Robust Control (1)

Robust control of MIMO systems

Olivier Sename

GIPSA-Lab

September 2013

Olivier Sename (GIPSA-Lab) Robust control September 2013 1 / 133

1 The H∞ norm and related definitionsH∞ norm as a measure of the system gain ?LTI systems and signals norms

2 Performance analysisDefinition of the sensitivity functionsFrequency-domain analysisStability issues

3 Performance specifications for analysis and/or designIntroductionSISO caseMIMO caseSome performance limitations

4 H∞ control designAbout the control structureSolving the H∞ control problemThe Riccati approachExample

5 Introduction to Linear Matrix InequalitiesBackground in OptimisationLMI in controlThe LMI approach for H∞ control designSome useful lemmas

6 Uncertainty modelling and robustness analysisIntroductionRepresentation of uncertaintiesDefinition of Robustness analysisRobustness analysis: the unstructured caseRobustness analysis: the structured caseTowards Robust control design

7 Introduction to LPV systems and controlLPV modellingLPV ControlO. Sename [GIPSA-lab] 2/133

Reference books

These texbooks are very close to the given course. Some of them are available on internet.

S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: analysis and design, JohnWiley and Sons, 2005. www.nt.ntnu.no/users/skoge/book

K. Zhou, Essentials of Robust Control, Prentice Hall, New Jersey, 1998.www.ece.lsu.edu/kemin

J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback control theory, MacmillanPublishing Company, New York, 1992.https://sites.google.com/site/brucefranciscontact/Home/publications

G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, NewJersey, 2001. csd.newcastle.edu.au

G. Duc et S. Font, Commande Hinf et -analyse: des outils pour la robustesse, Hermès,France, 1999.

D. Alazard, C. Cumer, P. Apkarian, M. Gauvrit, et G. Ferreres, Robustesse et commandeoptimale, Cépadues Editions, 1999.

O. Sename [GIPSA-lab] 3/133

The H∞ norm and related definitions

Nonlinear systems

In control theory, dynamical systems are mostly modeled and analyzed thanks to the use of a setof Ordinary Differential Equations (ODE)1.Nonlinear dynamical system modeling is the "most" representative model of a given system.Generally this model is derived thanks to system knowledge, physical equations etc. Nonlineardynamical systems are described by nonlinear ODEs.

Definition (Nonlinear dynamical system)

For given functions f : Rn ×Rnw 7→ Rn and g : Rn ×Rnw 7→ Rnz , a nonlinear dynamicalsystem (ΣNL) can be described as:

ΣNL :

x = f(x(t), w(t))z = g(x(t), w(t))

(1)

where x(t) is the state which takes values in a state space X ∈ Rn, w(t) is the input taking valuesin the input space W ∈ Rnw and z(t) is the output that belongs to the output space Z ∈ Rnz .

1Partial Differential Equations (PDEs) may be used for some applications such as irrigation channels, traffic flow,etc. but methodologies involved are more complex compared to the ones for ODEs.



Comments

The main advantage of nonlinear dynamical modeling is that (if it is correctly described) itcatches most of the real system phenomena.

On the other side, the main drawback is that there is a lack of mathematical andmethodological tools; e.g. parameter identification, control and observation synthesis andanalysis are complex and non systematic (especially for MIMO systems).

In this field, notions of robustness, observability, controllability, closed loop performance etc.are not so obvious. In particular, complex nonlinear problems often need to be reduced inorder to become tractable for nonlinear theory, or to apply input-output linearizationapproaches (e.g. in robot control applications).

As nonlinear modeling seems to lead to complex problems, especially for MIMO systemscontrol, the LTI dynamical modeling is often adopted for control and observation purposes.The LTI dynamical modeling consists in describing the system through linear ODEs.According to the previous nonlinear dynamical system definition, LTI modeling leads to a localdescription of the nonlinear behavior (e.g. it locally describes, around a linearizing point, thereal system behavior).

From now on only Linear Time-Invariant (LTI) systems are considered.



Definition of LTI systems

Definition (LTI dynamical system)

Given matrices A ∈ Rn×n, B ∈ Rn×nw , C ∈ Rnz×n and D ∈ Rnz×nw , a Linear Time Invariant(LTI) dynamical system (ΣLTI ) can be described as:

ΣLTI :

x(t) = Ax(t) +Bw(t)z(t) = Cx(t) +Dw(t)

(2)

where x(t) is the state which takes values in a state space X ∈ Rn, w(t) is the input taking valuesin the input space W ∈ Rnw and z(t) is the output that belongs to the output space Z ∈ Rnz .

The LTI system locally describes the real system under consideration and the linearizationprocedure allows to treat a linear problem instead of a nonlinear one. For this class of problem,many mathematical and control theory tools can be applied like closed loop stability, controllability,observability, performance, robust analysis, etc. for both SISO and MIMO systems. However, themain restriction is that LTI models only describe the system locally, then, compared to nonlinearmodels, they lack of information and, as a consequence, are incomplete and may not provideglobal stabilization.


The H∞ norm and related definitions Towards H∞ norm

Illustration of the gain computation

For a SISO system, y=Gd, the gain at a given frequency is simply

|y(ω)||d(ω)|

=|G(jω)d(ω)||d(ω)|

= |G(jω)|

The gain depends on the frequency, but since the system is linear it is independent of the inputmagnitudeFor a MIMO system we may select:

‖y(ω)‖2‖d(ω)‖2

=‖G(jω)d(ω)‖2‖d(ω)‖2

= ‖G(jω)‖2

Which is « independent » of the input magnitude. But this is not a correct definition. Indeed theinput direction is of great importance



A first ’bad’ approach

Let consider

G =

[5 43 2

]How to define and evaluate its gain ?Consider five different inputs:

u =

(10

)u =

(01

)u =

(0.7070.707

)u =

(0.707−0.707

)u =

(0.6−0.8

)Input magnitude :

‖u1‖2 = ‖u2‖2 = ‖u3‖2 = ‖u4‖2 = ‖u5‖2 = 1

But the corresponding outputs are

y =

(53

)y =

(42

)y =

(6.36303.5350

)y =

(0.70700.7070

)y =

(−0.20.2

)and

the ratio are ‖y‖2/‖u‖25.8310 4.4721 7.2790 0.9998 0.2828



Towards SVD


MAXIMUM SINGULAR VALUE

maxd 6=0

‖Gd‖2‖d‖2

= σ(G)

MINIMUM SINGULAR VALUE

mind 6=0

‖Gd‖2‖d‖2

= σ(G)

We see that, depending on the ratiod20/d10, the gain varies between 0.27and 7.34 .


What about eigenvalues ?

Eigenvalues are a poor measure of gain. Let

G =

[0 1000 0

]Eigenvalues are 0 and 0.But an input vector [

01

]leads to an output vector [

1000

].

Clearly the gain is not zero.Now, the maximal singular value is = 100It means that any signal can be amplified at most 100 timesThis is the good gain notion.



For MIMO systems

Finally, in the case of a transfer matrix G(s) : (m inputs, p outputs) u vector of inputs, y vector ofoutputs.

σ(G(jω)) ≤‖y(ω)‖2‖u(ω)‖2

≤ σ(G(jω))


Example of A two-mass/spring/damper system2 inputs: F1 and F2 2 outputs: x1 and x2

Singular Values

Frequency (rad/sec)S

ingu

lar V

alue

s (d

B)

10-1

100

101

102

-100

-80

-60

-40

-20

0

20

40

Hinf norm 11.4664 = 21.18 dB

smallest singular value

largest singular value

G=ss(A,B,C,D): LTI systemnormhinf(G): Compute Hinf normnorm(G,’inf’): Compute Hinf normsigma(G): plot max and min SV

The H∞ norm and related definitions LTI systems and signals norms

Signal norms

Reader is also invited to refer to the famous book of Zhou et al., 1996, where all the followingdefinitions and additional information are given.All the following definitions are given assuming signals x(t) ∈ C, then they will involve theconjugate (denoted as x∗(t)). When signals are real (i.e. x(t) ∈ R), x∗(t) = xT (t).

Definition (Norm and Normed vector space)

Let V be a finite dimension space. Then ∀ p ≥ 1, the application ||.||p is a norm, defined as,

||v||p =(∑

i

|vi|p)1/p (3)

Let V be a vector space over C (or R) and let ‖.‖ be a norm defined on V . Then V is anormed space.



L? norms

Definition (L1, L2, L∞ norms)

The 1-Norm of a function x(t) is given by,

‖x(t)‖1 =

∫ +∞

0|x(t)|dt (4)

The 2-Norm (that introduces the energy norm) is given by,

‖x(t)‖2 =√∫+∞

0 x∗(t)x(t)dt

=√

12π

∫+∞−∞ X∗(jω)X(jω)dω

(5)

The second equality is obtained by using the Parseval identity.

The∞-Norm is given by,‖x(t)‖∞ = sup

t|x(t)| (6)

‖X‖∞ = supRe(s)≥0

‖X(s)‖ = supω‖X(jω)‖ (7)

if the signals that admit the Laplace transform, analytic in Re(s) ≥ 0 (i.e. ∈ H∞).



About vector spaces

Definition (Banach, Hilbert, Hardy and Lp spaces)

A Banach space is a (real or complex) complete (i.e. all Cauchy sequences, of points in Khave a limit that is also in K) normed vector space B (with norm ‖.‖p).

A Hilbert space is a (real or complex) vector space H with an inner product < ., . > that iscomplete under the norm defined by the inner product. The norm of f ∈ H is then defined by,

‖f‖ =√< f, f > (8)

Every Hilbert space is a Banach since a Hilbert space is complete with respect to the normassociated with its inner product.

The Hardy spaces (Hp) are certain spaces of holomorphic functions (functions defined on anopen subset of the complex number plane C with values in C that are complex-differentiableat every point) on the unit disk or upper half plane.

The Lp space are spaces of p-power integrable functions (function whose integral exists,generally called Lebesgue integral), and corresponding sequence spaces.

For example, Rn and Cn with the usual spatial p-norm, ‖.‖p for 1 ≤ p <∞, are Banach spaces.This means that a Banach space is a vector space B over the real or complex numbers with anorm ‖.‖p such that every Cauchy sequence (with respect to the metric d(x, y) = ||x− y||) in Bhas a limit in B.



L∞ and H∞ spaces

Definition (L∞ space)

L∞ is the space of piecewise continuous bounded functions. It is a Banach space ofmatrix-valued (or scalar-valued) functions on C and consists of all complex bounded matrixfunctions f(jω), ∀ω ∈ R, such that,

supω∈R

σ[f(jω)] <∞ (9)

Definition (H∞ and RH∞ spaces)

H∞ is a (closed) subspace in L∞ with matrix functions f(jω), ∀ω ∈ R, analytic in Re(s) > 0(open right-half plane). The real rational subspace of H∞ which consists of all proper and realrational stable transfer matrices, is denoted by RH∞.



L∞ and H∞ spaces

Example

In control theorys+1

(s+10)(s+6)∈ RH∞

s+1(s−10)(s+6)

6∈ RH∞s+1

(s+10)∈ RH∞

(10)



H∞ norm

Definition (H∞ norm)

The H∞ norm of a proper LTI system defined as on (2) from input w(t) to output z(t) and whichbelongs to RH∞, is the induced energy-to-energy gain (L2 to L2 norm) defined as,

‖G(jω)‖∞ = supω∈R σ (G(jω))

= supw(s)∈H2

‖z(s)‖2‖w(s)‖2

= maxw(t)∈L2

‖z‖2‖w‖2

(11)

Remark

H∞ physical interpretations

This norm represents the maximal gain of the frequency response of the system. It is alsocalled the worst case attenuation level in the sense that it measures the maximumamplification that the system can deliver on the whole frequency set.

For SISO (resp. MIMO) systems, it represents the maximal peak value on the Bodemagnitude (resp. singular value) plot of G(jω), in other words, it is the largest gain if thesystem is fed by harmonic input signal.

Unlike H2 , the H∞ norm cannot be computed analytically. Only numerical solutions can beobtained (e.g. Bisection algorithm, or LMI resolution).



Recalls on Singular Value Definition

Let A ∈ Rm×n, There exists unitary matrices:

U = [u1, u2, . . . , um] and V = [v1, v2, . . . , vn]

such that

A = UΣV T , Σ =

[Σ1 00 0

],Σ1 =

σ1 0 . . . 00 σ2 . . . 0...

... · · · . . .0 0 . . . σp

with σ1 ≥ σ2 ≥ . . . ≥ σp ≥ 0 , p = minm,n.Singular values are good measures of the size of a matrix Singular vectors are good indications ofstr, ong/weak input or output directions. Note that:Avi = σiui and ATui = σivi. Then

AATui = σ2i ui

σ = σmax(A) = σ1 = the largest singular value of A.andσ = σmax(A) = σ1 = the smallest singular value of A.



L2 and H2 spaces

Definition (L2 space)

L2 is the space of piecewise continuous square integrable functions. It is a Hilbert space ofmatrix-valued (or scalar-valued) functions on C and consists of all complex matrix functions f(jω),∀ω ∈ R, such that,

||f ||2 =

√1

2π

∫ +∞

−∞Tr[f∗(jω)f(jω)]dω <∞ (12)

The inner product for this Hilbert space is defined as (for f, g ∈ L2)

< f, g >=

√1

2π

∫ +∞

−∞Tr[f∗(jω)g(jω)]dω (13)

Definition (H2 and RH2 spaces)

H2 is a subspace (Hardy space) of L2 with matrix functions f(jω), ∀ω ∈ R, analytic in Re(s) > 0(functions that are locally given by a convergent power series and differentiable on each point of itsdefinition set). In particular, the real rational subspace of H2, which consists of all strictly properand real rational stable transfer matrices, is denoted by RH2.



L2 and H2 spaces

Example

In control theorys+1

(s+10)(s+6)∈ RH2

s+1(s−10)(s+6)

6∈ RH2s+1

(s+10)6∈ RH2

(14)



H2 norm

Definition (H2 norm)

The H2 norm of a strictly proper LTI system defined as on (2) from input w(t) to output z(t) andwhich belongs to RH2, is the energy (L2 norm) of the impulse response g(t) defined as,

‖G(jω)‖2 =√∫+∞−∞ g∗(t)g(t)dt

=√

12π

∫+∞−∞ Tr[G∗(jω)G(jω)]dω

= supw(s)∈H2

||z(s)||∞||u(s)||2

(15)

The norm H2 is finite if and only if G(s) is strictly proper (i.e. G(s) ∈ RH2).

Remark

H2 physical interpretations and remarks

For SISO systems, it represents the area located below the so called Bode diagram.

For MIMO systems, the H2 norm is the impulse-to-energy gain of z(t) in response to a whitenoise input w(t) (satisfying W ∗(jω)W (jω) = I, i.e. uniform spectral density).

The H2 norm can be computed analytically (through the use of the controllability andobservability Grammians) or numerically (through LMIs).


Performance analysis

Outline







7 Introduction to LPV systems and controlLPV modellingLPV Control


Performance analysis

Introduction

Objectives of any control system [?]shape the response of the system to a given reference and get (or keep) a stable system inclosed-loop, with desired performances, while minimising the effects of disturbances andmeasurement noises, and avoiding actuators saturation, this despite of modelling uncertainties,parameter changes or change of operating point.This is formulated as:

Nominal stability (NS): The system is stable with the nominal model (no model uncertainty)

Nominal Performance (NP): The system satisfies the performance specifications with the nominalmodel (no model uncertainty)

Robust stability (RS): The system is stable for all perturbed plants about the nominal model, up tothe worst-case model uncertainty (including the real plant)

Robust performance (RP): The system satisfies the performance specifications for all perturbedplants about the nominal model, up to the worst-case model uncertainty(including the real plant).


Performance analysis Definition of the sensitivity functions

The control structure

Figure: Complete control scheme

In the SISO case, it leads to:

y = 11+G(s)K(s)

(GKr + dy −GKn+Gdi)

u = 11+K(s)G(s)

(Kr −Kdy −Kn−KGdi)

Loop transfer function L = G(s)K(s)

Sensitivity function S(s) = 11+L(s)

Complementary Sensitivity function T (s) =L(s)

1+L(s)

N.B. S is often referred to as the ’Output Sensitivity’.



Input/Output performances

Defining two new ’sensitivity functions’:Plant Sensitivity: SG = S(s).G(s) (often referred to as the ’Input Sensitivity’, e.g in Matlab)Controller Sensitivity: KS = K(s).S(s)



Definition of the sensitivity functions- The MIMO case

Figure: Complete control scheme

The output & the control input satisfy the following equations :

(Ip +G(s)K(s))y(s) = (GKr + dy −GKn+Gdi)(Im +K(s)G(s))u(s) = (Kr −Kdy −Kn−KGdi)

BUT : K(s)G(s) 6= G(s)K(s) !!



Definition of the sensitivity functions- The MIMO case

Definitions

Output and Output complementary sensitivity functions:

Sy = (Ip +GK)−1, Ty = (Ip +GK)−1GK, Sy + Ty = Ip

Input and Input complementary sensitivity functions:

Su = (Im +KG)−1, Tu = KG(Im +KG)−1, Su + Tu = Im

Properties

Ty = GK(Ip +GK)−1

Tu = (Im +KG)−1KGSuK = KSy



MIMO Input/Output performances

Defining two new ’sensitivity functions’:Plant Sensitivity: SyG = Sy(s).G(s)Controller Sensitivity: KSy = K(s).Sy(s)


Performance analysis Frequency-domain analysis

Frequency-domain analysis- Some classical analysis criteria (1)


The transfer function KSy(s) should beupper bounded so that u(t) does notreach the physical constraints, even fora large reference r(t)

The effect of the measurement noisen(t) on the plant input u(t) can bemade « small » by making thesensitivity function KSu(s) small (inHigh Frequencies)

The effect of the input disturbance di(t)on the plant input u(t) + di(t) (actuator)can be made « small » by making thesensitivity function Su(s) small (takecare to not trying to minimize Tu whichis not possible)


Frequency-domain analysis- Some classical analysis criteria (2)


The plant output y(t) can track thereference r(t) by making thecomplementary sensitivity functionTy(s) equal to 1. (servo pb)

The effect of the output disturbancedy(t) (resp. input disturbance di(t) ) onthe plant output y(t) can be made «small » by making the sensitivityfunction Sy(s) (resp. SyG(s) ) « small »

The effect of the measurement noisen(t) on the plant output y(t) can bemade « small » by making thecomplementary sensitivity functionTy(s) « small »


Trade-offs

ButS? + T? = I∗

Some trade-offs are to be looked for...These trade-offs can be reached if one aims :

to reject the disturbance effects in low frequencies

to minimize the noise effects in high frequencies

It remains to require:

Sy and SyG to be small in low frequencies to reduce the load (output and input) disturbanceeffects on the controlled output

Ty and KSy to be small in high frequencies to reduce the effects of measurement noises onthe controlled output and on the control input (actuator efforts)



Stability and robustness margins ...


Classical definitions:

Gain Margin:indicates the additionalgain that would take the closed loopto the critical stability condition

Phase margin: quantifies the purephase delay that should be added toachieve the same critical stabilitycondition

Delay margin: quantifies themaximal delay that should be addedin the loop to achieve the samecritical stability condition


Robustness margins ...


It is important to consider the modulemargin that quantifies the minimaldistance between the curve and thecritical point (-1,0j): this is a robustnessmargin.

∆M = 1/MSMS = max

ω|S(jω)| = ‖S‖∞

Good value MS < 2 (6dB)

The MODULE MARGIN is a robustnessmargin. Indeed, the influence of plantmodelling errors on the CL transfer function:

T =K(s)G(s)

1 +K(s)G(s)

is given by:

∆T

T=

1

1 +K(s)G(s)

∆G

G= S.

∆G

G

A good module margin implies good gain andphase margins:

GM ≥MS

MS − 1andPM ≥

1

MS

For MS = 2, then GM > 2 and PM > 30

Last:MT = max

ω|T (jω)|

A good value : MT < 1.5(3.5dB)


Frequency-domain analysis- Dynamical behavior

As mentioned in [?]:The concept of bandwidth is very important in understanding the benefits and trade-offs involvedwhen applying feedback control. Above we considered peaks of closed-loop transfer functions,which are related to the quality of the response. However, for performance we must also considerthe speed of the response, and this leads to considering the bandwidth frequency of the system.In general, a large bandwidth corresponds to a faster rise time, since high frequency signals aremore easily passed on to the outputs. A high bandwidth also indicates a system which is sensitiveto noise and to parameter variations. Conversely, if the bandwidth is small, the time response willgenerally be slow, and the system will usually be more robust.

Definition

Loosely speaking, bandwidth may be defined as the frequency range [ω1, ω2] over which control iseffective. In most cases we require tight control at steady-state so ω1 = 0, and we then simply callω2 the bandwidth. The word "effective" may be interpreted in different ways : globally it meansbenefit in terms of performance.



Frequency-domain analysis- Bandwidth definitions

Definition (ωS )

The (closed-loop) bandwidth, ωS , is the frequency where |S(jω)| crosses −3dB (1/sqrt2) frombelow.

Remark: |S| < 0.707, frequency zone, where e/r = −S is reasonably small

Definition (ωT )

The bandwidth (in term of T ), ωT , is the frequency where |T (jω)| crosses −3dB (1/sqrt2) fromabove.

Definition (ωc)

The bandwidth (crossover frequency), ωc, is the frequency where |L(jω)| crosses 1 (0dB), for thefirst time, from above.



Some remarks

Remark

Usually ωS < ωc < ωT

Remark

In most cases, the two definitions in terms of S and T yield similar values for the bandwidth. Inother cases, the situation is generally as follows. Up to the frequency ωS , |S| is less than 0.7, andcontrol is effective in terms of improving performance. In the frequency range [ωS , ωT ] control stillaffects the response, but does not improve performance. Finally, at frequencies higher than ωT ,we have S ' 1 and control has no significant effect on the response.

Remark

Usually ωS < ωc < ωT

Finally the following relation is very useful to evaluate the rise time:

tr =2.3

ωT



Examples


Performance analysis Stability issues

Well-posedness


Consider

G = −s− 1

s+ 2, K = 1

Therefore the control input is non proper:

u =s+ 2

3(r − n− dy) +

s− 1

3di

DEF: A closed-loop system is well-posed if all the transfer functions are proper

⇔ I +K(∞)G(∞) is invertible

In the example 1 + 1× (−1) = 0 Note that if G is strictly proper, this always holds.


Internal stability

DEF: A system is internally stable if all the transfer functions of the closed-loop system are stable

Figure: Control scheme

(yu

)=

((I +GK)−1GK (I +GK)−1G

K(I +GK)−1 −K(I +GK)−1G

)(rdi

)For instance :

G =1

s− 1, K =

s− 1

s+ 1,

(yu

)=

(1s+2

s+1(s−1)(s+2)

s−1s+2

− 1s+2

)(rdi

)There is one RHP pole (1), which means that this system is not internally stable. This is due hereto the pole/zero cancellation (forbidden!!).



Small Gain theorem

Consider the so called M −∆ loop.

Figure: M −∆ form

Theorem

Suppose M(s) in RH∞ and γ a positive scalar. Then the system is well-posed and internallystable for all ∆(s) in RH∞ such that ‖∆‖∞ ≤ 1/γ if and only if

‖M‖∞ < γ



Input-Output Stability

Definition (BIBO stability)

A system G (x = Ax+Bu; y = Cx) is BIBO stable if a bounded input u(.) (‖u‖∞ <∞) maps abounded output y(.) (‖y‖∞ <∞).

Now, the quantification (for BIBO stable systems) of the signal amplification (gain) is evaluated as:

γpeak = sup0<‖u‖∞<∞

‖y‖∞‖u‖∞

and is referred to as the PEAK TO PEAK Gain.

Definition (L2 stability)

A system G (x = Ax+Bu; y = Cx) is L2 stable if ‖u‖2 <∞ implies ‖y‖2 <∞.

Now, the quantification of the signal amplification (gain) is evaluated as:

γenergy = sup0<‖u‖2<∞

‖y‖2‖u‖2

and is referred to as the ENERGY Gain, and is such that:

γenergy = supω‖G(jω)‖ := ‖G‖∞

For a linear system, these stability definitions are equivalent (but not the quantification criteria).


Performance specifications

Outline









Performance specifications Introduction

Framework... templates

Objective : good performance specifications are important to ensure better control systemmean : give some templates on the sensitivity functionsFor simplicity, presentation for SISO systems first.Sketch of the method:

Robustness and performances in regulation can be specified by imposing frequentialtemplates on the sensitivity functions.

If the sensitivity functions stay within these templates, the control objectives are met.

These templates can be used for analysis and/or design. In the latter they are considered asweights on the sensitivity functions

The shapes of typical templates on the sensitivity functions are given in the following slides

Mathematically, these specifications may be captured by an upper bound, on the magnitude of asensitivity function, given by another transfer function, as for S:

|S(jω)| ≤1

|We(jω)|, ∀ω ⇔ ‖WeS‖∞ ≤ 1

where We(s) is a WEIGHT selected by the designer.


Performance specifications SISO case

Template on the sensitivity function - Weighted sensitivity

Typical specifications in terms of S include:1 Minimum bandwidth frequency ωS2 Maximum tracking error at selected frequencies.3 System type, or the maximum steady-state tracking error ε04 Shape of S over selected frequency ranges.5 Maximum peak magnitude of S, ||S||∞ < MS .

The peak specification prevents amplification of noise at high frequencies, and also introduces amargin of robustness; typically we select MS = 2.



Template on the sensitivity function S


S(s) =1

1 +K(s)G(s)

1

We(s)=

s+ ωbε

s/MS + ωb

Generally ε ' 0 is considered, MS < 2(6dB) or (3dB - cautious) to ensuresufficient module margin.ωb influences the CL bandwidth : ωb ↑

faster rejection of the disturbance

faster CL tracking response

better robustness w.r.t. parametricuncertainties

Template on the Sensitivity function S

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

10−4

10−2

100

102

104

106−60

−50

−40

−30

−20

−10

0

10Template on the sensitivity function S

MS = 2 (6dB)

ε = 1e − 3

ωb s.t. |1/We| = 0dB


Template on the function KS


KS(s) =K(s)

1 +K(s)G(s)

1

Wu(s)=

ε1s+ ωbc

s+ ωbc/Mu

Mu chosen according to LF behavior ofthe process (actuator constraints:saturations)ωbc influences the CL bandwidth : ωbc ↓

better limitation of measurementnoises

roll-off starting from ωbc to reducemodeling errors effects

Template on the Controller*Sensitivity KS

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

100

105

−60

−50

−40

−30

−20

−10

0

10

M u = 2

ω bc fo r |1 / W u| = 0 d B

ε c = 1 e -3


Template on the function T


T (s) =K(s)G(s)

1 +K(s)G(s)

1

WT (s)=

εT s+ ωbt

s+ ωbt/MT

Generally εT ' 0 is considered, MSG

allows to limit the overshoot in theresponse to input disturbances.ωSG influences the bandwidth hence thetransient behavior of the disturbancerejection properties: ωbt ↓

better noise effects rejection

better filtering of HF modelling errors

10−4

10−2

100

102

104

106−60

−50

−40

−30

−20

−10

0

10Template on the Complementary sensitivity function T

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B) MT =1.5

εT =1e-3

ωbT s.t. |1/WT | = 0 dB


Template on the function SG


SG(s) =G(s)

1 +K(s)G(s)

1

WSG(s)=

s+ ωSGεSG

s/MSG + ωSG

Generally εSG ' 0 is considered,MT < 1.5 (3dB) to limit the overshoot.ωSG influences the CL bandwidth :ωSG ↑ =⇒ faster rejection of thedisturbance.

10−4

10−2

100

102

104

106−40

−30

−20

−10

0

10

20 Template on the Sensitivity*Plant SG

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B) MSG = 10

εSG=1e-2

ωsg s.t. |1/WSG| = 0 dB

Performance specifications MIMO case

A first insight into the MIMO case

The direct extension of the performances objectives to MIMO systems could be formulated asfollows:

1 Disturbance attenuation/closed-loop performances:

σ(Sy) <1

W1(jω)

with W1(jω) > 1 for ω < ωb2 Actuator constraints:

σ(KSy) <1

W2(jω)

with W2(jω) > 1 for ω > ωh3 Robustness to multiplicative uncertainties:

σ(Ty) <1

W3(jω)

with W3(jω) > 1 for ω > ωt

However these objectives do not consider the specific MIMO structure of the system, i.e. theinput-output relationship between actuators and sensors.It is then better to define the objectives accordingly with the system inputs and outputs.



Towards MIMO systems

Let us consider a system with 2 inputs and 1 output and define:

G =(G1 G2

), K =

(K1

K2

)Therefore

GK = G1K1 +G2K2, KG =

(K1G1 K1G2

K2G1 K2G2

)and the sensitivity functions are:

Sy =1

1 +G1K1 +G2K2, KSy =

(K1

1+G1K1+G2K2K2

1+G1K1+G2K2

)

While a single template We is convenient for Sy it is straightforward that the following diagonaltemplate should be used for KSy :

Wu(s) =

(W 1u(s) 00 W 2

u(s)

)where W 1

u and W 2u are chosen in order to account for each actuator specificity (constraint).



The MIMO general case

Let us consider G with m inputs and p outputs.

In the MIMO case the simplest way is to defined the templates as diagonal transfer matrices,i.e. using (MSi

, ωbi , εi)

In that case, a weighting function should be dedicated for each input, and for each output.

These weighting functions may of course be different if the specifications on each actuator(e.g. saturation), and on each sensor (e.g. noise), are different.

In addition, during the performance analysis step, take care to plot, in addition to the MIMOsensitivity functions, the individual ones related to each input/output to check if the individualspecification is met. Hence, in the simplest case:

1 If the specifications are identical then it is sufficient to plot:σ(Sy(jω)) and 1

|We(jω)| , for all ωσ(KSy(jω)) and 1

|Wu(jω)| , for all ω

2 If the specifications are different, one should plotσ(Sy(i, :)) and 1

|Wie|

, for all ω, i = 1, . . . , p

σ(KSy(k, :)) and 1

|Wku |

, for all ω, k = 1, . . . ,m.

i.e. p plots for all output behaviors and m plots for the input ones.3 In a very general case, plot σ(Sy) with σ(1/We)



More on weighting functions

When tighter (harder) objectives are to be met .....the templates can be defined more accurately by transfer functions of order greater than 1, as

We(s) =

(s/MS + ωb

s+ ωbε

)k,

if a roll-off of −20× k dB per decade is required.Take care to the choice of the parameters (MS , ωb, ε) to avoid incoherent objectives!

100

101

102

103

104

105

106

−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

Mu=2, wbc=100, epsi1=0.01

Wu and Wu square without parameter modification

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

100

101

102

103

104

105

106

−50

−40

−30

−20

−10

0

10

20

Original weight (specifications) Mu=2, ε1 = 0.01, ωb c=100 for Wu

Mu =√

2, ε1 =√

0.01,ωb c=100 for 1/W 2

u case 1

Mu =√

2, ε1 =√

0.01,ωb c=200 for 1/W 2

u case 2

Mu =√

2, ε1 =√

0.01,ωb c=400 for 1/W 2

u case 3

$1/W_u$ and $1/Wu_^2$ with parameter modification

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)



Final objectives

In terms of control synthesis, all these specifications can be tackled in the following problem: findK(s) s.t. ∥∥∥∥∥∥∥∥

WeSWuKSWTT

WSGSG

∥∥∥∥∥∥∥∥∞

≤ 1 ⇒ ‖WeS‖∞ ≤ 1 ‖WSGSG‖∞ ≤ 1‖WuKS‖∞ ≤ 1 ‖WTT‖∞ ≤ 1

Often, the simpler following one (referred to as the mixed sensitivity problem) is studied:

Find K s.t.

∥∥∥∥ WeSWuKS

∥∥∥∥∞≤ 1

since the latter allows to consider the closed-loop output performance as well as the actuatorconstraints.


Performance specifications Some performance limitations

Introduction

Framework

Main extracts of this part: see Goodwin et al 2001. "Performance limitations in control are not onlyinherently interesting, but also have a major impact on real world problems."Objective : take into account the limitations inherent to the system or due to actuators constraints,before designing the controller..... Understanding what is not possible is as important asunderstanding what is possible!

Example of structural constraints

S + T = 1, ∀ω

We then cannot have, for any frequency ω0, |S(jω0)| < 1 and |T (jω0)| < 1. This implies that,disturbance and noise rejection cannot be achieved in the same frequency range.

Bode’s Sensitivity Integral for open-loop stable systems

It is known that, for open loop stable plant:∫ ∞0

log|S(jω)|dω = 0

Then the frequency range where |S(jω)| < 1 is balanced by the frenquencies where |S(jω)| > 1



Bode sensitivity

Nice interpretation of the balance between reduction and magnification of the sensitivity.

For unstable system we have stronger constraints:∫ ∞0

log|det(S(jω))|dω = π

Np∑i=1

Re(pi),

where pi design the Np RHP poles. Therefore, in the presence of RHP poles, the control effortnecessary to stabilize the system is paid in terms of amplification of the sensitivity magnitude.



The interesting case of systems with RHP zeros

Theorem

Let G(s) a MIMO plant with one RHP zero at s = z, and Wp(s) be a scalar weight. Then,closed-loop stability is ensured only if:

‖Wp(s)S(s) ‖≥ |Wp(s = z)|

To illustrate the use of that theorem, if Wp is chosen as:

Wp(s) =

(s/MS + ωb

s+ ωbε

),

and , if the controller meets the requirements, then

‖Wp(s)S(s) ‖∞≤ 1

Therefore a necessary condition is:

|z/MS + ωb

z + ωbε|≤ 1

To conclude, if z is real, and if the performance specifications are such that : MS = 2 and ε = 0,then a necessary condition to meet the performance requirements is :

ωb ≤z

2


H∞ control design

Outline









H∞ control design About the control structure

The General Control Configuration

This approach has been introduced by Doyle (1983). The formulation makes use of the generalcontrol configuration.

P is the generalized plant (contains the plant, the weights, the uncertainties if any) ; K is thecontroller. The closed-loop transfer function is given by:

Tew(s) = Fl(P,K) = P11 + P12K(I − P22K)−1P21

where Fl(P,K) is referred to as a lower Linear Fractional Transformation.



H∞ control design- Problem definition

The overall control objective is to minimize some norm of the transfer function from w to e , forexample, the H∞ norm.

Definition (H∞ suboptimal control problem)

H∞ control problem: Find a controller K(s) which based on the information in y, generates acontrol signal u which counteracts the influence of w on e, thereby minimizing the closed-loopnorm from w to e.

Definition (H∞ optimal control problem)

Given γ a pre-specified attenuation level, a H∞ sub-optimal control problem is to design astabilizing controller that ensures :

‖Tew(s)‖∞ = maxω

σ(Tew(jω)) ≤ γ

The optimal problem aims at finding γmin.



H∞ control design- Problem formulation

It will be shown how to formulate such a control problem using "classical" control tools. Theprocedure will be 2-steps:

Build a control scheme s.t. the closed-loop system matrix does correspond to the tackled H∞problem (for instance the mixed sensitivity problem). Use of Matlab, sysic tool.

Use an optimisation algorithm that finds the controller K solution of the considered problem.

Illustration on∥∥∥∥ WeSWuKS

∥∥∥∥∞≤ γ

In that case the closed-loop system must be ‖Tew(s)‖∞ =


∥∥∥∥∞≤ γ i.e a system with

1 input and 2 outputs. Since S = r−yr

and KS = ur

, the control scheme needs only one externalinput r.



How to consider performance specification in H∞ control ?

Note that: in practice the performance specification concerns at least two sensitivity functions (Sand KS) in order to take into account the tracking objective as well as the actuator constraints.Control objectives:

y = Gu = GK(r − y)⇒ tracking error : ε = Sru = K(r − y) = K(r −Gu)⇒ actuator force : u = KSr

To cope with that control objectives the following control scheme is considered:

Objective w.r.t sensitivity functions: ‖WeS‖∞ ≤ 1, ‖WuKS‖∞ ≤ 1.Idea: define 2 new virtual controlled outputs:

e1 = WeSre2 = WuKSr



About the control structure- Problem definition

The performance specifications on the tracking error & on the actuator, given as some weights onthe controlled output, then leads to the new control scheme:

The associated general control configuration is :



About the control structure- Problem definition

The corresponding H∞ suboptimal control problem is therefore to find a controller K(s) such that

‖Tew(s)‖∞ =


∥∥∥∥∞≤ γ with

Tew(s) = Fl(P,K) = P11 + P12K(I − P22K)−1P21

=

[We

0

]+

[−WeGWu

]K(I +GK)−1I

=

[WeSWuKS

]in Matlab



What about disturbance attenuation ?

To account for input disturbance rejection, the control scheme must include di:

This corresponds to the closed-loop system.

Tew =

[WeSy WeSyGWuKSy WuTu

]The new H∞ control problem therefore includes the input disturbance rejection objective, thanksto SyG that should satisfy the same template as S, i.e an high-pass filter!Remarks: Note that WuTu is an additional constraint that may lead to an increase of theattenuation level γ since it is not part of the objectives. Hopefully Tu is low pass, and Wu as well.The input weight has to be on u not u+ di which would lead to an unsolvable problem.



Improve the disturbance attenuation

The previous problem, allows to ensure the input disturbance rejection, but does not provide anyadditional d-o-f to improve it (without impacting the tracking performance). In order to ’decouple’both performance objectives, the idea is to add a disturbance model that indeed changes thedisturbance rejection properties.Let then consider : di(t) = Wd.d. In that case the closed-loop system is This corresponds to theclosed-loop system.

Tew =

[WeSy WeSyGWd

WuKSy WuTuWd

]and the template expected for SyG is now 1

Wd.We.

First interest: improve the disturbance weight as Wd = 100... but this has a price (see Fig. belowfor an example)

10-2 10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

Sensitivity*Plant SG

Frequency (rad/sec)

with Wd=1

with Wd=100

10-2 10-1 100 101 102 103-50

-40

-30

-20

-10

0

10

20

30

40

Controller*Sensitivity KS

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

with Wd=100

with Wd=1



More generally...

To include multiple objectives in a SINGLE H∞ control problem, there are 2 ways:1 add some external inputs (reference, noise, disturbance, uncertainties ...)2 add new controlled outputs

Of course both ways increase the dimension of the problem to be solved....thus the complexity aswell. Moreover additional constraints appear that are not part of the objectives ....General rule: first think simple !!


H∞ control design Solving the H∞ control problem

Solving the H∞ control problem

The solution of the H∞ control problem is based on a state space representation of P , thegeneralized plant, that includes the plant model and the performance weights.The calculation of the controller, solution of the H∞ control problem , can then be done using theRiccati approach or the LMI approach of the H∞ control problem [?] [?].Notations:

P

x = Ax+B1w +B2ue = C1x+D11w +D12uy = C2x+D21w +D22u

⇒ P =

A B1 B2

C1 D11 D12

C2 D21 D22

with x ∈ Rn, x ∈ Rnw , u ∈ Rnu , z ∈ Rnz et y ∈ Rny


H∞ control design Solving the H∞ control problem

Problem formulation

Let K(s) be a dynamic output feedback LTI controller defined as

K(s) :

xK(t) = AK xK(t) +BK y(t),u(t) = CK xK(t) +DK y(t).

where xK ∈ Rn, and AK , BK , CK and DK are matrices of appropriate dimensions.Remark. The controller will be considered here of the same order (same number of state variables)n than the generalized plant, which here, in the H∞ framework, the order of the optimal controller.With P (s) and K(s), the closed-loop system N(s) is:

N(s) :

xcl(t) = A xcl(t) + B w(t),z(t) = C xcl(t) +D w(t),

(16)

where xTcl(t) =[xT (t) xTK(t)

]andA =

(A+B2 DK C2 B2 CK

BK C2 AK

),

B =

(B1 +B2 DK D21

BK D21

),

C =(C1 +D12 DK C2, D12 CK

),

D = B1 +B2 DK D21.

The aim is of course to find matrices AK , BK , CK and DK s.t. the H∞ norm of the closed-loopsystem (16) is as small as possible, i.e. γopt = min γ s.t. ||N(s)||∞ < γ.


H∞ control design The Riccati approach

Asuumptions for the Riccati method

A1: (A,B2) stabilizable and (C2, A) detectable: necessary for the existence of stabilizingcontrollers

A2: rank(D12) = nu and rank(D21) = ny : Sufficient to ensure the controllers are proper, hencerealizable

A3: ∀ω ∈ R, rank(

A− jωIn B2

C1 D12

)= n+ nu

A4: ∀ω ∈ R, rank(

A− jωIn B1

C2 D21

)= n+ ny Both ensure that the optimal controller does

not try to cancel poles or zeros on the imaginary axis which would result in CL instability

A5:D11 = 0, D22 = 0, D12

T [ C1 D12 ] =[

0 Im2],

[B1

D21

]D21

T =

[0Ip2

]not necessary but simplify the solution (does correspond to the given theorem next but can beeasily relaxed)



The problem solvability

The first step is to check whether a solution does exist of not, to the optimal control problem.

Theorem (1)

Under the assumptions A1 to A5, there exists a dynamic output feedback controlleru(t) = K(.) y(t) such that the closed-loop system is internally stable and the H∞ norm of theclosed-loop system from the exogenous inputs w(t) to the controlled outputs z(t) is less than γ, ifand only if

i the Hamiltonian H =

(A γ−2 B1 BT1 −B2 BT2

−C1 CT1 −AT)

has no eigenvalues on the

imaginary axis.

ii there exists X∞ ≥ 0 t.q. AT X∞+X∞ A+X∞ (γ−2 B1 BT1 −B2 BT2 ) X∞+CT1 C1 = 0,

iii the Hamiltonian J =

(AT γ−2 CT1 C1 − CT2 C2

−B1 BT1 −A

)has no eigenvalues on the

imaginary axis.

iv there exists Y∞ ≥ 0 t.q. A Y∞ + Y∞ AT + Y∞ (γ−2 CT1 C1 − CT2 C2) Y∞ +B1 BT1 = 0,

v the spectral radius ρ(X∞ Y∞) ≤ γ2.



Controller reconstruction

Theorem (2)

If the necessary and sufficient conditions of the Theorem 1 are satisfied, then the so-called centralcontroller is given by the state space representation

Ksub(s) =

[A∞ −Z∞L∞F∞ 0

]with

A∞ = A+ γ−2B1BT1 X∞ +B2F∞ + Z∞L∞C2

F∞ = −BT2 X∞, L∞ = −Y∞CT2Z∞ =

(I − γ−2Y∞X∞

)−1

The Controller structure is indeed an observer-based state feedback control law, with

u2(t) = −BT2 X∞ x(t),

where x(t) is the observer state vector

˙x(t) = A x(t) +B1 w(t) +B2 u(t) + Z∞ L∞(C2 x(t)− y(t)

). (17)

and w(t) is defined as

w(t) = γ−2 BT1 X∞ x(t).

Remark. w(t) is an estimation of the worst case disturbance. Z∞ L∞ is the filter gain for the OEproblem of estimating x(t) in the presence of the worst case disturbanceO. Sename [GIPSA-lab] 71/133

Introduction to LMIs

Outline









Introduction to LMIs Background in Optimisation

Brief on optimisation

Definition (Convex function)

A function f : Rm → R is convex if and only if for all x, y ∈ Rm and λ ∈ [0 1],

f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y) (18)

Equivalently, f is convex if and only if its epigraph,

epi(f) = (x, λ)|f(x) ≤ λ (19)

is convex.

Definition ((Strict) LMI constraint)

A Linear Matrix Inequality constraint on a vector x ∈ Rm is defined as,

F (x) = F0 +m∑i=1

Fixi 0( 0) (20)

where F0 = FT0 and Fi = FTi ∈ Rn×n are given, and symbol F 0( 0) means that F issymmetric and positive semi-definite ( 0) or positive definite ( 0), i.e. ∀u|uTFu(>) ≥ 0.



Convex to LMIs

Example

Lyapunov equation. A very famous LMI constraint is the Lyapunov inequality of an autonomoussystem x = Ax. Then the stability LMI associated is given by,

xTPx > 0xT (ATP + PA)x < 0

(21)

which is equivalent to,

F (P ) =

[−P 00 ATP + PA

]≺ 0 (22)

where P = PT is the decision variable. Then, the inequality F (P ) ≺ 0 is linear in P .

LMI constraints F (x) 0 are convex in x, i.e. the set x|F (x) 0 is convex. Then LMI basedoptimization falls in the convex optimization. This property is fundamental because it guaranteesthat the global (or optimal) solution x∗ of the the minimization problem under LMI constraints canbe found efficiently, in a polynomial time (by optimization algorithms like e.g. Ellipsoid, InteriorPoint methods).



LMI problem

Two kind of problems can be handled

Feasibility: The question whether or not there exist elements x ∈ X such that F (x) < 0 iscalled a feasibility problem. The LMI F (x) < 0 is called feasible if such x exists,otherwise it is said to be infeasible.

Optimization: Let an objective function f : S → R where S = x|F (x) < 0. The problem todetermine

Vopt = infx∈Sf(x)

is called an optimization problem with an LMI constraint. This problem involvesthe determination of Vopt, the calculation of an almost optimal solution x (i.e., forarbitrary ε > 0 the calculation of an x ∈ S such that Vopt ≤ f(x) ≤ Vopt + ε, orthe calculation of a optimal solutions xopt (elements xopt ∈ S such thatVopt = f(xopt)).



Examples of LMI problem

Stability analysis is a feasability problem.LQ control is an optimization problem, formulated as:

LQ control

Consider a controllable system x = Ax+Bu. Find a state feedback u(t) = −Kx(t) s.tJ =

∫∞0 (xTQx+ uTRu)dt is minimum (given Q > 0 and R > 0) is an optimisation problem

whose solution is obtained solving the Riccati equation:

Find P > 0, s.t. ATP + PA− PBR−1BTP +Q = 0

and then the state feedback is given by:

u(t) = −R−1BTPx(t)

which is equivalent to: find P > 0 s.t[ATP + PA+Q PB

BTP R

]> 0



Semi-Definite Programming (SDP) Problem

LMI programming is a generalization of the Linear Programming (LP) to cone positivesemi-definite matrices, which is defined as the set of all symmetric positive semi-definite matricesof particular dimension.

Definition (SDP problem)

A SDP problem is defined as,

min cT xunder constraint F (x) 0

(23)

where F (x) is an affine symmetric matrix function of x ∈ Rm (e.g. LMI) and c ∈ Rm is a givenreal vector, that defines the problem objective.

SDP problems are theoretically tractable and practically:

They have a polynomial complexity, i.e. there exists an algorithm able to find the globalminimum (for a given a priori fixed precision) in a time polynomial in the size of the problem(given by m, the number of variables and n, the size of the LMI).

SDP can be practically and efficiently solved for LMIs of size up to 100× 100 and m ≤ 1000see ElGhaoui, 97. Note that today, due to extensive developments in this area, it may be evenlarger.


Introduction to LMIs LMI in control

The state feedback design problem

Stabilisation

Let us consider a controllable system x = Ax+Bu. The problem is to find a state feedbacku(t) = −Kx(t) s.t the closed-loop system is stable.

Using the Lyapunov theorem, this amounts at finding P = PT > 0 s.t:

(A−BK)TP + P(A−BK) < 0

⇔ ATP + PA−KTBTP − PBK < 0

which is obviously not linear...

Solution; use of change of variables

First, left and right multiplication by P−1 leads to

P−1AT +AP−1 − P−1KTBT −BKP−1 < 0

⇔ QAT +AQ + YTBT +BY < 0

with Q = P−1 and Y = −KP−1.

The problem to be solved is therefore formulated as an LMI ! and without any conservatism !



The Bounded Real Lemma

The L2-norm of the output z of a system ΣLTI is uniformly bounded by γ times the L2-norm ofthe input w (initial condition x(0) = 0).A dynamical system G = (A,B,C,D) is internally stable and with an ||G||∞ < γ if and only isthere exists a positive definite symmetric matrix P (i.e P = PT > 0 s.t AT P + P A P B CT

BT P −γ I DT

C D −γ I

< 0, P > 0. (24)

The Bounded Real Lemma (BRL), can also be written as follows (see Scherer)I 0A B0 IC D

T

0 P 0 0P 0 0 00 0 −γ2I 00 0 0 I

I 0A B0 IC D

≺ 0 (25)

Note that the BRL is an LMI if the only unknown (decision variables) are P and γ (or γ2).



Quadratic stability

This concept is very useful for the stability analysis of uncertain systems.Let us consider an uncertain system

x = A(δ)x

where δ is an parameter vector that belongs to an uncertainty set ∆.

Theorem

The considered system is said to be quadratically stable for all uncertainties δ ∈ ∆ if there exists a(single) "Lyapunov function" P = PT > 0 s.t

A(δ)TP + PA(δ) < 0, for all δ ∈ ∆ (26)

This is a sufficient condition for ROBUST Stability which is obtained when A(δ) is stable for allδ ∈ ∆.


Introduction to LMIs The LMI approach for H∞ control design

- Solvability

In this case only A1 is necessary. The solution is base on the use of the Bounded Real Lemma,and some relaxations that leads to an LMI problem to be solved [?].when we refer to the H∞ control problem, we mean: Find a controller C for system M such that,given γ∞,

||Fl(P,K)||∞ < γ∞ (27)

The minimum of this norm is denoted as γ∗∞ and is called the optimal H∞ gain. Hence, it comes,

γ∗∞ = min(AK ,BK ,CK ,DK)s.t.σA⊂C−

‖Tzw(s)‖∞ (28)

As presented in the previous sections, this condition is fulfilled thanks to the BRL. As a matter offact, the system is internally stable and meets the quadratic H∞ performances iff. ∃ P = PT 0such that, ATP + PA PB CT

BTP −γ22I DT

C D −I

< 0 (29)

where A, B, C, D are given in (16). Since this inequality is not an LMI and not tractable for SDPsolver, relaxations have to be performed (indeed it is a BMI), as proposed in [?].



- Problem solution

Theorem (LTI/H∞ solution [?])

A dynamical output feedback controller of the form C(s) =

[Ac BcCc Dc

]that solves the H∞

control problem, is obtained by solving the following LMIs in (X, Y, A, B, C and D), whileminimizing γ∞,

M11 (∗)T (∗)T (∗)TM21 M22 (∗)T (∗)TM31 M32 M33 (∗)TM41 M42 M43 M44

≺ 0

[X InIn Y

] 0

(30)

where,M11 = AX + XAT +B2C + C

TBT2 M21 = A +AT + CT2 D

TBT2

M22 = YA+ATY + BC2 + CT2 BT

M31 = BT1 +DT21DTBT2

M32 = BT1 Y +DT21BT

M33 = −γ∞Inu

M41 = C1X +D12C M42 = C1 +D12DC2

M43 = D11 +D12DD21 M44 = −γ∞Iny

(31)



Controller reconstruction

Once A, B, C, D, X and Y have been obtained, the reconstruction procedure consists in findingnon singular matrices M and N s.t. M NT = I −X Y and the controller K is obtained as follows

Dc = DCc = (C−DcC2X)M−T

Bc = N−1(B− YB2Dc)

Ac = N−1(A− YAX− YB2DcC2X−NBcC2X− YB2CcMT )M−T

(32)

where M and N are defined such that MNT = In −XY (that can be solved through a singularvalue decomposition plus a Cholesky factorization).Remark. Note that other relaxation methods can be used to solve this problem, as suggested by[?].


Introduction to LMIs Some useful lemmas

Schur lemma

Lemma

Let Q = QT and R = RT be affine matrices of compatible size, then the condition[Q SST R

] 0 (33)

is equivalent toR 0

Q− SR−1ST 0(34)

The Schur lemma allows to a convert a quadratic constraint (ellipsoidal constraint) into an LMIconstraint.



Kalman-Yakubovich-Popov lemma

Lemma

For any triple of matrices A ∈ Rn×n, B ∈ Rn×m, M ∈ R(n+m)×(n+m) =

[M11 M12

M21 M22

], the

following assessments are equivalent:1 There exists a symmetric K = KT 0 s.t.[

I 0A B

]T [0 KK 0

] [I 0A B

]+M < 0

2 M22 < 0 and for all ω ∈ R and complex vectors col(x,w) 6= 0

[A− jωI B

] [ xw

]= 0⇒

[xw

]TM

[xw

]< 0

3 If M = −[I 0A B

]T [0 KK 0

] [I 0A B

]then the second statement is equivalent to

the condition that, for all ω ∈ R with det(jωI −A) 6= 0,[I

C(jωI −A)−1B +D

]∗ [Q SST R

] [I

C(jωI −A)−1B +D

]> 0

This lemma is used to convert frequency inequalities into Linear Matrix Inequalities.O. Sename [GIPSA-lab] 85/133


Projection Lemma

Lemma

For given matrices W = WT , M and N , of appropriate size, there exists a real matrix K = KT

such that,W +MKNT +NKTMT ≺ 0 (35)

if and only if there exist matrices U and V such that,

W +MU + UTMT ≺ 0W +NV + V TNT ≺ 0

(36)

or, equivalently, if and only if,MT⊥WM⊥ ≺ 0NT⊥WN⊥ ≺ 0

(37)

where M⊥ and N⊥ are the orthogonal complements of M , N respectively (i.e. MT⊥M = 0).

The projection lemma is also widely used in control theory. It allows to eliminate variable by achange of basis (projection in the kernel basis). It is involved in one of the H∞ solutions [?]seee.g..



Completion Lemma

Lemma

Let X = XT , Y = Y T ∈ Rn×n such that X > 0 and Y > 0. The three following statements areequivalent:

1 There exist matrices X2, Y2 ∈ Rn×r and X3, Y3 ∈ Rr×r such that,[X X2

XT2 X3

] 0 and

[X X2

XT2 X3

]−1

=

[Y Y2

Y T2 Y3

](38)

2

[X II Y

] 0 and rank

[X II Y

]≤ n+ r

3

[X II Y

] 0 and rank [XY − I] ≤ r

This lemma is useful for solving LMIs. It allow to simplify the number of variables when a matrixand its inverse enter in a LMI.



Finsler’s lemma

This Lemma allows the elimination of matrix variables.

Lemma

The following statement are equivalent

xTAx < 0 for all x 6= 0 s:t: Bx = 0

BTAB < 0 where BB = 0

A+ λBTB < 0 for some scalar λ

A+XB +BTXT < 0 for some matrix X

B⊥TAB⊥ < 0


Robust analysis

Outline









Robust analysis Introduction

Introduction

A control system is robust if it is insensitive to differences between the actual system and themodel of the system which was used to design the controller

How to take into account the difference between the actual system and the model ?

A solution: using a model set BUT : very large problem and not exact yet

A method: these differences are referred as model uncertainty.The approach

1 determine the uncertainty set: mathematical representation2 check Robust Stability3 check Robust Performance

Lots of forms can be derived according to both our knowledge of the physical mechanism thatcause the uncertainties and our ability to represent these mechanisms in a way that facilitatesconvenient manipulation.Several origins :

Approximate knowledge and variations of some parameters

Measurement imperfections (due to sensor)

At high frequencies, even the structure and the model order is unknown (100

Choice of simpler models for control synthesis

Controller implementation

Two classes: parametric uncertainties / neglected or unmodelled dynamics


Robust analysis Representation of uncertainties

Example 1: uncertainties [?]

G(s) =k

1 + τse−sh, 2 ≤ k, h, τ ≤ 3

Let us choose the nominal parameters as, k = h = τ = 2.5 and G the according nominal model.We can define the ’relative’ uncertainty, which is actually referred as a MULTIPLICATIVEUNCERTAINTY, as


G(s) = G(s)(I +Wm(s)∆(s))

with Wm(s) = 3.5s+0.25s+1

and ‖∆‖∞ ≤ 1


Example 2: unmodelled dynamcis

Let us consider the system:

G(s) = G(s)1

1 + τs, τ ≤ τmax

This can be modelled as:

G(s) = G0(s)(I +Wm(s)∆(s)), Wm(s) =τmaxjω

1 + τmaxjω

with ‖∆‖∞ ≤ 1which can be represented as

with

N(s) =

[N11(s) N12(s)N21(s) N22(s)

]=

(0 I

G0Wm(s) G0(s)

)



Example 3: parametric uncertainties

Consider the first order system:

G(s) =1

s+ a, a0 − b < a < a0 + b

Define now:

a = a0 + δ.b

with |δ| < 1. Then it leads:

1

s+ a=

1

s+ a0 + δ.b=

1

s+ a0(1 +

δ.b

s+ a0)−1

This can then be represented as a Multiplicative Inverse Uncertainty:


withz = y∆ = 1

s+a0(w − bu∆)


Example 4: parametric uncertainties in state space equations

Let us consider the following uncertain system:

G :

x1 = (−2 + δ1)x1 + (−3 + δ2)x2

x2 = (−1 + δ3)x2 + uy = x1

(39)

In order to use an LFT, let us define the uncertain inputs:

u∆1= δ1x1, u∆2

= δ2x2, u∆3= δ3x2

Then the previous system can be rewritten in the following LFR:


where ∆ and y∆ are given as:

∆ =

δ1 0 00 δ2 00 0 δ3

, y∆ =

x1

x2

x2

and N given by the state space representation:

N :

x1

x2

==

−2x1 − 3x2 + u∆1+ u∆2

−x2 + u+ u∆3

y = x1

(40)


Towards LFR (LFT)

The previous computations are in fact the first step towards an unified representation of theuncertainties: the Linear Fractional Representation (LFR).Indeed the previous schemes can be rewritten in the following general representation as:

Figure: N∆ structure

This LFR gives then the transfer matrix from w to z, and is referred to as the upper LinearFractional Transformation (LFT) :

Fu(N,∆) = N22 +N21∆(I −N11∆)−1N12

This LFT exists and is well-posed if (I −N11∆)−1 is invertible



LFT definition

In this representation N is known and ∆(s) collects all the uncertainties taken into account for thestability analysis of the uncertain closed-loop system.∆(s) shall have the following structure:

∆(s) = diag ∆1(s), · · · ,∆q(s), δ1Ir1 , · · · , δrIrr , ε1Ic1 , · · · , εcIcc

with ∆i(s) ∈ RHki×ki∞ , δi ∈ R and εi ∈ C.Remark: ∆(s) includes

q full block transfer matrices,

r real diagonal blocks referred to as ’repeated scalars’ (indeed each block includes a realparameter δi repeated ri times),

c complex scalars εi repeated ci times.

Constraints: The uncertainties must be normalized, i.e such that:

‖∆‖∞ ≤ 1, |δi| ≤ 1, |εi| ≤ 1



Uncertainty types

We have seen in the previous examples the two important classes of uncertainties, namely:

UNSTRUCTURED UNCERTAINTIES: we ignore the structure of ∆, considered as a fullcomplex perturbation matrix, such that ‖∆‖∞ ≤ 1.We then look at the maximal admissible norm for ∆, to get Robust Stability and Performance.This will give a global sufficient condition on the robustness of the control scheme.This may lead to conservative results since all uncertainties are collected into a single matrixignoring the specific role of each uncertain parameter/block.

STRUCTURED UNCERTAINTIES: we take into account the structure of ∆, (always such that‖∆‖∞ ≤ 1).The robust analysis will then be carried out for each uncertain parameter/block.This needs to introduce a new tool: the Structured Singular Value. We then can obtain morefine results but using more complex tools.

The analysis is provided in what follows for both cases.In Matlabthis analysis is provided in the tools robuststab and robustperf.


Robust analysis Definition of Robustness analysis

Robustness analysis: problem formulation

Since the analysis will be carried you for a closed-loop system, N should be defined as theconnection of the plant and the controller. Therefore, in the framework of the H∞ control, thefollowing extended General Control Configuration is considered:

Figure: P −K −∆ structure

and N is such thatN = Fl(P,K)



Robust analysis: problem definition

In the global P −K −∆ General Control Configuration, the transfer matrix from w to z (i.e theclosed-loop uncertain system) is given by:

z = Fu(N,∆)w,

with Fu(N,∆) = N22 +N21∆(I −N11∆)−1N12.and the objectives are then formulated as follows:

Nominal stability (NS): N is internally stable

Nominal Performance (NP): ‖N22‖∞ < 1 and NS

Robust stability (RS): Fu(N,∆) is stable ∀∆, ‖∆‖∞ < 1 and NS

Robust performance (RP): ‖Fu(N,∆)‖∞ < 1 ∀∆, ‖∆‖∞ < 1 and NS



Towards Robust stability analysis

Robust Stability= with a given controller K, we determine wether the system remains stable for allplants in the uncertainty set.According to the definition of the previous upper LFT, when N is stable, the instability may onlycome from (I −N11∆). Then it is equivalent to study the M −∆ structure, given as:

Figure: M −∆ structure

This leads to the definition of the Small Gain Theorem

Theorem (Small Gain Theorem)

Suppose M ∈ RH∞. Then the closed-loop system in Fig. 7 is well-posed and internally stablefor all ∆ ∈ RH∞ such that :

‖∆‖∞ ≤ δ(resp. < δ) if and only if ‖M(s)‖∞ < 1/δ(resp. ‖M(s)‖∞ ≤ 1)


Robust analysis Robustness analysis: the unstructured case

Definition of the uncertainty types

Figure: 6 uncertainty representations



Robust stability analysis: additive case

Objective: applying the Small Gain Theorem to these unstructured uncertainty representations.


Let us consider the following simple controlscheme as:


The objective is to obtain:

Additive case:G(s) = G(s) +WA(s)∆A(s).Computing the N −∆ form gives

N(s) =

(−WAKSy WAKSy

Sy Ty

)Output Multiplicative uncertainties:G(s) = (I +WO(s)∆O(s))G(s).Then it leads

N(s) =

(−WOTy WOTySy Ty

)


General results

Theorem (Small Gain Theorem)

Consider the different uncertainty types, and assume that NS is achieved, i.e M ∈ RH∞ for eachtype. Then the closed-loop system is robustly stable, i.e. internally stable for all ∆k ∈ RH∞ (fork =A, 0, I, iO, iI) such that :

Additive : ‖WAKSy‖∞ ≤ 1Additive Inverse: ‖WiASy‖∞ ≤ 1Output Multiplicative: ‖WOTy‖∞ ≤ 1Input Multiplicative: ‖WITu‖∞ ≤ 1Output Inverse Multiplicative: ‖WiOSy‖∞ ≤ 1Input Inverse Multiplicative: ‖WiISu‖∞ ≤ 1

This gives some robustness templates for the sensitivity functions. However this may beconservative.



Illustration on the SISO case

Here Robust Stability is analyzed through the Nyquist plot. For illustration, let us consider the caseof Multiplicative uncertainties (Input and Output case are identical for SISO systems), i.e

G = G(I +Wm∆m)

Then the loop transfer function is given as:

L = GK = GK(I +Wm∆m) = L+WmL∆m;


According to the Nyquist theorem, RS isachieved the the closed-loop system isstable for any L should not encircle, i.e Lshould not encircle -1 for all uncertainties.According to the figure, a sufficient conditionis then:

|WmL| < |1 + L| , ∀ω⇔∣∣∣WmL

1+L

∣∣∣ < 1, ∀ω⇔ |WmT | < 1 ∀ω


A first insight in Robust Performance

Objective: applying the Small Gain Theorem to these unstructured uncertainty representations.


Let us consider the following simple controlscheme as:


Case of Output Multiplicative uncertainties:G(s) = (I +WO(s)∆O(s))G(s).Computing the N −∆ form gives

N(s) =

[N11(s) N12(s)N21(s) N22(s)

]=

(−WOTy WOTy−WeSy WeSy

)

We wish to get:The objectives are then formulated asfollows:

NS: N is internally stable

NP: ‖WeSy‖∞ < 1 and NS

RS: ‖WOTy‖∞ < 1 and NS

RP: ‖Fu(N,∆)‖∞ < 1 ∀∆, ‖∆‖∞ < 1,Sufficient condition: NS andσ(WOTy) + σ(WeSy) < 1, ∀ω


Illustration on the SISO case

Here Robust Performance is analyzed through the Nyquist plot. For illustration, let us consider thecase of Multiplicative uncertainties (Input and Output case are identical for SISO systems), i.e

G = G(I +Wm∆m)

Then the loop transfer function is given as:

L = GK = GK(I +Wm∆m) = L+WmL∆m;


First NP is achieved when:|WeS| < 1 ∀ω, ⇔ |We| <|1 + L| , ∀ω.Therefore RP is achieved if∣∣∣WeS

∣∣∣ < 1, ∀S,∀ω

⇔ |We| <∣∣∣1 + L

∣∣∣ , ∀L, ∀ωSince

∣∣∣1 + L∣∣∣ ≥ |1 + L| − |WmL∆m|, a

sufficient condition is actually:

|We|+ |WmL| < |1 + L| , ∀ω⇔ |WeS|+ |WmT | < 1, ∀ω

Robust analysis Robustness analysis: the structured case

The structured case

∆ = diag∆1, · · · ,∆q , δ1Ir1 , · · · , δrIrr , ε1Ic1 , · · · , εcIcc ∈ Ck×k (41)

with ∆i ∈ Cki×ki , δi ∈ R, εi ∈ C,where ∆i(s), i = 1, . . . , q, represent full block complex uncertainties, δi(s), i = 1, . . . , r, realparametric uncertainties, and εi(s), i = 1, . . . , c, complex parametric uncertainties.Taking into account the uncertainties leads to the following General Control Configuration,

---

-

∆(s)

K(s)

y

e

z∆r

ω

v

P (s)

v∆r

Figure: General control configuration with uncertainties

where ∆ ∈ ∆.



The structured singular value

To handle parametric uncertainties, we need to introduce µ, the structured singular value, definedas:

Definition (µ)

For M ∈ Cn×n, the structure singular value is defined as:

µ∆(M) :=1

minσ(∆) : ∆ ∈ ∆, det(I −∆M) 6= 0

In other words, it allows to find the smallest structured ∆ which makes det(I −M∆) = 0.

Theorem (The structured Small Gain Theorem)

Let M(s) be a MIMO LTI stable system and ∆(s) a LTI uncertain stable matrix, (i.e. ∈ RH∞).The system in Fig. 7 is stable for all ∆(s) in (41) if and only if:

∀ω ∈ R µ∆ (M(jω)) ≤ 1, with M(s) := Nzv(s)

More generally both following statements are equivalent

For µ ∈ R, N(s) and ∆(s) belong to RH∞, and

∀ω ∈ R, µ∆ (M(jω)) ≤ µ

the system represented in figure 7 is stable for any uncertainty ∆(s) of the form (41) suchthat :

||∆(s)| |∞ < 1/µO. Sename [GIPSA-lab] 108/133


Build the whole control scheme



Introduction of a fictive block

Usually only real parametric uncertainties (given in ∆r) are considered for RS analysis. RPanalysis also needs a fictive full block complex uncertainty, as below,

- -

-

?

∆f

∆(s)

∆r

N(s)ω e

v∆rz∆r

N(s)

Figure: N∆

where N(s) =

[N11(s) N12(s)N21(s) N22(s)

], and the closed-loop transfer matrix is:

Tew(s) = N22(s) +N21(s)∆(s)(I −N11(s))−1N12(s) (42)



Robust analysis theorem

For RS, we shall determine how large ∆ (in the sense of H∞) can be without destabilizing thefeedback system. From (42), the feedback system becomes unstable if det(I −N11(s) = 0 forsome s ∈ C,<(s) ≥ 0. The result is then the following.

Theorem

Assume that the nominal system New and the perturbations ∆ are stable. Then the feedbacksystem is stable for all allowed perturbations ∆ such that ||∆(s)| |∞ < 1/β if and only if∀ω ∈ R, µ∆ (N11(jω)) ≤ β.

Assuming nominal stability, RS and RP analysis for structured uncertainties are therefore suchthat:

NP ⇔ σ(N22) = µ∆f(N22) ≤ 1, ∀ω

RS ⇔ µ∆r (N11) < 1, ∀ω

RP ⇔ µ∆(N) < 1, ∀ω, ∆ =

[∆f 00 ∆r

]Finally, let us remark that the structured singular value cannot be explicitly determined, so that themethod consists in calculating an upper bound and a lower bound, as closed as possible to µ.



Summary

The steps to be followed in the RS/RP analysis for structured uncertainties are then:

Definition of the real uncertainties ∆r and of the weighting functions

Evaluation of µ(N22)∆f, µ(N11)∆r

and µ(N)∆

Computation of the admissible intervals for each parameter

Remark: The Robust Performance analysis is quite conservative and requires a tight definition ofthe weighting functions that do represent the performance objectives to be satisfied by theuncertain closed-loop system. Therefore it is necessary to distinguish the weighting functionsused for the nominal design from the ones used for RP analysis.


Robust analysis Robust control design

Brief overview on robust control design

In order to design a robust control, i.e. a controller for which the synthesis actually accounts foruncertainties, some of the methods are:

Unstructured uncertainties: Consider an uncertainty weight (unstructured form), andinclude the Small Gain Condition through a new controlled output. For example, robustnessface to Ouptut Multiplicative Uncertainties can be considered into the design procedureadding the controlled output ey = WOy, which, when tracking performance is expected, leadsto the condition ‖WOTy ‖∞≤ 1.

Structured uncertainties: the design of a robust controller in the presence of suchuncertainties is the µ− synthesis. It is handled through an interactive procedure, referred toas the DK iteration. This procedure is much more involved than a "simple" H∞ controldesign and often leads to an increase of the order of the controller (which is already the sumof the order of the plant and of the weighting functions).

Use other mathematical representation of parametric uncertainties, [?], as for instance thepolytopic model. In that case the set of uncertain parameters is assumed to be a polytope(i.e. a convex) set. The stability issue in that framework is referred to as the ’Quadraticstability’ i.e find a single Lyapunov function for the uncertainty set. While in the general casethis is an unbounded problem, in the polytopic case (or in the affine case), the stability is to beanalyzed only at the vertices of the polytope, which is a finite dimensional problem.This approach can then be applied to find a single controller, valid over the potyopic set. Notethat this approach gives rise to the LPV design for polytopic systems, as described next.


LPV systems

Outline









LPV systems LPV modelling

The LPV approach

Definition of an Linear Parameter Varying system

Σ(ρ) :

xzy

=

A(ρ) B1(ρ) B2(ρ)C1(ρ) D11(ρ) D12(ρ)C2(ρ) D21(ρ) D22(ρ)

xwu

x(t) ∈ Rn, ...., ρ = (ρ1(t), ρ2(t), . . . , ρN (t)) ∈ Ω, is a vector of time-varying parameters (Ωconvex set)

System (ρ)

ExogeneousInputs

Controlledoutputs

Measuredoutputs

z

ControlInputs

w

u y

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



Different Models

According to the dependency on the parameter set, we may have several classes of models:1 Affine parameter dependency: A(ρ) = A0 +A1ρ1 + ...+ANρN2 Polynomial dependency: A(ρ) = A0 +A1ρ+A2ρ2 + ...+ASρ

S

3 Rationaldependency:A(ρ) = [An0 +An1ρn1 + ...+AnNρnN ][I +Ad1ρd1 + ...+AdNρdN ]−1

If ρ = ρ(x(t), t) then the system is referred to as quasi-LPV.For instance:

x(t) = x2(t) = ρ(t)x(t)

with ρ = x.



Polytopic models

Let us denote N the number of parameters.The parameter are assumed to be bounded : ρi ∈

[ρi

ρi].

The vector of parameters evolves inside a polytope represented by Z = 2N vertices ωi, as

ρ ∈ Coω1, . . . , ωZ (43)

It is then written as the convex combination:

ρ =Z∑i=1

αiωi, αi ≥ 0,Z∑i=1

αi = 1 (44)

where the vertices are defined by a vector ωi = [νi1, . . . , νiN ] where νij equals ρj or ρj .A system is then represented as

Σ(ρ) =Z∑k=1

αk(ρ)

[Ak BkCk Dk

], with

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

where[Ak BkCk Dk

]is a LTI system corresponding to a vertex k.



Poytopic models

For a LPV system with 2 parameters, boundend[ρ

1,2ρ1,2

], the corresponding polytope

onws 4 vertices as:Pρ =

(ρ

1, ρ

2), (ρ

1, ρ2), (ρ1, ρ2

), (ρ1, ρ2)

(45)

The polytopic coordinates are (αi) are obtained as:

ω1 = (ρ1, ρ

2), α1 =

(ρ1 − ρ1

ρ1 − ρ1

)×(ρ2 − ρ2

ρ2 − ρ2

)

ω2 = (ρ1, ρ2), α2 =

(ρ1 − ρ1

ρ1 − ρ1

)×(ρ2 − ρ2

ρ2 − ρ2

)

ω3 = (ρ1, ρ2), α3 =

(ρ1 − ρ1

ρ1 − ρ1

)×(ρ2 − ρ2

ρ2 − ρ2

)

ω4 = (ρ1, ρ2), α4 =

(ρ1 − ρ1

ρ1 − ρ1

)×(ρ2 − ρ2

ρ2 − ρ2

)(46)

where ρ1 and ρ2 are the instantaneous values of the parameters (ρ(k)i in the implementation step).

The LPV system is then rewritten under the polytopic representation:(A(ρ1,2) B(ρ1,2)C(ρ1,2) D(ρ1,2)

)= α1

(A(ω1) B(ω1)C(ω1) D(ω1)

)+ α2

(A(ω2) B(ω2)C(ω2) D(ω2)

)+ α3

(A(ω3) B(ω3)C(ω3) D(ω3)

)+ α4

(A(ω4) B(ω4)C(ω4) D(ω4)

)(47)



LFT models

z∆w∆

zw P

∆

u y

Figure: System under LFT form

The equation of the system under LFR representation is asfollows:

xz∆zy

=

A B∆ B1 B2

C∆ D∆∆ D∆1 D∆2

C1 D1∆ D11 D12

C2 D2∆ D21 D22

xw∆

wu



Towards a LFT/LPV model of an AUV

Considered the NL model with 2 state variables (the pitch angle θ and velocity q):

θ =cos(φ)q − sin(φ)r

Mq =− p× r(Ix − Iz)−m[Zg(q × w − r × v)]− (Zqm− ZfµV )gsin(θ)

− (Xqm−XfµV )gcos(θ)cos(φ) +Mwqw|q|+Mqqq|q|+ Ffins (48)

where M interia matrix, m AUV mass, V volume and µ mass density. The other parameters(Ix, Iz , Zg , Zq , Zf , Xq , Xf ,Mwq) appear in the dynamical and hydrodynamical functions.Ffins : forces and moments due to the fins.

Tangent linearization around Xeq = [0 ueq 0 0 0 0 0 0 θeq 0 0 0]:

˙θ = q

M ˙q = [−(Zgm− ZfµV )gcos(θeq) + (Xgm−XfµV )gsin(θeq)]θ + Ffinseq

where θ and q are the variations of θ and q.



A LFR model

An LFR model of the vehicle described in the form:

z∆w∆

zkwk P (z)

∆(ρ)

uk yk

The ∆ block contains the varying part of the model, which depends on the linearization point (θeq).The LFR form is defined by:

xk+1 = Axk +[B∆ B1

] (u∆

u

)(y∆

y

)=

[C∆C1

]xk +

[D∆ 00 D1

](u∆

u

) (49)



A LFR model

with

A =

[0 10 0

], B∆ =

[0 0

−(Zqm− ZfρV )g (Xqm−XfρV )g

]B1 =

[0

Ffins

], C∆ =

[1 10 0

], C1 =

[1 00 1

]D∆ =

[0 00 0

], D1 =

[0 00 0

](50)

z∆ =

[∆θ∆θ

]; u∆ = ∆z∆; ∆ =

[ρ1 00 ρ2

](51)

with: ρ1 = cos(θeq)ρ2 = sin(θeq)

(52)


LPV systems LPV Control

Towards LPV control

The "gain scheduling" approach

System (ρ)Controller (ρ)

Adaptationstrategy

Measured or estimatedParameters

References ControlInputs

Outputs

Externalparameters

Some references

Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth)

Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker ...)

Stability, stabilization (Scherer, Wu, Blanchini ...)

Geometric analysis (Bokor & Balas)



The H∞/LPV control problem

Definition

Find a LPV controller C(ρ) s.t the closed-loop system is stable and for γ∞ > 0, sup ‖z‖2‖w‖2< γ∞,

Unbounded set of LMIs (Linear Matrix Inequalities) to be solved (ρ ∈ Ω)

Some approaches: polytopic, LFT, gridding. See Arzelier [HDR, 2005], Bruzelius [Thesis,2004], Apkarian et al. [TAC, 1995]...

A solution: The "polytopic" approach [C. Scherer et al. 1997]

Problem solved off line for each vertex of a polytope (convex optimisation) (using here asingle Lyapunov function i.e. quadratic stabilization).

On-line the controller is computed as the convex combination of local linear controllers

C(ρ) =

2N∑k=1

αk(ρ)

[Ac(ωk) Bc(ωk)Cc(ωk) Dc(ωk)

],

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)

Easy implementation !!



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)

Problem on standard form

CL (ρ)

w: exagenous input u: control input

z: output to minimize y: measurement

Dynamical LPV generalized plant:

Σ(ρ) :

xzy

=

A(ρ) B1(ρ) B2(ρ)C1(ρ) D11(ρ) D12(ρ)C2(ρ) D21(ρ) D22(ρ)

xwu

(53)

LPV controller structure:

S(ρ) :

[xc

u

]=

[Ac(ρ) Bc(ρ)Cc(ρ) Dc(ρ)

] [xc

y

](54)

LPV closed-loop system:

CL(ρ) :

[ξz

]=

[A(ρ) B(ρ)C(ρ) D(ρ)

] [ξw

](55)



LPV control design

H∞ criteria Apkarian et al. [TAC, 1995]

Stabilize system CL(ρ) (find K > 0) while minimizing γ∞. A(ρ)TK +KA(ρ) KB∞(ρ) C∞(ρ)T

B∞(ρ)TK −γ2∞I D∞(ρ)T

C∞(ρ) D∞(ρ) −I

< 0

Infinite set of LMIs to solve (ρ ∈ Ω) (Ω is convex)

LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004]

LFT, Gridding, Polytopic



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N∑k=1

αk(ρ)

[Ac(ωk) Bc(ωk)Cc(ωk) Dc(ωk)

]where,

αk(ρ) =

∏Nj=1 |ρj − Cc(ωk)j |∏N

j=1(ρj − ρj),

where Cc(ωk)j = ρj if (ωk)j = ρj

or ρj otherwise.

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)



LPV/H∞ control synthesis

Proposition - feasibility (brief) Scherer et al. (1997)

Solve the following problem at each vertices of the parametrized points (illustration with 2parameters):

γ∗ = min γs.t. (57) |ρ1,ρ2s.t. (57) |ρ1,ρ2s.t. (57) |ρ1,ρ2s.t. (57) |ρ1,ρ2

(56)

AX +B2C(ρ1, ρ2) + (?)T (?)T (?)T (?)T

A(ρ1, ρ2) +AT YA+ B(ρ1, ρ2)C2 + (?)T (?)T (?)T

BT1 BT1 Y +DT21B(ρ1, ρ2)T −γI (?)T

C1X +D12C(ρ1, ρ2) C1 D11 −γI

≺ 0

[X II Y

] 0

(57)



LPV/H∞ control synthesis

Proposition - reconstruction (brief) Scherer et al. (1997)

Reconstruct the controllers as,solve (59) |ρ1,ρ2

(59) |ρ1,ρ2(59) |ρ1,ρ2(59) |ρ1,ρ2

(58)

Cc(ρ1, ρ2) = C(ρ1, ρ2)M−T

Bc(ρ1, ρ2) = N−1B(ρ1, ρ2)

Ac(ρ1, ρ2) = N−1(A(ρ1, ρ2)− Y AX −NBc(ρ1, ρ2)C2X

− Y B2Cc(ρ1, ρ2)MT)M−T

(59)

where M and N are defined such that MNT = I −XY which may be chosen by applying asingular value decomposition and a Cholesky factorization.



Interest of the LPV approach

LPV is a key tool to the control of complex systems.

Some examples :

Modelling of complex systems (non linear)

Use of a quasi-LPV representation to include non linearities in a linear state space model(even delays)

Transformation of constraints (e.g. saturation) into an ’external’ parameter

Modelling of LTV, hybrid (e.g. switching control)

BUT :

A q-LPV system is not equivalent to the non linear one:

stability: ρ = ρ(x(t), t) is assumed to be bounded... so are the state trajectories

controllability: some non controllable modes of a non linear system may vanish according tothe LPV representation



Interest of the LPV approach

Some of works using LPV approaches - former PhD students

Gain-scheduled control

Account for various operating conditions using a variable "equilibrium point": (Gauthier 2007)

Control with real-time performance adaptation using parameter dependent weightingfunctions from endogenous or exogenous parameters (Poussot 2008, Do 2011)

Analysis and control of LPV Time-Delay Systems: delay-scheduled control Briat 2008

Control under computation constraints: H∞ variable sampling rate controller with samplingdependent performances (Robert 2007, Roche 2011, Robert et al., IEEE TCST 2010))

Coordination of several actuators for MIMO systems

An LPV structure for control allocation Poussot et al. (CEP 2011)

Selection of a specific parameter for the control activation Poussot et al. (VSD 2011)



Some PhD students on robust and/or LPV control

Maria Rivas, "Modeling and Control of a Spark Ignited Engine for Euro 6 European Normative", PhD, GIPSA-lab / RENAULT, Grenoble INP, 2012.

Ahn-Lam Do, "LPV Approach for Semi-active Suspension Control & Joint Improvement of Comfort and Security", PhD, GIPSA-lab, Grenoble INP,2011.

David Hernandez, "Robust control of hybrid electro-chemical generators", PhD, GIPSA-lab / G2Elab, Grenoble INP, 2011.

Emilie Roche, "Commande Linéaire à Paramètres Variants discrète à échantillonnage variable : application à un sous-marin autonome", PhD,GIPSA-lab, Grenoble INP, 2011.

Sébastien Aubouet, "Semi-active SOBEN suspensions modelling and control", PhD, GIPSA-lab / SOBEN, INP Grenoble, 2010.

Charles Poussot-Vassal, "Robust LPV Multivariable Global Chassis Control", PhD , GIPSA-lab, INP Grenoble, 2008.

Corentin Briat, "Robust control and observation of LPV time-delay systems", PhD, GIPSA-lab, INP Grenoble, 2008.

Christophe Gauthier, "Commande multivariable de la pression d’injection dans un moteur Diesel Common Rail", PhD, LAG / DELPHI, Grenoble INP,2007.

David Robert, "Contribution à l’interaction commande/ordonnacement", PhD, LAG, Grenoble INP, 2007.

Marc Houbedine, "Contribution pour l’amélioration de la robustesse et du bruit de phase des synthétiseurs de fréquences" , PhD, LAG / STMicrolectronics, Grenoble INP, 2006.

Alessandro ZIN, "Sur la commande robuste de suspensions automobiles en vue du contrôle global de châssis", PhD, LAG / Grenoble INP, 2005.

Julien Brely, " Régulation multivariable de filières de production de fibre de verre", PhD, LAG / ST Gobain Vetrotex, Grenoble INP, 2003.

Giampaolo Filardi, "Robust Control design strategies applied to a DVD-video player", PhD, LAG / ST Microlectronics, Grenoble INP, 2003.

Damien Sammier, "Modélisation et commmande de véhicules automobiles: application aux éléments suspension", PhD, LAG / Grenoble INP, 2001.

Anas Fattouh, "Observabilité et commande des systèmes linéaires à retards", PhD, LAG / Grenoble INP, 2000.



Some references

Doumiati, Moustapha; Sename, Olivier; Dugard, Luc; Martinez Molina, John Jairo; Gaspar, Peter; Szabo, Zoltan, «Integrated vehicle dynamics controlvia coordination of active front steering and rear braking", to appear in European Journal of Control, n/c (2013)

Dugard, Luc; Sename, Olivier; Aubouet, Sébastien; Talon, Benjamin, "Full vertical car observer design methodology for suspension controlapplications", Control Engineering Practice, vol 20, Issue 9, pp 832-845, 2012.

C. Poussot-Vassal, O. Sename, L. Dugard, P. Gaspar, Z. Szabo & J. Bokor, "Attitude and Handling Improvements Through Gain-scheduledSuspensions and Brakes Control", Control Engineering Practice (CEP), Vol. 19(3), March, 2011, pp. 252-263.

C. Poussot-Vassal, O. Sename, L. Dugard, S.M. Savaresi, "Vehicle Dynamic Stability Improvements Through Gain-Scheduled Steering and BrakingControl", Vehicle System Dynamics (VSD), vol 49, Nb 10, pp 1597-1621, 2011

Briat, C.; Sename, O., "Design of LPV observers for LPV time-delay systems: an algebraic approach", International Journal of Control, vol 84, nb 9,pp 1533-1542, 2011

S.M. Savaresi, C. Poussot-Vassal, C. Spelta, O. Sename & L. Dugard. "Semi-Active Suspension Control Design for Vehicles", Elsevier,Butterworth-Heinmann, 2010.

Robert, D., Sename, O., and Simon, D. (2010). An H∞ LPV design for sampling varying controllers: experimentation with a T inverted pendulum.IEEE Transactions on Control Systems Technology, 18(3):741–749.

C. Briat, O. Sename, and J.-F. Lafay, ""Memory-resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays"Systems & Control Let., vol. 59, pp. 451-459, 2010.

C. Briat, O. Sename, and J. Lafay, "Hinf delay-scheduled control of linear systems with time-varying delays," IEEE Transactions in Automatic Control,vol. 42, no. 8, pp. 2255-2260, 2009.

Martinez Molina J. J., Sename O., Voda A., "Modeling and robust control of Blu-ray disc servo-mechanisms", Mechatronics, (2009) vol 19, n5, pp715-725

Poussot-Vassal, C.; Sename, O.; Dugard, L.; Gaspar, P.; Szabo, Z. & Bokor, J. "A New Semi-active Suspension Control Strategy Through LPVTechnique", Control Engineering Practice, 2008, 16, 1519-1534

Zin A., Sename O., Gaspar P., Dugard L., Bokor J. "Robust LPV - Hinf Control for Active Suspensions with Performance Adaptation in view of GlobalChassis Control", Vehicle System Dynamics, Vol. 46, No. 10, 889-912, October 2008

Gauthier C., Sename O., Dugard L., Meissonnier G. "An LFT approach to Hinf control design for diesel engine common rail injection system", Oil &Gas Science and Technology, vol 62, nb 4, pp. 513-522 (2007)

Sammier D. ; Sename O. ; Dugard L , "Skyhook And Hinf Control Of Semi-Active Suspensions: Some Practical Aspects", Vehicle Systems Dynamics,Vol. 39, No 4, 2003, Pp. 279-308


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