z

approaches a steady-state component z,, for any periodic input w E Wr. This led us

to choosing the power-nom of this steady-state component, which is in Wr itself, to

Chapter 5. Sampled-Data Repetitive Control: Un known Periodic Inputs 85

be the tracking measure. However, this rneasure directly depends on what the input

is. For an unknown periodic input we need a tracking measure that considers al1 the

possibilities that the input can assume. One such measure that treats al1 the periodic

inputs in WT equally is defined by

Definition 5.1 The induced power-non of the system in Figure 5.1 is

That is, the induced power-norm equals the power of the error vector r in the worst

case for inputs of unit power.


Now

is wil

within the class WT are al1 pure tones of arbitrarily large f requenc~ !$. 1 bound the indiiced-power n o m from below if there is a feed-through path

from the periodic input w to the tracking error z, that is, if the DI1-matrix in the

statespace representation for G is nonzero. So WT may be too large. In fact, WT

is too large for practical applications where the exogenous periodic input is rich only

in a few harmonics. In this respect, in Definition 5.1 WT perhaps should be replaced

with the set of a11 pmctically possible periodic inputs. However, since this would be

too application dependent, a better remedy is to introduce a fictitious low-pass filter

F at the input as in Figure 5.2 and let WT be the class of periodic inputs to the

filter. In this way, by choosing F one can shape WT to get the desired spectrum at

the input to the plant G.

Chapter 5. Sampled-Data Repetitive Controi: Unknown Periodic Inputs 86

Figure 5.2: Fictitious filter F shapes the spectrum of the exogenous input W.

.4bsorbing F in the plant G, we can equivalently assume that GI1 and Gzi are

lowpass throughout.


The induced power-nom provides us with a tracking measure that can be used for

designing repetitive controllers for unknown periodic inputs.

Problem 5.1 (Optimal sampled-data repetitive control for unknown

periodic inputs)

minimize Jzw (K) . KEC

Problem 5.1 involves bot h continuaus- time and discrete-tirne signals and hence is

not amenable to a simple analysis. Following our approach in Chapter 4, we formulate

a problem close to it instead which, as we shall see, has a state-space solution.

Note in Figure 5.1 that there are two time periods: T is the period of the periodic

input w ; h is the sampling penod. We still assume that T = Mh, where M is an

integer.

To state the approximate formulation, ive move to the setup of Figure 5.3. This

setup is the same as that in Figure 4.8, that is, we mode1 the input w as the output

of a fast zero-order hold HN of period h / N and the output z is looked a t through a

fast sampler SN that is synchronized with HN. The integer N is a design parameter.

Chapter 5. Sampled-Data Repetitive Control: Unknown Periodic Inputs 87

In contrast with Section 4.3 where w was fiued, here we let it span over the space of

al1 discrete-time periodic signals that would make w periodic of period T. Recalling

that T , h and h/N are related by

this would mean that we take w to be in f i h f N . In this way, we may not be able to

cover al1 WT at the output of H X , but by choosing N large enough, we expect that

the setup of Figure 5.3 will emulate the sampled-data setup of Figure 5.1.

Figure 5.3: Fast discretization to emulate Figure 4.1.

Now, let's take the samplers and holds into the plant G as we did in Section 4.3.

This is shown in Figure 5.4, where

Recalling Theorem 4.3, we see that for E RMN and a stabilizing controller K, the

output C can be uniquely decomposed into two cornponents, G, in Run. and Ct, in

&(Z+), so that C = Cs, + Ct,. Also, the rnap from w to Cs, is linear on RMN. This

allows us to define an induced power-norm for the system in Figure 5.4.

Definition 5.2 The induced power-nom of the system in Figure 5.4 is defined by



And we have a counterpart to Problem 5.1:

minimize Jc, ( K) . K EC

Note that the class of controllers over which the rninimization is performed is the

same as before.

Nexto introduce the discrete-time lifting operator and its inverse in the two-rate

system of Figure 5.4 to arrive at the LTI system of Figure 5.5a and take them into the

plant P to get Figure 5.5b, where P is given by (2.41). A state-space representation

for f is obtained through (2.42-2.48). Since - C = LC and g = Lw, there is a natural

decomposition for - C when g E RM, namely, - C = Ls + Lr where Ls E R M and

E &(Z+). kloreover, the mapping from g to LS is linear. Hence the following -tr

definition is allowed.

Definition 5.3 In Figure 55b, the induced power-norm is

The corresponding minimization problem would then be

minimize Jl,(hf) - . KEC

Chapter 5. Sampled-Data Re~etitive Control: Unknown Periodic l n ~ u t s 89

Figure 5.3: (a) Lifting. (b) Single-rate LTI systern.

Since L scales the power-norm by a factor of by the analysis on page 31, (5.5)

and (5.7) are equivalent. The advantage of (5.7) is that it is stated on a system that

is LTI. This allows us to employ frequency-domain formulas which help to obtain a

closed-form solution for an optimal controller. We see this in the next section.

In what follows it will be convenient to use the following notation:

: operator frorn w to z in Figure 5.1

oPK : operator from g := Lcw to g := L,z

: operator frorn w to in Figure 5.4

gK : operator from g to - < in Figure 3.5.

and g,, !?& are LTI with transfer functions $,(A)! &(A); the former is operator-

valued, the Iatter matrix-valued.

5.3 Design Procedure

In this section we present the solution to (5.7). From Lemma 2.2, (5.7) can be written

as the minimax problem

rninimize max [I& (wk)] , KEC O l k S M - i

Chapter 5. Sarnpled-Data Repetitive Control: Unknown Periodic Inputs 90

where we recall that W = ëjs . To find the solution to this problem, we parametrize

al1 controllers in C exactly in the same way as was done

to the state-space realization of P in (2.48), namely

Cld E l l d on* C2 o 2 l d 0 1

al1 the stabilizing controllers can be represented as the

in Section 4.3. With respect

7 (5-9)

input-output system of the

block diagram in Figure 5.6, where Q is stable FDLTI and J has the realization

in which F and H are matrices such that *dd + B2dF and & + HC2 are stable.

The advantage of this characterization is that the input-output transfer matrix in

Figure 5.5b from g to C is an affine function of q(X), that is, -

where I.&(x), &(A) and &(A) are given in (4.31). Then (5.8) becomes

The solution to this problem is as follows.

Lemma 5.1 A minimizing Q-parameter for (5.12) is


Figure 5 -6: Stabilizing controllers

where

[that is, ~ ( n ) - DFT-l i ( k ) ] , and T(k) solves

Proof The minimum in (5.12) is in general greater than or equal to

min

Specifically, they are equal if there is a stable Q(X) that satisfies

which there surely is: Formulas (5.13) and (5.14) give such 4(X).

Furthermore. since the optimization variables are independent. (5.16) is equivalent

to (5.15). W

Polynomial (5.13) is the unique one of degree M - 1 chat interpolates the points

f ( k ) . Of course, there are other polynomials of higher degree that interpolate, as

well as, conceivably, lower order rational functions that interpolate. The minimizing

Chapter 5 . Sampled-Data Re~etitive Control: Unknown Periodic l n ~ u t s 92

Q-parameter obtained from this theorem can be plugged back in the general repre-

sentation of al1 stabilizing controllers, Figure 5.6, to get the controller itself.

So, one only has to find the solution to M rninimization problems in (5.15). These

minimization problems share the general form

minirnize a ( A + BXC), (5.17) X

where A, B, C, X are complex matrices. This problem has a closed-form solution. The

nevt lemma takes care of this. Let t denote the Moore-Penrose generalized inverse.

Lemma 5.2 Let

and

C = & [O O] v; be t.he singular value decompositions of B and C. Define

and let

be natural partitions induced by the product

Chapter 5. Sam pled-Data Repetitive Control: Un known Periodic Inputs 93

and a minimizing solution is given by

- -

Proof Set Y := A + BXC and Y := UiYVc. Since matrices Us and Vc are unitary,

we have

a ( i ) = a(y). (5.27)

On the other hand,

where

From the main result in [DKW82],

min e(I) = <Y z

and a minimizing solution is

Z = -A2A;(d - &A;)+&.


So an optimal for P satisfies

This implies chat one minimizing is given by

Recalling (5.21); we arrive a t (5.25).

5.4 Summary of Design Procedure

In this section, we summarize the controller design procedure. The procedure takes

a minimal realization of G, the period T of periodic inputs to be tracked or rejected,

the sampling period It and the number of intersample points N. By assumption, Tlh

is an integer, M . The design steps are as follows:

Step 1: Starting with the realization of G, obtain a realization of P in Fig-

ure 5.5b (formulas ((2.42-2.48)).

Step 2: Parametrize ail the stabilizing controllers as in Figure 5.6. Get matrices

&(A) (i = 1 ,2 ,3 ) in (4.31). -I

Step 3: Using Lernma 5.2, find the solutions T(k) to the following M mini-

mization pro blems:

Chapter 5. Sampled-Data Repetitive Control: Unknown Periodic lnouts 95

The maximum of these minimums is the optimal induced power-nom in Fig-

ure 5.5b.

Step 4: Take the DFT-L of i ( k ) to get r (k ) . -4 minirnizing Q-parameter is

given by

Step 5: Insert Q in Figure 5.6 to arrive at the optimal controller.

This procedure has been coded in M.4TLAB.

5.5 Convergence Issues

As in the case of a fiwed known periodic input, Section 4.5, the method of fast dis-

cretization gives rise to some convergence issues in the case of unknown periodic inputs

as well. The first issue, the anulysis problem, is if for a given repetitive controller,

the induced power-norm of a sampled-data system can be computed to any desired

degree of acçuracy by fast discretization. The other issue, the design problem, is if a

sequence of controllers can be designed by fast discretization which in performance

converge to the optimal controller. In this section we will look only a t the analysis

problem.

To see the analysis problem, we focus on Figure 5.7. The following result shows

that the induced power-norm from w to < approaches that from w to z, as larger and

larger nurnber of intersample points N are taken in the fast-discretization process.

Theorem 5.1

Remark 5.1 As we saw in Section 5.2, Jcw(K) equals Jcg(K). Since the system -

from g to - C, &, is an LTI discrete-tirne system, JC, - (K) is readily cornputable from - Lemma 2.2. So in fact this theorem provides us with an algorithm for computing


Figure 5.7: Computing the induced power-norm of QK by fast discretization.

The proof of Theorem 5.1 requires some preliminaries. The first result, which is

the counterpart of Lemma 2.2, gives a frequency-domain expression for Jzw(K) .

Lemma 5.3

Proof Exactly as in Lemma 2.2, we can derive that

To show that Jzw(K) achieves this upper bound, iet km, denote the index at

which &(Wk) takes its maximum value. For given E > O, there exists w E K2 of I I - II unit norm so that

Set I

Then from above

which shows that Ills 11 can be made arbitrarily close to the upper bound.

The next lemma is the core of the proof to Theorem 5.1.

Chapter 5. Samded-Data Re~etitive Control: Unknown Periodic l n ~ u t s 97

Before çeeing the proof, we note that in the statement of this lemma, >,(A) is

operator-valued whereas 4 (A) is matku-valued. Nevert heless, t hese two transfer -K

functions are related in an interesting way. First write

Then define

This operator is basically the restriction of the discrete-time lifting operator to finite

sequences of length 1V. The inverse of L exists and is given by

It is straightforward to check that L and L-' are norm-preserving.


Now for L,HNL-', from the block diagram

Also from the same diagram

and

Hence

~ ( n ) = ~ , ~ - l ~ ( n ) .

The A-transforms of g and g are then related by

Similarly, we can derive a relation between the A-transforms of the input 2 and

the output - C of LSNLc-l, shown in the following block diagram:


For n in L+ me can write

Also

and

Therefore

And from here

<(A) = LS,Z(X)* - (3.37)

The following lemma is then a direct result of combining (5.33), (5.36) and (5.37).

The relation between the norrns of &(A) and &.(A) is even sirnpler. %y invoking

the norrn-preserving property of L and L-' in (5.38), we get

We need one more result to prove Lemma 5.4.

Lemma 5.6


Proof For 2 in EN

Therefore

Proof of Lemma 5.4 Immediate from (5.39), Lemma 5.6 and Corollary 4.6. . Proof of Theorem 5.1 Immediate from Lernmas 2.2, 5.3 and 5.4. R

5.6 Fan Noise Suppression in an Acoustic Duct

For Our design example, we will consider suppressing the periodic noise that is gen-

erated by a fan in an exhaust duct. The physical setup we study is an experimental

setup a t the University of Michigan [HAV+96]. In this first attempt in applying

sampled-data repetitive control to noise attenuation in ducts, we use the same an-

alytical rnodel of an open-ended duct derived in [HAV+96]. Besides the derivat ion,

that paper also considers the case of Hz control applied to the experimental duct.

Some other parameters, such as the speaker parameters, are borrowed from that

paper as well.

We start by a brief description of the setup considered here and a derivation of

its state-space model.


The Acoustic Duct

The experimental setup in [HS4V+96] is essentially a rectangular acoustic duct as

shown in Figure 5.8. The duct is open a t both ends, with transversal dimensions II

and l2 and length L,. It is assumed that lI/L,,12/L, < 1. With this açsumption,

the duct acts as a one-dimensional acoustic waveguide with waves traveling along its

longitudinal auis x. The disturbance in the duct is a pressure signal that is injected

into the duct through a disturbance speaker located a t x,. In an industrial setting,

this disturbance might be actually originating from, Say, the fan in an air conditioning

system, in which case it would have a significant periodic component. The disturbance

speaker is excited by a voltage W. A measuring microphone is installed on the duct

wall at some point x,. The output y of this microphone is then used to compute a

control signal u that excites an actuatzng speaker placed at x,. The acoustic signal z

a t a point x, is the signal to be minimized. In an industrial setting, x, would be a t the

point where duct enters the workspace. One could take the feedback measurement

exactly a t the same point where minimization of the acoustic signal is desired, in

which case x,, and x, will be equal.

Now let's see a more detailed description. First denote by p(x, t ) the acous-

tic pressure at point x along the x-axis at tirne t. In fact, the signals that we

are interested in are p(x,, t ) and ~ ( x , ~ , t). Also let's respectively denote by a,(t)

and a,(t) the baffle accelerations of the actuating and the disturbance speakers.

The linearized partial differential equation t hat governs the ideal dynamics of p ( x , t )

along with the corresponding boundary conditions at the two open ends is then given

by [H.4iT+96]

with c being the velocity of the acoustic wave (343 m/s in air under standard condi-

tions of temperature and altitude) and po being the equilibrium density (1.21 kg/m3

for air under standard conditions). This setup along with a digital irnplernentation

Chapter 5. Sarnpled-Data Repetitive Control: Unknown Periodic Inputs 102

Figure 5.8: -4coustic duct.

of the controller is summarized in the block diagram of Figure 5.9.

Next ive shall find a state-space description for the system from w and u to z and

y in this diagram. This systern is denoted by G. First for i E IV, define the spatial

frequency

ki := i n / L , (5.41)

and the time frequency

Wi := cki.

Let C: : [O, L,] + R be the function defined by

V,(X) = c \ /2 /LD sin kix

and let


Measurement Microphone

Digital Controller

Figure 5.9: Block diagram for active noise control in duct.

& i o let qi : lE& + R be a function that satisfies

4 ( t ) + ~ T q i ( t ) = bpaD ( t) + bfa, ( t )

q (0 ) = qi(0) = o.

Then the solution to the differential equation (5.40) can be derived by the method of

separation of variables as 00

Corresponding to the infinite set of second-order differential equations in (5.46),

t here is an infinite-dimensional st ate space representation. However, we limit our-

selves here to finite dimensions. Preserving only n modes, define the state vector

X P := [ql QI . . . qn qn]'- We also use a pressure type microphone, whose output is

proportional to the input pressure. Assuming a normalized gain of 1 for the micro-

phone, we get


where

1 A, = block-diagonal

Notice that damping coefficients CI,. . . , G have been introduced here for practical

considerations. In practice, these coefficients are obtained via identification algo-

rithms.

The dynamics of the control and the disturbance speakers should be included in

the state space description as well. The transfer functions from w to a , and u to a,

are assumed to be given by

where hfA and KD are the speaker gains, w, and wD are the speaker natural frequen-

cies: and C and CD are the darnping ratios. Corresponding t o these transfer functions

are state-space descriptions given by


where

By combining the state vectors x,, x, and xD into one vector x, Le., x = [x', x: x',]',

and combining the state-space descriptions for the speakers and the duct, we arrive

at the state-space description for G:

Here

Chapter 5. Sampled-Data Repetitive Control: Unknown Periodic l n ~ u t s 106

With this state-space description, we arrive at Figure 5.10, where the sample

and hold model, respectively, the A/D and DIA of Figure 5.9, and K is a FDLTI

discrete-time controller to be designed. Denote the system from w to z by < P K o as

before.

Figure 5.10: Sarnpled-data setup for the duct control system.

Simulation

We are interested in suppressing the noise that is generated by a 3600rpm fan, installed

at one end of a duct of length L, = 1.0195m. Hence, the periodic noise is of period

S = l/ f = 60Hz. In Our setup in Figure 5.8, we model this noise by a periodic signal

that is injected into the duct through a properly positioned speaker. In doing so,

we require al1 the modes that are preserved in the finite-dimensional mode1 of the

duct to be controllable from the disturbance input. This specification makes sure


that al1 the modes of interest are excited by the disturbance. As a result of this

requirement, x, = O is immediately excluded from our choices since by examining

(5.43), (5.44) and (5.52), we realize that this choice makes B,, = O, which translates

into having no disturbance a t all. Also, the disturbance speaker cannot be atl Say,

x, = LD/8 = 0.502m, since from (5.44) it makes b,D = O. A possible location not

too far in the duct for the disturbance speaker is x, = 0.114m, for which at least

eight modes are controllable. For the disturbance speaker, we borrow the parameters

K, = 1: w, = 2r x 67rad/sec and CD = 74% from [HAVC96].

Our goal is to minimize the effect of the noise at one point near the end of the

duct. The point picked for this simulation is x, = 3.66111. We choose a CO-located

arrangement for the actuating speaker and measuring microphone, that is, x, = x,.

In selecting this common location, we require that the system frorn the control input u

to the rneasured output y in (5.48-5.49) be controllable and observable. By examining

these equations, we note that this requirement is met if we set x, and x , to be the

same as x,.

For simplicity, we consider only 8 modes in the duct model. The natiiral frequen-

cies for these modes, which are obtained from (5.42), are brought in Table 5.1. The

damping ratios are randomly generated in a range that covers those of the identified

model in [H.4Vf 96). The Bode plot for the system from the disturbance w to the

output z is shown in Figure 5.11. From the magnitude plot, the open-loop induced

power-norm equals 2.34. We also consider the Bode plot of the system from w to t

with many modes included in the duct model, to verify that the duct is of a broadband

frequency response. This plot is shown in Figure 5.12 for 80 modes.

Table 5.1: The damping ratios and naturai frequencies of the modes preserved in the finite-dimensional model for the duct.

For designing the controller, we consider two methods, the induced power-norm


-100 1 1 1 o2 10' 1 o4

Frequency (radlsec)

Figure 5.11: Bode plot for the system from w to z for 8 modes.

720

O - u m Y>

2 -720 n

-1440

- 1 o o l 1 1 1

I cl2 1 O= 1 04 1 os Frequency (radlsec)

-

-

- 1

1 op 1 0' 1 o4 Frequency ( rad lsec )

Figure 5.12: Bode plot for the systern from w to z for 80 modes.

2 8 8 0

O ' u

2 - 2 8 8 0 C O

-5760

-8640

-

- - -

1 1

1 o2 1 O' 1 o4 Frequency (radlsec)

Chapter 5. Sampted-Data Repetitive Control: Unknown Periodic Inputs 109

minimization of this chapter and the power-norm minimization of Chapter 4. In order

to study the intersample behavior of the system, we choose to sample only hl = 6

times during each period of the periodic input for the control operation. This gives

h = T/hf = l/( f M) = 1/360sec, or a sampling rate of 360Hz for the controller.

From the Bode plot of the mode1 with 8 modes, this sampling rate compares with the

bandwidth of the plant and hence is considered to be slow.

Method 1: Induced Power-Norm Minimization By following the design pro-

cedure of Section 5.4 for 1V = 1,2,4 and 10, we obtain four different controllers,

which minimize the induced power-norm of the sampled-data system, fast-discretized

respectively a t N = 1,2,4 and 10 times the sampling rate of the controller. The case

N = 1 represents the conventional discrete-time design, where the emphasis is put

on the error signal only a t the sampling instants, whereas for N = 2, N = 4, and

iV = 10, the error signal is computed and minimized for one, three and nine extra

points in between the sampling instants, respectively. Thus one expects to see better

tracking in the latter cases.

The resulting induced power-norms with these controllers in place are listed in

Table 5.2. This table also includes the open-loop, or uncompensated induced power-

norm, which is obtained from the magnitude bode plot in Figure 5.11. From this

table, we see that the conventional discretetime design, i.e., iV = 1, yields a very

poor performance, considering that the corresponding induced power-norm is even

larger than that of the uncornpensated system. However, we observe that by taking

just one intersample point. i.e., !V = 2, the induced power-norm reduces drastically

of the induced power-norm of

Method 2: Power-Norm

the uncompensated system.

Minimization In this method, we pick a periodic

disturbance w of fundamental frequency 60Hz and design the controller so that the

power-norm of the corresponding steady-state output noise a t x,, z,,, is rninimized.

The disturbance w that is used in this example is shown in Figure 5.13. This signal

Cha~ter 5. Sarn~led-Data Re~etit ive Control: Unknown Periodic ln~u ts 110

Table 5.2: Induced power-nom in Figure 5.10 for different controllers obtained by blethod 1.

N 1

was generated in MATL-4B by a random choice of the Fourier coefficients in

These coefficients are given in Table 5.3. We calculate the power-norm of w to be

Jzw (K) 3.1451

-2 , 0.234 0.236 0.238 0.24 0.242 0.244 0.246 0.248

Tirne (sec)

- .

Open-loop induced power-nom

2.3401

Figure 5.13: Input to the duct at x,.

As in Method 1, we use fast discretization and design the controller for zero, one,

three, and nine intersample points, that is, for N = 1,2,4, and 10. One period of z,,

resulting from the disturbance w for these controllers is shown in Figure 5.14. Also,

Chapter 5. Sampled-Data Re~etitive Control: Unknown Periodic l n ~ u t s 111

Table 5.3: Fourier coefficients for the disturbance w in Method 2.

plotted on the same gaph is the uncompensated steady-state noise in response to the

same disturbance input. This noise is denoted by zUmm,. -4s can be seen, with N = 1:

the steady-state noise is forced to zero at the sampling instants. However, this is done

a t the cost of a large intersample behavior. We also observe that, by taking just one

intersample point into the design, Le., N = 2, the noise is attenuated quite a bit. As

ive expect, a lower level of noise is achieved by taking more and more points into the

design. The power-norm of z,, for different values of N dong with the that of zU,,,,

are listed in Table 5.4. This table shows that by taking the intersample behavior into

account, one can achieve considerable reduction in the output noise, over that of the

discrete-time design. We also note that with the sampled-data design, the noise can

be reduced to approximately 25% of its uncompensated level, or equivalently, a noise

reduction of 12dB. This level of attenuation is significant from a practical point of

view [H.Nf 961.

Table 3.4: The power-norm of z,, for different values of N .

Cornparison and Discussion Now let's compare Methods 1 and 2. Table 5.5 lists

the induced power-norm of the sampled-data setup for the two sets of controllers that

were designed by these methods. We see that the discrete-time design, i.e., N = 1,

yields the same induced power-norm in both of these methods; thus they are as poor

in attenuating the noise at the output. From this table, we also observe that for


Figure 5.14: One period of the steady-state noise at x, (Method 2).

other values of IV? the induced power-noms of the sampled-data system with the

controlIers designed by Method 2 are very close to those achieved by controllers that

are designed by hlethod 1. This is very interesting because in Method 2: the design is

not based on minimization of the induced power-nom. In other words, ewn though

it is one arbitrary disturbance that we base the design on, the resulting controllers

performs very close to the controllers that are based on al1 of WT.

Table 5.5: Comparison between the induced power-norms for the controllers obtained by Methods 1 and 2.

!V

Lire have also applied the disturbance w of Method 2 to the sampled-data setup,

under controllers that are designed by Method 1. One period of the resulting steady-

J*, (KI blethod 1

Jzw ( K ) Method 2

7

Open-loop induced power-norm

Chapter 5. Sarn~led-Data Repetitive Control: Unknown Periodic Inputs 113

state noise, z,,, is plotted in Figure 5.15, along with one penod of zunrmp- The

power-noms of these signds are listed in Table 5.6. Our first observation is that

the steady-state output noise for N = 1 is the sarne as in Method 2. This can

be attributed to the fact the resulting controllers for N = 1 are both discrete-time

intemal models of the periodic signals of interest. Finally, we observe that for al1

other values of N, the steady-state output noise is at a higher level compared to

Figure 5.14. This can be explained by noting that we are now considering controllers

that are not designed for the specific input considered here; they are worst-case based.

0.234 0.236 0.238 0.24 0.242 0.244 0.246 0.248 Tirne (sec)

Figure 5.lS: One penod of the steady-state noise at x,, with controllers designed by Method 1.

Table 5.6: Cornparison between the llzssll for the controllers obtained by Methods 1 and 2.

1 1 ~uncmnp 1 1 IIzssII Method 1

I l h Il Method 2

Chapter 6

Robustness Analysis of

Sampled-Data Repetitive Control

Systems

This chapter formulates and analyzes a robust tracking problem for sampled-data

repetitive control systems in the presence of structured linear periodically time-

wrying perturbations. Two pertinent issues are addressed. One is stability robust-

ness. t hat is, if stability will be retained under such perturbations. The other is robust

tracking, that is, with stability robustness guaranteed, whether the tracking criterion

d l be met for al1 variations of the plant under the given class of perturbations.

We take the tracking measure to be the induced power-norm of Chapter 5 and the

tracking criterion is that only induced power-norms which are less than a prespecified

bound are acceptable. Dullerudys generalized notion of structured singular value is

used to answer these questions in the form of a necessary and sufficient condition.

6.1 Stability and Tracking Robustness

Consider the sampled-data setup of Figure 6.1. The generalized plant G is FDLTI

and w E WT. The controller K is a discrete-time FDLTI repetitive controller that

might have been obtained through different design methodologies, for example, by

C h a ~ t e r 6. Robustness Analvsis of Samded-Data Re~etit ive Control Svstems 115

doing a digital implementation of an analog design [NH86], by discretizing the plant

and performing a discrete-time design [TTC89, HS931, or by a sampled-data approach

as we have seen in Chapters 4 and 5. in any case, the assumption is that K stabilizes

the system in Figure 6.1 and offers a good level of tracking. This, however, might

not be the case when K is implemented in the actual system of which G represents

only a model. By implementing K on the real system, we might end up having poor

tracking or even instability. What we plan to do in this chapter is to create some tools

that could assist us in expanding our knowledge about the performance of K when

the plant is subject to uncertainty. This issue has also been the subject of study of

previous works, [TT94, Hi1941, but the major effort has been in the context of digital

control where the plant has been discretized. Since in practice, the discrete-time

controller is connected to an analog plant, a sampled-data approach to this issue is

more precise.

Figure 6.1: A sampled-data repetitive control system.

To state our robust tracking problem, we move to the setup of Figure 6.2, where

with abuse of notation, G is still allowed to denote the FDLTI generalized plant,

which now accommodates perturbations. Inputs pl and pz are introduced for stability

definition purposes, as we will see later. Perturbation A represents the uncertainty

in the plant. Typically, A rnight be LTI. More generally, to discuss what type of

A should be allowed, let us first note that obviously, for output z to track periodic

signals, it has to get to steady state first. Not al1 perturbations allow this. Therefore,

to have a well-defined problem, one has to constrain perturbation A so that a steady

state c m be guaranteed for the output z. One class of such perturbations that makes

Chapter 6. Robustness Analwis of Sampled-Data Repetitive Control Svstems 116

this possible is the class of stable linear h-periodic perturbations, as we will see in a

later section. Recailing that d l LTI perturbations are h-periodic, we see that they

constitute another candidate class of perturbations. For simplicity, we formulate

and solve the problem only for h-periodic perturbations. However, this would be too

conservative if we don't need more than LTI perturbations to describe the uncertainty

in the plant [Du196]. Also, we would like to work with structured perturbations

since the solution to the full-block perturbation case will then be immediate. As a

by-product, necessary and sufficient conditions for robust stability and tracking are

obtained. (The conditions would be only sufficient for LTI perturbations.)

Figure 6.2: Sampled-data repetitive control system with uncertainty.

We assume that I< stabilizes and meets the tracking criterion for the nominal

system (A = O). The tracking measure is the induced power-nom of the preceding

chapter. As we saw, this measure is the power of the steady-state tracking error z for

the worst periodic input w of unit power. As well, this measure features the largest

amount of power that can build up in the error in between the sampling points, that

is, in the intersample behavior. The main question is robust stability, that is, if the

system will retain its stability under such perturbations. Also it is important to have

a way of testing if under perturbations of the given class the tracking criterion is met,

that is, if the induced power-norm will stay less than a prespecified bound.

To answer these questons, we need some preliminaries.

Chapter 6. Robustness Analysis of Sarnpled-Data Repetitive Control Systems 117

6.2 Robustness Analysis Setup

We begin with the plant G. In dealing with G, we group ql and w together as one

input and q2 and z together as one output. Corresponding to this, we have the

partition

We dso assume that G has a minimal realization of the form

The matrices Dll? D12, and 022 are set to zero for well-posedness and because w is

assumed to be Iowpass filtered.

The controller h' is assumed to have minimal realization

and is assumed to internally stabilize the nominal plant, that is, for no input present,

i.e., w = pl = pz = 0, and for any initial conditions x,(O) and x,[O] of plant G and

controller K , x , ( t ) + O and x,[k] + O as t + m, k + M.

With regard to the uncertainty, we assume that perturbation 4 represents a stable

linear structured h-periodic system. -4s Ive have seen in Section 2.4, by applying

continuous lifting to an h-periodic system, one obtains an LTI associated sp t em that

is easier to work with. To be precise, we consider only those h-periodic systems that

are in LA(D,L(K2)). From (2.38) we recall that the operator-valued transfer functions

for the lifted associates of members of La(o,L(x2)) are continuous on fi. This continuity

brings some convenience to the analysis. To have A structured, take A in

Cha~ter 6. Robustness Analvsis of Samded-Data Repetitive Control Svstems 118

-4s can be seen, this imposes a structure on 4 with respect to its Euclidean dimensions.

In summary, we take h to be in

Note that by setting I = 1 in A',, we find the solution for the full-block perturbation,

Le., unstructured case. Corresponding to each A E Xpry there is A = Li1ALC with

where & is in A(D, L(li2)). So for h E Xmv, A is a mapping from D to

Finally, the input w is in WT and we retain the assumption that there is an integer

number of sampling periods in T, i.e, T = Mh.


In this section we give a precise formulation of the robust tracking analysis problem.

First, let's see if the problem is well-defined, that is, if for A E XpTV in Figure 6.2

and a controller K that stabilizes the perturbed system, the output z gets to a steady

state for w E WT. In Figure 6.2, set pi = pz = O and bring in L, and L;' to get the

system in Figure 6.3. Then absorb them together with the sample and hold in the

plant G and the perturbation A to arrive a t Figure 6.4.

Chapter 6. Robustness Analysis of Sarnpled-Data Repetitive Control Systerns 119

Figure 6.3: Continuous-time lifting of the perturbed repetitive control system.

Figure 6.4: The lifted system is single-rate discrete-time.

Chapter 6. Robustnes Analysis of Sampled-Data Repetitive Control Systems 120

The lifted plant G is

As we saw in Section 2.4, G is an LTI system with a state space representation

given by (2.25-2.30). The lifted perturbation A, L,AL;', is LTI too since 4 is

h-periodic. So the lifted system is LTI. Now denote the mapping w H z by a4. Since the controller K stabilizes the perturbed system, is bounded on L2. By

Proposition 2.2, the LTI lifted map from - to 4, 9, := L,04 L;', defines a bounded

map on t2(1C2). Thus, by Proposition 2.5, there exists 6 E '&(ID, L(&)) so that -A

eA = A%- A. kloreover, from the state space representation for G we see that ij -A -

is continuous. So is since it is in XpTV and hence in A(D, L(K2)). Therefore 4 is -A

continuous on b. Also, w E WT and T = Mh imply that g E RA,&) and that its

A- transform is given by (2.20). Therefore

where

and

Noting that $, is of the form in (2.21), it is a periodic signal of period M with DFT

coefficients

2 (k) = 4 (wk) @(k). 4 . 9 -A (6.10)

-41~0 in the expression for &,,

Chaoter 6. Robustness Analvsis of Sarn~led-Data Re~etit ive Control Svstems 121

is in R,(D,L(IC2)) because it is analytic and bounded on D. Therefore, &, E

3t2 (ID, K2). By Proposition 2.1, &r E e2(K2) .

Now, define z,, := L;'&, and q, = L;l&,. Thus, z = zss + zt,, z,, E WT and by

Proposition 2.2, Gr E L2. This shows for h E XmV and a controller that stabilizes

the perturbed system, the output z approaches a steady-state component, z,,. Also,

from above, it is seen that the mapping from w to z,, is linear.

The following theorem is a summary of the above analysis.

Theorem 6.1 In Figure 6.2, suppose that 4 E LA(QL(Kz)) and that K stabilizes the

perturbed system. Then for pl = O, p* = O and ui E W , there exist unique signals z,,

in WT and zt, in L2 such that r = z,, + Zr,-. Moreover, the mapping from w to z,, is

Iinear on W.

Linearity of the map from w to z,, allows the following definition for the stable

perturbed system.

Definition 6.1 The induced power-norm of the perturbed system in Figure 6.2 is

For each perturbation A, me have a measure of tracking given

wer-norm. To state Our robust tracking analysis problern, ive ha

by the induced

,ve t o consider a

class of perturbations. Let a pre&~ U denote the open unit ball. We have the following

defini tions.

Definition 6.2 The system in Figure 6.2 is robustly stable with respect to UXmv

if the map

Chapter 6. Robustness Analysis of Sampled-Data Repetitive Control Systems 122

is bounded on L2 for each A E UXPTV.

Definition 6.3 The system in Figure 6.2 has robust tracking of periodic signals with

respect to UXmV if it is robustly stable with respect to UXmv and JJ4) 5 1 for

each 4 E UX-.

The problem of robustness analysis can then be stated as:

Problem 6.1 Find necessary and sufficient conditions that guarantee robust tracking

of the setup in Figure 6.2 with respect to UXprv.

For the solution, we need the notion of structured singular value of an opertator.

6.4 Generalized Structured Singular Values

The notion of structured singular value was first defined for matrices in [Doy82] for the

analysis of LTI feedback systems with structured uncertainties. For sarnpled-data sys-

tems, we need a generalization of this notion for operators on Hilbert spaces [DG93].

Let E and F be Hilbert spaces. Then the space of al1 bounded linear transformations

frorn E to 3 is denoted by L(E, 7). If E and F are identical, we write simply L(ê),

as before.

Definition 6.4 Suppose P E L(E, 3) and A is a subspace of L ( 3 , E). The struc-

tured singular value pA(P) of P with respect to the perturbation set A is the inpieme

of the largest 6 such that

I - P A is invertible in L ( 3 ) VA E A, 11411 5 6.

Thus (P)-l is a stability margin for structured perturbations.

The Main Loop Theorem from [ZGDSS] may be generalized as rvell. First, sorne

definitions. Let E l , E2, FI, and F2 represent four arbitrary Hilbert spaces and let @

denote the direct sum. Suppose that

Chapter 6. Robustness Analvsis of Sampled-Data Repetitive Control Svstems 123

are given. Assuming that I - PllAl is invertible, define the upper lznear-fractional

transformation (upper LFT) by

The rationale for this terminology cornes from Figure 6.5, in which Al appears in the

upper loop, and the corresponding input-output map is given by Fu(P7 Al) .

Figure 6.5: Upper LFT of A l equals the input-output map.

The following result is obtained.

Theorem 6.2 (Main Loop Theorem)

Proof (-) Let A E A, llAll < 1. We must show I - P A is invertible. The

inequality IIAII < 1 implies llAlll < 1 and IlA211 < 1, since IlAl1 = max(ll&ll, 11A211).

Cha pter 6. Robustness Analvsis of Sampled-Data Repetitive Control Svstems 124

Since I - Pll Al is invertible, one c m write

I - P A = 1 1 - pi141 -p12A2 1

v

invertible

[I-;'"' ; ] . 1

in& ible

invertible

So I - P A is invertible.

(*) pa,(Pll) 5 1 since othemise for some Al E Ai with llAill i 1. 1 - PliAl

is not invertible which means that for

that satisfies II4II = I l A l I I < 1, I - P A is non-invertible. This contradicts ( P ) 5 1 .

Since I - Pl lAl is invertible for llAlll < 1, from (6.15) one can see that for each

Al E UAI , I - Fu(P, Al)A2 is invertible for al1 A2 E A2 with Ilh211 < 1. Therefore


This section closes with a lemma which is the Small Gain Theorern in a generd

setting. This lemma is used in the proof of the main result of the chapter.

Lemma 6.1 Suppose that P E L(E) and that A = L(E). Shen pa(P) = ilPII.

Proof If 11 Al1 < A, then 1 1 PA11 < 1 which implies the invertibility of I - PA. This

implies that ,ua(P) 4 IIPII.

On the other hand, for a given c > O, there is unit vector u E & so that llPull >

llPll - C. Set u := PvIIIPvII and define the perturbation 4 E L(E) by

One observes t hat

(1 - PA)u = U - - llPvll < U ' U '

which shows that I - P A is not invertible. -41so

< sup 1

I < u , w > I I!wll=L IIPII - É

which gives pa(P) < IlPl[.

6.5 The Main Result

For the solution of Problem 6.1, absorb the controller K into G to get the system in

Figure 6.6, where P is LTI and has state space representation given in (2.32-2.36).


With respect to its inputs and outputs, P c m be partitioned as in

The following lemma then gives a frequency-domain expression for J z w ( A )

lemrna can be proved exactly as in Lemma 5.3.

This

We aIso need to bring in the following two results from [Du196]. The first result states

that one can check for the robust stability of the setup in Figure 6.2 by performing a

test on the LTI lifted system in Figure 6.6.

Figure 6.6: Lifted uncertain system.

Lemma 6.3 The system in Figure 6.2 is robustly stable with respect to UXprv iff

for A E UXprv, ( I -Pi& is invertible in L(&(lc,)).

The second result allows this test to be carried out in the frequency domain.

Theorem 6.3 The system in Figure 6.2 is robustly stable with respect to UXmv iff

Chapter 6. Robustness Analvsis of Samded-Data Re~etitive Control Svstems 127

So the maximum value that p*,,, fiI, (A)] takes an the unit cirde is the determining [- factor for robust stability.

Now, define the robust tracking uncertainty set A,, by

Art := {A,~ E L(K2) : A,, = diag(& Ai), where E Amy, At E ~ ( l i ~ ) ) .

(6.18)

We note from (6.6) that in fact

The solution to Problem 6.1 can then be stated as foIlows.

Theorem 6.4 The system in Figure 6.2 has robust tracking with respect to UXmv

iff it has robust stablization with respect to UXmv and

Proof (e) By Definition 6.3, one only has to show that

Condition (6.20) implies that for O 5 k 5 l\.I - 1,

By Theorem 6.2 then

or by Lemrna 6.1

Chapter 6 . Robustness Analvsis of Samded-Data Re~etitive Control Svstems 128

So by Lernma 6.2,

( 1 : Robust tracking with respect to UXmv implies robust stabilization with

respect to UXprv. So one only has to show that condition (6.20) holds. Since in

Figure 6.2

by Lernma 6.2

So for each k,O 5 k 5 LW - 1, and for each A E UXmv,

This means that for each k, O 5 k 5 M - 1 and for al1 A E UAprv

since if for E UAav this is not true, then for &(A) = XI.V-'& one obtains a

contradiction. Therefore by Lemma 6.1

for O 5 k 5 Al - 1. On the other hand, frorn Theorem 6.3, robust stabilization with

respect to UXPTv implies that

Apply Theorem 6.2 to get (6.20).

Cha pter 6. Robustness Analysis of Sarnpled-Data Repetitive Control Systems 129

6.6 Computational Aspects

Theorems 6.3 and 6.4, state the necessary and sufficient conditions for robust stability

and robust tracking of the system in Figure 6.2 in terms of the generalized structured

singular values of some operators. Computing these values is not an easy problem.

In fact, there is no general solution to the mat* case yet. However, there are upper

bounds that can be used to estimate an acceptable size for the perturbations to the

plant. For instance, from the Srnall Gain Theorem, an upper bound for the singular

value of an operator is its norrn. In the case of structured perturbations, though,

this upper bound leads to conaervative results, due to the fact that, information on

the perturbation structure is ignored. Similar to the matrix case, this information

could be utilized to derive tighter and, hence, less conservative upper bounds. Our

intention in this section is to introduce one such upper bound. This upper bound is

in fact a well-established bound from the matrix case that is generalized to operators

in [Dul96]. As we will see in this section, this upper bound together with our algorithm

for computing the induced-norm of operators in Section 5.5 make a nice complement

to Our robustness analysis machinary.

Let E be a Hilbert space and let A be a subspace of L(E). Denote by Da the set

of al1 invertible operators in L(E) that commute with al1 members of A: that is.

V~ := { D E L ( E ) : D is invertible and DA = AD: VA E A ) . (6.21)

The upper bound of our interest is then provided by the following result from [DulSG].

Proof of this result is similar to the matrix case, see e.g. [ZGD95].

Proposition 6.1 Suppose that P is in L(E). Then

pa(P) 5 inf ( 1 D P D-l ( 1 . DEVA

This result states that, the infimum over a set of weighted norms of P is a bound for

its structured singular value pA(P). Noting that the identity operator I is in fiA, one can see why the bound given by this result may in general be smaller than 11 PI1 ,

Chapter 6. Robustness Analysis of Sampled-Data Re~etitive Control Svstems 130

which is the bound given by the Small Gain Theorem.

For APTv and Ae, we have

and

Thuç finding the upper bounds for PA,,, [&,(A)] in (6.17) and ,UA,, (IVk)] in

(6.20) reduces to two minimization problems over 1 and 1 + 1 complex variables,

respectively. It is even possible to work with real optimization variables as opposed

to complex. Define

Dullerud shows in [Du1961 that

With this identity, we get

Similarly, define

to get

Cha~ter 6. Robustness Analvsis of Samded-Data Re~etit ive Control Svstems 131

To use optimization algorithms to find the upper bounds in (6.26) and (6.28), we

need to compute the functionals that appear in the infimums in those upper bounds.

Let's concentrate on

for now, where X and D = diag(dlI,. . . ,d l I ,d l+lI) , d* > 0, , i = 1 , 2 , . - -,ZJ + 1 are

given. First recall from Chapter 2 that, given the state-space representations for j ( s )

and k(X), (6.2) and (6.3), respectively, that of the lifted closed-loop system p ( X ) - is

where &, &, & and & are obtained through (2.26-2.30) and (2.33-2.36). Rom

here

$(A) - = a + A& (1 - X A ~ ) - ' al. (6.31)

We saw in Section 5.5 that Lemma 5.4 gives us a rnethod for computing Ilp(A)ll - to

any degree of accuracy, by means of fast discretization. As we will see below, this

lemma can be used to compute the functional in (6.29) as well.

Define the scaling matrix

where Ini , i = 1, . . . , 1 + 1, are identity matrices with the same Euclidean dimension

as that of the uncertainty block number i in (6.19). Then multiply (6.31) from left

by and from right by 0 - I l 2 . This gives


By evamining the equations for &, & and & in (2.342.36), we note that

is the transfer function of the lifted closed-loop system, with B I , Cl and D12 in

the state-space representation for G replaced with B~ D;~'*, D ~ / ~ c ~ and 0 0 I 2 D ~ ~ ,

respectively. Therefore, fast discret ization can be used to corn pute the weighted

norm (6.29) as well. The same argument is valid for the functional in (6.26); we just

need to start with block (1,l) of @(A), - Pl, (A).

Example: Consider the uncertain sampled-data setup of Figure 6.2. The plant G

has s t a t e-space realization

and the periodic input w is unknown of period T = l0sec. In the absence of pertur-

bation A, the controller

that receives its measurements from the plant every h = 0.5sec7 stabilizes the closed-

loop system and achieves an induced power-norm of 0.1818 for the systern from the

input w to the output z. The perturbation A is diagonal, Le., A = dzag(hl , A*),

with Al and A2 bounded on L2 and h-periodic. We would like to find a bound for the

induced power-nom of the system with plant being subject to such perturbations.

We start by examining the stability of the system. From Theorem 6.3; we need to

Cha pter 6. Robustness Analysis of Sampled-Data Repetitive Control Systems 133

8

bound PA,,, [il, (dw)] for w E [O, 27~). In fact, we need to do this only for w E [O, x],

since al1 the transfer functions are real rational. First, we concentrate on the upper

bound in (6.26), 114th

By using the technique that was just explained and the constr function in MATLAB

optimization toolbox, we minimize the functional in (6.26) for a vector of frequencies

from O to n. The result is the graph shown in soild in Figure 6.6. From this graph,

we obtain .1

Thus perturbations with size

will not destabilize the system. .41so, on the same plot, but shown in dashed, is the

j p p h of I l f ~ , , (gw) 11 versus frequency w. This graph gives

which bounds us to perturbations mith

In this case, we see that the bound provided by the Small Gain Theorem is only

slightly more conservative than the bound obtained by minimizing the weighted norms

in (6.26).

Now we can use Theorem 6.4 to find a bound for the induced power-norm of the

system. First we note that W = ë j t , since M = T / h = 20. So we only need to

bound Pa,, (A)] for X = CVk, k = O, 1 , . . . ,19. Again, due to symmetry, we would

Chapter 6. Robustness Analvsis of Sam~led-Data Re~etitive Control Svstems 134

Figure 6.7: Plot of upper bound for p Î, (du) versus w , predicted by the Srnall [-Il 1

Gain Theorem (dashed) and weight ed norm minimization (solid).

need to do this only for k = 0,1, . . . ,IO. The bound provided in (6.28) is obtained

by minimizing the weighted norms of f i (w') over -

These bounds, are listed in Table 6.1. From this table, we gather that

Thus for A with

the induced power-norm from w to z will be less than 0.6942. Note that for such

perturbations, the closed-loop system will be stable as predicted by (6.34).

Table 6.1 also has the bounds predicted by the Small Gain Theorem, which give


If we were to use the Small gain theorem, we would have

1

That is, the Small Gain Theorem restncts the perturbation to a much smaller size

and yields a much larger bound for the induced power-nom.

Table 6.1: Different upper bounds for p (wk)] , k = 0' 1, . . . , 19.

k 112 (wk) II inf 1 1 ~ ' / * 2 (w") D - ' / * I ~ -=A,,

Chapter 7

Conclusions

In this final chapter, we give an overview of the work presented in this thesis along

with some directions for future research.

Summary

In this thesis, ive introduce a performance rneasure for repetitive contro! systems,

referred to as the induced power-norm. This measure, which represents the maxi-

mum power-norm of the steady-state error vector in the system for the worst-case

periodic input of unit power-norm, accounts for three major contributing factors to

the steady-state error: the low-pass nature of most physical systems (as was reviewed

in Chapter 3), the intersample behavior associated with digital irnplementations and

last but not least the plant uncertainties. Thus, the induced power-norm is a useful

tool for comparing different repetitive controllers.

Moreover, ive establish a framework for design of sampled-data repetitive con-

trollers by proposing two methodologies. The first is based on minimizing the power-

n o m of the steady-state error vector for a given periodic input. We show that, under

certain mild conditions in the SIS0 case, the resulting controller is optimal for a wide

class of periodic inputs. The second methodology considers an optimal design for

unknown periodic inputs through the rninimization of the induced power-norm. We

verify that fast discretization is a useful computational tool for obtaining suboptimal

Chapter 7. Conclusions 137

cont rollers and evaluating t heir performance under bot h techniques. We also demon-

strate the effectiveness of t hese techniques by studying active suppression of the fan

noise present in an acoustic duct.

Also, in order to study the steady-state error arising from plant uncertainties,

we formulate a robust tracking problem for sampled-data repetitive control systems.

S pecifically, ive invest igate whet her the induced power-nom of the closed-loop system

remains below a given bound in the presence of stmctured linear periodically tirne-

varying perturbations. We show that the result can be stated in terms of a necessary

and sufficient condition which involves Dullerud's generalized notion of structured

singular values for operators. We illustrate some of the computational aspects by a

numerical example.

7.2 Future Research

The methods developed in this thesis may be extended in order to address some other

issues in repetitive control. Below we list a few possible extensions.

O Design for minimum transient response.

The methods developed in Chapters 4 and 5 can be modified to incorporate

the transient response of the closed-loop system into the design specifications.

For example, ive see in Chapter 4 that controller parametrization in terrns of a

stable FDLTI system Q can be used to convert Prnblem 4.1 to the equivalent

but simpler problem stated by (4.10). .4 solution to this problern is given by

Lemma 4.1. However, as mentioned in the discussion surrounding the lemma,

the solution is not unique. If we define

Q := {Q : Q solves (4.10)),

then a problem for optimal sarnpled-data repetitive control systems with mini-

mum transient response can be stated as follows:

Chapter 7. Conclusions 138

Recalling from Theorem 4.1 that z,, is an L2 signal, we note that (7.2) together

with (4.10) pose a mixed t2 - WT problem for sampled-data repetitive con-

trol systems with known periodic inputs. Similarly, a mixed problem can be

posed for such systems with unknown periodic inputs, by minimizing the WT

to L2 induced-norm of the mapping from w to ztr over the set of al1 controllers

that solve (5.1).

Design for rejection of non-periodic disturbances.

Repetitive control, which aims a t rej ecting periodic disturbances, does not guar-

antee any performance in rejecting non-periodic disturbances. In fact, Tenney

and Tomizuka [TT961 show that repetitive control systems can perform poorly in

response to disturbances of short duration. In order to consider non-periodic dis-

turbances in our formulation, we propose extending the definition of zt, in (7.2)

such that it includes the transient responses arising from both periodic and

non-periodic disturbances.

Robustness analysis wit h respect to LTI perturbations.

In our robustness analysis of sampled-data repetitive control systems, we con-

sider periodically time-varying plant perturbations. Such perturbations form

a much larger class than LTI perturbations, which are a natural candidate for

rnodeling the uncertainties in LTI plants. Thus the conditions in our analysis

of tracking robustness would be too conservative. A good research problem is

to see if less conservative conditions may be found.

a Design for robust performance.

Our final suggestion on future research is developing techniques which incorpo-

rate the tracking robustness criterion directly into the design.

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