Linear Matrix Inequality(LMI)

A linear matrix inequality is an expression of

the form

F (x) , F0 + x1F1 + · · ·+ xmFm > 0 (1)where

• x = (x1, · · · , xm) ∈ 0 in (1) means positivedefinite, i.e., uTF (x)u > 0 for all u ∈

Definition[Linear matrix inequality(LMI)]

A linear matrix inequality is

F (x) > 0 (2)

where F is an affine function mapping a finite

dimensional vector space to the set Sn , {M :M = MT ∈ 0, of real matrices.

remark Recall, from definition, that an affine

mapping F : V→ Sn necessarily takes the formF (x) = F0+T (x) where F0 ∈ Sn and T : V→ Snis a linear transformation. Thus if V is of di-mension m, and {e1, · · · , em} constitutes a basisfor V, then we can write

T (x) =m∑

j=1

xjFj

where the elements {x1, · · · , xm} are such thatx =

∑mj=1 xjej and Fj = T (ej) for j = 1, · · · , m.

Hence we obtain (1) as a special case.

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Remark. The same remark applies to map-

pings F : 0 .

Here, A, Q ∈

Remark. The LMI

F (x) = F0 + xF1 + · · ·+ xmFmdefines a convex constraint on x = (x1, · · · , xm).i.e., the set

F , {x : F (x) > 0}is convex. Indeed, if x1, x2 ∈ F and α ∈ (0,1)then

F (αx1+(1−α)x2) = αF (x1)+(1−α)F (x2) > 0

Convexity has an important consequence: even

though the LMI has no analytical solution in

general, it can be solved numerically with guar-

antees of fining a solution when one exists. Al-

though the LMI may seem special, it turns out

that many convex sets can be represented in

this way.

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1. Note that a system of LMIs (i.e. a finite

set of LMIs) can be written as a single LMI

since

F1(x) < 0...

FK(x) < 0

is equivalent to

F (x) , diag[F1(x), · · · , FK(x)] < 0

2. Combined constraints (in the unknown x)

of the form{

F (x) > 0Ax = b

or

{F (x) > 0x = Ay + b for some y

where the affine function F :

where M is an affine subset of

that the dimension of x̄ is smaller than the

dimension of x.

3. (Schur Complement) Let F : V→ Sn be anaffine function partitioned to

F (x) =

[F11(x) F12(x)F21(x) F22(x)

]

where F11(x) is square. Then

F (x) > 0 iff{F11(x) > 0

F22(x)− F21(x)F−111 (x)F12(x) > 0(4)

Note that the second inequality in (4) is

a nonlinear matrix inequality in x. It fol-

lows that nonlinear matrix inequalities of

the form (4) can be converted to LMIs,

and nonlinear inequalities (4) define a con-

vex constraint on x.

Types of LMI problems

Suppose that F, G : V → Sn1 and H : V → Sn2are affine functions.

Feasibility: The test whether or not there ex-

ist solutions x of F (x) > 0 is called a fea-

sibility problem. The LMI is called non-

feasible if no solutions exist.

Optimization: Let f : S → < and supposethat S = {x|F (x) > 0}. The problem to de-termine Vopt = infx∈S f(x) is called an op-timization problem with an LMI constraint.

Generalized eigenvalue problem: Minimize a

scalar λ ∈ < subject to

λF (x)−G(x) > 0F (x) > 0H(x) > 0

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What are LMIs good for?

Many optimization problems in control design,

identification, and signal processing can be for-

mulated using LMIs.

Example. Asymptotic stability of the LTI sys-

tem

ẋ = Ax , A ∈ 0, ATX + XA < 0

i.e. equivalent to feasibility of the LMI[

X 00 −ATX −XA

]> 0

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Example. Determine a diagonal matrix D such

that ||DMD−1|| < 1 where M is some givenmatrix. Since

||DMD−1|| < 1 ⇐⇒ D−TMTDTDMD−1 < I⇐⇒ MTDTDM < DTD⇐⇒ X −MTXM > 0

where X := DTD > 0 we see that the existence

of such a matrix means the feasibility of LMI.

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Example. Let F be an affine function and

consider the problem of minimizing

f(x) , σmax(F (x)) over x.

λmax(FT (x)F (x)) < γ

⇐⇒ γI − FT (x)F (x) > 0⇐⇒

[γI FT (x)

F (x) I

]> 0

if we define

x̄ ,[

xγ

], F̄ (x̄) ,

[γI FT (x)

F (x) I

], f̄(x̄) , γ ,

then F̄ is an affine function of x̄ and the prob-

lem to minimize the maximum eigenvalue of

F (x) is equivalent to determining inf f̄(x̄) sub-

ject to the LMI F̄ (x̄) > 0. Hence, this is an

optimization problem with a linear objective

function f̄ and an LMI constraint.

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Example(Simultaneous stabilization)

Consider k LTI systems with n-dim state space

and m-dim input space:

ẋ = Aix + Biu

where Ai ∈

the existence of a joint Lyapunov function, i.e.

Xi = · · · = Xk = X. The joint stabilizationproblem has a solution if this system of LMIs

is feasible.

H∞ nominal performance

Consider

x = Ax + Bu (7)

y = Cx + Du (8)

with state space X = 0 that satisfy the LMI.

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H2 nominal performance

We take impulsive inputs of the form u(t) =

δ(t)ei with ei the ith basis vector in the standard

basis of the input space 0Deiδ(t) for t = 00 for t < 0.

.

Only if D = 0, the sum of the squared norms

of all such impulse responses∑m

i=1 ||yi||22 is welldefined and given by

m∑

i=1

||yi||22 = trace∫ ∞0

BT exp(At)CTC exp(At)B dt

= trace∫ ∞0

C exp(At)BBT exp(AT t)CT dt

= trace∫ ∞−∞

G(jω)G∗(jω) dω

where G is the transfer function of the system.

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proposition Suppose that the system (7) is

asymptotically stable (and D = 0), then the

following statements are equivalent.

(a) ||G||2 < γ

(b) there exists K = KT > 0 and Z such that[

ATK + KA KBBTK −I

]< 0;

[K CT

C Z

]> 0; (10)

trace(Z) < γ2 (11)

(c) there exists K = KT > 0 and Z such that[

AK + KAT KCT

CK −I

]< 0;

[K B

BT Z

]> 0; (12)

trace(Z) < γ2 (13)

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pf. note that ||G||2 < γ is equivalent to re-quiring that the controllability gramian Wc :=∫∞0 exp(At)BB

T exp(AT t) dt satisfies

trace(CWCT ) < γ2.

Since the controllability gramian is the unique

positive definite solution to the Lyapunov equa-

tion

AW + WAT + BBT = 0

this is equivalent to saying that there exists

X > 0 such that

AX + XAT + BBT < 0; trace(CXCT ) < γ2.

With a change of variables K := X−1, this isequivalent to the existence of K > 0 and Z

such that

ATK + KA + KBBTK < 0; CK−1CT < Z;

and

trace(Z) < γ2.

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Now, using Schur complements for the first

two inequalities yields that ||G||2 < γ is equiv-alent to the existence of K > 0 and Z such

that[

ATK + KA KBBTK I

]< 0;

[K CT

C Z

]> 0;

and

trace(Z) < γ2 .

The equivalence with (12) is obtained by the

observation that ||G||2 = ||GT ||2.

Therefore, the smallest possible upper bound

of the H2-norm of the transfer function can be

calculated by minimizing the criterion trace(Z)

over the variables K > 0 and Z that satisfy

the LMIs defined by the first two inequalities

in (10) or (12).

Controller Synthesis

Let

ẋ = Ax + B1w + B2u

z∞ = C∞x + D∞1w + D∞2uz2 = C2x + D21w + D22u

y = Cyx + Dy1w

and

ẋK = AKxK + BKy

u = CKxK + DKy

be state-space realizations of the plant P (s)

and the controller K(s) respectively.

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Denoting by T∞(s) and T2(s) the CL TF fromw to z∞ and z2, respectively, we consider thefollowing multi-objective synthesis problem:

Design an output feedback controller u = K(s)y

such that

• H∞ performance: maintains the H∞ normof T∞ below γ0.

• H2 performance: maintains the H2 normof T2 below ν0.

• Multi-objective H2/H∞ controller design:minimizes the trade-off criterion of the form

α||T∞||2∞ + β||T2||22 with some α, β ≥ 0.

• Pole placement: places the CL poles insome prescribed LMI region D.

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Let the following denote the corresponding CL

state-space eqns,

ẋcl = Aclxcl + Bclw

z∞ = Ccl1xcl + Dcl1wz2 = Ccl2xcl + Dcl2w

then our design objectives can be expressed as

follows:

• H∞ performance: the CL RMS gain fromw to z∞ does not exceed γ iff there existsa symmetric matrix X∞ such that

AclX∞ + X∞ATcl Bcl X∞CTcl1

BTcl −I DTcl1Ccl1X∞ Dcl1 −γ2I

< 0

X∞ > 0

• H2 performance: the LQG cost from w toz2 does not exceed ν iff Dcl2 = 0 and there

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exists a symmetric matrices X2 and Q such

that[

AclX2 + X2ATcl Bcl

BTcl −I

]< 0

[Q CTcl2X2

X2CTcl2 X2

]> 0

trace(Q) < ν2

• Pole placement: the CL poles lie in theLMI region D := {z ∈ C : L + Mz + MT z̄ <0} with L = LT = [λij]1≤i,j≤m and M =[µij]1≤i,j≤m iff there exists a symmetric ma-trix Xpol such that

[λijXpol + µijAclXpol + µjiXpolATcl]1≤i,j≤m < 0

Xpol > 0 .

For tractability, we seek a single Lyapunov ma-

trix X := X∞ = X2 = Xpol that enforces allthree sets of constraints. Factorizing X as

X =

[R I

MT 0

] [0 SI NT

]−1

and introducing the transformed controller vari-

ables:

BK := NBK + SB2DKCK := CKMT + DKCyRAK := NAKMT + NBKCyR + SB2CKMT

+S(A + B2DKCy)R ,

the inequality constraints on X are turned into

LMI constraints in the variables R, S, Q,AK,BK, CKand DK. And we have the following subop-

timal LMI formulation of our multi-objective

synthesis problem:

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Minimize αγ2+βtrace(Q) over R, S, Q,AK,BK, CK, DKand γ2 satisfying:

AR + RAT + B2CK + CTKBT2 AK + A + B2DKCy B1 + B2DKDy1 FF ATS + SA + BKCy + CTy BTK SB1 + BKDy1 FF F −I F

C∞R + D∞2CK C∞ + D∞2DKCy D∞1 + D∞2DKDy1 −γ2I

< 0

Q C2R + D22CK C2D22DKCyF R IF I S

> 0

[λij

[R II S

]+ µij

[AR + B2CK A + B2DKCy

AK SA + BKCy

]+

µji

[(AR + B2CK)T ATK(A + B2DKCy)

T (SA + BKCy)T]]

1≤i,j≤m< 0

trace(Q) < ν20γ2 < γ20

D21 + D22DKDy1 = 0 .

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Given optimal solutions γ∗, Q∗ of this LMI prob-lem, the closed loop performances are bounded

by

||T ||∞ ≤ γ∗, ||T ||2 ≤√

trace(Q∗) .

This has been implemented by the matlab com-

mand “hinfmix”.

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Reference

• Boyd S, El Ghaoui L, Feron E, Balakrish-nan V. Linear matrix inequalities in system

and control theory, vol. 15 ed.

• Scherer C, Weiland S. Linear matrix in-equalities in control. Lecture notes of DISC

Course

• LMI Control Toolbox, Gahinet, Nemirovski,Laub, Chilali, Mathworks

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Affine combinations of linear systems

Often models uncertainty about specific pa-

rameters is reflected as uncertainty in specific

entries of the state space matrices A, B, C, D.

Let p = (p1, ..., pn) denote the parameter vec-

tor which expresses the uncertain quantities in

the system and suppose that this parameter

vector belongs to some subset P ⊂

time varying and coincide with state compo-

nents then (14) is better viewed as a nonlinear

system.

Of particular interest will be those systems in

which the system matrices affinely depend on

p. This means that

A(p) = A0 + p1A1 + · · ·+ pnAn (16)B(p) = B0 + p1B1 + · · ·+ pnBn (17)C(p) = C0 + p1C1 + · · ·+ pnCn (18)D(p) = D0 + p1D1 + · · ·+ pnDn . (19)

Or, written in a more compact form

S(p) = S0 + p1S1 + ... + pnSn

where

S(p) =

[A(p) B(p)C(p) D(p)

]

is the system matrix associated with (14). We

call these models affine parameter dependent

models. In MATLAB such a system is repre-

sented with the routines psys and pvec. For

n = 2 and a parameter box

P , {(p1, p2)| p1 ∈ [pmin1 , pmax1 ], p2 ∈ [pmin2 , pmax2 ]}the syntax is

affsys = psys( p, [s0, s1, s2] );

p = pvec( ‘box’, [p1min p1max ; p2min

p2max])

where p is the parameter vector whose i-th

component ranges between pimin and pimax.

Bounds on the rate of variations, ṗi(t) can be

specified by adding a third argument “rate”

when calling “pvec”.

See also the following routines:

• pdsimul for time simulations of affine pa-rameter models

• aff2pol to convert an affine model to anequivalent polytopic model

• pvinfo to inquire about the parameter vec-tor

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