robust and lpv control of mimo systems part 1: tools for ...o.sename/docs/tools.pdf · part 1:...

Some definitions Signal and system norms Stability issues Introduction to LMIs

Robust and LPV control of MIMO systemsPart 1: Tools for analysis and control of dynamical systems

Olivier Sename

GIPSA-Lab

Tecnologico de Monterrey - July 2016


Some definitionsNonlinear dynamical systemsLTI dynamical systems

Signal and system normsSignal normsSome topological spaces recallsSystem norms: H2 andH∞ perfomances

Stability issuesWell-posednessInternal stabilityInput-Output stability

Introduction to LMIsBackground in OptimisationLMI in controlTo go further: some useful lemmas

O. Sename [GIPSA-lab] 2/38


Reference books

To be studied during the courseI 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

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

I 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

I Carsten Scherer’s courseshttp://www.dcsc.tudelft.nl/ cscherer/. , Lecture slides available (MScCourse "Robust Control", MSc Course "Linear Matrix Inequalities in Control")

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



Outline


Signal and system normsSignal normsSome topological spaces recallsSystem norms: H2 and H∞ perfomances





Nonlinear systemsIn control theory, dynamical systems are mostly modeled and analyzed thanks to theuse of a set of Ordinary Differential Equations (ODE)1.Nonlinear dynamical system modeling is the "most" representative model of a givensystem. Generally this model is derived thanks to system knowledge, physicalequations etc. Nonlinear dynamical systems are described by nonlinear ODEs.

Definition (Nonlinear dynamical system)

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

ΣNL :

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

(1)

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

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



Comments

I The main advantage of nonlinear dynamical modeling is that (if it is correctlydescribed) it catches most of the real system phenomena.

I On the other side, the main drawback is that there is a lack of mathematical andmethodological tools; e.g. parameter identification, control and observationsynthesis and analysis are complex and non systematic (especially for MIMOsystems).

I In this field, notions of robustness, observability, controllability, closed loopperformance etc. are not so obvious. In particular, complex nonlinear problemsoften need to be reduced in order to become tractable for nonlinear theory, or toapply input-output linearization approaches (e.g. in robot control applications).

I As nonlinear modeling seems to lead to complex problems, especially for MIMOsystems control, the LTI dynamical modeling is often adopted for control andobservation purposes. The LTI dynamical modeling consists in describing thesystem through linear ODEs. According to the previous nonlinear dynamicalsystem definition, LTI modeling leads to a local description of the nonlinearbehavior (e.g. it locally describes, around a linearizing point, the real systembehavior).

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



Definition of LTI systems

Definition (LTI dynamical system)

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

ΣLTI :

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

(2)

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

The LTI system locally describes the real system under consideration and thelinearization procedure allows to treat a linear problem instead of a nonlinear one. Forthis class of problem, many mathematical and control theory tools can be applied likeclosed loop stability, controllability, observability, performance, robust analysis, etc. forboth SISO and MIMO systems. However, the main restriction is that LTI models onlydescribe the system locally, then, compared to nonlinear models, they lack ofinformation and, as a consequence, are incomplete and may not provide globalstabilization.



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

Definition (Norm and Normed vector space)I Let V be a finite dimension space. Then ∀ p ≥ 1, the application ||.||p is a norm,

defined as,

||v||p =(∑

i

|vi|p)1/p (3)

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



L? norms

Definition (L1, L2, L∞ norms)I The 1-Norm of a function x(t) is given by,

‖x(t)‖1 =

∫ +∞

0|x(t)|dt (4)

I 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.I 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∞).



About vector spaces

Definition (Banach, Hilbert, Hardy and Lp spaces)I 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).I A Hilbert space is a (real or complex) vector space H with an inner product< ., . > that is complete under the norm defined by the inner product. The norm off ∈ H is then defined by,

‖f‖ =√< f, f > (8)

Every Hilbert space is a Banach since a Hilbert space is complete with respect tothe norm associated with its inner product.

I The Hardy spaces (Hp) are certain spaces of holomorphic functions (functionsdefined on an open subset of the complex number plane C with values in C thatare complex-differentiable at every point) on the unit disk or upper half plane.

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

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



L2 and H2 spaces

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

||f ||2 =

√1

2π

∫ +∞

−∞Tr[f∗(jω)f(jω)]dω <∞ (9)

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

< f, g >=

√1

2π

∫ +∞

−∞Tr[f∗(jω)g(jω)]dω (10)

Definition (H2 and RH2 spaces)H2 is a subspace (Hardy space) of L2 with matrix functions f(jω), ∀ω ∈ R, analytic inRe(s) > 0 (functions that are locally given by a convergent power series anddifferentiable on each point of its definition set). In particular, the real rational subspaceof H2, which consists of all strictly proper and real rational stable transfer matrices, isdenoted by RH2.



L2 and H2 spaces

ExampleIn control theory

s+1(s+10)(s+6)

∈ RH2s+1

(s−10)(s+6)6∈ RH2

s+1(s+10)

6∈ RH2

(11)



L∞ and H∞ spaces

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

supω∈R

σ[f(jω)] <∞ (12)

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



L∞ and H∞ spaces

ExampleIn control theory

s+1(s+10)(s+6)

∈ RH∞s+1

(s−10)(s+6)6∈ RH∞

s+1(s+10)

∈ RH∞(13)



H2 norm

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

‖G(jω)‖2 =√∫+∞−∞ g∗(t)g(t)dt

=√

12π

∫+∞−∞ Tr[G∗(jω)G(jω)]dω

= supw(s)∈H2

||z(s)||∞||u(s)||2

(14)

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

RemarkH2 physical interpretations and remarks

I For SISO systems, it represents the area located below the so called Bodediagram.

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

I The H2 norm can be computed analytically (through the use of the controllabilityand observability Grammians) or numerically (through LMIs).



H∞ norm

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

‖G(jω)‖∞ = supω∈R σ (G(jω))

= supw(s)∈H2

‖z(s)‖2‖w(s)‖2

= maxw(t)∈L2

‖z‖2‖w‖2

(15)

RemarkH∞ physical interpretations

I 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 themaximum amplification that the system can deliver on the whole frequency set.

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

I Unlike H2 , the H∞ norm cannot be computed analytically. Only numericalsolutions can be obtained (e.g. Bisection algorithm, or LMI resolution).



Recalls on Singular Value DefinitionLet A ∈ Rm×n, There exists unitary matrices:

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

such that

A = UΣV T , Σ =

[Σ1 00 0

],Σ1 =

σ1 0 . . . 00 σ2 . . . 0...

... · · · . . .0 0 . . . σp

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

AATui = σ2i ui

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



Outline







Well-posedness


Consider

G = −s− 1

s+ 2, K = 1

Therefore the control input is non proper:

u =s+ 2

3(r − n− dy) +

s− 1

3di

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

⇔ I +K(∞)G(∞) is invertible

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


Internal stabilityDEF: A system is internally stable if all the transfer functions of the closed-loop systemare stable

Figure: Control scheme

(yu

)=

((I +GK)−1GK (I +GK)−1G

K(I +GK)−1 −K(I +GK)−1G

)(rdi

)For instance :

G =1

s− 1, K =

s− 1

s+ 1,

(yu

)=

(1s+2

s+1(s−1)(s+2)

s−1s+2

− 1s+2

)(rdi

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



Small Gain theoremConsider the so called M −∆ loop.

Figure: M −∆ form

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

‖M‖∞ < γ



Input-Output Stability

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

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

γpeak = sup0<‖u‖∞<∞

‖y‖∞‖u‖∞

and is referred to as the PEAK TO PEAK Gain.

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

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

γenergy = sup0<‖u‖2<∞

‖y‖2‖u‖2

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

γenergy = supω‖G(jω)‖ := ‖G‖∞

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



Outline







Brief on optimisation

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

f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y) (16)

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

epi(f) = {(x, λ)|f(x) ≤ λ} (17)

is convex.

Definition ((Strict) LMI constraint)A Linear Matrix Inequality constraint on a vector x ∈ Rm is defined as,

F (x) = F0 +m∑i=1

Fixi � 0(� 0) (18)

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



Convex to LMIs

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

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

(19)

which is equivalent to,

F (P ) =

[−P 00 ATP + PA

]≺ 0 (20)

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

LMI constraints F (x) � 0 are convex in x, i.e. the set {x|F (x) � 0} is convex. ThenLMI based optimization falls in the convex optimization. This property is fundamentalbecause it guarantees that the global (or optimal) solution x∗ of the the minimizationproblem under LMI constraints can be found efficiently, in a polynomial time (byoptimization algorithms like e.g. Ellipsoid, Interior Point methods).



LMI problemTwo kind of problems can be handled

Feasibility: The question whether or not there exist elements x ∈ X such thatF (x) < 0 is called a feasibility problem. The LMI F (x) < 0 is calledfeasible 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}. Theproblem to determine

Vopt = infx∈Sf(x)

is called an optimization problem with an LMI constraint. This probleminvolves the determination of Vopt, the calculation of an almostoptimal solution x (i.e., for arbitrary ε > 0 the calculation of an x ∈ Ssuch that Vopt ≤ f(x) ≤ Vopt + ε, or the calculation of a optimalsolutions xopt (elements xopt ∈ S such that Vopt = f(xopt)).



Examples of LMI problemStability analysis is a feasability problem.LQ control is an optimization problem, formulated as:

LQ controlConsider a controllable system x = Ax+Bu. Find a state feedback u(t) = −Kx(t)s.t J =

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

problem whose solution is obtained solving the Riccati equation:

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

and then the state feedback is given by:

u(t) = −R−1BTPx(t)

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

BTP R

]> 0



Semi-Definite Programming (SDP) ProblemLMI programming is a generalization of the Linear Programming (LP) to cone positivesemi-definite matrices, which is defined as the set of all symmetric positivesemi-definite matrices of particular dimension.

Definition (SDP problem)A SDP problem is defined as,

min cT xunder constraint F (x) � 0

(21)

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

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

global minimum (for a given a priori fixed precision) in a time polynomial in the sizeof the problem (given by m, the number of variables and n, the size of the LMI).

I SDP can be practically and efficiently solved for LMIs of size up to 100× 100 andm ≤ 1000 see ElGhaoui, 97. Note that today, due to extensive developments inthis area, it may be even larger.



The state feedback design problem

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

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

(A−BK)TP + P(A−BK) < 0

⇔ ATP + PA−KTBTP − PBK < 0

which is obviously not linear...

Solution; use of change of variables

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

P−1AT +AP−1 − P−1KTBT −BKP−1 < 0

⇔ QAT +AQ + YTBT +BY < 0

with Q = P−1 and Y = −KP−1.

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



The Bounded Real LemmaThe L2-norm of the output z of a system ΣLTI is uniformly bounded by γ times theL2-norm of the input w (initial condition x(0) = 0).A dynamical system G = (A,B,C,D) is internally stable and with an ||G||∞ < γ ifand only is there exists a positive definite symmetric matrix P (i.e P = PT > 0 s.t

AT P + P A P B CT

BT P −γ I DT

C D −γ I

< 0, P > 0. (22)

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

T

0 P 0 0P 0 0 00 0 −γ2I 00 0 0 I

I 0A B0 IC D

≺ 0 (23)

Note that the BRL is an LMI if the only unknown (decision variables) are P and γ (orγ2).



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

x = A(δ)x

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

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

A(δ)TP + PA(δ) < 0, for all δ ∈ ∆ (24)

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



Schur lemma

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

Q SST R

]� 0 (25)

is equivalent to

R � 0Q− SR−1ST � 0

(26)

The Schur lemma allows to a convert a quadratic constraint (ellipsoidal constraint) intoan LMI constraint.



Kalman-Yakubovich-Popov lemma

LemmaFor any triple of matrices A ∈ Rn×n, B ∈ Rn×m,

M ∈ R(n+m)×(n+m) =

[M11 M12

M21 M22

], the following assessments are equivalent:

1. There exists a symmetric K = KT � 0 s.t.[I 0A B

]T [0 KK 0

] [I 0A B

]+M < 0

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

[A− jωI B

] [ xw

]= 0⇒

[xw

]TM

[xw

]< 0

3. If M = −[I 0A B

]T [0 KK 0

] [I 0A B

]then the second statement is

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

C(jωI −A)−1B +D

]∗ [Q SST R

] [I

C(jωI −A)−1B +D

]> 0

This lemma is used to convert frequency inequalities into Linear Matrix Inequalities.



Projection Lemma

LemmaFor given matrices W = WT , M and N , of appropriate size, there exists a real matrixK = KT such that,

W +MKNT +NKTMT ≺ 0 (27)

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

W +MU + UTMT ≺ 0W +NV + V TNT ≺ 0

(28)

or, equivalently, if and only if,

MT⊥WM⊥ ≺ 0NT⊥WN⊥ ≺ 0

(29)

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

The projection lemma is also widely used in control theory. It allows to eliminatevariable by a change of basis (projection in the kernel basis). It is involved in one of theH∞ solutions ?)see e.g..



Completion Lemma

LemmaLet X = XT , Y = Y T ∈ Rn×n such that X > 0 and Y > 0. The three followingstatements are equivalent:

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

XT2 X3

]� 0 and

[X X2

XT2 X3

]−1

=

[Y Y2Y T2 Y3

](30)

2.[X II Y

]� 0 and rank

[X II Y

]≤ n+ r

3.[X II Y

]� 0 and rank [XY − I] ≤ r

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



Finsler’s lemmaThis Lemma allows the elimination of matrix variables.

LemmaThe following statement are equivalent

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

I BTAB < 0 where BB = 0

I A+ λBTB < 0 for some scalar λI A+XB +BTXT < 0 for some matrix XI B⊥

TAB⊥ < 0



Interest of LMIsLMIs allow to formulate complex optimization problems into "Linear" ones, allowing theuse of convex optimization tools.Usually it requires the use of different transformations, changes of variables ... in orderto linearize the considered problmems: Congruence, Schur complement, projectionlemma, Elimination lemma, S-procedure, Finsler’s lemme ...

Examples of handled criteriaI stabilityI H∞, H2, H2/H∞ performancesI robustness analysis: Small gain theorem, Polytopic uncertianties, LFT

representations...I Robust control and/or observer designI pole placementI stability, stabilization with input constraintsI Passivity constraintsI Time-delay systems



References

I Boyd et al., 1994 Boyd, S., El-Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). LinearMatrix Inequalities in System and Control Theory. SIAM Studies in applied mathematics.

I Briat, 2015 Briat C. (2015). Linear Parameter-Varying and Time-Delay Systems: Analysis,Observation, Filtering & Control. Springer Verlag (Advances in Delays and Dynamics Series),2015. www.briat.info

I Doyle et al. 92 J.C. Doyle, B.A. Francis, and A.R. Tannenbaum. Feedback control theory.Macmillan Publishing Company, New York, 1992. www.control.utoronto.ca/ francis.

I El-Ghaoui(1997) L. El-Ghaoui., Commande Robuste, Ecole Nationale des TechniquesAvancées / Ecole Polytechnique, 1997.

I Poussot-Vassal, 2008 Poussot-Vassal, C. (2008). Commande Robuste LPV Multivariable deChâssis Automobile. PhD thesis, Grenoble INP.sites.google.com/site/charlespoussotvassal

I Scherer and Wieland, 2004 Scherer, C. and Wieland, S. (2004). Linear Matrix inequalities inControl. lecture support, DELFT University.

I Skogestad and Postlethwaite, 2005 S. Skogestad and I. Postlethwaite. Multivariable FeedbackControl: analysis and design. John Wiley and Sons, 2005.www.nt.ntnu.no/users/skoge.

I Zhou, 98 K. Zhou. Essentials of Robust Control. Prentice Hall, New Jersey, 1998.www.ece.lsu.edu/kemin.

I Zhou et al., 1996 Zhou, K., Doyle, J., and Glover, K. (1996). Robust and Optimal Control.Prentice hall.


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