Linear Matrix Inequalities (LMI) Positive Cone on Sn Positiveness on ...

<strong>Positive</strong> <strong>Cone</strong> on S n
<strong>Linear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> (<strong>LMI</strong>)
Γ := {X : X ∈ S n , x T Xx ≥ 0, ∀x ∈ R n },
int Γ = {X : X ∈ S n , x T Xx > 0, ∀x ∈ R n , x = Θ}.
<strong>Positive</strong>ness on S n Given X ∈ S n the following statements are equivalent:
a) X is positive semidefinite.
b) X ∈ Γ (X Θ).
c) λmin(X) ≥ 0 (λmax(−X) ≤ 0).
a) X is positive definite.
b) X ∈ int Γ (X ≻ Θ).
c) λmin(X) > 0 (λmax(−X) < 0).
Negativeness on S n A matrix X ∈ S n is negative (semi)definite if (−X Θ) −X ≻ Θ.
Notation: X ≺ Θ (X Θ).
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 1

<strong>Linear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> (<strong>LMI</strong>)
Affine functional The functional f : X → S n is affine if
f(αx + (1 − α)y) = αf(x) + (1 − α)f(y), ∀x, y ∈ X, α ∈ R,
Proposition The set {x : f(x) Θ} (or {x : f(x) ≺ Θ}), where f : X → S n is an
affine functional, is a convex set.
Proposition The set {x : f(x) Θ} (or {x : f(x) ≻ Θ}), where f : X → S n is an
affine functional, is a convex set.
Proof An affine functional is simultaneously convex and concave.
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 2

<strong>Linear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> (<strong>LMI</strong>)
Definition Let f : X → S n be an affine functional. The functional inequalities
f(x) Θ or f(x) Θ
are known as <strong>Linear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> (<strong>LMI</strong>). The functional inequalities
are strict <strong>LMI</strong>.
Examples
f(x) ≺ Θ or f(x) ≻ Θ
1. f : S n → S n , f(X) := A T X + XA + Q Θ.
2. f : S n → S n , f(X) := A T X + XA ≺ Θ.
3. f : Sn → Sn , f(X) := AT XA − X + Q Θ.
4. f : Sn → S2n ⎡
, f(X) := ⎣ X AT ⎤
XA
X
⎦ ≻ Θ.
X
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 3

Semidefinite programming (SDP)
Proposition All affine functional F : X → S n where X is a finite dimensional space with
F = R can be put in the canonical form
Example
F (x) Θ, F (x) := F0 +
m
i=1
Fixi, Fi ∈ S n , x ∈ R m .
1. G(X) := AT X + XA + Q Θ, X ∈ S2 .
⎡
X = ⎣ x1
⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x2 1
⎦ = ⎣
0
0 0
⎦ x1 + ⎣
0 1
1 0
⎦ x2 + ⎣
0 0
0
⎦ x3 =
1
x2 x3
=⇒ −G(X) = −Q
F0
+
3
i=1
−(A T Xi + XiA)
2. G(µ, X) := µ − tr CXC T Θ, µ ∈ R, X ∈ S 2 .
G(µ, X) = µ − tr CXC T = 1
Fi
µ +
3
i=1
Fi
3
Xixi, x ∈ R 3 .
i=1
− tr T
CXiC
Fi
xi = F (x)
xi = F (µ, x)
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 4

Canonical Semidefinite Program
Remarks
Semidefinite programming (SDP)
min
x∈R m
cT x
s.t. F (x) Θ
• Several <strong>LMI</strong> Fj(x) Θ, j = 1, . . . , N can be handled by stacking Fj(x) as a block
diagonal constraint
F (x) = block diag [F1(x), . . . , Fj(x), . . . , FN ] Θ.
• Most solvers require the user to generate Fis (SeDuMi, SDP, SDPHA).
• Some solvers automatically generate Fis from the original high level matrix description
(<strong>LMI</strong>Tool, <strong>LMI</strong>Solve).
• <strong>LMI</strong>Lab (Matlab) requires the user to input data in “intermediate level” matrix form
(F (X) =
i AiXBi).
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 5

<strong>LMI</strong> with equality constraints (LME)
Semidefinite programming (SDP)
1. f : R n×n → S n , f(X) := A T X + XA + Q Θ, X = X T .
2. f : R n×n → S n , f(X) := A T X + XA ≺ Θ, CX = B T .
Canonical SDP with equality constraints
min
x∈R m
cT x
s.t. F (x) Θ
Ax = b
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 6

Elimination of equality constraints
Solve
Ax = b ⇒
Semidefinite programming (SDP)
¯x(y) = A † b + A ⊥ y, y ∈ R p
(p = dim A ⊥ ),
c T ¯x(y) = ˜c0 + ˜c T y, ˜c0 := c T A † b, ˜c := A ⊥T c,
˜F (y) := F (¯x(y)),
and substitute back into the original problem! Notice that
min
x∈Rn ,Ax=b cT x = min
y∈Rp cT ¯x(y) = min
y∈Rp This provides the standard SDP
min
y∈R p
˜cT y
s.t. ˜F (y) Θ.
˜c0 + ˜c T y = ˜c0 + min
y∈R p ˜cT y
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 7

<strong>Linear</strong> Programming
Equivalent SDP
Scope of Semidefinite Programming
min
x∈R m
min
x∈R m
cT x
s.t. Ax b
cT x
s.t. F (x) Θ,
where F (x) := block diag [Aix − bi], Ai is the ith row of A.
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 8

Scope of Semidefinite Programming
Congruence Transformation Two matrices X, Y ∈ S n are said to be congruent if there
exists a nonsingular matrix T ∈ R n×n such that Y = T T XT .
Proposition If X and Y are congruent then Y ≻ Θ if, and only if, X ≻ Θ.
Proof If X ≻ Θ then x T Xx > 0, ∀x ∈ R n , x = Θ. Since X and Y are congruent there
exists T nonsingular such that Y = T T XT . Hence, using the fact that T is nonsingular, for all
x = Θ, the vector y := T −1 x = Θ and
X ≻ Θ ⇔ x T Xx = y T T T XT y = y T Y y > 0 ⇔ Y ≻ Θ
Generalization For any T ∈ R m×n , the matrix Y = T T XT Θ if X Θ. The
converse is false when T does not have full row rank; example: t = 0, x = −1, y = t 2 x ≥ 0.
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 9

Scope of Semidefinite Programming
Schur Complement For all X ∈ S n , Y ∈ R m×n , Z ∈ S m , the following statements are
equivalent:
a) Z ≻ Θ, X − Y T Z−1 ⎡
Y ≻ Θ.
⎤
b)
X
⎣
Y
T Y
⎦ ≻ Θ.
Z
Proof Assume Z ≻ Θ. The nonsingular matrix
⎡
I
T = ⎣
−Z
Θ
−1 ⎤
Y
⎦
I
a) Z ≻ Θ, X − Y T Z−1Y Θ.
⎡ ⎤
b) Z ≻ Θ,
X
⎣
Y
T Y
⎦ Θ.
Z
establishes the congruence transformation
T T
⎡ ⎤ ⎡
X
⎣
Y
T Y
⎦ T = ⎣
Z
X − Y T Z−1 ⎤
Y
Θ
Θ
⎦ ≻ Θ( Θ).
Z
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 10

Quadratic Programming (convex)
Using Schur complement
Equivalent SDP
Scope of Semidefinite Programming
min
x∈R m
xT Q T Qx + c T x
s.t. Ax b
γ ≥ x T Q T Qx + c T x ⇔ FQ(γ, x) :=
min
γ∈R,x∈R m
γ
⎡
s.t. F (x) Θ,
⎣ γ − cT x x T Q T
Qx I
where F (γ, x) = block diag [FQ(γ, x), Aix − bi], Ai is the ith row of A.
⎤
⎦ Θ
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 11

Minimum <strong>Matrix</strong> Norm
min
X∈R m×n
Scope of Semidefinite Programming
A − BXC 2 , where M := λ 1/2
max [M T M]
Notice that
T
γ ≥ λmax (A − BXC) (A − BXC) ⇔
T
γI (A − BXC) (A − BXC)
Using Schur complement
γI (A − BXC) T (A − BXC) ⇔ F (γ, X) :=
Equivalent SDP
min
γ∈R,X∈R m×n
γ
s.t. F (γ, X) Θ,
⎡
γI (A − BXC)
⎣
T
⎤
⎦ Θ.
(A − BXC) I
280B - <strong>Linear</strong> Control Design Maurício de Oliveira Introduction to <strong>LMI</strong> – p. 12