convex optimization and linear matrix inequalitiessistawww/smc/...convex optimization and linear...

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Convex Optimization

and

Linear Matrix Inequalities

Carsten Scherer

Delft Center for Systems and Control (DCSC)

Delft University of Technology

The Netherlands

Outline

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• From Optimization to Convex Semi-Definite Programming

• Linear Matrix Inequalities (LMIs)

• Truss-Topology Design

• LMIs and Stability

• A First Glimpse at Robustness

• LMI Regions

Optimization

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The ingredients of any optimization problem are:

• A universe of decisions x ∈ X

Optimization

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• A subset S ⊂ X of feasible decisions

Optimization

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• A cost function or objective function f : S → R

Optimization

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• A cost function or objective function f : S → R

Optimization Problem/Programming Problem

Find a feasible decision x ∈ S with the least possible cost f(x).

In short such a problem is denoted as

minimize f(x)

subject to x ∈ S

Questions in Optimization

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• What is the least possible cost? Compute the optimal value

fopt := infx∈S

f(x) ≥ −∞.

Convention: If S = ∅ then fopt = +∞.

If fopt = −∞ the problem is said to be unbounded from below.


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• What is the least possible cost? Compute the optimal value

fopt := infx∈S

f(x) ≥ −∞.

Convention: If S = ∅ then fopt = +∞.

If fopt = −∞ the problem is said to be unbounded from below.

• How can we compute almost optimal solutions? For any chosen

positive absolute error ε, determine

xε ∈ S with fopt ≤ f(xε) ≤ fopt + ε.

By definition of the infimum such an xε does exist for all ε > 0.


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• Is there an optimal solution? Does there exist

xopt ∈ S with f(xopt) = fopt?


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If yes, the minimum is attained and we write

fopt = minx∈S

f(x).

Set of all optimal solutions is denoted as

arg minx∈S

f(x) = {x ∈ S : f(x) = fopt}.


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If yes, the minimum is attained and we write

fopt = minx∈S

f(x).

Set of all optimal solutions is denoted as

arg minx∈S

f(x) = {x ∈ S : f(x) = fopt}.

• Is optimal solution unique?

Nonlinear Programming (NLP)

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With decision universe X = Rn, the feasible set S is often defined by

constraint functions g1 : X → R, . . . , gm : X → R as

S = {x ∈ X : g1(x) ≤ 0, . . . , gm(x) ≤ 0} .


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S = {x ∈ X : g1(x) ≤ 0, . . . , gm(x) ≤ 0} .

The general nonlinear optimization problem is formulated as

minimize f(x)

subject to x ∈ X , g1(x) ≤ 0, . . . , gm(x) ≤ 0.


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S = {x ∈ X : g1(x) ≤ 0, . . . , gm(x) ≤ 0} .

The general nonlinear optimization problem is formulated as

minimize f(x)

subject to x ∈ X , g1(x) ≤ 0, . . . , gm(x) ≤ 0.

By exploiting particular properties of f, g1, . . . , gm (e.g. smoothness),

optimization algorithms typically allow to iteratively compute locally

optimal solutions x0: There exists an ε > 0 such that x0 is optimal on

S ∩ {x ∈ X : ‖x− x0‖ ≤ ε}.

Linear Programming (LP)

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With the decision vectors x = (x1 · · · xn)T ∈ Rn consider the problem

minimize c1x1 + · · ·+ cnxn

subject to a11x1 + · · ·+ a1nxn ≤ b1

...

am1x1 + · · ·+ amnxn ≤ bm

The cost is linear and the set of feasible decisions is defined by finitely

many affine inequality constraints.


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...




Simplex algorithms or interior point methods allow to efficiently

1. decide whether the constraint set is feasible (value < ∞).


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...






2. decide whether problem is bounded from below (value > −∞).


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...






2. decide whether problem is bounded from below (value > −∞).

3. compute an (always existing) optimal solution if 1. & 2. are true.


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Major early contributions in 1940s by

• George Dantzig (simplex method)

• John von Neumann (duality)

• Lenoid Kantorovich (economics applications)


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Leonid Khachiyan proved polynomial-time solvability in 1979.


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Narendra Karmakar introduce an interior point method in 1984.


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Has numerous applications for example in economical planning problems

(business management, flight-scheduling, resource allocation, finance).


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Has numerous applications for example in economical planning problems

(business management, flight-scheduling, resource allocation, finance).

LPs have spurred the development of optimization theory and appear

as subproblem in many optimization algorithms.

Recap

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For a real or complex matrix A the inequality A 4 0 means that A

is Hermitian and negative semi-definite

Recap

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is Hermitian and negative semi-definite.

• A is defined to be Hermitian if A = A∗ = AT . If A is real then this

amounts to A = AT and A is then called symmetric.

Recap

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All eigenvalues of Hermitian matrices are real.

Recap

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• By definition A is negative semi-definite if A = A∗ and

x∗Ax ≤ 0 for all complex vectors x 6= 0.

A is negative semi-definite iff all its eigenvalues are non-positive.

Recap

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• A 4 B means by definition: A, B are Hermitian and A−B 4 0.

Recap

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• A 4 B means by definition: A, B are Hermitian and A−B 4 0.

• A ≺ B, A 4 B, A < B, A � B defined/characterized analogously.

Recap

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Let A be partitioned with square diagonal blocks as

A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.

Recap

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A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.

Prototypical Proof. Choose any vector zi 6= 0 of length compatible

with the size of Aii.

Recap

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A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.


with the size of Aii. Define z = (0, . . . , 0, zTi , 0, . . . , 0)T 6= 0 with the

zeros blocks compatible in size with A11, . . . , Amm.

Recap

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A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.



zeros blocks compatible in size with A11, . . . , Amm. This construction

implies zT Az = zTi Aiizi.

Recap

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A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.




implies zT Az = zTi Aiizi. Since A ≺ 0, we infer zT Az < 0.

Recap

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A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.




implies zT Az = zTi Aiizi. Since A ≺ 0, we infer zT Az < 0. Therefore,

zTi Aiizi < 0.

Recap

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A =

A11 · · · A1m

.... . .

...

Am1 · · · Amm

.

Then

A ≺ 0 implies A11 ≺ 0, . . . , Amm ≺ 0.




implies zT Az = zTi Aiizi. Since A ≺ 0, we infer zT Az < 0. Therefore,

zTi Aiizi < 0. Since zi 6= 0 was arbitrary we infer Aii ≺ 0.

Semi-Definite Programming (SDP)

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Let us now assume that the constraint functions G1, . . . , Gm map Xinto the set of symmetric matrices, and define the feasible set S as

S = {x ∈ X : G1(x) 4 0, . . . , Gm(x) 4 0} .


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S = {x ∈ X : G1(x) 4 0, . . . , Gm(x) 4 0} .

The general semi-definite program (SDP) is formulated as

minimize f(x)

subject to x ∈ X , G1(x) 4 0, . . . , Gm(x) 4 0.


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S = {x ∈ X : G1(x) 4 0, . . . , Gm(x) 4 0} .


minimize f(x)


• This includes NLPs as a special case.


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S = {x ∈ X : G1(x) 4 0, . . . , Gm(x) 4 0} .


minimize f(x)



• Is called convex if f and G1, . . . , Gm are convex.


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S = {x ∈ X : G1(x) 4 0, . . . , Gm(x) 4 0} .


minimize f(x)



• Is called convex if f and G1, . . . , Gm are convex.

• Is called linear matrix inequality (LMI) optimization problem

or linear SDP if f and G1, . . . , Gm are affine.

Recap: Affine Sets and Functions

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The set S in the vector space X is affine if

λs1 + (1− λ)s2 ∈ S for all s1, s2 ∈ S, λ ∈ R.


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The matrix-valued function F defined on S is affine if the domain

of definition S is affine and if

F (λs1 + (1− λ)s2) = λF (s1) + (1− λ)F (s2)

for all s1, s2 ∈ S, λ ∈ R.


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The matrix-valued function F defined on S is affine if the domain

of definition S is affine and if

F (λs1 + (1− λ)s2) = λF (s1) + (1− λ)F (s2)

for all s1, s2 ∈ S, λ ∈ R.

For points s1, s2 ∈ S recall that

• {λs1 + (1− λ)s2 : λ ∈ R} is the line through s1, s2

• {λs1 + (1− λ)s2 : λ ∈ [0, 1]} is the line segment between s1, s2

Recap: Convex Sets and Functions

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The set S in the vector space X is convex if

λs1 + (1− λ)s2 ∈ S for all s1, s2 ∈ S, λ ∈ (0, 1).


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The symmetric-valued function F defined on S is convex if the

domain of definition S is convex and if

F (λs1 + (1− λ)s2) 4 λF (s1) + (1− λ)F (s2)

for all s1, s2 ∈ S, λ ∈ (0, 1).


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F (λs1 + (1− λ)s2) 4 λF (s1) + (1− λ)F (s2)

for all s1, s2 ∈ S, λ ∈ (0, 1).

F is strictly convex if 4 can be replaced by ≺.


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F (λs1 + (1− λ)s2) 4 λF (s1) + (1− λ)F (s2)

for all s1, s2 ∈ S, λ ∈ (0, 1).

F is strictly convex if 4 can be replaced by ≺.

If F is real-valued, the inequalities 4 and ≺ are the same as the usual

inequalities ≤ and < between real numbers. Therefore our definition

captures the usual one for real-valued functions!

Relevance of Convexity?

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• Solvers for general nonlinear programs typically determine local op-

timal solutions. There are no guarantees for global optimality.


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• Main feature of convex programs:

Locally optimal solutions are globally optimal.


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Convexity alone neither guarantees that the optimal value is finite,

nor that there exists an optimal solution/efficient solution algorithms.


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• In general convex programs can be solved with guarantees on accuracy

if one can compute (sub)gradients of objective/constraint functions.


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• In general convex programs can be solved with guarantees on accuracy

if one can compute (sub)gradients of objective/constraint functions.

• Strictly feasible linear semi-definite programs are convex and can be

solved very efficiently, with guarantees on accuracy at termination.

Outline

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• LMI Regions

Linear Matrix Inequalities (LMIs)

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With the decision vectors x = (x1 · · · xn)T ∈ Rn a system of LMIs is

A10 + x1A

11 + · · ·+ xnA

1n 4 0

...Am

0 + x1Am1 + · · ·+ xnA

mn 4 0

where Ai0, Ai

1, . . . , Ain, i = 1, . . . ,m, are real symmetric data matrices.


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A10 + x1A

11 + · · ·+ xnA

1n 4 0

...Am

0 + x1Am1 + · · ·+ xnA

mn 4 0

where Ai0, Ai


LMI feasibility problem: Test whether there exist x1, . . . , xn that

render the LMIs satisfied.


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A10 + x1A

11 + · · ·+ xnA

1n 4 0

...Am

0 + x1Am1 + · · ·+ xnA

mn 4 0

where Ai0, Ai


LMI feasibility problem: Test whether there exist x1, . . . , xn that

render the LMIs satisfied.

LMI optimization problem: Minimize c1x1 + · · · + cnxn over all

x1, . . . , xn that satisfy the LMIs.

Only simple cases can be treated analytically → Numerical techniques.

LMI Optimization Problems

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subject to Ai0 + x1A

i1 + · · ·+ xnA

in 4 0, i = 1, . . . ,m


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i1 + · · ·+ xnA

in 4 0, i = 1, . . . ,m

• Natural generalization of LPs with inequalities defined by the cone of

positive semi-definite matrices. Considerable richer class than LPs.


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i1 + · · ·+ xnA

in 4 0, i = 1, . . . ,m



• The i-th constraint can be equivalently expressed as

λmax(Ai0 + x1A

i1 + · · ·+ xnA

in) ≤ 0.

This scalar constraint is defined with a non-differentiable function.


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i1 + · · ·+ xnA

in 4 0, i = 1, . . . ,m



• The i-th constraint can be equivalently expressed as

λmax(Ai0 + x1A

i1 + · · ·+ xnA

in) ≤ 0.

This scalar constraint is defined with a non-differentiable function.

• Interior-point or bundle methods allow to effectively decide about

feasibility/boundedness and to determine almost optimal solutions.


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Developments

• Bellman/Fan initialized derivation of optimality conditions (1963)

• Jan Willems coined terminology LMI and revealed relation to dissi-

pative dynamical systems (1971/72)

• Nesterov/Nemirovski exhibited essential feature (self-concordance)

for existence of polynomial-time solution algorithm (1988)

• Interior-point methods: Alizadeh (1992), Kamath/Karmarkar (1992)


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Developments

• Bellman/Fan initialized derivation of optimality conditions (1963)

• Jan Willems coined terminology LMI and revealed relation to dissi-

pative dynamical systems (1971/72)

• Nesterov/Nemirovski exhibited essential feature (self-concordance)

for existence of polynomial-time solution algorithm (1988)

• Interior-point methods: Alizadeh (1992), Kamath/Karmarkar (1992)

Suggested books: Boyd/El Ghaoui (1994), El Ghaoui/Niculescu (2000),

Ben-Tal/Nemiorvski (2001), Boyd/Vandenberghe (2004)

What are LMIs good for?

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• Many engineering optimization problem can be (often but not always

easily) translated into LMI problems.


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• Various computationally difficult optimization problems can be effec-

tively approximated by LMI problems.


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• In practice the description of the data is affected by uncertainty.

Robust optimization problems can be handled/approximated by

standard LMI problems.


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• In practice the description of the data is affected by uncertainty.

Robust optimization problems can be handled/approximated by

standard LMI problems.

In this course

How can we translate and solve (robust) control problems with LMIs?

Outline

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• LMI Regions

Truss Topology Design

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0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

Example: Truss Topology Design

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• Connect nodes with N bars of length l = col(l1, . . . , lN) (fixed) and

cross-sections x = col(x1, . . . , xN) (to-be-designed).


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• Impose bounds ak ≤ xk ≤ bk on cross-section and lT x ≤ v on total

volume (weight). Abbreviate a = col(a1, . . . , aN), b = col(b1, . . . , bN).


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• If applying external forces f = col(f1, . . . , fM) (fixed) on nodes the

construction reacts with the node displacement d = col(d1, . . . , dM).


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Mechanical model: A(x)d = f where A(x) is the stiffness matrix

which depends linearly on x and has to be positive semi-definite.


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Mechanical model: A(x)d = f where A(x) is the stiffness matrix

which depends linearly on x and has to be positive semi-definite.

• Goal is to maximize stiffness, for example by minimizing the elastic

stored energy fT d.


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Find x ∈ RN which minimizes fT d subject to the constraints

A(x) < 0, A(x)d = f, lT x ≤ v, a ≤ x ≤ b.


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A(x) < 0, A(x)d = f, lT x ≤ v, a ≤ x ≤ b.

Features

• Data: Scalar v, vectors f , a, b, l, and symmetric matrices A1, . . . , AN

which define the linear mapping A(x) = A1x1 + · · ·+ ANxN .


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A(x) < 0, A(x)d = f, lT x ≤ v, a ≤ x ≤ b.

Features



• Decision variables: Vectors x and d.


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A(x) < 0, A(x)d = f, lT x ≤ v, a ≤ x ≤ b.

Features




• Objective function: d → fT d which happens to be linear.


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A(x) < 0, A(x)d = f, lT x ≤ v, a ≤ x ≤ b.

Features




• Objective function: d → fT d which happens to be linear.

• Constraints: Semi-definite constraint A(x) < 0, nonlinear equality

constraint A(x)d = f , and linear inequality constraints lT x ≤ v,

a ≤ x ≤ b. Latter interpreted elementwise!

From Truss Topology Design to LMI’s

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Render LMI inequality strict. Equality constraint A(x)d = f allows to

eliminate d which results in

minimize fT A(x)−1f

subject to A(x) � 0, lT x ≤ v, a ≤ x ≤ b.


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Push objective to constraints with auxiliary variable:

minimize γ

subject to γ > fT A(x)−1f, A(x) � 0, lT x ≤ v, a ≤ x ≤ b.


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minimize γ


Trouble: Nonlinear inequality constraint γ > fT A(x)−1f .

Recap: Congruence Transformations

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Given a Hermitian matrix A and a square non-singular matrix T ,

A → T ∗AT

is called a congruence transformation of A.


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A → T ∗AT


If T is square and non-singular then

A ≺ 0 if and only if T ∗AT ≺ 0.


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A → T ∗AT


If T is square and non-singular then

A ≺ 0 if and only if T ∗AT ≺ 0.

The following more general statement is also easy to remember.

If A is Hermitian and T is nonsingular, the matrices A and T ∗AT

have the same number of negative, zero, positive eigenvalues.

What is true if T is not square? ... if T has full column rank?

Recap: Schur-Complement Lemma

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The Hermitian block matrix

(Q S

ST R

)is negative definite

if and only if

Q ≺ 0 and R− ST Q−1S ≺ 0

if and only if

R ≺ 0 and Q− SR−1ST ≺ 0.


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(Q S

ST R


if and only if

Q ≺ 0 and R− ST Q−1S ≺ 0

if and only if

R ≺ 0 and Q− SR−1ST ≺ 0.

Proof. First equivalence follows from(I 0

−ST Q−1 I

)(Q S

ST R

)(I −Q−1S

0 I

)=

(Q 0

0 R− ST Q−1S

).


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(Q S

ST R


if and only if

Q ≺ 0 and R− ST Q−1S ≺ 0

if and only if

R ≺ 0 and Q− SR−1ST ≺ 0.

Proof. First equivalence follows from(I 0

−ST Q−1 I

)(Q S

ST R

)(I −Q−1S

0 I

)=

(Q 0

0 R− ST Q−1S

).

The proof reveals a more general relation between the number of

negative, zero, positive eigenvalues of the three matrices.


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minimize γ


Linearize with Schur lemma to equivalent LMI problem

minimize γ

subject to

(γ fT

f A(x)

)� 0, lT x ≤ v, a ≤ x ≤ b.

Yalmip-Coding: Truss Toplogy Design

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minimize γ

subject to

(γ fT

f A(x)

)� 0, lT x ≤ v, a ≤ x ≤ b.

Suppose A(x)=∑N

k=1 xkmkmTk with vectors mk collected in matrix M .

Yalmip-Coding: Truss Toplogy Design

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minimize γ

subject to

(γ fT

f A(x)

)� 0, lT x ≤ v, a ≤ x ≤ b.

Suppose A(x)=∑N

k=1 xkmkmTk with vectors mk collected in matrix M .

The following code with Yalmip commands solves LMI problem:

gamma=sdpvar(1,1); x=sdpvar(N,1,’full’);

lmi=set([gamma f’;f M*diag(x)*M’]);

lmi=lmi+set(l’*x<=v);

lmi=lmi+set(a<=x<=b);

options=sdpsettings(’solver’,’sedumi’);

solvesdp(lmi,gamma,options); s=double(x);

Result: Truss Toplogy Design

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0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

Quickly Accessible Software

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General purpose Matlab interface Yalmip:

• Free code developed by J. Lofberg and accessible at

http://control.ee.ethz.ch/~joloef/yalmip.msql

• Can use usual Matlab-syntax to define optimization problem.

Is extremely easy to use and very versatile. Highly recommended!

• Provides access to a whole suite of public and commercial optimiza-

tion solvers, including fastest available dedicated LMI-solvers.

Quickly Accessible Software

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General purpose Matlab interface Yalmip:

• Free code developed by J. Lofberg and accessible at

http://control.ee.ethz.ch/~joloef/yalmip.msql

• Can use usual Matlab-syntax to define optimization problem.

Is extremely easy to use and very versatile. Highly recommended!

• Provides access to a whole suite of public and commercial optimiza-

tion solvers, including fastest available dedicated LMI-solvers.

Matlab LMI-Toolbox for dedicated control applications. Has recently

been integrated into new Robust Control Toolbox.

Outline

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• LMI Regions

General Formulation of LMI Problems

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Let X be a finite-dimensional real vector space. Suppose the mappings

f : X → R, G : X → {symmetric matrices of fixed size} are affine.


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LMI feasibility problem: Test existence of X ∈ X with G(X) 4 0.


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LMI optimization problem: Minimize f(X) over all X ∈ X that

satisfy the LMI G(X) 4 0


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satisfy the LMI G(X) 4 0.

Practical Hints. For all algorithms to work properly, make sure that

the constraint is strictly feasible: There exists some X ∈ X such that

G(X) ≺ 0.


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satisfy the LMI G(X) 4 0.

Practical Hints. For all algorithms to work properly, make sure that

the constraint is strictly feasible: There exists some X ∈ X such that

G(X) ≺ 0.

If G is genuinely linear, fopt is either +∞ or −∞. Impose additional

bounds/constraints on the decision variables to render problem sensible.

Example: Spectral Norm Approximation

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For real data matrices A, B, C and some unknown X consider

minimize ‖AXB − C‖subject to X ∈ X

where X is a matrix subspace reflecting structural constraints.


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Key equivalence with Schur:

‖M‖ < γ ⇐⇒ MT M ≺ γ2I ⇐⇒

(γI M

MT γI

)� 0.


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Key equivalence with Schur:

‖M‖ < γ ⇐⇒ MT M ≺ γ2I ⇐⇒

(γI M

MT γI

)� 0.

Norm minimization hence equivalent to following LMI problem:

minimize γ

subject to X ∈ X ,

(γI AXB − C

(AXB − C)T γI

)� 0

Stability of Dynamical Systems

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For dynamical systems one can distinguish many notions of stability.

We will mainly rely on definitions related to the state-space descriptions

x(t) = Ax(t), x(t) = A(t)x(t), x(t) = f(t, x(t)), x(t0) = x0

which capture the behavior of x(t) for t →∞ depending on x0.


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Exponential stability means that there exist real constants a > 0

(decay rate) and K (peaking constant) such that

‖x(t)‖ ≤ ‖x(t0)‖Ke−a(t−t0) for all trajectories and t ≥ t0.

K and α are assumed not to depend on t0 or x(t0) (uniformity).


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Exponential stability means that there exist real constants a > 0

(decay rate) and K (peaking constant) such that

‖x(t)‖ ≤ ‖x(t0)‖Ke−a(t−t0) for all trajectories and t ≥ t0.

K and α are assumed not to depend on t0 or x(t0) (uniformity).

Lyapunov theory provides the background for testing stability.

Stability of LTI Systems

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The linear time-invariant dynamical system

x(t) = Ax(t)

is exponentially stable if and only if there exists K with

K � 0 and AT K + KA ≺ 0.

Stability of LTI Systems

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The linear time-invariant dynamical system

x(t) = Ax(t)

is exponentially stable if and only if there exists K with

K � 0 and AT K + KA ≺ 0.

Two inequalities can be combined as(−K 0

0 AT K + KA

)≺ 0.

Since the left-hand side depends affinely on the matrix variable K, this

is indeed a standard strict feasibility test!

Matrix variables are fully supported by Yalmip and LMI-toolbox!

Trajectory-Based Proof of Sufficiency

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Choose ε > 0 such that AT K + KA + εK ≺ 0.


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Choose ε > 0 such that AT K + KA + εK ≺ 0. Let x(.) be any state-

trajectory of the system.


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trajectory of the system. Then

x(t)T (AT K + KA)x(t) + εx(t)T Kx(t) ≤ 0 for all t ∈ R


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and hence (using x(t) = Ax(t))d

dtx(t)T Kx(t) + εx(t)T Kx(t) ≤ 0 for all t ∈ R


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and hence (integrating factor eεt)

x(t)T Kx(t) ≤ x(t0)T Kx(t0)e−ε(t−t0) for all t ∈ R, t ≥ t0.


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and hence (integrating factor eεt)

x(t)T Kx(t) ≤ x(t0)T Kx(t0)e−ε(t−t0) for all t ∈ R, t ≥ t0.

Since λmin(K)‖x‖2 ≤ xT Kx ≤ λmax(K)‖x‖2 we can conclude that

‖x(t)‖ ≤ ‖x(t0)‖

√λmax(K)λmin(K)

e−ε(t−t0) for t ≥ t0.

Algebraic Proof

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Sufficiency. Let λ ∈ λ(A).

Algebraic Proof

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Sufficiency. Let λ ∈ λ(A). Choose a complex eigenvector x 6= 0 with

Ax = λx.

Algebraic Proof

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Ax = λx. Then the LMI’s imply x∗Kx > 0 and

0 > x∗(AT K + KA)x = λx∗Kx + x∗Kxλ = 2Re(λ)x∗Kx.

Algebraic Proof

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This guarantees Re(λ) < 0.

Algebraic Proof

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This guarantees Re(λ) < 0. Therefore all eigenvalues of A are in C−.

Necessity if A is diagonizable. Suppose all eigenvalues of A are in C−.

Algebraic Proof

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Since A is diagonizable there exists a complex nonsingular T with TAT−1 =

Λ = diag(λ1, . . . , λn).

Algebraic Proof

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Λ = diag(λ1, . . . , λn). Since Re(λk) < 0 for k = 1, . . . , n we infer

Λ∗ + Λ ≺ 0 and hence (T ∗)−1AT T ∗ + TAT−1 ≺ 0.

Algebraic Proof

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If we left-multiply with T ∗ and right-multiply with T (congruence) we infer

AT (T ∗T ) + (T ∗T )A ≺ 0.

Algebraic Proof

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If we left-multiply with T ∗ and right-multiply with T (congruence) we infer

AT (T ∗T ) + (T ∗T )A ≺ 0.

Hence K = T ∗T � 0 satisfies the LMI’s.

Algebraic Proof

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Necessity if A is not diagonizable. If A is not diagonizable it can be

transformed by similarity into its Jordan form: There exists a nonsingular

T with TAT−1 = Λ + J where Λ is diagonal and J has either ones or zeros

on the first upper diagonal.

Algebraic Proof

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For any ε > 0 one can even choose Tε with TεAT−1ε = Λ + εJ .

Algebraic Proof

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For any ε > 0 one can even choose Tε with TεAT−1ε = Λ + εJ . Since Λ has

the eigenvalues of A on its diagonal we still infer Λ∗ + Λ ≺ 0.

Algebraic Proof

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the eigenvalues of A on its diagonal we still infer Λ∗ + Λ ≺ 0. Therefore it

is possible to fix a sufficiently small ε > 0 with

0 � Λ∗ + Λ + ε(JT + J) = (Λ + εJ)∗ + (Λ + εJ).

Algebraic Proof

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the eigenvalues of A on its diagonal we still infer Λ∗ + Λ ≺ 0. Therefore it

is possible to fix a sufficiently small ε > 0 with

0 � Λ∗ + Λ + ε(JT + J) = (Λ + εJ)∗ + (Λ + εJ).

As before we can conclude that K = T ∗ε Tε satisfies the LMI’s.

Outline

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• LMI Regions

A First Glimpse at Robustness

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With some compact set A ⊂ Rn×n consider the family of LTI systems

x(t) = Ax(t) with A ∈ A.


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A is said to be quadratically stable if there exists K such that

K � 0 and AT K + KA ≺ 0 for all A ∈ A.


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Why name? V (x) = xT Kx is quadratic Lyapunov function.


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Why relevant? Implies that all A ∈ A are Hurwitz.


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Why relevant? Implies that all A ∈ A are Hurwitz.

Even stronger: Implies, for any piece-wise continuous A : R → A,

exponential stability of the time-varying system

x(t) = A(t)x(t).

Computational Verification

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If A has infinitely many elements, testing quadratic stability amounts

to verifying the feasibility of an infinite number of LMIs.


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Key question: How to reduce to a standard LMI problem?


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Let A be the convex hull of {A1, . . . , AN}: For each A ∈ A there

exist coefficients λ1 ≥ 0, . . . , λN ≥ 0 with λ1 + · · ·+ λN = 1 such that

A = λ1A1 + · · ·+ λNAN .


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Let A be the convex hull of {A1, . . . , AN}: For each A ∈ A there

exist coefficients λ1 ≥ 0, . . . , λN ≥ 0 with λ1 + · · ·+ λN = 1 such that

A = λ1A1 + · · ·+ λNAN .

If A is the convex hull of {A1, . . . , AN} then A is quadratically

stable iff there exists some K with

K � 0 and ATi K + KAi ≺ 0 for all i = 1, . . . , N.

Proofs. Exercises.

Outline

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• LMI Regions

LMI Regions

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For a real symmetric 2m×2m matrix P the set of complex numbers

LP :=

{z ∈ C :

(I

zI

)∗

P

(I

zI

)≺ 0

}

is called an LMI region.

LMI Regions

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LP :=

{z ∈ C :

(I

zI

)∗

P

(I

zI

)≺ 0

}


Half-plane: Re(z) < α ⇔ z + z − 2α ≺ 0 ⇔ z ∈ LP1 .

LMI Regions

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LP :=

{z ∈ C :

(I

zI

)∗

P

(I

zI

)≺ 0

}



Circle: |z| < r ⇔ zz − r2 ≺ 0 ⇔ z ∈ LP2

LMI Regions

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Carsten Scherer


LP :=

{z ∈ C :

(I

zI

)∗

P

(I

zI

)≺ 0

}



Circle: |z| < r ⇔ zz − r2 ≺ 0 ⇔ z ∈ LP2

Sector: Re(z) tan(φ) < −|Im(z)| ⇔ z ∈ LP3

where

P1 =

(−2α 1

1 0

), P2 =

(−r2 0

0 1

), P3 =

0 0 sin(φ) cos(φ)

0 0 − cos(φ) sin(φ)

∗ ∗ 0 0

∗ ∗ 0 0

.

Properties of LMI Regions

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The intersection of finitely many LMI regions is an LMI region.

We can hence describe rather general subsets of C.

Properties of LMI Regions

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The intersection of finitely many LMI regions is an LMI region.

We can hence describe rather general subsets of C.

Proof. Just follows from usual stacking property of LMI’s:(I

zI

)∗(Q1 S1

ST1 R1

)(I

zI

)≺ 0,

(I

zI

)∗(Q2 S2

ST2 R2

)(I

zI

)≺ 0

if and only ifI 0

0 I

zI 0

0 zI

∗

Q1 0 S1 0

0 Q2 0 S2

ST1 0 R1 0

0 ST2 0 R2

I 0

0 I

zI 0

0 zI

≺ 0.

Recap

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Recall the definition of the Kronecker product

A⊗B =

A11B · · · A1nB...

...

Am1B · · · AmnB

for A ∈ Cm×n, B ∈ Ck×l.

Recap

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Recall the definition of the Kronecker product

A⊗B =

A11B · · · A1nB...

...

Am1B · · · AmnB

for A ∈ Cm×n, B ∈ Ck×l.

Easy to prove following properties (for compatible dimensions):

• 1⊗ A = A = A⊗ 1.

• (A + B)⊗ C = (A⊗ C) + (B ⊗ C)

• (A⊗B)(C ⊗D) = (AC)⊗ (BD).

• (A⊗B)∗ = A∗ ⊗B∗ and (A⊗B)−1 = A−1 ⊗B−1.

Generalization Standard Stability Characterization

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All eigenvalues of A ∈ Rn×n are contained in the LMI-region{z ∈ C :

(I

zI

)∗(Q S

ST R

)(I

zI

)≺ 0

}if and only if there exists a K � 0 such that(

I

A⊗ I

)T (K ⊗Q K ⊗ S

K ⊗ ST K ⊗R

)(I

A⊗ I

)≺ 0

Generalization Standard Stability Characterization

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All eigenvalues of A ∈ Rn×n are contained in the LMI-region{z ∈ C :

(I

zI

)∗(Q S

ST R

)(I

zI

)≺ 0

}if and only if there exists a K � 0 such that(

I

A⊗ I

)T (K ⊗Q K ⊗ S

K ⊗ ST K ⊗R

)(I

A⊗ I

)≺ 0

Beautiful generalization of standard Lyapunov stability characterization!

Gahinet, Chilali (1996)

Proof. Along lines of algebraic proof of standard result and by exploiting

the properties of the Kronecker product.

Discrete-Time versus Continuous-Time

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• xk+1 = Axk is exponentially stable iff A is a Schur matrix.

• x = Ax is exponentially stable iff A is Hurwitz.

The unit disk/the open left half-plane are LMI regions:(1

z

)∗(−1 0

0 1

)(1

z

)≺ 0,

(1

z

)∗(0 1

1 0

)(1

z

)≺ 0.

A is a Schur matrix/A is Hurwitz iff there exists K � 0 with(I

A

)T (−K 0

0 K

)(I

A

)≺ 0,

(I

A

)T (0 K

K 0

)(I

A

)≺ 0.

Lessons to be Learnt

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• Many interesting engineering problems are LMI problems.

• Variables can live in arbitrary vector space.

In control: Variables are typically matrices.

Can involve equation and inequality constraints. Just check whether

cost function and constraints are affine & verify strict feasibility.

• Translation to input for solution algorithm by parser (e.g. Yalmip).

Can choose among many efficient LMI solvers (e.g. Sedumi).

• Main trick in removing nonlinearities so far: Schur Lemma.

convex optimization and linear matrix inequalitiessistawww/smc/...convex optimization and linear...

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