Fractional-order Calculus based Design and Imple ... · Fractional-order Calculus based Design and...

Fractional-order Calculus based Design and Imple-

mentation of Robust Industrial Control

Aleksei Tepljakov

October 30, 2013

Talk overview

Aleksei Tepljakov 2 / 86

• Mathematical basis of fractional-order calculus;

• Identification of fractional-order systems:

Model classification, stability, time-domain identification;

Transition from conventional process models to fractional-order ones;

• Tuning of fractional-order controllers:

PIλDµ controllers, robustness characteristics and design specifications;

Transition from conventional controllers to those of fractional-order;

• Analog and digital implementations of fractional-order systems and controllers:

An updated continuous-time approximation method;

A framework for the synthesis of electrical network approximations;

Digital implementation for embedded control applications.

• Applications of fractional-order controllers: A case study.

Introduction: Mathematical Basis ofFractional-order Calculus


Introduction: Historical facts


• The concept of the differentiation operator D = d/dx is awell-known fundamental tool of modern calculus. For a suitablefunction f the n-th derivative is well defined as

Dnf(x) = d nf(x)/dxn, (1)

where n is a positive integer.

• What happens if we extend this concept to a situation, whenthe order of differentiation is arbitrary, for example, fractional?

• That was the very same question L’Hôpital addressed to Leibnizin a letter in 1695. Since then the concept of fractional calculushas drawn the attention of many famous mathematicians,including Euler, Laplace, Fourier, Liouville, Riemann, Abel.

Fractional derivative of a power function:An approach based on intuition


For the power function f(x) = xk the fractional derivative can beshown to be

dαf(x)

dxα=

Γ(k + 1)

Γ(k − α+ 1)xk−α, (2)

where Γ(·) is the Gamma function—generalization of the factorialfunction—defined as

Γ(x) =

∫

∞

0tx−1e−tdt, x > 0. (3)

For example,d1/2(x2)

dx1/2=

Γ(3)

Γ(5/2)x

3/2 =8x3/2

3√π.

Example: fractional-order derivative of afunction f(x) = x


Fractional derivative of a trigonometricfunction: An approach based on intuition


We observe, what happens when we repeatedly differentiate thefunction f(x) = sinx:

d

dxsinx = cosx,

d2

dx2sinx = − sinx,

d3

dx3sinx = − cosx, . . .

The pattern can be deduced: for the nth derivative, the functionsinx is shifted by nπ/2 radians. This is clear when observing agraph of the function. Thus, if we replace n by α ∈ R

+, we have

dα

dxαsinx = sin

(

x+απ

2

)

. (4)

Obviously, a similar equation holds for the cosine function as well.

Example: Half derivative of a sine function


0 1 2 3 4 5 6−1.5

−1

−0.5

0

0.5

1

1.5

t

Am

plitu

de

sin(t)

d0.5/dt0.5 (sin(t))π / 4

The generalized operator


Fractional calculus is a generalization of integration anddifferentiation to non-integer order operator aD

αt , where a and t

denote the limits of the operation and α denotes the fractionalorder such that

aDαt =

dα

dtα ℜ(α) > 0,

1 ℜ(α) = 0,∫ ta (dτ)

−α ℜ(α) < 0,

(5)

where generally it is assumed that α ∈ R, but it may also be acomplex number. We restrict our attention to the former case.

Fractional-order derivative: Definitions


Definition 1. (Riemann-Liouville)

Ra D

αt f(t) =

1

Γ(m− α)

(

d

dt

)m[

∫ t

a

f(τ)

(t− τ)α−m+1dτ

]

, (6)

where m− 1 < α < m, m ∈ N,α ∈ R+.

Definition 2. (Caputo)

C0 D

αt f(t) =

1

Γ(m− α)

∫ t

0

f (m)(τ)

(t− τ)α−m+1dτ, (7)

where m− 1 < α < m, m ∈ N.

Definition 3. (Grünwald-Letnikov)

GLD

αf(t)|t=kh = limh→0

1

hα

k∑

j=0

(−1)j(

α

j

)

f(kh− jh). (8)

Properties of fractional-order differentiation


Fractional-order differentiation has the following properties:

1. If α = n and n ∈ Z+, then the operator 0D

αt can be

understood as the usual operator dn/dtn.

2. Operator of order α = 0 is the identity operator:

0D0t f(t) = f(t).

3. Fractional-order differentiation is linear; if a, b are constants,then

0Dαt

[

af(t) + bg(t)]

= a 0Dαt f(t) + b 0D

αt g(t). (9)

4. If f(t) is an analytic function, then the fractional-orderdifferentiation 0D

αt f(t) is also analytic with respect to t.

Properties of fractional-order differentiation(continued)


5. For the fractional-order operators with ℜ(α) > 0,ℜ(β) > 0,and under reasonable constraints on the function f(t) (fordetails see [2]) it holds the additive law of exponents:

0Dαt

[

0Dβt f(t)

]

= 0Dβt

[

0Dαt f(t)

]

= 0Dα+βt f(t) (10)

6. The fractional-order derivative commutes with integer-orderderivative

dn

dtn(

aDαt f(t)

)

= aDαt

(

dnf(t)

dtn

)

= aDα+nt f(t), (11)

and if t = a we have f (k)(a) = 0, (k = 0, 1, 2, ..., n− 1).

Fractional-order derivative definitions:Laplace transform


Definition 4. (Riemann-Liouville)

L

[

RD

αf(t)]

= sαF (s)−m−1∑

k=0

sk[

Dα−k−1f(t)

]

t=0. (12)

Definition 5. (Caputo)

L

[

CD

αf(t)]

= sαF (s)−m−1∑

k=0

sα−k−1f (k)(0). (13)

Definition 6. (Grünwald-Letnikov)

L

[

GLD

αf(t)]

= sαF (s). (14)

For the first two definitions we have (m− 1 6 α < m).

On the meaning of the fractional-orderderivative


We shall call F (ft(·), t) a hereditary operator acting on a causeprocess ft(·) to produce a time-shifted effect g(t) which dependson the history of the process ft(τ); τ < t:

g(t) = F[

ft(·); t]

. (15)

We can replace g(t) by the function f(t) or its derivatives, i.e.

df(t)

dt= F

[

ft(·); t]

(16)

and so on.

Some hereditary process examples from physics: Brownian motion;Viscoelasticity.

Part I: Identification of Fractional-orderSystems


Identification of Fractional-order Systems


Author’s contribution:

• Research of time-domain (non)commensurate-order fractional modeloutput error based identification methods;

• Study of fractional-order characteristics of industrial plant models.

Publications:

1. A. Tepljakov, E. Petlenkov, and J. Belikov, “FOMCON: a MATLABtoolbox for fractional-order system identification and control,”International Journal of Microelectronics and Computer Science, vol. 2,no. 2, pp. 51–62, 2011

2. A. Tepljakov, E. Petlenkov, J. Belikov, and M. Halás, “Design andimplementation of fractional-order PID controllers for a fluid tank system,”in Proc. 2013 American Control Conference (ACC), Washington DC, USA,June 2013, pp. 1780–1785

Linear, time invariant fractional-ordersystem classification


Fractional-order transfer functions


A linear, fractional-order continuous-time dynamic system can be expressed bya fractional differential equation of the following form

anDαny(t) + an−1D

αn−1y(t) + · · ·+ a0Dα0y(t) = (17)

bmDβmu(t) + bm−1D

βm−1u(t) + · · ·+ b0Dβ0u(t),

We apply the Laplace transform with zero initial conditions and obtain thefractional-order transfer function:

G(s) =Y (s)

U(s)=

bmsβm + bm−1sβm−1 + · · ·+ b0s

β0

ansαn + an−1sαn−1 + · · ·+ a0sα0

. (18)

In the case of a system with commensurate order q we have

G(s) =

m∑

k=0

bk (sq)k

n∑

k=0

ak (sq)k. (19)

Stability


Theorem 1. (Matignon’s stability theorem) The fractionaltransfer function G(s) = Z(s)/P (s) is stable if and only if thefollowing condition is satisfied in σ-plane:

∣

∣arg(σ)∣

∣ > qπ

2, ∀σ ∈ C, P (σ) = 0, (20)

where σ := sq. When σ = 0 is a single root of P (s), the systemcannot be stable. For q = 1, this is the classical theorem of polelocation in the complex plane: no pole is in the closed right planeof the first Riemann sheet.

Algorithm summary: Find the commensurate order q of P (s), finda1, a2, . . . an in (19) and solve for σ the equation

∑nk=0 akσ

k = 0.If all obtained roots satisfy the condition (1), the system is stable.

Stability regions


Time-domain identification: Methoddescription


Given the transfer function model in (18)

G(s) =bmsβm + bm−1s

βm−1 + · · ·+ b0sβ0

ansαn + an−1sαn−1 + · · ·+ a0sα0

we search for a parameter set θ = [ ap αp bz βz ], such that

ap = [ an an−1 · · · a0 ], αp = [ αn αn−1 · · · α0 ],

bz = [ bm bm−1 · · · b0 ], βz = [ βn βn−1 · · · β0 ],

by employing numerical optimization with an objective functiongiven by an output error norm

∥

∥e (t)∥

∥

2

2, where e(t) = y(t)− y(t) is

obtained by taking the difference of the original model output y(t)and simulated model output y(t).

Time-domain identification: Process models


Consider the following generalizations of conventional process models used inindustrial control design.

(FO)FOPDT G(s) = K1+Ts

e−Ls G(s) = K1+Tsα

e−Ls

(FO)IPDT G(s) = Kse−Ls G(s) = K

sαe−Ls

(FO)FOIPDT G(s) = Ks(1+Ts)

e−Ls G(s) = Ks(1+Tsα)

e−Ls

Therefore, due to additional parameters K (gain) and L (delay) we mayupdate the optimized parameter set discussed previously to

θ = [ K L ap αp bz βz ].

Time-domain identification: FOMCONtoolbox


Identification example: Method validation


Suppose a fractional-order system is given by

G2 (s) =1

0.8s2.2 + 0.5s0.9 + 1.

In order to generate basic validation test data, the followingMATLAB commands can be used:

G2 = fotf(’1’,’0.8s^2.2+0.5s^0.9+1’);t = (0:0.01:20)’;u = zeros(length(t),1);u(1:200) = ones(200,1);u(1000:1500) = ones(501,1);y = lsim(G2,u,t)’;iddata1 = fidata(y,u,t);

Identification example: Method validation(continued)


In the time-domain identification tool, generate an initial model

Ginit(s) =1

s3 + s1.5 + 1.

After 82 iterations the following result is obtained with the errornorm enorm = 6.7310 · 10−10:

G(s) =1

0.8s2.2 + 0.5s0.90002 + 1s2.717·10−6 .

It is easy to see, that this model corresponds to the original one.To recover the model completely, choose from the identificationtool menu Model→Truncate and use the default accuracy of 0.001.

Identification example: Original response vs.Identified model (prior to truncation)


0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

2

Sys

tem

out

put y

(t)

Initial dataIdentified model

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Sys

tem

inpu

t u(t

)

0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

8x 10

−5

Time [s]

Out

put e

rror

Identification example: Identified modelstability check (prior to truncation)


Part II: Tuning of Fractional-order Controllers


Tuning of Fractional-order Controllers



• Development of a flexible, time-domain simulation based PIλDµ controller tuningmethod incorporating robustness specifications derived from open-loopfrequency-domain analysis;

• Research of auto-tuning methods for PIλDµ controllers and implementationthereof in a hardware prototype (results to be published);

• Case studies of applications of PIλDµ controllers to various industrial plantmodels.

Publications:

1. A. Tepljakov, E. Petlenkov, and J. Belikov, “A flexible MATLAB tool for optimalfractional-order PID controller design subject to specifications,” in Proceedings ofthe 31st Chinese Control Conference, W. Li and Q. Zhao, Eds., Hefei, Anhui,China, 2012, pp. 4698–4703

2. A. Tepljakov, E. Petlenkov, and J. Belikov, “Development of analytical tuningmethods for fractional-order controllers,” in Proc. of the Sixth IKTDK Informationand Communication technology Doctoral School Conf., 2012, pp. 93–96

Fractional-order controllers


The fractional PIλDµ controller, where λ and µ denote the orders of theintegral and differential components, respectively, is given by

C(s) = Kp +Ki

sλ+Kd · sµ. (21)

The transfer function, corresponding to the fractional lead-lagcompensator of order α, has the following form:

CL(s) = K

(

1 + bs

1 + as

)α

. (22)

When α > 0 we have the fractional zero and pole frequencies ωz = 1/b,ωh = 1/a and the transfer function in (22) corresponds to a fractionallead compensator. For α < 0, a fractional lag compensator is obtained.

Basics of fractional control: fractionalcontrol actions


Let a basic fractional control action be defined as C(s) = K · sγ .The control actions in the time domain for γ ∈ [−1, 1] with K = 1under different input signals are given below.

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [s]

u(t)

γ=0

γ=−0.5

γ=−0.7

γ=−1

Fractional integrator s−γ

0 2 4 6 8 10−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time [s]u(

t)

γ=0

γ=0.5

γ=0.7

γ=1

Fractional differentiator sγ

PID controller vs. PI0.5D0.5 controller:frequency-domain characteristics


10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

105

−90

−45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

0

20

40

60

80

100

Mag

nitu

de (

dB)

Classical PID

Fractional PID

Classical PID

Fractional PID

Optimization based PIλDµ tuning


Optimization provides general means of tuning a fractional-orderPID controller given a cost function and suitable optimizationconstraints. There are several aspects to the problem of designinga proper controller using constrained optimization:

• The type of plant to be controlled (integer or noninteger order,nonlinear);

• Optimization criterion (cost function);

• Fractional controller design specifications;

• Specific parameters to optimize;

• Selection of initial controller parameters.

Optimization based PIλDµ tuning: Costfunction


In case of a linear model we use time-domain simulation of atypical negative unity feedback loop

Gcs(s) =Gc(s)G(s)

1 +Gc(s)G(s). (23)

For the cost function we consider performance indicies:

• integral square error ISE =∫ τ0 e2(t)dt,

• integral absolute error IAE =∫ τ0

∣

∣e(t)∣

∣dt,

• integral time-square error ITSE =∫ τ0 te2(t)dt,

• integral time-absolute error ITAE =∫ τ0 t

∣

∣e(t)∣

∣dt.

Optimization based PIλDµ tuning:Constraints


The design specifications include:

• Gain margin Gm and phase margin ϕm specifications;

• Complementary sensitivity function T (jω) constraint, providingA dB of noise attenuation for frequencies ω > ωt rad/s;

• Sensitivity function S(jω) constraint for output disturbancerejection, providing a sensitivity function of B dB forfrequencies ω < ωs rad/s;

• Robustness to plant gain variations: a flat phase of the systemis desired within a region of the system critical frequency ωcg;

• For practical reasons, a constraint on the control effort u(t)may also be set.

Optimization based PIλDµ tuning: Initialparameter selection


The choice of initial PIλDµ parameters (Kp, Ki, Kd, λ and µ)depends on the selected set of parameters to optimize. There areseveral options available which depend on the particular controllerdesign strategy:

• Optimize all parameters;

• Optimize gains only;

• Optimize orders only.

It is possible to obtain the initial parameters to optimize by usingclassical, integer-order PID design methods, and then enhance theresulting controller by introducing the fractional orders with respectto design specifications.

Optimization based PIλDµ tuning:FOMCON toolbox


Optimization based PIλDµ tuning example


The task is to design a controller for a heating furnace described bya transfer function

G(s) =1

14494s1.31 + 6009.5s0.97 + 1.69.

First we approximate this model by a conventional FOPDT model:

GFOPDT (s) =0.4043

1 + 3440.71se−66.9311s.

Using the Ziegler-Nichols tuning formula the integer-order PIDparameters are obtained such that

Kp = 152.596,Ki = 1.13995,Kd = 5106.72.

Optimization based PIλDµ tuning example(continued)


Next, we take these parameters, fix the gains and optimize only theorders. Model approximation parameters for the Oustaloup refinedfilter are ω = [0.0001, 10000] rad/s, N = 5. The specifications areas follows

• Gain and phase margins Gm ≥ 10 dB, γ ≥ 75;

• Control law saturation ulim = [0; 750].

Using the ITSE performance index for optimization yields thefractional orders λ = 0.76145 and µ = 0.33145. A phase marginϕm = 102.84 is achieved.

The obtained result can be in part explained by relatively largedifferences in the model and the FOPDT approximation.

Optimization based PIλDµ tuning example:Closed-loop transient response comparison


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time [s]

Am

plitu

de

Ziegler−Nichols tuning

Optimized PIλDµ controller

Part III: Analog and Digital Implementationsof Fractional-order Controllers


Analog and Digital Implementations ofFractional-order Controllers



• Study and update of an existing continuous-time method for fractional-orderoperator approximation;

• Research and development of a unified framework for analog approximations offractional-order systems and controllers;

• Development of a method for digital implementation of fractional-order filterapproximation for embedded control applications (to be published).

Publications:

1. A. Tepljakov, E. Petlenkov, and J. Belikov, “Application of Newton’s method toanalog and digital realization of fractional-order controllers,” International Journalof Microelectronics and Computer Science, vol. 2, no. 2, pp. 45–52, 2012

2. A. Tepljakov, E. Petlenkov, and J. Belikov, “Efficient analog implementations offractional-order controllers,” in Proc. of the 14th International Carpathian ControlConference (ICCC), 2013, pp. 377–382

Carlson’s approximation of fractionalcapacitors


This particular approximation of nth roots was considered byCarlson and the method was generalized to obtain rational transferfunction approximations of fractional capacitors in the form(1/s)1/n. The method utilizes the following equation:

Gk+1(s) = Gk(s)(n− 1)

(

Gnk(s)

)

+ (n+ 1)(

H(s))

(n+ 1)(

Gnk(s)

)

+ (n− 1)(

H(s)) , (24)

whereH(s) = 1/s

and the initial estimate is G0(s) = 1. The method works for botheven n = 2m and odd n = 2m+ 1 roots.

Carlson’s method: drawbacks


Several drawbacks of the approximation method can be outlined:

• The initial estimate for each particular approximation could beproperly chosen;

• The method only allows to obtain approximations for transferfunctions of order 1/n;

• Resulting approximations can be of a very high order;

• The limited frequency range where the approximation is valid.

In fact, the application of this method could be slightly different.That is, it would be beneficial to approximate

G(s) =

(

bs+ 1

as+ 1

)α

. (25)

The revised Carlson approximation method:Algorithm summary


1. Use a suitable decomposition method to obtain fractional powers1/m1, 1/m2, . . . , 1/mk;

2. Use the base transfer function Gbase(s) =bs+1as+1

and for each fractionalpower αk = 1/mk find the initial estimate

G0,k =

∣

∣

∣

∣

jbωm + 1

jaωm + 1

∣

∣

∣

∣

αk

,

where ωm = (ab)−0.5;

3. Obtain the approximation using

Gα(s) =

k∏

j=1

G1/mj

base (s) (26)

applying order reduction, if necessary;

4. Apply the order reduction technique to obtain the final approximation.

The revised Carlson approximation methodexample


Consider a problem of implementing a fractional-order leadcompensator for a position servo described by an integer-ordermodel G(s):

G(s) =1.4

s(0.7s+ 1)e−0.05s.

Design specifications are as follows: phase margin ϕm = 80, gaincrossover frequency ωcg = 2.2 rad/s. The controller C(s) to fulfillthese specifications may be designed such that

C(s) =

(

2.0161s+ 1

0.0015s+ 1

)0.702

.

Let us observe the convergence of the approximation obtainedusing the described method to the ideal frequency response.

The revised Carlson approximation methodexample: initial estimate


0

5

10

15

20

25

30

35

40

45

Mag

nitu

de (

dB)

10−3

10−2

10−1

100

101

102

103

104

105

0

30

60

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

The revised Carlson approximation methodexample: iteration 1 (2nd order system)


0

5

10

15

20

25

30

35

40

45

Mag

nitu

de (

dB)

10−3

10−2

10−1

100

101

102

103

104

105

0

30

60

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

The revised Carlson approximation methodexample: iteration 2 (11th order system)


0

5

10

15

20

25

30

35

40

45

Mag

nitu

de (

dB)

10−3

10−2

10−1

100

101

102

103

104

105

0

30

60

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

The revised Carlson approximation methodexample: iteration 3 (56th order system)


0

5

10

15

20

25

30

35

40

45

Mag

nitu

de (

dB)

10−3

10−2

10−1

100

101

102

103

104

105

0

30

60

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Network approximation of a fractional-orderoperator: Fractance


By choosing Z1(s) and Z2(s) in the active filter configuration it ispossible to implement a fractional-order integrator or differentiator.The corresponding network approximations can be computed usinga suitable method. A unified framework adopting multiple methodscan be designed.

General fractance object: FOMCON toolbox


Our goal was to obtain a general enough approach tosystematization of existing network topologies and their generation.In FOMCON, fractance network synthesis and analysis is handledby means of a central component—an object, containing completecircuit information. The object contains references to

• particular network structures (MATLAB functions), howevercomplex, which return computed network transfer functions(impedance values),

• corresponding implementations (also MATLAB functions).

The idea is that a single structure can have several differentimplementations, including, e.g., optimization based ones.

General fractance object relations:FOMCON toolbox


For generality, we also consider the possibility of using inductivecomponents in network structures.

Fractance class properties: FOMCONtoolbox


Class frac_rcl() properties:

• model — a model of the fractional-order system (fotf object);

• structure — network structure (Cauer, Foster, etc.);

• implementation — function that carries out the actual computationof the network component values;

• ω — frequency points used for model validation;

• params — parameters used for implementation and/or in thestructure;

• K — network gain compensation factor(s);

• R,C,L — cell array with component value vectors (size depends onthe number of substructures);

• results — implementation/validation results.

Fractance class methods: FOMCON toolbox


The following particular methods are implemented:

• tf(), zpk() — return the impedance Z(s) in transfer function orzero-pole-gain format, corresponding to the fractance circuit, fornetwork analysis. This can be used to automate, e.g. frequencyresponse analysis.

• prefnum() — locates closest component values according to thepreferred series and replaces network components accordingly. It isalso possible to provide custom values of components so that thealgorithm will choose the closest matches among a custom set ofvalues. This can be necessary to analyze the changes of the systemfrequency characteristics due to variation of component values.

• zscale() — implementation of impedance scaling used to shift thevalues of discrete electronic components into the feasible domain.

Approximation of fractional operators: TheOustaloup filter


The Oustaloup recursive filter gives a very good approximation offractional operators in a specified frequency range and is widelyused in fractional calculus. For a frequency range (ωb, ω) and oforder N the filter for an operator sγ , 0 < γ < 1, is given by

sγ ≈ K

N∏

k=−N

s+ ω′

k

s+ ωk, K = ωγ

h, ωr =ωh

ωb, (27)

ω′

k = ωb(ωr)k+N+1

2 (1−γ)

2N+1 , ωk = ωb(ωr)k+N+1

2 (1+γ)

2N+1 .

The resulting model order is 2N + 1.

A modified Oustaloup filter has been proposed in literature [3].

Oustaloup filter approximation example


Recall the fractional-order transfer function

G(s) =1

14994s1.31 + 6009.5s0.97 + 1.69,

and approximation parameters ω = [10−4; 104], N = 5.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Step Response

Time (sec)

Am

plitu

de

Grunwald−Letnikov

Oustaloup filter

Refined Oustaloup filter

10−5

100

105

−135

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

−250

−200

−150

−100

−50

0

Mag

nitu

de (

dB)

Oustaloup filter

Refined Oustaloup filter

Original plant

Oustaloup filter

Discrete-time approximation: The zero-polematching equivalents method


Continuous zeros and poles, obtained using the Oustaloup recursivefilter, are directly mapped to their discrete-time counterparts bymeans of the relation

z = esTs , (28)

where Ts is the desired sampling interval. The gain of the resultingdiscrete-time system H(z) must be corrected by a proper factor.

This implementation method has been successfully used in ourprevious work. We remark, that for the synthesis of continuouszeros and poles in (27) with the intent to obtain a discrete-timeapproximation the transitional frequency ωh may be chosen suchthat

ωh 62

Ts. (29)

Discrete-time approximation offractional-order controllers


After acquiring a set of discrete-time zeros and poles by means of(28), the fractional-order controller may be implemented in form ofa IIR filter represented by a discrete-time transfer function H(z−1).In general, one has two choices:

1. Implement each fractional-order component approximation ofthe controller in (21) separately as Hλ(z−1) and Hµ(z−1); thismethod offers greater flexibility, since the components may bereused in the digital signal processing chain, but requires morememory and is generally more computationally expensive;

2. Compute a single LTI object approximating the wholecontroller; this method is suitable when there is a need for astatic description of a fractional-order controller.

Digital controller implementation: IIR filters


In this particular work we choose the second option, that is we seeka description of the controller in the form

H(z−1) = Kb0 + b1z

−1 + b2z−2 + · · ·+ bmz−m

a0 + a1z−1 + a2z−2 + · · ·+ anz−n. (30)

For practical reasons, the equivalent IIR filter should be comprisedof second-order sections. This allows to improve computationalstability when the target signal digital processing hardware haslimited DSP capabilities. Thus, the discrete-time controller mustbe transformed to yield

H(z−1) = b0

N∏

k=1

1 + b1kz−1 + b2kz

−2

1 + a1kz−1 + a2kz−2. (31)

Part IV: Applications of Factional-orderControl


Applications of Fractional-order Control



• Tuning and digital implementation of controllers for simulated and physicallaboratory models;

• Implementation of an analog controller for a simulated model.

Publications:

1. A. Tepljakov, E. Petlenkov, J. Belikov, and M. Halás, “Design and implementationof fractional-order PID controllers for a fluid tank system,” in Proc. 2013American Control Conference (ACC), Washington DC, USA, June 2013, pp.1780–1785

2. A. Tepljakov, E. Petlenkov, J. Belikov, and J. Finajev, “Fractional-order controllerdesign and digital implementation using FOMCON toolbox for MATLAB,” inProc. of the 2013 IEEE Multi-Conference on Systems and Control conference,2013, pp. 340–345

3. A. Tepljakov, E. Petlenkov, J. Belikov, and S. Astapov, “Digital fractional-ordercontrol of a position servo,” in Proc. 20th Int Mixed Design of Integrated Circuitsand Systems (MIXDES) Conference, 2013, pp. 462–467

Case study (1): Fractional-order control ofthe coupled tank system


The system is modeled in continuous time inthe following way:

x1 =1

Au1 − d12 − w1c1

√x1, (32)

x2 =1

Au2 + d12 − w2c2

√x2,

where x1 and x2 are levels of fluid, A is thecross section of both tanks; c1, c2, and c12 areflow coefficients, u1 and u2 are pump powers;valves are denoted by wi : wi ∈ 0, 1 and

d12 = w12 · c12·sign(x1 − x2)√

|x1 − x2|.

Case study (1): Fractional-order control ofthe coupled tank system (continued)


Our task is to control the level in the first tank. We identify thereal plant from a step experiment with w1 = w12 = 1, w2 = 0 in(32). The resulting fractional-order model is described by a transferfunction

G2 =2.442

18.0674s0.9455 + 1e−0.1s. (33)

We notice, that this model does not tend to exhibit integer-orderdynamics. Due to the value of the delay term the basic tuningformulae for integer-order PID tuning do not provide feasibleresults. It is possible to select some starting point manually andrun optimization several times. However, it is important to choosethe correct frequency domain specifications to ensure controlsystem stability.

Case study (1): Experiments with controllerimplementation: Hardware platform




In our case the goal is to minimize the impact of disturbance, soconstraints on the sensitivity functions could be imposed. Ourchoice is such that

∣

∣T (jω)∣

∣ ≤ −20 dB, ∀ω ≥ 10 rad/s and∣

∣S(jω)∣

∣ ≤ −20 dB, ∀ω ≤ 0.1 rad/s. The gain and phase marginsare set to Gm = 10dB and ϕ = 60, respectively. Additionally, inorder to limit the overshoot, the upper bound of control signalsaturation was lowered from 100% to 60%. Using the IAEperformance metric we finally arrive at the following PIλDµ

controller parameters by optimizing the response of the nonlinearsystem in Simulink:

Kp = 6.9514, Ki = 0.13522, Kd = −0.99874,

λ = 0.93187, µ = 0.29915. (34)



0 10 20 30 40 50 60 70 80 90 100 110 1200

0.05

0.1

0.15

0.2

0.25

Leve

l [m

]

0 10 20 30 40 50 60 70 80 90 100 110 1200

0.2

0.4

0.6

0.8

1

Con

trol

law

u(t

) [%

]

0 10 20 30 40 50 60 70 80 90 100 110 1200

1

t [s]

w2

Real plant, controller in SimulinkDiscrete−time simulationReal controller, model in Simulink

Real plant

Real plant

Case study (2): Fractional-order control ofa servo system


Case study (2): Fractional-order control ofa servo system (continued)


The following transfer function is identified:

G(s) =192.1638

s(1.001s+ 1).

The generic PD controller parameters provided by INTECO areKp = 0.1,Kd = 0.01. We shall use these parameters as the initialones for the optimization.

The results of optimization are such, that after 100 iterations thegains of the PD controller have been found as Kp = 0.055979 andKd = 0.025189.

After fixing the gains and manually perturbing the value of µ to0.5, the optimized PDµ controller is obtained with µ = 0.88717.Phase margin of the open loop control system is ϕm = 65.3.



Experimental setup for evaluating the digital implementation of thefractional-order PID controller:



0 10 20 30 40 50 60−50

0

50

100

150

200

Ang

le [r

ad]

0 10 20 30 40 50 60−1

−0.5

0

0.5

1

Con

trol

law

u(t

)

0 10 20 30 40 50 600

0.2

0.4

Time [s]

Dis

turb

ance

System with integer−order PD controller

System with PDµ controller (in Simulink)

System with PDµ controller (external)Reference angle

Case study (3): Network approximation of aFO lead compensator


Recall the example, where our goal was to obtain an analogimplementation a fractional controller for a model of a positionservo

G(s) =1.4

s(0.7s+ 1)e−0.05s.

We now provide the results of approximating the controller

C(s) =

(

2.0161s+ 1

0.0015s+ 1

)0.7020

by an electrical network by using a deterministic method,implemented as part of the unified network generation frameworkin FOMCON, for obtaining the parameters of the network.

Case study (3): Electrical networkapproximations


In order to implement it, the following steps are carried out:

• We choose R1 = 200kΩ and C1 = 1µF due to the timeconstant τ .

• The basic structure is the Foster II form RC network and theimplementation is done by means of the mentioned algorithm.

• To obtain the differentiator, we use the propertyZd(s) = 1/Zi(s), where Zd(s) and Zi(s) correspond toimpedances of a differentiator and an integrator, respectively.

• This is done by setting the impedances in the active filtercircuit such that Z1(s) = Zi(s) and Z2(s) = Rk, where Rk

serves as a gain correction resistor.

Case study (3): Electrical networkapproximations (continued)


b = 2.0161; wz = 1/b;alpha = 0.702;Gc = fotf(’s’)^alpha / wz^alpha;

params = struct; params.R1 = 200e3;params.C1 = 1e-6; params.N = 4;params.varphi = 0.01;

imp2 = frac_rcl(1/Gc, ...’frac_struct_rc_foster2’, ...’frac_imp_rc_foster2_abgen’, ...logspace(-2,3,1000), ...params);



The controller is obtained from the object using

C = 1/zpk(imp2);

Now we set the resistor values to the preferred series with 5% tolerance,and the capacitor values are substituted for closest components out ofthe 10%-series:

imp2 = imp2.prefnum(’5%’,’10%’,[],’5%’);

Finally, the bill of materials can be generated using engnum():

[vals, str] = engnum(imp2.R);

The variable str will contain the following:

’360 k’ ’200 k’ ’75 k’ ’27 k’ ’9.1 k’

The gain setting resistor Rk has the preferred value of 390kΩ.



−40

−30

−20

−10

0

10

20

30

40

50

60

Mag

nitu

de (

dB)

10−3

10−2

10−1

100

101

102

103

0

30

60

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

Fractional differentiator

Ideal response of lead compensator

Network approximation



Bode Diagram

Frequency (rad/s)

−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

System: CompensatedGain Margin (dB): 24.9At frequency (rad/s): 23.3Closed loop stable? Yes

System: UncompensatedGain Margin (dB): 23.2At frequency (rad/s): 5.28Closed loop stable? Yes

Mag

nitu

de (

dB)

100

101

102

−270

−225

−180

−135

−90

−45

0 System: CompensatedPhase Margin (deg): 85.5Delay Margin (sec): 0.71At frequency (rad/s): 2.1Closed loop stable? Yes

System: UncompensatedPhase Margin (deg): 49.1Delay Margin (sec): 0.774At frequency (rad/s): 1.11Closed loop stable? Yes

Pha

se (

deg)

UncompensatedCompensated

Case study (3): Frequency response aroundωcg = 2.2 rad/s


Case study (3): Frequency response aroundωcg = 2.2 rad/s


Bode Diagram

Frequency (rad/s)100

101

0

30

60

90

Pha

se (

deg)

0

2

4

6

8

10

12

14

16

18

20

Mag

nitu

de (

dB)

Fractional lead compensatorElectrical network approximation (real)

Case study (3): Electrical networkapproximations: Results


0 20 40 60 80 100 120−10

−5

0

5

10

15

Am

plitu

de

0 20 40 60 80 100 120−5

0

5

Time [s]

Con

trol

law

u(t

)

Response with simulated model and real controllerSimulated responseSet point

Analog controllerSimulated controller

FOMCON project: Fractional-orderModeling and Control


• Official website: http://fomcon.net/

• Toolbox for MATLAB available;

• An interdisciplinary project supported by the Estonian DoctoralSchool in ICT and Estonian Science Foundation grant nr. 8738.

http://fomcon.net/

Acknowledgments


References


[1] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus andFractional Differential Equations, Wiley, New York, 1993.

[2] I. Podlubny, Fractional Differential Equations, Academic Press, SanDiego, CA, 1999.

[3] C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, V. Feliu,Fractional-order Systems and Controls Fundamentals andApplications, Springer-Verlag, London, 2010.

[4] R. Hilfer, Applications of Fractional Calculus in Physics, WorldScientific, Singapore, 2000.

[5] I. Podlubny, Geometric and Physical Interpretation of FractionalIntegration and Fractional Differentiation, Fractional Calculus andApplied Analysis, vol. 5, no. 4, pp. 367-386, 2002.

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