SDP Duality and LTI Systems - Research University | Purdue University

Semidefinite Programming Duality andLinear Time-invariant Systems

Venkataramanan (Ragu) BalakrishnanSchool of ECE, Purdue University

2 July 2004Workshop on Linear Matrix Inequalities in Control

LAAS-CNRS, Toulouse, France

Semidefinite Programming Duality andLinear Time-invariant Systems

Venkataramanan (Ragu) BalakrishnanSchool of ECE, Purdue University

2 July 2004Workshop on Linear Matrix Inequalities in Control

LAAS-CNRS, Toulouse, France

Joint work with Lieven Vandenberghe, UCLA

SDP DUALITY AND LTI SYSTEMS 1

Basic ideas

• Many control constraints yield LMIs, many control problems are SDPs


Basic ideas


• LMIs are convex constraints, SDPs are convex optimization problems

• From duality theory in convex optimization:

? Theorem of alternatives for LMIs

? SDP duality


Basic ideas


• LMIs are convex constraints, SDPs are convex optimization problems

• From duality theory in convex optimization:

? Theorem of alternatives for LMIs

? SDP duality

• Explore implication of convex duality theory on underlying control problem:

? New (often simpler) proofs for classical results

? Some new results


LMIs and Semidefinite Programming

• V is a finite-dimensional Hilbert space, S is a subspace of Hermitianmatrices, F : V → S is a linear mapping, F0 ∈ S

• Inequality F(x) + F0 ≥ 0 is an LMI

• SDP is an optimization of the form:

minimize: 〈c, x〉Vsubject to: F(x) + F0 ≥ 0


A theorem of alternatives for LMIs

Exactly one of the following statements is true

1. F(x) + F0 > 0 is feasible

2. There exists Z ∈ S s.t. Z 0, Fadj(Z) = 0, 〈F0, Z〉S ≤ 0

(Fadj(·) denotes adjoint map, i.e., ∀ x ∈ V, Z ∈ S, 〈F(x), Z〉S = 〈x,Fadj(Z)〉V)


A theorem of alternatives for LMIs

Exactly one of the following statements is true

1. F(x) + F0 > 0 is feasible

2. There exists Z ∈ S s.t. Z 0, Fadj(Z) = 0, 〈F0, Z〉S ≤ 0

(Fadj(·) denotes adjoint map, i.e., ∀ x ∈ V, Z ∈ S, 〈F(x), Z〉S = 〈x,Fadj(Z)〉V)

• Variants available for nonstrict inequalities such as F(x) + F0 0 andF(x) + F0 ≥ 0, and with additional linear equality constraints F(x) = 0

• Typically get weak alternatives, need additional conditions (constraintqualifications) to make them strong


Proof of theorem of alternatives

LMI F(x) + F0 > 0 infeasible iff

F0 6∈ C ∆= {C | F(x) + C > 0 for some x ∈ V}





C is open, nonempty and convex, so there exists hyperplane strictly separtingF0 and C:

∃Z 6= 0 s.t. 〈F0, Z〉S < 〈C,Z〉S for all C ∈ C







∃Z 6= 0 s.t. 〈F0, Z〉S < 〈−F(x) + X, Z〉S for all x ∈ V, X > 0








∃Z 6= 0 s.t. 〈F0, Z〉S < 〈x,−Fadj(Z)〉V + 〈X, Z〉S for all x ∈ V, X > 0








∃Z 6= 0 s.t. 〈F0, Z〉S < 〈x,−Fadj(Z)〉V + 〈X, Z〉S for all x ∈ V, X > 0

Thus, there exists Z ∈ S s.t. Z 0, Fadj(Z) = 0, 〈F0, Z〉S ≤ 0


Application: A Lyapunov inequality

• LMI A∗P + PA < 0 is feasible, or

• There exists Z s.t. Z 0, AZ + ZA∗ = 0

Factoring Z = UU∗, can show

AU = US, S has pure imaginary eigenvalues

Thus:

LMI A∗P + PA < 0 is infeasible if and only if A has a pure imaginaryeigenvalue


Other results

• “P > 0, A∗P + PA < 0” is infeasible iff λi(A) ≥ 0 for some i

• “A∗P + PA � 0” is infeasible iff A is similar to a purely imaginary diagonalmatrix

• “A∗P + PA ≤ 0, P 0” is infeasible iff λi(A) ≥ 0 for all i

• “A∗P + PA � 0, PB = 0” is infeasible iff all uncontrollable modes of (A,B)are nondefective and correspond to imaginary eigenvalues

• “P 0, A∗P + PA ≤ 0, PB = 0” is infeasible iff all uncontrollable modes of(A,B) correspond to eigenvalues with positive real part

• “P 6= 0, A∗P + PA ≤ 0, PB = 0” is infeasible iff (A,B) is controllable


Other results

• “P > 0, A∗P + PA < 0” is infeasible iff λi(A) ≥ 0 for some i

• “A∗P + PA � 0” is infeasible iff A is similar to a purely imaginary diagonalmatrix

• “A∗P + PA ≤ 0, P 0” is infeasible iff λi(A) ≥ 0 for all i

• “A∗P + PA � 0, PB = 0” is infeasible iff all uncontrollable modes of (A,B)are nondefective and correspond to imaginary eigenvalues

• “P 0, A∗P + PA ≤ 0, PB = 0” is infeasible iff all uncontrollable modes of(A,B) correspond to eigenvalues with positive real part

• “P 6= 0, A∗P + PA ≤ 0, PB = 0” is infeasible iff (A,B) is controllable


Frequency-domain inequalities: The KYP Lemma

Inequalities of the form[(jωI −A)−1B

I

]∗M

[(jωI −A)−1B

I

]> 0

are commonly encountered in systems and control:

• Linear system analysis and design

• Digital filter design

• Robust control analysis

• Examples of constraints: |H(jω)| < 1 (small gain), <H(jω) > 0 (passivity),H(jω) + H(jω)∗ + H(jω)∗H(jω) < 1 (mixed constraints)


The Kalman-Yakubovich-Popov Lemma

FDI [(jωI −A)−1B

I

]∗M

[(jωI −A)−1B

I

]> 0

holds for all ω iff LMI [A∗P + PA PB

B∗P 0

]−M < 0

is feasible

• Infinite-dimensional constraint reduced to finite-dimensional constraint

• No sampling in frequency required


Control-theoretic proof of KYP Lemma

Suppose LMI [A∗P + PA PB

B∗P 0

]−M < 0

is feasible

Then

0 <

[(jωI −A)−1B

I

]∗(M −

[A∗P + PA PB

B∗P 0

]) [(jωI −A)−1B

I

]=

[B∗(−jωI −A∗)−1 I

]M

[(jωI −A)−1B

I

]


Control-theoretic proof of KYP Lemma

Suppose LMI [A∗P + PA PB

B∗P 0

]−M < 0

is feasible

Then

0 <

[(jωI −A)−1B

I

]∗(M −

[A∗P + PA PB

B∗P 0

]) [(jωI −A)−1B

I

]=

[B∗(−jωI −A∗)−1 I

]M

[(jωI −A)−1B

I

]

Converse much harder; based on optimal control theory


New KYP lemma proof

More general version of the KYP Lemma:

Suppose M22 > 0 [A∗P + PA PB

B∗P 0

]−M < 0,

is feasible iff

(jωI −A)u = Bv, (u, v) 6= 0 =⇒[

u∗ v∗]M

[uv

]> 0

• A can have imaginary eigenvalues

• If A has no imaginary eigenvalues, recover classical version


Duality-based KYP Lemma proof

Infeasibility of [A∗P + PA PB

B∗P 0

]−M < 0

equivalent to existence of Z s.t.

Z =[

Z11 Z12

Z∗12 Z22

] 0, Z11A

∗ + AZ11 + Z12B∗ + BZ∗12 = 0, TrZM ≤ 0

• Must have Z11 0. Hence, factor Z as

[Z11 Z12

Z∗12 Z22

]=

[U 0V V

] [U∗ V ∗

0 V ∗

],

where U has full rank


• Can show

US −AU = BV, Tr([

U∗ V ∗]M

[UV

])≤ 0,

with S + S∗ = 0

• Take Schur decomposition of S: S =∑m

i=1 jωiqiq∗i , with

∑i qiq

∗i = I

• Then

q∗k[

U∗ V ∗]M

[UV

]qk ≤ 0

for some k

• Define u = Uqk, v = V qk. Then

[u∗ v∗

]M

[uv

]≤ 0 and (jωI −A)u = Bv


Outline

• Theorem of alternatives for LMIs, and their applications

• SDP duality, and its application


Primal and dual SDPs

Primal SDP:minimize: 〈c, x〉Vsubject to: F(x) + F0 ≥ 0

Dual SDPmaximize −〈F0, Z〉Ssubject to Fadj(Z) = c, Z ≥ 0

• If Z is dual feasible, then −TrF0Z ≤ p∗

• If x is primal feasible, then cTx ≥ d∗

• Under mild conditions, p∗ = d∗

• At optimum, (F(xopt) + F0) Zopt = 0


Application of duality: Bounds on H∞ normStable LTI system

x = Ax + Bu, x(0) = 0, y = Cx

• Transfer function H(s) = C(sI −A)−1B

• H∞ norm of H defined as

‖H‖∞ = sup<s>0

σmax(H(s))

• ‖H‖2∞ equals maximum energy gain

‖H‖2∞ = max

u

∫yTy∫uTu


‖H‖∞ computation as an SDP

minimize: β

subject to:

[A∗P + PA + C∗C PB

B∗P −βI

]≤ 0

(‖H‖2∞ = βopt)

Dual problem

maximize: TrCZ11C∗

subject to: Z11A∗ + AZ11 + Z12B

∗ + BZ∗12 = 0[Z11 Z12

Z∗12 Z22

]≥ 0, TrZ22 = 1


Control-theoretic interpretation of dual problem

• Suppose u(t) any input that steers state from x(T1) = 0 to x(T2) = 0, forsome T1, T2. Let y(t) be the corresponding output

• Define

Z11 =∫ T2

T1

x(t)x(t)∗ dt, Z12 =∫ T2

T1

x(t)u(t)∗ dt, Z22 =∫ T2

T1

u(t)u(t)∗ dt

Can show Z11, Z12 and Z22 are dual feasible

• TrZ22 =∫ T2

T1u(t)∗u(t) dt = 1 normalizes input energy

• Dual objective is corresponding output energy, gives lower bound:

TrCZ11C∗ =

∫ T2

T1

y(t)∗y(t) dt


New upper bounds on ‖H‖∞Recall primal problem:

minimize: β

subject to:

[A∗P + PA + C∗C PB

B∗P −βI

]≤ 0

A primal feasible point is

P = 2Wo, β = 4λmax(WoBB∗Wo, C∗C)

where Wo is observability Gramian, obtained by solvingWoA + A∗Wo + C∗C = 0

Thus, new upper bound on ‖H‖∞ is given by

2√

λmax(WoBB∗Wo, C∗C)


New lower bounds on ‖H‖∞Recall dual problem

maximize: TrCZ11C∗

subject to: Z11A∗ + AZ11 + Z12B

∗ + BZ∗12 = 0[Z11 Z12

Z∗12 Z22

]≥ 0, TrZ22 = 1

A dual feasible point is

Z11 = Wc/α, Z12 = B/(2α), Z22 = B∗W−1c B/(4α),

where α = Tr(B∗W−1c B/4)

Thus new lower bound is

2√

TrCWcC∗/(TrB∗W−1c B)


Application of duality: LQR problem

Primal

minimize: TrQZ11 + TrZ22

subject to: AZ11 + BZ∗12 + Z11A∗ + Z12B

∗ + x0x∗0 ≤ 0,[

Z11 Z12

Z∗12 Z22

]≥ 0

Dualmaximize: x∗0Px0

subject to:

[A∗P + PA + Q PB

B∗P I

]≥ 0, P ≥ 0


The Linear-Quadratic Regulator problem

x = Ax + Bu, x(0) = x0,

find u that minimizes J =∫ ∞

0

(x(t)∗Qx(t) + u(t)∗u(t)) dt,

s.t. limt→∞ x(t) = 0

Well-known solution: Solve Riccati equation

ATP + PA + Q− PBBTP = 0

such that P > 0. Then,uopt(t) = −BTPx(t)

(Proof using quadratic optimal control theory)


Duality-based proof: Basic ideas

• Primal problem gives upper bound on LQR objective

• Dual problem gives lower bound on LQR objective

• Optimality condition gives LQR Riccati equation


Primal problem interpretation

Assume u = Kx, s.t. x(t) → 0 as t →∞

Then LQR objective reduces to

JK =∫ ∞

0

x(t)∗ (Q + K∗K) x(t) dt

and is an upper bound on the optimum LQR objective

• Condition x(t) → 0 as t →∞ equivalent to

(A + BK)Z + Z(A + BK)∗ + x0x∗0 = 0, Z ≥ 0

• LQR objective is TrZ(Q + K∗K)


Best upper bound using state-feedback:

minimize: TrZ(Q + K∗K)subject to: Z ≥ 0

(A + BK)Z + Z(A + BK)∗ + x0x∗0 = 0

With Z11 = Z, Z12 = ZK∗, Z22 = KZK∗:

minimize: TrQZ11 + TrZ22

subject to: AZ11 + BZ∗12 + Z11A∗ + Z12B

∗ + x0x∗0 ≤ 0,[

Z11 Z12

Z∗12 Z22

]≥ 0

(Same as primal problem)


Dual problem interpretation

Suppose for P ≥ 0 , ddtx(t)∗Px(t) ≥ − (x(t)∗Qx(t) + u(t)∗u(t)), for all t ≥ 0,

and for all x and u satisfying x = Ax + Bu, x(T ) = 0. Then,

x∗0Px0 ≤∫ T

0

(x(t)∗Qx(t) + u(t)∗u(t)) dt,

So Jopt ≥ x∗0Px0

Derivative condition equivalent to LMI[A∗P + PA + Q PB

B∗P I

]≥ 0

So lower bound to LQR objective given by dual problem


Optimality conditions

• Stabilizability of (A,B) guarantees strict primal feasibility

• Detectability of (Q,A) guarantees strict dual feasibility

• Recall, at optimality (F(xopt) + F0) Zopt = 0. This becomes[Z11 Z12

Z∗12 Z22

] [A∗P + PA + Q PB

B∗P I

]= 0

Reduces to [I K∗ ] [

A∗P + PA + Q PBB∗P I

]= 0,

or K = −B∗P , with all the eigenvalues of A + BK having negative realpart, and

A∗P + PA + Q− PBB∗P = 0(Classical LQR result, derived using duality)


Conclusions

• SDP duality theory has interesting implications systems and control

• Implications for numerical computation:

? Dual problems sometimes have fewer variables

? Most efficient algorithms solve primal and dual together; control-theoreticinterpretation can help increase efficiency

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SDP Duality and LTI Systems - Research University | Purdue University