Robust Control of MIMO Systems

8/15/2019 Robust Control of MIMO Systems

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Robust control of MIMO systems

Olivier Sename

Grenoble INP / GIPSA-lab

September 2015

Olivier Sename (Grenoble INP / GIPSA-lab) Robust control September 2015 1 / 138

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1. The H ∞ norm and related denitionsSystem denitionsH ∞ norm as a measure of the systemgain ?LTI systems and signals norms

2. Performance analysisDenition of the sensitivity functionsFrequency-domain analysisStability issues

3. Performance specications for analysisand/or design

IntroductionSISO caseMIMO caseSome performance limitations

4. Introduction to Linear Matrix InequalitiesBackground in OptimisationLMI in control

Some useful lemmas5. H ∞ control design

About the control structureSolving the H ∞ control problemThe Riccati approachThe LMI approach for H ∞ control designExample

6. Uncertainty modelling and robustnessanalysis

IntroductionRepresentation of uncertaintiesDenition of Robustness analysisRobustness analysis: the unstructuredcaseRobustness analysis: the structured caseTowards Robust control design

7. Introduction to LPV systems and controlLPV modellingLPV Control

O. Sename GIPSA-lab 2/138

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Reference booksTo be studied during the course

• S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: analysis and design, John

Wiley and Sons, 2005.www.nt.ntnu.no/users/skoge/book , chap 1 to 3 available• K. Zhou, Essentials of Robust Control, Prentice Hall, New Jersey, 1998.

www.ece.lsu.edu/kemin , book slides available• J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback control theory, Macmillan

Publishing Company, New York, 1992.

https://sites.google.com/site/brucefranciscontact/Home/publications , ,book available• Carsten Scherer’s courses

http://www.dcsc.tudelft.nl/ cscherer/ . , Lecture slides available (MSc Course"Robust Control", MSc Course "Linear Matrix Inequalities in Control")

• + all the MATLAB demo, examples and documentation on the ’Robust Control toolbox’(mathworks.com/products/robust )

Other references (some in french)

• G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, New Jersey, 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, Robu st es se e t comma nde op timale,Cépadues Editions, 1999.


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1 The H ∞ norm and related denitions System denitions

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.

Denition (Nonlinear dynamical system)

For given functions f : R n × R n w → R n and g : R n × R n w → R n z , a nonlinear dynamicalsystem ( Σ N L ) can be described as:

Σ N L : ẋ = 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 ∈R n , w(t ) is the input taking valuesin the input space W ∈R n w and z(t ) is the output that belongs to the output space Z ∈R n z .

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


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Comments

• The main advantage of nonlinear dynamical modeling is that (if it is correctly described) it

catches most of the real system phenomena.• On the other side, the main drawback is that there is a lack of mathematical and

methodological tools; e.g. parameter identication, control and observation synthesis andanalysis are complex and non systematic (especially for MIMO systems).

• In this eld, 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 denition, 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.


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Denition of LTI systems

Denition (LTI dynamical system)

Given matrices A ∈R n × n , B ∈R n × n w , C ∈R n z × n and D ∈R n z × n w , a Linear Time Invariant(LTI) dynamical system ( Σ LT I ) can be described as:

Σ LT I : ẋ (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 ∈R n w and z(t ) is the output that belongs to the output space Z ∈R n z .

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, the

main 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.


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1 The H ∞ norm and related denitions Towards H ∞ norm

A rst ’bad’ approach

Let consider

G = 5 43 2

How to dene and evaluate its gain ?Consider ve different inputs:

u = 10 u = 01 u =

0.7070.707 u =

0.707− 0.707 u =

0.6− 0.8

Input magnitude :u 1 2 = u 2 2 = u 3 2 = u 4 2 = u 5 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


1 Th d l d d i i T d

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Towards SVD


MAXIMUM SINGULAR VALUE

maxd =0

Gd 2d 2

= σ̄ (G)

MINIMUM SINGULAR VALUE

mind =0

Gd 2d 2

= σ(G)

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

1 The H norm and related denitions To ards H norm

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What about eigenvalues ?

Eigenvalues are a poor measure of gain. Let

G = 0 1000 0

Eigenvalues are 0 and 0.But an input vector

0

1leads to an output vector

1000 .

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


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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≤ σ̄ (G( jω ))


Example of A two-mass/spring/damper system2 inputs: F 1 and F 2 2 outputs: x1 and x2

Singular Values

Frequency (rad/sec)

S i n g u

l a r

V a

l u e s

( d B )

10-1

100

101

102

-100

-80

-60

-40

-20

0

20

40

Hinf norm 11.4664 = 21.18 dB

smallest singularvalue

largest sin gularvalue

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

1 The H ∞ norm and related denitions LTI systems and signals norms

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Signal norms

Reader is also invited to refer to the famous book of Zhou et al., 1996 , where all the followingdenitions and additional information are given.

All the following denitions 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 ) .

Denition (Norm and Normed vector space)

• Let V be a nite dimension space. Then ∀ p ≥ 1, the application || .|| p is a norm, dened as,

|| v|| p =i

|vi |p 1/p

(3)

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



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L norms

Denition ( L 1 , L 2 , 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 ) ∞ = supt

|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∞

).



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∞ y g

About vector spaces

Denition (Banach, Hilbert, Hardy and Lp spaces)

• A Banach space is a (real or complex) complete (i.e. all Cauchy sequences, of points in K have 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 dened by the inner product. The norm of f ∈H is then dened 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 ( H p ) are certain spaces of holomorphic functions (functions dened 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 L p space are spaces of p-power integrable functions (function whose integral exists,generally called Lebesgue integral), and corresponding sequence spaces.

For example, R n and C n 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 .



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L ∞ and H ∞ spaces

Denition ( 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)

Denition ( 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 ∞ .

ExampleIn control theory

s +1( s +10)( s +6) ∈ RH ∞

s +1( s − 10)( s +6) ∈ RH ∞

s +1( s +10) ∈ RH ∞

(10)



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H ∞ norm

Denition ( H ∞ norm)

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

G( jω ) ∞ = sup ω∈R σ (G( jω ))= sup w ( s )∈H 2

z ( s ) 2w ( s ) 2

= max w ( t )∈L 2z 2w 2

(11)

In MATLAB, use norm(sys,inf)

Remark

H ∞ physical interpretations • This norm represents the maximal gain of the frequency response of the system. It is also

called the worst case attenuation level in the sense that it measures the maximum amplication that the system can deliver on the whole frequency set.• For SISO (resp. MIMO) systems, it represents the maximal peak value on the Bode

magnitude (resp. singular value) plot of G( jω ) , in other words, it is the largest gain if the system is fed by harmonic input signal.

• Unlike H 2 , the H ∞ norm cannot be computed analytically. Only numerical solutions can be

obtained (e.g. Bisection algorithm, or LMI resolution).O. Sename GIPSA-lab 17/138


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Recalls on Singular Value Denition

Let A ∈R m × n , There exists unitary matrices:

U = [ u1 , u2 , . . . , u m ] and V = [v1 , v2 , . . . , v n ]such that

A = U Σ V T , Σ = Σ 1 00 0 , Σ 1 =

σ1 0 . . . 00 σ2 . . . 0...

... · · · . . .0 0 . . . σp

with σ1 ≥ σ2 ≥ . . . ≥ σp ≥ 0 , p = min {m, 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:Av i = σi u i and AT u i = σi vi . Then

AA T u i = σ2i u i

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



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L 2 and H 2 spaces

Denition ( L 2 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 = 12π + ∞−∞ Tr[f ∗( jω )f ( jω )]dω < ∞ (12)Denition ( H 2 and RH 2 spaces)

H 2 is a subspace (Hardy space) of L 2 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 itsdenition set). In particular, the real rational subspace of H 2 , which consists of all strictly properand real rational stable transfer matrices , is denoted by RH 2 .

Example

In control theory

s +1( s +10)( s +6) ∈ RH 2

s +1( s − 10)( s +6) ∈ RH 2

s +1( s +10) ∈ RH 2

(13)



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H 2 norm

Denition ( H 2 norm)

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

G( jω ) 2 = + ∞−∞ g∗(t )g(t )dt= 12π

+ ∞−∞ Tr[G

∗( jω )G( jω )]dω

= supw ( s )∈H 2

|| z ( s ) || ∞

|| u ( s ) || 2

(14)

The norm H 2 is nite if and only if G(s) is strictly proper (i.e. G(s) ∈ RH 2 ).

In MATLAB, use norm(sys,2)

Remark

H 2 physical interpretations and remarks • For SISO systems, it represents the area located below the so called Bode diagram.• For MIMO systems, the H 2 norm is the impulse-to-energy gain of z(t ) in response to a white

noise input w(t ) (satisfying W ∗( jω )W ( jω ) = I , i.e. uniform spectral density).• The H 2 norm can be computed analytically (through the use of the controllability and

observability Grammians) or numerically (through LMIs).


2 Performance analysis

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Outline

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

2. Pe rformance analysisDenition of the sensitivity fu nctionsFrequency-dom ain analysisStability issues

3. Pe rformance sp ecications for analysis and/or designIntroductionSISO caseMIMO caseSome performance limitations

4. Introduction to Linear Matrix InequalitiesBackground in OptimisationLMI in controlSome useful lemmas

5. H ∞ control designAbout the control structureSolving the H ∞ con trol problemThe Riccati approachThe LMI approach for H ∞ control designExample

6. Uncertainty mo delling and robustness analysisIntroductionRepresentation of uncertaintiesDenition of Robustness analysisRobustness analysis: the unstructured ca seRobustness analysis: the struct ured caseTowards Robust control design



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Introduction

Objectives of any control system [Skogestad and Postlethwaite(1996)]shape the response of the system to a given reference and get (or keep) a stable system in

closed-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 satises the performance specications with the nominal

model (no model uncertainty)Robust stability (RS): The system is stable for all perturbed plants about the nominal model, up to

the worst-case model uncertainty (including the real plant)

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


2 Performance analysis Denition of the sensitivity functions

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The control structure

Figure: Complete control scheme

In the SISO case, it leads to:

y = 11+ G ( s ) K ( s ) (GKr + dy − GKn + Gd i )u = 11+ K ( s ) G ( s ) (Kr − Kd y − Kn − KGd i )

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’.O. Sename GIPSA-lab 23/138


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Stability and robustness margins ...


Classical denitions:

• Stability ⇔ | L( jω π ) |< 1, where ωπis the phase crossover frequencydened by φ(L( jω π )) = − π .

• Gain Margin : indicates the additionalgain that would take the closed loopto the critical stability condition(GM (dB ) = − [|L( jω π |]dB

)• Phase margin : quanties the pure

phase delay that should be added toachieve the same critical stabilitycondition(ΦM = 180 o + arg [L( jω c )], where|L( jω c |) = 1(0 dB ))

• Delay margin : quanties themaximal delay that should be addedin the loop to achieve the samecritical stability condition, P M/ω c


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Robustness margins ...


It is important to consider the modulemargin that quanties the minimal

distance between the curve and thecritical point (-1,0j): this is a robustnessmargin.

∆ M = 1/M SM S = maxω |S ( jω ) | = S ∞

Good value M S < 2 (6dB )

• The MODULE MARGIN is a robustnessmargin. Indeed, the inuence 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)∆ GG

= S.∆ GG

• A good module margin implies good gain andphase margins:

GM ≥ M SM S − 1

and P M ≥ 1M S

For M S = 2 , then GM > 2 and P M > 30◦• Last:

M T = maxω |T ( jω ) |

A good value : M T < 1.5(3 .5dB )


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Input/Output performances

Dening 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)



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Performance analysis: answer to ....

Analysis of S :• Steady state error in tracking and output disturbance rejection : S (ω = 0) = 0 ?• Maximum peak criterion (Module Margin): S ∞ < 2 ?

Analysis of T :• Steady state error in tracking : T (ω = 0) = 1 ?.• Attenuation of measurement noise: |T ( jω ) | small when ω → ∞ ?• Maximum of T , T ∞ < 1.5 ?

Analysis of KS :• Input saturation: |u(t ) | < |u max | ? (where |umax | < KS ∞ |r max |).• Attenuation of measurement noise: |KS ( jω ) | small when ω → ∞ ?

Analysis of SG :• Steady state error in input disturbance rejection : SG (ω = 0) = 0 ?

• Attenuation of input disturbance effet: |SG ( jω ) | small in the frequency range of interest ? .Plus: all the transient behaviors (bandwidths), as seen later.



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Denition of the sensitivity functions - The MIMO case

Figure: Complete control scheme

The output & the control input satisfy the following equations :

(I p + G(s)K (s)) y(s) = ( GKr + dy − GKn + Gd i )

(I m + K (s)G(s)) u(s) = ( Kr − Kd y − Kn − KGd i )

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



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Using Matlab

% Determination of the sensitivity fucntionsL=series(G,Khinf) % Loop transfer function L=GKS=inv(1+L); % S= 1 /(1+L)

poleS=pole(S)T= feedback(L,1)

poleT=pole(T)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SG=S*G;

poleSG=pole(SG)KS=Khinf*S;

poleKS=pole(KS)%%%%w=logspace(-3,3,500); %% to be adjusted

subplot(2,2,1), sigma(S,w), title('Sensitivity function')subplot(2,2,2), sigma(T,w), title('Complementary sensitivity function')subplot(2,2,3), sigma(SG,w), title('Sensitivity*Plant')subplot(2,2,4), sigma(KS,w), title('Controller*Sensitivity')



f h f h

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Denition of the sensitivity functions - The MIMO case

Denitions

Output and Output complementary sensitivity functions:

S y = ( I p + GK )− 1 , T y = ( I p + GK )− 1 GK, S y + T y = I p

Input and Input complementary sensitivity functions:

S u = ( I m + KG )− 1 , T u = KG (I m + KG )− 1 , S u + T u = I m

Properties

T y = GK (I p + GK )− 1T u = ( I m + KG )− 1 KGS u K = KS y



MIMO I /O f

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MIMO Input/Output performances

Dening two new ’sensitivity functions’:Plant Sensitivity: S y G = S y (s).G (s)

Controller Sensitivity: KS y = K (s).S y (s)


2 Performance analysis Frequency-domain analysis

F d i l i S l i l l i it i (1)



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


• The transfer function KS y (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 KS u (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 S u (s) small (takecare to not trying to minimize T u whichis not possible)


F q d i l i S l i l l i it i (2)

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


• The plant output y(t ) can track thereference r (t ) by making thecomplementary sensitivity functionT y (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 S y (s) (resp. S y G(s) ) « small »

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


Trade offs

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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:•

S y and S y G to be small in low frequencies to reduce the load (output and input) disturbanceeffects on the controlled output• T y and KS y to be small in high frequencies to reduce the effects of measurement noises on

the controlled output and on the control input (actuator efforts)



Frequency domain analysis Dynamical behavior

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Frequency-domain analysis - Dynamical behavior

As mentioned in [ Skogestad and Postlethwaite(1996)] :The concept of bandwidth is very important in understanding the benets and trade-offs involved

when 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 .

Denition

Loosely speaking, bandwidth may be dened 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 meansbenet in terms of performance.



Frequency domain analysis Bandwidth denitions

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Frequency-domain analysis - Bandwidth denitions

Denition ( ωS )

The (closed-loop) bandwidth, ωS , is the frequency where |S ( jω ) | crosses − 3dB (1/sqrt 2) frombelow.

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

Denition ( ωT )

The bandwidth (in term of T ), ωT , is the frequency where |T ( jω ) | crosses − 3dB (1/sqrt 2) fromabove.

Denition ( ωc )

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



Some remarks

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Some remarks

Remark

Usually ωS < ω c < ω T

Remark

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

Remark

Usually ωS < ω c < ω T

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

t r = 2.3ωT



Examples

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Examples


2 Performance analysis Stability issues

Well-posedness

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Well posedness


Consider

G = −s − 1s + 2

, K = 1

Therefore the control input is non proper:

u = s + 2

3 (r − n − dy ) +

s − 13

di

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

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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 )− 1 GK (I + GK )− 1 GK (I + GK )− 1 − K (I + GK )− 1 G

rdi

For instance :

G = 1s − 1

, K = s − 1s + 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

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Consider the so called M − ∆ loop.

Figure: M − ∆ form

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

M ∞ < γ



Input-Output Stability

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p p y

Denition (BIBO stability)

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

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

γ peak = sup0< u ∞ < ∞

y ∞u ∞

and is referred to as the PEAK TO PEAK Gain .

Denition ( L 2 stability)

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

Now, the quantication of the signal amplication (gain) is evaluated as:

γ energy = sup0< u 2 < ∞

y 2u 2

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

γ energy = sup ω G( jω ) := G ∞

For a linear system, these stability denitions are equivalent (bu t not the q ua ntica tion cr iteria) .



Synthesis

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y

Provide a clear and detailed frequency-domain performance analysis using the sensitivityfunctions in order to explain the trade-off performance/robustness/actuator constraints.

Qualitative analysis

Use of S y , T y , KS y , S y G to:• predict the behavior of the output w.r.t different external inputs (reference, disturbance, noise)• predict the behavior of the control input w.r.t different external inputs (reference, disturbance,

noise)

Quantitative analysis

Use of S y , T y , KS y , S y G to:• Stability analysis and margins.• Compute the steady-state errors in tracking, output and input disturbance attenuations.• Give the maximum of the input/output gains to analyze the transient behaviors of the output

and control input (incl. saturation).• Give all the bandwidths of the sensitivity functions• Evaluate the rise time in tracking• Evaluate the robustness margins


3 Performance specications

Outline

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4. Introduction to Linear Matrix InequalitiesBackground in OptimisationLMI in controlSome useful lemmas





3 Performance specications Introduction

Framework... templates



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

• Robustness and performances in regulation can be specied 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 as

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

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

|S ( jω ) | ≤ 1

|W e ( jω ) | , ∀ω ⇔ W e S ∞ ≤ 1

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


3 Performance specications SISO case

Template on the sensitivity function - Weighted sensitivity

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Typical specications 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 04 Shape of S over selected frequency ranges.5 Maximum peak magnitude of S , || S || ∞ < M S .

The peak specication prevents amplication of noise at high frequencies, and also introduces a

margin of robustness; typically we select M S = 2 .



Template on the sensitivity function S

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S (s) = 1

1 + K (s)G(s)

1W e (s)

= s + ωbεs/M S + ωb

Generally ε 0 is considered, M S < 2(6dB ) or (3dB - cautious) to ensuresufcient module margin.ωb inuences the CL bandwidth : ωb ↑

• faster rejection of the disturbance• faster CL tracking response• better robustness w.r.t. parametric

uncertainties

Template on the Sensitivity function S

Frequency (rad/sec)

S i n g u

l a r

V a

l u e s

( d B )

10−4

10−2

100

102

104

106−60

−50

−40

−30

−20

−10

0

10Template on the sensitivity function S

M S = 2 (6dB)

ε = 1e − 3

ωb s.t. |1/W e | = 0dB


Template on the function KS

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KS (s) = K (s)

1 + K (s)G(s)

1W u (s)

= ε1 s + ωbcs + ωbc /M u

M u chosen according to LF behavior ofthe process (actuator constraints:saturations)ωbc inuences 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)

S i n g u

l a r

V a

l u e s

( d B )

100

105

−60

−50

−40

−30

−20

−10

0

10

M u = 2

ω bc f o r |1 / W u | = 0 d B

c = 1 e - 3


Template on the function T

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T (s) = K (s)G(s)1 + K (s)G(s)

1W T (s)

= εT s + ωbts + ωbt /M T

Generally εT 0 is considered,M

T < 1.5 (3dB ) to limit the overshoot.

ωbt inuences the bandwidth hence thetransient behavior of the disturbancerejection properties: ωbt ↓

• better noise effects rejection• better ltering 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)

S i n g u

l a r

V a

l u e s

( d B ) M T = 1.5

T =1e-3

ωbT s.t. |1 /W T | = 0 dB


Template on the function SG

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SG (s) = G(s)

1 + K (s)G(s)

1W SG (s)

= s + ωSG εSGs/M SG + ωSG

M SG allows to limit the overshoot in theresponse to input disturbances.Generally εSG 0 is considered,ωSG inuences 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)

S i n g u

l a r

V a

l u e s

( d B ) M SG = 10

SG =1e-2

ωsg s.t. |1 /W SG | = 0 dB

3 Performance specications MIMO case

A rst insight into the MIMO case

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

1

Disturbance attenuation/closed-loop performances:

σ̄ (S y ) < 1

|W 1 ( jω ) |

with |W 1 ( jω ) | > 1 for ω < ω b2 Actuator constraints:

σ̄ (KS y ) < 1|W 2 ( jω ) |

with |W 2 ( jω ) | > 1 for ω > ω h3 Robustness to multiplicative uncertainties:

σ̄ (T y ) < 1

|W 3 ( jω ) |with |W 3 ( jω ) | > 1 for ω > ω t

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



Towards MIMO systems

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

G = G1 G2 , K = K 1K 2

Therefore

GK = G1 K 1 + G2 K 2 , KG = K 1 G1 K 1 G2K 2 G1 K 2 G2

and the sensitivity functions are:

S y = 1

1 + G1 K 1 + G2 K 2, KS y =

K 11+ G 1 K 1 + G 2 K 2

K 21+ G 1 K 1 + G 2 K 2

While a single template W e is convenient for S y it is straightforward that the following diagonaltemplate should be used for KS y :

W u (s) = W 1u (s) 00 W 2u (s)

where W 1u and W 2u are chosen in order to account for each actuator specicity (constraint).



The MIMO general case

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Let us consider G with m inputs and p outputs.• In the MIMO case the simplest way is to dened the templates as diagonal transfer matrices,

i.e. using ( M S i , 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 specications 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 individual

specication is met. Hence, in the simplest case:1 If the specications are identical then it is sufcient to plot:

• σ̄ (S y ( jω )) and 1| W e ( jω ) | , for all ω• σ̄ (KS y ( jω )) and 1| W u ( jω ) | , for all ω

2 If the specications are different, one should plot• σ̄ (S y ( i, :)) and 1| W i

e|

, for all ω , i = 1 , . . . , p

• σ̄ (KS y (k, :)) and 1| W ku |, 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 σ̄ (S y ) with σ̄ (1/W e )



More on weighting functions

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

W e (s) =s/M S + ωb

s + ωbε

k,

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

0 1 2 3 4 5 6−80

−70

−60

−50

−40

−30

−20−10

0

10

20

Mu=2, wbc=100, epsi1=0.01

Wu and Wu square without parameter modification

S i n g u

l a r

V a

l u e s

( d B )

0 1 2 3 4 5−50

−40

−30

−20

−10

0

10

20

Original weight (specications) M u =2, 1 = 0 .01, ωb c =100 for W u

M u = √ 2, ε1 = √ 0.01,ωbc =100 for 1 /W 2u case 1

M u = √ 2, ε1 = √ 0.01,ωbc =200 for 1 /W 2u case 2

M u = √ 2, ε1 = √ 0.01,ωb c =400 for 1 /W 2u case 3

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

S i n g u

l a r

V a

l u e

s ( d B )



Final objectives

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

W e S W u KS W T T

W SG SG ∞

≤ 1 ⇒ W e S ∞ ≤ 1 W SG SG ∞ ≤ 1W u KS ∞ ≤ 1 W T T ∞ ≤ 1

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

Find K s.t. W e S W u KS ∞≤ 1

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


3 Performance specications Some performance limitations

Introduction

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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 = 0Then the frequency range where |S ( jω ) | < 1 is balanced by the frenquencies where |S ( jω ) | > 1



Bode sensitivity

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

For unstable system we have stronger constraints:

∞0 log |det (S ( jω )) |dω = π N pi =1 Re ( pi ),where pi design the N p RHP poles. Therefore, in the presence of RHP poles, the control effortnecessary to stabilize the system is paid in terms of amplicatio n of the se nsitivity ma gn itude.



The interesting case of systems with RHP zeros

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Theorem

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

W p (s)S (s) ≥ | W p (s = z) |

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

W p (s) =s/M

S + ωb

s + ωbε ,

and , if the controller meets the requirements, then

W p (s)S (s) ∞ ≤ 1

Therefore a necessary condition is:

| z/M S + ωbz + ωbε |≤ 1

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

ωb ≤ z2


4 Introduction to LMIs

Outline

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4. Introduction to Linear Matrix Inequalities

Background in OptimisationLMI in controlSome useful lemmas





4 Introduction to LMIs Background in Optimisation

Brief on optimisation

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Denition (Convex function)

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

f (λx + (1 − λ )y) ≤ λf (x) + (1 − λ )f (y) (15)

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

epi (f ) = {(x, λ ) |f (x) ≤ λ} (16)

is convex.

Denition ((Strict) LMI constraint)

A Linear Matrix Inequality constraint on a vector x ∈R m is dened as,

F (x) = F 0 +

m

i =1F i x i 0( 0) (17)

where F 0 = F T 0 and F i = F T i ∈R

n × n are given, and symbol F 0( 0) means that F issymmetric and positive semi-denite ( 0) or positive denite ( 0), i.e. {∀u |u T F u (> ) ≥ 0}.



Convex to LMIs

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Example

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

xT P x > 0xT (AT P + P A )x < 0 (18)

which is equivalent to,

F (P ) = − P 00 AT P + P A ≺ 0 (19)

where P = P T 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 efciently, in a polynomial time (by optimization algorithms like e.g. Ellipsoid, InteriorPoint methods).



LMI problem

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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

V opt = inf x∈S f (x)

is called an optimization problem with an LMI constraint. This problem involves

the determination of V opt , the calculation of an almost optimal solution x (i.e., forarbitrary > 0 the calculation of an x ∈S such that V opt ≤ f (x) ≤ V opt + , orthe calculation of a optimal solutions xopt (elements xopt ∈S such thatV opt = f (xopt )) .



Examples of LMI problem



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

LQ controlConsider a controllable system ẋ = Ax + Bu . Find a state feedback u(t ) = − Kx (t) s.tJ = ∞0 (xT Qx + u T Ru )dt is minimum (given Q > 0 and R > 0) is an optimisation problemwhose solution is obtained solving the Riccati equation:

F ind P > 0, s.t. A T P + P A − P BR − 1 B T P + Q = 0

and then the state feedback is given by:

u(t ) = − R − 1 B T P x (t)

which is equivalent to: nd P > 0 s.t

AT P + P A + Q P BB T P R > 0



Semi-Denite Programming (SDP) Problem

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

Denition (SDP problem)

A SDP problem is dened as,

min cT xunder constraint F (x) 0 (20)

where F (x) is an afne symmetric matrix function of x ∈R m (e.g. LMI) and c ∈R m is a givenreal vector, that denes the problem objective.

SDP problems are theoretically tractable and practically:• They have a polynomial complexity, i.e. there exists an algorithm able to nd the global

minimum (for a given a priori xed 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 efciently 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.


4 Introduction to LMIs LMI in control

The state feedback design problem

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Stabilisation

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

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

(A − B K )T P + P (A − B K ) < 0⇔ AT P + P A − K T B T P − P B K < 0

which is obviously not linear...

Solution; use of change of variables

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

P − 1 AT + AP − 1 − P − 1 K T B T − B K P − 1 < 0⇔ QAT + AQ + YT B T + B Y < 0

with Q = P − 1 and Y = − K P − 1 .

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



The Bounded Real Lemma

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The L2 -norm of the output z of a system Σ LT I is uniformly bounded by γ times the L 2 -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 ifthere exists a positive denite symmetric matrix P (i.e P = P T > 0 s.t

AT P + P A P B C T B T P − γ I D T

C D − γ I < 0, P > 0. (21)

The Bounded Real Lemma (BRL), can also be written as follows (see Scherer)

I 0A B0 I C D

T 0 P 0 0P 0 0 00 0 − γ 2 I 00 0 0 I

I 0A B0 I C D

≺ 0 (22)

Note that the BRL is an LMI if the only unknown (decision variables) are P and γ (or γ 2 ).



Quadratic stability

Thi i f l f h bili l i f i

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

ẋ = 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 = P T > 0 s.t

A(δ)T P + P A(δ) < 0, for all δ ∈∆ (23)

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


4 Introduction to LMIs Some useful lemmas

Schur lemma



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Lemma

Let Q = QT and R = R T be afne matrices of compatible size, then the condition

Q S S T R 0 (24)

is equivalent to

R 0Q − SR − 1 S T 0 (25)

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



Kalman-Yakubovich-Popov lemma

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Lemma

For any triple of matrices A ∈R n × n

, B ∈R n × m

, M ∈R ( n + m ) × ( n + m )

= M 11 M 12M 21 M 22 , the

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

I 0A B

T 0 K K 0 I 0A B + M < 02 M 22 < 0 and for all ω ∈R and complex vectors col(x, w ) = 0

A − jωI B xw = 0 ⇒ xw

T M xw < 0

3

If M = − I 0A B

T

0 K K 0

I 0A B then the second statement is equivalent to

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

I C ( jωI − A)− 1 B + D

∗ Q S S T R

I C ( jωI − A)− 1 B + D > 0

This lemma is used to convert frequency inequalities into Linear Ma trix Ine qualities.O. Sename GIPSA-lab 69/138


Projection Lemma

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Lemma

For given matrices W = W T , M and N , of appropriate size, there exists a real matrix K = K T such that,

W + MKN T + NK T M T ≺ 0 (26)

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

W + MU + U T M T ≺ 0W + NV + V T N T ≺ 0 (27)

or, equivalently, if and only if,

M T ⊥

W M ⊥ ≺ 0N T ⊥

W N ⊥ ≺ 0 (28)

where M ⊥ and N ⊥ are the orthogonal complements of M , N respectively (i.e. M T ⊥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[Doyle et al.(1989)Doyle, Glover, Khargonekar, and Francis] see e.g..



Completion Lemma

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Lemma

Let X = X T , Y = Y T ∈R n × n such that X > 0 and Y > 0. The three following statements are equivalent:

1 There exist matrices X 2 , Y 2 ∈R n × r and X 3 , Y 3 ∈R r × r such that,

X X 2X T 2 X 3

0 and X X 2X T 2 X 3

− 1= Y Y 2Y T 2 Y 3

(29)

2 X I I Y 0 and rank X I I Y ≤ n + r

3 X I I Y 0 and rank [XY − I ] ≤ r

This lemma is useful for solving LMIs. It allow to simplify the number of variables when a matrix

and its inverse enter in a LMI.



Finsler’s lemma

This Lemma allows the elimination of matrix variables

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This Lemma allows the elimination of matrix variables.

Lemma

The following statement are equivalent• xT Ax < 0 for all x = 0 s:t: Bx = 0• B̃ T A B̃ < 0 where B B̃ = 0• A + λB T B < 0 for some scalar λ• A + XB + B T X T < 0 for some matrix X

• B⊥T AB ⊥ < 0



Interest of LMIs

LMIs allow to formulate complex optimization problems into "Linear" ones, allowing the use of

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LMIs allow to formulate complex optimization problems into Linear ones, allowing the use ofconvex optimization tools.Usually it requires the use of different transformations, changes of variables ... in order to linearize

the considered problmems: Congruence, Schur complement, projection lemma, Eliminationlemma, S-procedure, Finsler’s lemme ...

Examples of handled criteria• stability• H ∞ , H 2 , H 2 /H ∞ performances• robustness analysis: Small gain theorem, Polytopic uncertianties, LFT representations...• Robust control and/or observer design• pole placement• stability, stabilization with input constraints• Passivity constraints

• Time-delay systems


5 H ∞ control design

Outline

1. The H ∞ norm and related denitionsS t d iti



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System denitionsH ∞ norm as a measure of the system gain ?LTI systems and signals norms



4. Introduction to Linear Matrix Inequalities

Background in OptimisationLMI in controlSome useful lemmas


6. Uncertainty mo delling and robustness analysis

IntroductionRepresentation of uncertaintiesDenition of Robustness analysisRobustness analysis: the unstructured ca seRobustness analysis: the struct ured caseTowards Robust control design



5 H ∞ control design About the control structure

The General Control Conguration

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

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pp y y ( ) gcontrol conguration.

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:

T ew (s) = F l (P, K ) = P 11 + P 12 K (I − P 22 K )− 1 P 21

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



H ∞ control design - Problem denition

The overall control objective is to minimize some norm of the transfer function from w to e , for

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jexample, the H ∞ norm.

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

Denition ( H ∞ suboptimal control problem)

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

T ew (s) ∞ = maxω σ̄ (T ew ( jω )) ≤ γ

The optimal problem aims at nding γ min (done using hinfsyn in MATLAB).



A rst case: the state feedback control problem

Let consider the system:



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ẋ (t ) = Ax(t) + B 1 w(t) + B 2 u(t ) (30)

y(t ) = Cx (t )

The objective is to nd a state feedback control law u = − Kx s.t:

T yw (s) ∞ ≤ γ

The method consists in applying the Bounded Real Lemma to the closed-loop system, and then

try to obtain some convex solutions (LMI formulation).



H ∞ control design - Problem formulation

We consider here the case of Dynamic Output Feedback controllers K (s) dened as

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K (s) : ẋK (t ) = AK xK (t ) + B K y(t ),u (t ) = C K xK (t ) + D K y(t ).

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

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

Compute K : Use an optimisation algorithm that nds the controller K solution of theconsidered problem.

Illustration on W e S W u KS ∞

≤ γ

In that case the closed-loop system must be T ew (s) ∞ =W e S W u KS ∞

≤ γ i.e a system with

1 input and 2 outputs. Since S = r − yr and KS = ur , the control scheme needs only one external

input r .



How to consider performance specication in H ∞ control ?

Note that: in practice the performance specication concerns at least two sensitivity functions ( S

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and 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: W e S ∞ ≤ 1, W u KS ∞ ≤ 1.Idea : dene 2 new virtual controlled outputs :

e1 = W e Sre2 = W u KSr



About the control structure - Problem denition

The performance specications on the tracking error & on the actuator, given as some weights onh ll d h l d h l h



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the controlled output, then leads to the new control scheme:

The associated general control conguration is :



About the control structure - Problem denition

The corresponding H ∞ suboptimal control problem is therefore to nd a controller K (s) such thatW S

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T ew (s) ∞ =W e S W u KS ∞

≤ γ with

T ew (s) = F l (P, K ) = P 11 + P 12 K (I − P 22 K )− 1 P 21

= W e0 + − W e G

W uK (I + GK )− 1 I

= W e S W u KS

in Matlab



What about disturbance attenuation ?

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

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This corresponds to the closed-loop system.

T ew = W e S y W e S y GW u KS y W u T u

The new H ∞ control problem therefore includes the input disturbance rejection objective, thanksto S y G that should satisfy the same template as S , i.e an high-pass lter!Remarks: Note that W u T u is an additional constraint that may lead to an increase of the attenuation level γ since it is not part of the objectives. Hopefully T u is low pass, and W u 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’

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additional 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 the

disturbance rejection properties.Let then consider : di (t ) = W d .d . In that case the closed-loop system is This corresponds to theclosed-loop system.

T ew = W e S y W e S y GW dW u KS y W u T u W d

and the template expected for S y G is now 1W d .W e .First interest: improve the disturbance weight as W d = 100 ... but this has a price (see Fig. belowfor an example)

10 -2 10 -1 10 0 10 1 10 2 10 3-120

-100

-80

-60

-40

-20

0

20

M a g n

i t u

d e

( d B )

Sensitivity*Plant SG

with W d=1

with W d=100

10 -2 10 -1 10 0 10 1 10 2 10 3-50

-40

-30

-20

-10

0

10

20

30

40

Controller*Sensitivity KS

S i n g u

l a r

V a

l u e s

( d B )

with W d=100

with W d=1



More generally...

To include multiple objectives in a SINGLE H ∞ control problem, there are 2 ways:1

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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: rst think simple !!


5 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

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generalized 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 the

Riccati approach or the LMI approach of the H ∞ control problem[Zhou et al.(1996)Zhou, Doyle, and Glover] [Skogestad and Postlethwaite(1996)] .Notations:

P ẋ = Ax + B 1 w + B 2 ue = C 1 x + D 11 w + D 12 uy = C 2 x + D 21 w + D 22 u

⇒ P =A B 1 B2C 1 D11 D12C 2 D21 D22

with x ∈R n , x ∈R n w , u ∈R n u , z ∈R n z et y ∈R n y




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5 H ∞ control design The Riccati approach

Asuumptions for the Riccati method

A1: (A, B 2 ) stabilizable and (C 2 , A) detectable: necessary for the existence of stabilizing


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controllers

A2: rank (D 12 ) = n u and rank (D 21 ) = n y : Sufcient to ensure the controllers are proper, hencerealizable

A3: ∀ω ∈R , rank A − jωI n B2C 1 D12 = n + n u

A4: ∀ω ∈R , rank A − jωI n B1C 2 D21 = n + n y 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:

D 11 = 0 , D 22 = 0 , D 12 T [ C 1 D12 ] = 0 I n u ,

B1

D 21D 21 T =

0

I n ynot necessary but simplify the solution (does correspond to the given theorem next but can beeasily relaxed)

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The problem solvability

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

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Theorem (1)

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

i the Hamiltonian H = A γ − 2 B 1 B T 1 − B 2 B

T 2

− C T 1 C 1 − AT has no eigenvalues on the

imaginary axis.ii there exists X ∞ ≥ 0 t.q. AT X ∞ + X ∞ A + X ∞ (γ − 2 B 1 B T 1 − B 2 B

T 2 ) X ∞ + C

T 1 C 1 = 0 ,

iii the Hamiltonian J = AT γ − 2 C T 1 C 1 − C

T 2 C 2

− B 1 B T 1 − Ahas no eigenvalues on the

imaginary axis.

iv there exists Y ∞ ≥ 0 t.q. A Y ∞ + Y ∞ AT + Y ∞ (γ − 2 C T 1

C 1 − C T 2

C 2 ) Y ∞ + B 1 B T 1

= 0 ,v the spectral radius ρ(X ∞ Y ∞ ) ≤ γ 2 .

O Sename GIPSA-lab 88/138 5 H ∞ control design The Riccati approach

Controller reconstruction

Theorem (2)

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If the necessary and sufcient conditions of the Theorem 1 are satised, then the so-called central

controller is given by the state space representation

K sub (s) = Â∞ − Z ∞ L∞F ∞ 0

with

Â∞ = A + γ − 2 B 1 B T 1 X ∞ + B 2 F ∞ + Z ∞ L∞ C 2F ∞ = − B T 2 X ∞ , L∞ = − Y ∞ C

T 2

Z ∞ = I − γ − 2 Y ∞ X ∞− 1

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

u 2 (t ) = − B T 2 X ∞ x̂ (t ),where x̂ (t ) is the observer state vector

˙̂x(t ) = A x̂(t ) + B 1 ŵ(t ) + B 2 u(t ) + Z ∞ L∞ C 2 x̂ (t ) − y(t ) . (32)

and ŵ(t ) is dened asO Sename GIPSA-lab 89/138

5 H ∞ control design The LMI approach for H ∞ control design

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 [Scherer(1990)] .

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p ( )when we refer to the H ∞ control problem, we mean: Find a controller C for system M such that,

given γ ∞ ,||F l (P, K ) || ∞ < γ ∞ (33)

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

γ ∗∞ = min( A K ,B K ,C K ,D K ) s.t.σ A⊂ C −

T zw (s) ∞ (34)

As presented in the previous sections, this condition is fullled thanks to the BRL. As a matter offact, the system is internally stable and meets the quadratic H ∞ performances iff. ∃ P = P T 0such that,

A T P + PA PB C T BT P − γ 22 I D

T

C D − I < 0 (35)

where A , B, C, D are given in (31 ). 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[Scherer et al.(1997b)Scherer, Gahinet, and Chilali] .

O Sename GIPSA-lab 90/138 5 H ∞ control design The LMI approach for H ∞ control design

The LMI approach for H ∞ control design - Problem solution

Theorem (LTI/ H ∞ solution [Scherer et al.(1997a)Scherer, Gahinet, and Chilali] )



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

C c Dcthat solves the H ∞

control problem, is obtained by solving the following LMIs in ( X , Y , A, B , C and D ), while minimizing γ ∞ ,M 11 (∗)T (∗)T (∗)T M 21 M 22 (∗)T (∗)T M 31 M 32 M 33 (∗)T

M 41 M 42 M 43 M 44

≺ 0

X I nI n Y

0

(36)

where,

M 11 = AX

+X

AT

+ B 2C

+ C

T

BT 2 M 21 =

A+ A

T

+ C T 2

D T

BT 2

M 22 = Y A + AT Y + B C 2 + C T 2 B T

M 31 = B T 1 + DT 21 D

T B T 2

M 32 = B T 1 Y + DT 21 B

T M 33 = − γ ∞ I n u

M 41 = C 1 X + D 12 C M 42 = C 1 + D 12 D C 2M 43 = D 11 + D 12 D D 21 M 44 = − γ ∞ I n y

(37)

O Sename GIPSA-lab 91/138 5 H ∞ control design The LMI approach for H ∞ control design

Controller reconstruction

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

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D c = DC c = ( C − D c C 2 X)M − T B c = N − 1 (B − YB 2 D c )Ac = N − 1 (A − YAX − YB 2 D c C 2 X − NB c C 2 X − YB 2 C c M T )M − T

(38)

where M and N are dened such that M N T = I n − 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 [Gahinet(1994)] .

O Sename GIPSA lab 92/138 6 Robust analysis

Outline


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LTI systems and signals norms

2. Pe rformance analysis

Denition of the sensitivity fu nctionsFrequency-dom ain analysisStability issues


4. Introduction to Linear Matrix InequalitiesBackground in Optimisation

LMI in controlSome useful lemmas




O Sename GIPSA lab 93/138 6 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

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model 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 unmodelle d dyna mics

O Sename GIPSA lab 94/138 6 Robust analysis Representation of uncertainties

Example 1: uncertainties [ ? ]

G̃(s) = k

1 +e− sh , 2 ≤ k, h, τ ≤ 3

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1 + τs

Let us choose the nominal parameters as, k = h = τ = 2 .5 and G the according nominal model.We can dene the ’relative’ uncertainty, which is actually referred as a MULTIPLICATIVEUNCERTAINTY, as

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G̃(s) = G(s)( I + W m (s)∆( s))

with W m (s) = 3. 5s +0 .25s +1and ∆ ∞ ≤ 1

6 Robust analysis Representation of uncertainties

Example 2: unmodelled dynamcis

Let us consider the system:

G̃(s) = G(s) 1

1 +, τ ≤ τ max

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( ) ( )1 + τs

This can be modelled as:

G̃(s) = G0 (s)( I + W m (s)∆( s)) , W m (s) = τ max jω1 + τ max jω

with ∆ ∞ ≤ 1which can be represented as

withN (s) = N 11 (s) N 12 (s)N 21 (s) N 22 (s)

= 0 I G0 W m (s) G0 (s)

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Example 4: parametric uncertainties in state space equations

Let us consider the following uncertain system:

ẋ1 = ( − 2 + δ1 )x1 + ( − 3 + δ2 )x2


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G :x1 ( 2 + δ1 )x1 + ( 3 + δ2 )x2ẋ2 = ( − 1 + δ3 )x2 + uy = x1

(39)

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

u ∆ 1 = δ1 x1 , u∆ 2 = δ2 x2 , u∆ 3 = δ3 x2

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

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where ∆ and y∆ are given as:

∆ =δ1 0 00 δ2 00 0 δ3

, y∆ =x1x2x2

and N given by the state space representation:

N :ẋ1ẋ2

==

− 2x1 − 3x2 + u ∆ 1 + u− x2 + u + u ∆ 3

y = x1


Towards LFR (LFT)

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



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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 LinearFr

Robust Control of MIMO Systems

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