LMI-Linear Matrix Inequality

Dep. of Information Engineering

University of Pisa (Italy)

From Convex Optimization toLinear Matrix Inequalities

eng. Sergio Grammatico

[email protected]

Class of Identification of Uncertain Systems 2012/13

held by prof. Andrea Caiti

S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 1 / 30

Outline

1 Introduction

2 Semi-Definite ProgrammingFrom Convex Optimization to Semi-DefiniteProgramming

3 Linear Matrix InequalitiesDefinitionBrief Historical PerspectiveFirst simple problems

4 SoftwareSeDuMi and CVXExamples

5 ConclusionS. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 2 / 30

Outline

1 Introduction





Convex Optimization and Linear Matrix Inequalities(LMIs)Thanks to Stephen Boyd, Carsten Scherer et al.

This presentation is mainly based on

S. Boyd and Lieven Vandenberghe, Convex Optimization,Cambridge University Press, 2004.

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan,Linear Matrix Inequalities in System and Control Theory,Society for Industrial and Applied Mathematics (SIAM), 1994.

S. Boyd, Lecture notes in Convex Optimization,Standford University.


Outline

1 Introduction





Convex OptimizationGeneral problem

Convex Optimization Problem

minxRn

f(x)

sub. to gi(x) 0, i = 1, 2, ...,mAx = b

(1)

f , g1, ..., gm convex functions, A Rpn, b Rp.

Global Optimality

Any local optimal point of a convex problem is globally optimal


Special CaseLinear Programming (LP)

Linear Programming

minxRn

c>x

sub. to Gx h, Ax = b(2)


Special CaseQuadratic Programming (QP)

Quadratic Programming

minxRn

x>Px+ q>x

sub. to Gx h, Ax = b(3)

P positive semi-definite P < 0 (P = P>, eig(P ) 0).


Another Special CaseSemi-Definite Programming (SDP)

Semi-Definite Programming

minxRn

c>x

sub. to F0 + x1F1 + x2F2 + ...+ xnFn 4 0,Ax = b

(4)

Fi = F>i Rnn


Outline

1 Introduction





Linear Matrix InequalitiesDefinition

Standard LMI

F (x) := F0 +ni=1

xiFi 4 0 (5)

The inequality constraint of an SDP is an LMI.

Multiple LMIs as a unique LMI:[F0 0

0 F0

]+ x1

[F1 0

0 F1

]+ ...+ xn

[Fn 0

0 Fn

]4 0 (6)


Linear Matrix InequalitiesFeasibility and Optimization

Feasibility LMI Problem

Find x Rn such that F (x) < 0

Strict Feasibility LMI Problem

Find x Rn and R>0 such thatF (x) < In

LMI Optimization Problem

Find x Rn such thatF (x) < 0c>x is minimized


Linear Matrix InequalitiesA trick for NonLinear MIs

Schur Complement

Given Q(x) = Q(x)>, R(x) 0, S(x) affinely dependent on x,the NonLinear MI

Q(x) S(x)R(x)1S(x)> < 0 (7)

is equivalent to the LMI[Q(x) S(x)

S(x)> R(x)

]< 0. (8)


Brief Historical PerspectiveBack to Lyapunov

1 First LMI in Control Theory (A.M. Lyapunov)

Lyapunov Inequality (1890)

x = Ax A.S. P 0 : A>P + PA 0 (9)

Lyapunov equality: A>P + PA = Q, P ,Q 0Matrices as variables

2 Further contributions

Lure (Stability under Saturations via LMIs);Kalman, Yakubovich, Popov (Positive Real Lemma); ...


Brief Historical PerspectiveRiccati revisited

3 On the LQ Optimal Control (J.C. Willems)

Algebraic Riccati (LM)Inequality (1971)

A>P + PA+Q (PB + C>)R1 (PB + C>)> < 0 (10)is equivalent (by Schur Complement) to[

A>P + PA+Q PB + C>(PB + C>

)>R

]< 0. (11)

4 Interior-point algorithms for LMIs(Y.E. Nesterov and A. Nemirovski)


Brief Historical PerspectiveToday

LMIs are used for:

1 Stability analysis of uncertain systems

2 Robust control design for uncertain systems

3 Robust optimization

4 System identification

5 ...


Simple LMI problemsStability Analysis of Uncertain Linear Systems

Linear Differential Inclusions (Time-varying Uncertain Linear Systems)

x conv {Aix | i [1, s]} (12)

Quadratic Stability (Sufficient Condition)

P 0 : A>i P + PAi 0 i = (12) A.S. (13)


Simple LMI problemsRobust Linear State-Feedback Control design (1)

Controlled Linear Differential Inclusions

x conv {Aix+Biu | i [1, s]} , (14)

1 u(x) = Kx

2 x conv {(Ai +BiK)x}3 NonLinear MI in P , K:

(Ai +BiK)> P + P (Ai +BiK) 0 i (15)


Simple LMI problemsRobust Linear State-Feedback Control design (2)

Change of Variables for the LMI design of K:

4 Multiply (15) both on the left and on the right by P1

5 Q := P1

6 Y := KQ

7 LMI in Y , Q:

AiQ+QA>i +BiY + Y

>B>i 0 i (16)

8 P := Q1, K := Y Q1


Outline

1 Introduction





Solver and Matlab interfaceSeDuMi and CVX

Solver: SeDuMi (by J.F. Sturm)

http://sedumi.ie.lehigh.edu/

Matlab interface: CVX (by S. Boyd)

http://cvxr.com/cvx/

CVXOPT in Python

CVXGEN for C-code generation


Example 1Variable Mass-Spring System as Uncertain Linear System (1)

mx = kx bxk [k, k]

m

k

kx bx

x0

1 := x, 2 := x, w := k



Variable Mass-Spring System as Uncertain Linear System

=

[0 1

w 1] (17)

w [0.1, 1]

A1 =

[0 1

0.1 1], A2 =

[0 11 1

]Robust LMI Analysis

A>i P + PAi 0, i = 1, 2



Matlab code using CVX

A{1} = [0 1; -0.1 -1]; A{2} = [0 1; -1 -1];n = 2; s = 2;

cvx begin

variable P(n,n) symmetric

variable a

maximize(a)

P == semidefinite(n);

a >= 0;

for i=1:s

A{i}*P + P*A{i} + a*eye(n) == -semidefinite(n);end

cvx end


Example 2Quadratic Stabilizability is only Sufficient for Uncertain Linear Systems

Counterexample

x =

[0 11 0

]x+

[w1

]u (18)

w [w,w]

A1 = A2 = A, B1 =

[w1

], B2 =

[w1

]

Conservatism of Quadratics

(18) quadratic stabilizable (searching P, K) only if w [1, 1]

(18) stabilizable w R


Example 2Quadratic Stabilizability is only Sufficient for Uncertain Linear Systems

Counterexample

x =

[0 11 0

]x+

[w1

]u (18)

w [w,w]

A1 = A2 = A, B1 =

[w1

], B2 =

[w1

]

Conservatism of Quadratics

(18) quadratic stabilizable (searching P, K) only if w [1, 1](18) stabilizable w R


Example 3Nonlinear Inverted Pendulum as Uncertain Linear System (1)

I = mgl sin() +

mg

2 ,l I

x1 := , x2 := , u = = x2 = mglI

sin(x1)

x1x1 +

1

Iu



Inverted Pendulum as Uncertain Linear System

x =

[0 1

aw(x) 0

]x+

[0b

]u (19)

w(0) = 1

x1 [pi4 , pi4 ] w(x1) [0.9, 1]

A1 =

[0 1

0.9 a 0], A2 =

[0 1

1 a 0]

B1 = B2 =

[0b2

]



Robust LMI Control design from (16)

AiQ+QA>i +BiY + Y

>B>i 0, P = Q1, K = Y Q1

Matlab code using CVXcvx begin

variable Q(n,n) symmetric

variable Y(m,n)

maximize(log det(Q))

Q == semidefinite(n);

for i=1:s

A{i}*Q + Q*A{i} + B{i}*Y + Y*B{i} == -semidefinite(n);end

cvx end

P = Q^(-1); K = Y*P


Outline

1 Introduction





ConclusionLMIs in Systems and Control Theory

1 LMIs Robust Analysis and Control design

2 LMIs Uncertain Linear Systems


IntroductionSemi-Definite ProgrammingFrom Convex Optimization to Semi-Definite Programming

Linear Matrix InequalitiesDefinitionBrief Historical PerspectiveFirst simple problems

SoftwareSeDuMi and CVXExamples

Conclusion

LMI-Linear Matrix Inequality

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