Linear Matrix Inequalities in System and Control Theory

Solmaz Sajjadi KiaAdviser: Prof. Jabbari

System, Dynamics and Control SeminarUCI, MAE Dept.April 14, 2008

Linear Matrix Inequality (LMI)

miRFFR

FFF

nnTii

m

m

iii

,...,2,1,0,

)0(0)(1

0

x

xx

Set of n polynomial inequalities in x, e.g.,

Convex constraint on x

0)2()3)(1(

01

032

210

11

11

01

11

32

21)(

221221

21

221

212121

xxxxx

xx

xxx

xxxxxxxF

Matrices as Variable

0,0 XXX AAT

010

00

01

10

00

01

.,.

33221132132

21

22

EEE

Rge

xxxxxxxx

xxX

X

0)()()( 333222111 AEEAAEEAAEEA TTT xxx

00

0)(3

1

i i

iiT

iE

AEEAx

Multiple LMIs

)0)(....,,0)(,0)((

,0)(....,,0)(,0)()()2()1(

)()2()1(

xFxFxFDiag

xFxFxFn

nmiRFFR

FFxF

nnTii

m

m

iii

,...,2,1,0,

)0(0)(1

0

x

x

LMI Problems

Feasibility

)0(0)( xF

Minimization Problem

)0(0)(

tosubjectmin

xx

F

cT

How do we cast our control problems in LMI form?

We rely on quadratic function V(x)=x’Px

Three Useful Properties to Cast Problems in Convex LMI From

Congruent Transformation

S-Procedure

Schur Complement

Congruent transformation

invertible and square is where)0(0)(

)0(0)(

PPxTP

xT

T

Stable State Feedback Synthesis Problem

0))()((0x(x)V

0x,xV(x)

Stability Lyapunov

T

T

xBABAxxxx TTTT KPPKPP

PP

xBAxxu

BuAxx)( K

K

0PKKPPP TTT BBAA

1

0

0

FQKFFQ

QTTT BBAQA

TTTT BBAA QQPQQPKKPPPQ , where0)( 1

0TTT BBAA QKKQQQ

KQFFFQQ where0TTT BBAA

S Procedure

12

0)()(such that,..2,1,0:2

,..2,10)(such that for 0)(:1

10

0

SS

TTNkS

NkTVTS

k

N

kkk

k

xx

xxx

12

0)()(such that,..2,1,0:2

,..2,10)(such that for 0)(:1

10

0

SS

TTNkS

NkTVTS

k

N

kkk

k

xx

xxx

Three Useful Properties to Cast Problems in Convex LMI From

Congruent Transformation

S-Procedure

Schur Complement

Reachable Set/Invariant Set for Peak Bound Disturbance

The reachable set (from zero): is the set of points the state vector can reach with zero initial condition, given some limitations on the disturbance.

The invariant set: is the set that the state vector does not leave once it is inside of it, again given some limits on the disturbance.

Reachable Set/Invariant Set for Peak Bound Disturbance

Ellipsoidal Estimate

wBxAx cl 1Peak Bound Disturbance

2max)()( wtwtwT

}:{),( 2cPxxxcP T

0))()()(()(,)()( 2max twtwxVxVwtwtw TT

0*

1

w

x

I

BAAwx cl

TclTT

PPPP

0,)( PPxxxV T

}:{),( 2maxmax wxxxwP T P

0))(()( 2max wxVxV

2max)(for)0)((0)( wxxxVxVxV T P

0))(()( 2max wxVxV

0)( 11 wwxBwwBxxAAx TTTTcl

Tcl

T PPPPP

0*

1

I

BAA clTcl

PPPP

xxSxRxRxQxQxRxS

xSxQ

xSxQxSxRxQ

TTT

T

onaffinely depend)( and),()(),()( where0)()(

)()(

0)()()()(,0)( 1

Linear (thus convex) Verses Nonlinear Convex inequality

Nonlinear (convex) inequalities are converted to LMI form using Schur Complement

Three Useful Properties to Cast Problems in Convex LMI From

Congruent Transformation

S-Procedure

Schur Complement

H∞ or L2 Gain

0)1(*

1

DCD

CBAAT

TT

2γ

PPP

00

21 such that smalles i.e., system, theofgain L2or Energy (t)w(t)wγ(t)z(t)zDwCxz

wBAxx TT

00

2

00

2 0)0())(( (t)w(t)wγ(t)z(t)z(t)w(t)wγ(t)z(t)zVxV TTTT

0(x(t))V

0x,xV(x) :Lyapunov Quadratic2

T

(t)w(t)wγ(t)z(t)z TTPP

0)()()()( 211 (t)w(t)wγDwCxDwCxPxBwwPBxxPAPAx TTTTTTT

0*

0*

11

DD

DCBCCAA

w

x

DD

DCBCCAAwx

T

TTT

T

TTTTT

22 γ

PPP

γ

PPP

0

**

*1

I

D

CBAAT

TT

2γ

PPP

Norm of a vector in an ellipsoid

Find Max of ||u||=||Kx|| for x in {x| xTPx≤c2 }

max2max

22max

2

2max

2

2max

2

2

2max

||||||||)(

0))((0)(,00

uKxuuKxKxcPxxu

cKxKx

xKu

cKPxK

u

cKPP

c

uK

KP

TTTTT

TTT

T

0

**

* 11

1

I

D

CBAAT

Tclcl

Tcl

2γ

PPP

I

I

Q

00

00

00

0

**

* 112

1

I

D

CBAAT

Tcl

Tclcl

QQQ

0*

1

I

BAA clTcl

PPPP

wDuDxCz

wBuBAxx

11121

12

wDxKDCz

wBxKBAx

11121

12

)(

)(Kxu

0 QQ clTcl AA

Q0 cl

Tcl AA PP

I

Q

0

0

0*

1

I

BAA clTcl

QQQ

0*

2

2max

c

u

TQKQ0

2

2max

c

u

T

K

KP

I

Q

0

0

FKQ

A Saturation Problem

11121

112111

21

,

,

DDKDCC

BBBKAAwhere

wDxCz

wBxAx

Kxu

uDwDxCzuBwBAxx

clcl

clcl

clcl

clclcl

Problem: Synthesis/Analysis of a Bounded State Feedback Controller (||u||<umax) exposed wT(t)w(t)≤w2

max

Analysis: What is the largest disturbance this system can tolerate with K

Synthesis: Find a K such that controller never saturates

xT Px<wT

max

Analysis: What is the largest disturbance (e.g. wmax) the system can tolerate ?

umax=Kx-umax=Kx

}:{),( 2maxmax wxxxwP T P

0*

1

I

BAA clTcl

PPPP

0

0*2max

P

w

P

2max

uK T

2maxw

β1

βminimizeβ

wmax

1

xT Px<wT

max

Synthesis: Find a K such that controller never saturates

Kx=umaxKx=-umax

}:{),( 2maxmax wxxxwP T P

0*

1

I

BAA clTcl

QQQ

0

0*

2max

Q

QKQ

2maxw

u

T

FKQ

ilityfeasibfor ckeck

1FQK

0&0

0

min

2max

2max

21222

11

1111

Iw

uI

BAA

I

DI

CBAA

TTclcl

T

Tcl

Tclcl

KQQQQQ

γ

QQQ

2

2

0&0

0

min

2max

2max

21222222

11

1211121211

Iw

uI

BBBAA

I

DI

DCBBBAA

TTT

T

TTTTTT

T2

2

211

KQQQKQKQQQ

γ

KQQKQKQQQ

1

FQK

FKQ,QQ 21

Solution

Good Reference

Prof. Jabbari’s Note on LMIs

S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, “Linear Matrix Inequalities in Systems and Control Theory”