1

L INEAR M ATRIX I NEQUALITIES AND S EMI D EFINITE P ROGRAMMING I MPACTS ON C ONTROL S YSTEM D ESIGN P. Apkarian

Université Paul Sabatier & ONERA-CERT Mathématiques pour l’Industrie et la Physique CNRS UMR 5640

invitation by Professor Giuseppe Franze - Calabria University

– p. 1/129

Seminar Outline ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦

2

Linear Matrix Inequalities and SDP Tricks to reformulate into LMIs System concepts via LMIs Multi-channel/objective with LMIs Uncertain systems analysis Gain-scheduling and LPV synthesis Hard non-LMI problems Conclusions, perspectives.

– p. 2/129

Linear Matrix Inequalities and SDP

• • • • • •

3

Definitions, manipulations Schur’s complements Classes of convex optimization problems Semi- Definite Programming Algorithms to solve SDP, duality, complexity Software, links.

– p. 3/129

linear matrix inequalities

4

an LMI is a constraint on a vector x ∈ Rn : F (x) := F0 + x1 F1 + . . . + xn Fn  0, where F0 , F1 , . . . , Fn are symmetric matrices ➠

 is inequality on symmetric matrix cone

➠

LMI equivalent to λmin (F (x)) ≥ 0

➠

F (x)  0 iff η ′ F (x)η ≥ 0, ∀η

➠

F (x)  0 iff det {ppal mat.} ≥ 0

➠

F (x) ≻ 0 iff η ′ F (x)η > 0, ∀η 6= 0

– p. 4/129

geometry of LMIs ➠

5

an LMI de fine a convex set F (λx+(1−λ)y) = λF (x)+(1−λ)F (y)  0 whenever F (x)  0, F (y)  0

➠

➠

set with non necessarily smooth boundary (corners)

LMI plane and curved faces

describe wide variety of constraints

– p. 5/129

LMI - diagonal augmentation

6

LMI constraints F1 (x)  0, . . . , Fq (x)  0 are equivalent to single LMI constraint 

F1 (x)  0 .. .

0 ... 0



... 0 0 Fq (x)

LMI 2

LMI 1

LMI 3

– p. 6/129

linear constraints

7

finite set of scalar linear (affine) constraints a′i x ≤ bi , i = 1, . . . , m can be represented as LMI F (x)  0, with F (x) = diag(a′1 x − b1 , . . . , a′m x − bm )

polyhedral LMI

– p. 7/129

Schur complements

8

partitioned symmetric matrix   P1 P2 P := P2′ P3 S = P3 − P2′ P1−1 P2 is the Schur complement of P1 in P (provided P1 invertible) Schur complement lemmas ➠

P ≻ 0 if and only if P1 ≻ 0 and S ≻ 0

➠

if P1 ≻ 0, then P  0 if and only if S  0

– p. 8/129

Schur complement consequence

9

complicate constraint in variable x P3 (x) − P2 (x)′ P1 (x)−1 P2 (x) ≻ 0 is turned into simpler one   P1 (x) P2 (x) ≻ 0. ′ P2 (x) P3 (x) provided that P1 (x) ≻ 0.

– p. 9/129

ellipsoidal constraints

10

an ellipsoid can be described in different ways • as kAx + bk ≤ 1, iff   I Ax + b 0 ′ (Ax + b) 1 • as (x−x0 )′ W (x−x0 ) ≤ 1, with W > 0 iff   1 (x − x0 )′ 0 −1 (x − x0 ) W

LMI

– p. 10/129

fractional constraints

11

consider fractional constraints (c′ x)2 d′ x

≤ t

Ax + b ≥ 0 (assume d′ x > 0, whenever Ax + b ≥ 0) can be represented as   ′ t cx  0 ′ ′ cx dx Ax + b ≥ 0 – p. 11/129

convex quadratic constraints

12

Convex quadratic constraints can be rewritten (Ax + b)′ (Ax + b) − c′ x − d ≤ 0 has the LMI representation   I Ax + b 0 ′ ′ (Ax + b) c x + d • can be used to show that convex quadratic programming can be solved via SDP

– p. 12/129

classes of convex optimization problems

13

• linear prog. (LP) minimize c′ x, Ax  b (componentwise) • convex quadratic prog. (CQP) Qj  0 minimize x′ Q0 x + b′0 x + c0 s.t. x′ Qi x + b′i x + ci ≤ 0 All (and others) are generalized by SDP !:

– p. 13/129

LMIs in control with P variable • Lyapunov inequality A′ P + P A ≺ 0 can be represented in canonical form F0 +

n X i=1

xi Fi ≺ 0

14

pick a basis (Pi )i of the symmetric matrices, X P = xi Pi i

hence recover the canonical form with F0 = 0,

Fi = A′ Pi +Pi A

– p. 14/129

symmetric matrix expressions are LMIs

15

• Any (symmetric) linear constraints in the variables X, Y AY B + (AY B)′ + X + . . .  0 can be represented in the canonical form F (x) = F0 + x1 F1 + . . . + xn Fn  0 by appropriate selection of the Fi ’s.

– p. 15/129

Riccati and quadratic matrix inequality

16

quadratic matrix inequality in P A′ P + P A + P BR−1 B ′ P + Q  0 where R > 0, is equivalent to LMI  ′  A P + PA + Q PB 0 ′ BP −R (proof by Schur complements) Riccati-based control method can be solved via LMIs

– p. 16/129

classes of semidefinite programs

17

• I feasibility problem: find x : F0 + x1 F1 + . . . + xn Fn  0 • II linear objective minimization subject to LMIs minimize c′ x, s.t. F0 + x1 F1 + . . . + xn Fn  0 • III generalized eigenvalue minimization minimize subject to

λ A(x) − λB(x)  0, B(x)  0, C(x)  0

(A, B, C affine symmetric expressions in x) – p. 17/129

solving LMI - a rich set of algorithms

18

much work and progress since 1990 ! ➠

primal interior-point method (method of centers)

➠

primal-dual interior-point method

➠

non-differentiable methods (bundle, ...)

Primal-dual methods very efficient. other fast algorithms under development (aug. Lagrangian)

– p. 18/129

central property

19

because of structure and convexity algorithms are guaranteed to find global solutions !

– p. 19/129

primal-dual IPMs

20

ideas: ➠

instead of working in primal space, formulate problem in “primal-dual” space

➠

target objective is duality gap, and is zero at optimum

➠

try to solve (Lagrange) optimality conditions

– p. 20/129

SDP duality • primal

21

• dual

min c′ x s.t. F (x)  0

max − Tr (F0 Z) s. t.Z ≻ 0, Tr Fi Z = ci • optimality cond. if (x, Z) is primal-dual feasible c′ x =

n X i=1

≥0

z }| { xi Tr ZFi =Tr ZF (x) −Tr ZF0 ≥ −Tr ZF0

hence global optimality pairs (x, Z) such that Tr ZF (x) = 0 since primal and dual obj. coincide at solution

– p. 21/129

mechanisms of primal-dual algorithms solve subject to

22

Tr ZF (x) = 0 Pn F0 + i=1 xi Fi  0

Z ≻ 0, Tr Fi Z = ci , i = 1, . . . , n • Actually, one tries to solve Tr ZF (x) = µI for decreasing value of µ (µ −→ 0) • Newton steps for the linearization of Tr ZF (x). • superlinear convergence can be guaranteed kxk+1 − xopt k ≤ kxk − xopt kq , q > 1 very efficient in practice ! – p. 22/129

SDP software ➠

MATLAB LMI toolbox by Gahinet, Chilali, Laub, Nemirovski

➠

DSDP by Benson, Ye

➠

SDPpack by Alizadeh, Haeberly, Nayakkankuppam, Overton

➠

SeDuMi by Sturm

➠

Imitool-2.0 by Boyd et al.

➠

Cutting plane methods by Helmberg, Oustry, Kiwiel, etc.

➠

Many others ...

23

– p. 23/129

ftp addresses for SDP

24

• ftp addresses, codes, papers, courses on SDP

http://orion.math.uwaterloo.ca:80/ hwolkowi/henry/software/readme.html# http://www.zib.de/helmberg/semidef.html http://rutcor.rutgers.edu/ alizadeh/sdp.html

– p. 24/129

tricks to turn hard problems into LMIs • • • • • • •

25

Schur’s complements (see previous) LMIs and quadratic forms multi-convexity, monotonicity, etc. Finsler’s lemmas Projection lemmas changes of variables augmentation by slack

– p. 25/129

S-Procedure and quadratic inequalities ➠

26

S-Procedure transforms quadratic problems into LMIs(possibly conservative)

given Qi ’s symmetric or hermitian matrices, define F0 (x) = x′ Q0 x, F1 (x) = x′ Q1 x, . . . , FL (x) = x′ QL x, F0 (x) < 0 over the set F1 (x) ≤ 0, . . . , FL (x) ≤ 0 whenever ∃s1 ≥ 0, . . . , sL ≥ 0 (slacks), such that F0 (x) −

L X i=1

si Fi (x) < 0 or LMI Q0 −

L X

si Qi  0

i=1

– p. 26/129

Finsler’s Lemma

27

• converts checking the sign of a quadratic form over a subspace into solving an LMI problem x′ Qx < 0, ∀x 6= 0, M x = 0 if and only there exists a scalar σ such that Q − σM ′ M ≺ 0 M x = 0 can also be formulated as x′ M ′ M x = 0 • proof via convexity of numerical ranges

– p. 27/129

generalized Finsler’s Lemma

28

• convert family of constrained quadratic inequalities into an LMI feasibility problem Q = Q′ and M given, and a compact subset of real matrices U we have the equivalence • for all U ∈ U, x′ Qx < 0, ∀x 6= 0 with U M x = 0, iff there exists Θ s.t. Q + M ′ ΘM ≺ 0 ∀U ∈ U NU′ ΘNU  0, where NU is basis of nullspace of U – p. 28/129

multi-convexity

29

given a function f (δ1 , . . . , δK ) • it is multi-convex function if separately convex along each direction δi • multi-convexity is weaker than convexity • convexity iff • multi-convexity iff ∂2 f (δ)]1≤i,j≤K  0 [ ∂δi δj

∂2 f (δ) ≥ 0, 2 ∂δi

i = 1, . . . , K

Turn parameter-dependent LMIs into finite set of LMIs.

– p. 29/129

projection Lemma - two-Sided

30

given Ψ = Ψ′ ∈ Rm×m , P, Q of column dim. m find X such that Ψ + P ′ X ′ Q + Q′ XP ≺ 0 let columns of NP , NQ form bases of the null spaces of P and Q inequality is solvable for X if and only if NP′ Ψ NP ≺ 0 NQ′ Ψ NQ ≺ 0 (Gahinet & Apkarian 1993)

– p. 30/129

System concepts via LMIs

• • • • •

31

Stability L2 gain or H∞ norm H2 norm Pole clustering ...

– p. 31/129

stability & equilibria

32

• Equilibrium points x˙ = f (x) are defined as the solutions x∗ of 0 = f (x∗ ). system has trajectory x(t) = x∗ , ∀t ≥ 0 if initialized at x∗ From now on, we assume x∗ = 0.

– p. 32/129

types of stability

33

• stability (simple) ∀R > 0, ∃r > 0, kx(0)k < r ⇒ ∀t ≥ 0, kx(t)k < R • asymptotic stability if it is stable and ∃r > 0, kx(0)k < r ⇒ x(t) → 0, as t → ∞ • exponentially stable if ∃ α > 0 and λ > 0 s. t. ∀t > 0, kx(t)k ≤ αkx(0)ke−λt in some ball. λ rate of conv. – p. 33/129

positive-definite functions of the state

34

Assume D is open region containing x∗ = 0. • A function V (x) from Rn into R is positive semi-definite on a domain D if (1) V (0) = 0 (2) V (x) ≥ 0, ∀x ∈ D • A function V (x) from Rn into R is positive definite on a domain D if (1) V (0) = 0 (2) V (x) > 0, ∀x ∈ D, x 6= 0

– p. 34/129

postive-definite functions: level curves

35

x2 V (x) = c2 V (x) = c1 O V (x) = c3

x1

c3 < c2 < c1

Typical level curves of positive-definite functions

– p. 35/129

Lyapunov derivatives

36

• if x is state of system x˙ = f (x), then V (x) is implicitly a function of time. Its time derivative is ′

′

∂V dV (x) ∂V ˙ = x˙ = f (x) V (x) = dt ∂x ∂x since x is constrained to satisfy x˙ = f (x). • it is referred to as derivative of V along the system trajectories (also Lyapunov’s derivative).

– p. 36/129

Lyapunov function: definition

37

• V (x) is a Lyapunov function of the system x˙ = f (x) if ➠

it is C 1 with respect to x on D

➠

it is positive definite (see earlier) on D

➠

its derivative on the system trajectories is negative semi-definite, that is, V˙ (x) ≤ 0, on D

as a function of x.

– p. 37/129

Lyapunov theorem for local stability

38

• if in a ball around the origin (= x∗ ), there exists V (x) in C 1 such that V (x) is positive definite ➠ V˙ (x) is negative semi-definite ➠

then the equilibrium point x∗ = 0 is (loc.) stable. It is asymptotically stable if V (x) is negative definite, i. e., V˙ (x) < 0, ∀x 6= 0, x ∈ ball as a function of x. • global stability if ball = Rn .

– p. 38/129

Lyapunov geometry

39

∂V (x) ∂x

x(t) ˙ x(t) 0

V˙ (x) < 0

angle between derivatives is greater than 90 deg.

– p. 39/129

stability for linear systems

40

the system d x = Ax dt is exponentially stable if and only if there exists X with X ≻ 0, A′ X + XA ≺ 0 why ? ′

V (x) = x Xx is a quadratic Lyapunov function

– p. 40/129

proof of exponential stability

41

perturb Lyapunov LMI to A′ X + XA + εX ≺ 0

for any state trajectory x(t), we infer x(t)′ (A′ X + XA)x(t) + εx(t)′ Xx(t) ≤ 0 and thus d ′ ′ x(t) Xx(t) + εx(t) Xx(t) ≤ 0 dt – p. 41/129

trick !

42

• note that solution of d x(t)′ Xx(t) + εx(t)′ Xx(t) = z(t) with z(t) ≤ 0 dt is V (x(t)) = x(0)′ Xx(0)e−εt +

Z

t

e−ε(t−τ ) z(τ )dτ

0

hence x(t)′ Xx(t) ≤ x(0)′ Xx(0)e−εt , ∀t ≥ 0 .

– p. 42/129

proof of exponential stability - continued

43

we have, with initial condition x(0) yields ′

′

x(t) Xx(t) ≤ x(0) Xx(0)e

−εt

finally, using ′

2

λmin (X)kxk ≤ x Xx ≤ λmax (X)kxk

2

gives kx(t)k ≤ kx(0)k

s

λmax (X) −εt/2 e for t ≥ 0 λmin (X)

system is exponentially stable ! – p. 43/129

necessity of Lyapunov inequalities

44

Assume A is stable (Re λi (A) < 0) and consider for Q ≻ 0, the (well-defined) integral Z ∞ d A′ t −Q = (e QeAt )dt dt 0 =

Z

∞ ′ A′ t

(A e

Qe

At

+e

A′ t

QeAt A)dt

0

′

= A P + P A with P :=

Z

∞ A′ t

e QeAt dt ≻ 0 0

– p. 44/129

necessity continued

45

finally, we have A′ P + P A = −Q ≺ 0,

P ≻ 0.

LMI problem has a solution whenever A is stable. • condition is iff • for linear systems quadratic Lyapunov functions are rich enough

– p. 45/129

energy gain or L2 gain

46

Energy gain not larger than γ: with w ∈ L2 and x(0) = 0, every trajectory of d x = Ax + Bw dt z = Cx + Dw should satisfy kzk2 ≤ γkwk2 , or

Z

∞ 0

z(t)′ z(t) dt ≤ γ 2

∀w ∈ L2 Z

∞

w(t)′ w(t) dt

0

– p. 46/129

H∞ norm

47

• stable and the L2 gain w −→ z is smaller than γ if and only if there exists X ≻ 0 " ′ ′ # A X + XA XB C B ′X −γI D′ ≺ 0 C D −γI • freq. domain kC(sI − A)−1 + Dk∞ < γ via KYP. • similarly, H2 norm, LQ, LQG, many others ...

– p. 47/129

H∞ norm

48

• necessity call for general LQ theory. • we shall only prove sufficiency.

– p. 48/129

proof of sufficiency

49

• Note first that the (1, 1) block of the LMI implies that A is stable • By Schur complement, LMI is rewritten  ′   ′ A X + XA XB −1 C +γ D] ≺ 0 ′ ′ [C BX −γI D   x(t) Left- and right-multiply with yields ... w(t)

– p. 49/129

proof of sufficiency cont.

50

d V dt}| { z x′ (A′ X + XA)x + x′ XBw + w′ B ′ Xx −γw′ w + γ −1 z ′ z ≤ 0

– p. 50/129

proof of sufficiency cont.

51

integrate over [0, T ] and exploit x(0) = 0: Z T x(T )′ Xx(T ) + γ −1 kz(t)k2 − γkw(t)k2 dt ≤ 0 0

Recall X ≻ 0 and take T → ∞ (w ∈ L2 ): Z ∞ Z ∞ kz(t)k2 dt ≤ γ 2 kw(t)k2 dt ≤ 0 0

0

Can perturb γ to γ − ε to get strict inequality

– p. 51/129

relation to frequency domain

52

For ω ∈ R, left- and right multiply with   (jω − A)−1 B I to get γ −1 T (jω)∗ T (jω) − γI ≺ 0 hence kT (jω)k < γ,

∀ω ∈ R

From the right-lower block, we also get   ′ −γI D ≺ 0 or kDk < γ D −γI – p. 52/129

relation to frequency domain

53

finally, kT (jω)k < γ for ω ∈ R ∪ {∞} hence, kT k∞ :=

sup

kT (jω)k < γ.

ω∈R∪{∞}

– p. 53/129

H2 performance

54

• H2 norm of T defined as s Z ∞ 1 Tr kT k2 := T (jω)∗ T (jω) dω 2π −∞ • in the time domain (via Parseval) sZ ∞

Tr (CeAt B)′ (CeAt B) dt

kT k2 :=

0

– p. 54/129

H2 performance computation

55

Easily computed by solving linear equation AP0 + P0 A′ + BB ′ = 0 ⇒ kT k22 = Tr (CP0 C ′ ) A′ Q0 + Q0 A + C ′ C = 0 ⇒ kT k22 = Tr (B ′ Q0 B) Why ? see stability notes. • note that D = 0 for H2 norm to be well defined.

– p. 55/129

stochastic interpretation of H2 norm

56

w white noise, x˙ = Ax + Bw, x(0) = 0, z = Cx. Recall: with solution of P˙ (t) = AP (t) + P (t)A′ + BB ′ ,

P (0) = 0

we have E(x(t)x(t)′ ) = P (t). Hence lim E(z(t)′ z(t)) = lim E(x(t)′ C ′ Cx(t))

t→∞

t→∞

= lim Tr E(Cx(t)x(t)′ C ′ ) t→∞

= Tr (CP0 C ′ ) = kT k22 • asymptotic variance of output of system. – p. 56/129

deterministic interpretation of H2 norm

57

let zj be impulse response to Bej δ(t) with standard unit vector ej of x˙ = Ax, x(0) = x0 , z = Cx Z

∞

′

zj (t) zj (t) dt = 0

P

Z

∞ 0

′ A′ t ′ Bj e C CeAt Bj

dt

v v = Tr (vv ) and j Bj Bj′ = BB ′ implies XZ ∞ kzj (t)k2 dt = kT k22 . ′

′

j

0

– p. 57/129

How to get LMI characterization ?

58

With A stable, it is easy to see that Tr (CP0 C ′ ) < γ 2 for AP0 + P0 A′ + BB ′ = 0 if and only if there exists X with Tr (CXC ′ ) < γ 2 and AX + XA′ + BB ′ ≺ 0 . • for ⇐ take difference of Lyapunov conditions

– p. 58/129

How to get LMI characterization ?

59

• for ⇒ since trace inequality is strict and by continuity there exists ε > 0 and X such that AX + XA′ + BB ′ + εI = 0,

Tr (CXC ′ ) < γ 2 .

Note that AX + XA′ + BB ′ ≺ 0 and Z ∞ At ′ A′ t X= e (BB + εI)e dt ≻ P0 0

Hence, kC(sI−A)−1 Bk2H2 := Tr (CP0 C ′ ) < Tr (CXC ′ ) < γ 2 .

– p. 59/129

LMI characterization for H2 norm

60

♦ A is stable and kT k22 < γ if and only if Y ≻ 0 with Tr (CY C ′ ) < γ,

AY + Y A′ + BB ′ ≺ 0

or if and only if X ≻ 0 with Tr (B ′ XB) < γ,

A′ X + XA + C ′ C ≺ 0

– p. 60/129

regional pole constraints

61

• to shape transient responses of closed-loop system • damping, settling time, rise time related to location of poles • useful regions: vertical strips, disks, conic sectors, etc • An LMI region R is defined as R = {z ∈ C : U + zV + z¯V ′ ≺ 0} .

– p. 61/129

LMI region intersections

62

• a large variety of regions can be represented this way • intersections of LMI regions are LMI regions

– p. 62/129

a short catalog of useful LMI regions LMI Regions

α

β

63

Characterization

fR (z) =

"

1 (z + z −α + 2 ¯)

0

0

(z + z ¯) β− 1 2

fR (z) =

"

i (z − z −α − 2 ¯)

0

0

¯) β + 2i (z − z

fR (z) =

»

−r q + z q+z ¯ −r

fR (z) =

»

sin θ(z + z ¯) cos θ(z − z ¯) cos θ(¯ z − z) sin θ(z + z ¯)

α

#

#

β

r −q

θ

–

–

– p. 63/129

Lyapunov theorem for LMI regions

64

• System d x = Ax has all its poles in LMI region R dt iff there exists X ≻ 0 s. t. U ⊗ X + V ⊗ (A′ X) + V ′ ⊗ (XA) ≺ 0 . is an LMI with respect to X. (⊗ is Kronecker product A ⊗ B := ((Aij B))) ➠

classical Lyapunov theorem with U = 0, V = 1

➠

intersection by diagonal augmentation of U , V .

other specs. can be combined by just merging LMI constraints – p. 64/129

LMI regions- proof of sufficiency

65

condition is X ≻ 0,

U ⊗ X + V ⊗ (A′ X) + V ′ ⊗ (XA) ≺ 0 .

pick an eigenpair of A, (λ, v), Av = λv, and pre- and post-multiply inequality by I ⊗ v ∗ , I ⊗ v, gives >0

Hence,

z }| { (v ∗ Xv) (U + λ∗ V + λV ′ ) < 0 U + λ∗ V + λV ′ < 0.

Implies λ∗ , λ are in R. – p. 65/129

multi-objective/channel controller synthesis

• • • • •

66

formulation linearizing change of variables state-feedback synthesis output-feedback synthesis projected form.

– p. 66/129

controller synthesis

67

• synthesis structure z1 z2

.. .

.. .

P

y

w1 w2

u K

• given P (s), find K(s) to achieve a set of specifications for channels w1 → z1 , w2 → z2 , ...

– p. 67/129

example of multi-channel/objective problem 68 min kTw21 ←z21 k2 1 ←z 1 k∞ < γ1 , kTw∞ ∞

kTw22 ←z22 k2 < γ2

poles in LMI region R .

– p. 68/129

controller synthesis - data

69

• synthesis interconnection  d x = Ax + B w + B u, A ∈ Rn×n  1 2  dt P (s) z = C1 x + D11 w + D12 u   y = C2 x + D21 w • controller ( d x = A x + B y, K K K K d t K(s) u = C K xK + D K y

AK ∈ Rn×n

• Stability, Perfo.: H∞ , H2 , pole plac. on various channels – p. 69/129

derivation ➠

compute closed-loop data

➠

write stability/performance (ineq.) conditions in closed loop

➠

apply congruence transformations

➠

use suitable linearizing transformations

70

– p. 70/129

controller synthesis - state-feedback

71

• turns out to be very simple problem  n×n  x ˙ = Ax + B w + B u, A ∈ R 1 2  P (s) z = C1 x + D11 w + D12 u   y = x ←− measurable state vector

and

u = Kx ←− state-feedback

closed-loop data are x˙ = (A + B2 K)x + B1 w z = (C1 + D12 K)x + D11 w – p. 71/129

state-feedback H∞ synthesis • characterization is X ≻ 0 and  ∗ (A + B2 K)′ X + ∗  −γI B1′ X C1 + D12 K D11

72



∗ ∗ ≺0 −γI

perform congruence transformation diag(Y = X −1 , I, I) to get Y ≻ 0 and  (A + B2 K)Y + Y (A + B2 K)′ ∗  (C1 + D12 K)Y −γI ′ D11 B1′



∗ ∗  ≺ 0, −γI – p. 72/129

state-feedback H∞ synthesis continued

73

note Y is invertible perform change of variable W = KY to get LMI !: Y ≻ 0 and   AY + Y A′ + B2 W + (B2 W )′ ∗ ∗  C1 Y + D12 W −γI ∗  ≺ 0. ′ −γI D11 B1′

• note change of variable is without loss (NSC) • when solved, deduce (state-feedback) controller using K = W Y −1 .

– p. 73/129

trick

74

• ⇐ (Y, KY ) solution → (Y, W ) easy • ⇒ (Y, W ) solution → (Y, KY ) note that term B2 W is B2 W Y −1 Y hence (Y, K = W Y

−1

) is a solution.

– p. 74/129

state-feedback H2 synthesis

75

• similar derivation • characterization (A + B2 K)′ X + ∗ + (C1 + D12 K)′ (C1 + D12 K) ≺ 0, Tr (B1′ XB1 ) < η 2 become via Schur complements   ′ (A + B2 K) X + ∗ ∗ ≺0 (C1 + D12 K) −I   ′ Z B1 2  0, Tr Z < η B1 X −1 – p. 75/129

state-feedback H2 synthesis

76

• perform congruence transformations diag(Y = X −1 , I) and diag(I, Y ) to get   AY + B2 KY + ∗ ∗ ≺0 C1 Y + D12 KY −I   Z B1′ Y  0, Tr Z < η 2 Y B1 Y

– p. 76/129

state-feedback H2 synthesis

77

• change of variable W = KY yields LMIs !   AY + B2 W + ∗ ∗ ≺ 0, C1 Y + D12 W −I with



Z Y B1

B1′ Y Y



 0, Tr Z < η 2

– p. 77/129

state-feedback pole clustering

78

• similarly Y ≻ 0 and U ⊗ Y + V ⊗ (A + B2 K)Y + V ′ ⊗ Y (A + B2 K)′ ≺ 0 . change of variable W = KY leads to LMI!: Y ≻0 U ⊗ Y + V ⊗ (AY + B2 W ) + V ′ ⊗ (AY + B2 W )′ ≺ 0 .

– p. 78/129

multiple constraints

79

• the Y ’s are not the same for all perfs. • hard problem is relaxed by taking a single Y for all perfs. • technique is constantly refined to exploit different Y ’s by spec. (active area).

– p. 79/129

output feedback case - closed-loop data

80

• 2 4

A

B1

C1

D11

3

2

6 5 := 6 4

A

0

B1

0

0

0

C1

0

D11

3 2 7 6 7 +6 5 4

0

B2

I

0

0

D12

3

»

7 AK 7 5 CK

BK DK

–

2 4

0

I

0

C2

0

D21

3

5,

• Above analysis condition must be satisfied in closed-loop. Synthesis conditions in 3 steps 1- introduce a single variable P common specification/channel (conservative step), 2- perform adequate congruence transformations, 3- use linearizing changes of variables to end up with LMI synthesis conditions. – p. 80/129

linearizing change of variable

81

Introduce notation 2

P=4

X N′

N ⋆

3

5,

P

−1

2

=4

Y

M

M′

⋆

3 5

From PP −1 = I infer 2

PΠY = ΠX with ΠY := 4

Y M′

I 0

3

2

5 , ΠX := 4

I

X

0

N′

3

5.

Define change of variable (wlog N , M are invertible) 8 bK < A

:=

N AK M ′ + N BK C2 Y + XB2 CK M ′ + X(A + B2 DK C2 )Y,

:=

b K := CK M ′ + DK C2 Y, D b K := DK . N BK + XB2 DK , C

(1)

: b BK

and, perform congruence transformations to get

bK , B bK, C bK, D bK ! linear terms in the new variables X, Y, A – p. 81/129

LMI for H∞ specification 2

L11 6 6 A b K C2 )′ 6 bK + (A + B2 D 6 6 b K D21 )′ (B1 + B2 D 4 bK C1 Y + D12 C

where

b′ + (A + B2 D b K C2 ) A K L22 b K D21 )′ (XB1 + B b K C2 C1 + D12 D

82

∗

∗

∗

∗

−γI

∗

b K D21 D11 + D12 D

−γI

3

7 7 7 7≺0 7 5

b K + (B2 C b K )′ , L22 := A′ X+ XA + B b K C2 + (B b K C2 )′ . L11 := AY + YA′ + B2 C

• similarly for H2 and LMI region specs. • for multi- channel/objective just stack together various LMI specs.

– p. 82/129

LMI for H2 specification 2 6 6 4

b K + (B2 C b K )′ + B2 C ∗ b K C2 )′ b K C2 + (B b K C2 )′ + (A + B2 D A′ X+ XA + B bK b K C2 C1 Y + D12 C C1 + D12 D

83

∗

YA′

AY + bK A 2 6 6 4

I

X b K D21 )′ (XB1 + B b K D21 )′ (B1 + B2 D b K D21 = 0. Tr (Q) < ν, D11 + D12 D

"

Y I I X

3

−I

b K D21 B1 + B2 D 7 b XB1 + BK D21 7 5 ≻ 0, Q

I

Y

∗

#

3

7 7 ≺ 0, 5

≻0

– p. 83/129

LMI region constraint specification

84

• congruence diag(ΠY , . . . , ΠY ) yields h i i   h b b AY +B2 CK A+B2 DK C2 Y I + ∗ ≺ 0. λjk I X + µjk bK b K C2 A XA+ B "

Y I I X

#

≻0

– p. 84/129

multiple constraints

85

• again for multiple constraints take the same X, Y bK B b K , . . . for all LMIs. and A

• controller construction: just reverse the change of variables

– p. 85/129

pure H∞ synthesis: projected characterization86 For a single objective, LMI can be simplified, Projection Lemma yields 2 4 2 4

NY

0

0

I

NX

0

0

I

3′

2

6 5 6 4 3′

2

6 5 6 4

YC1′

B1

C1 Y

−γI

D11

B1′

′ D11

−γI

XB1

C1′ ′ D11

AY +

A′ X+

YA′

XA

B1′ X

−γI

C1

D11

−γI

3

2

7 N 74 Y 5 0 3

2

0

Y

I

I

X

4 NY and NX null spaces of

B2′

′ D12

i

and

h

C2

I

7 N 74 X 5 0 2

h

0

D21

I

3

≺

0

3 5

≺

0

3 5

≻

0.

5

i ,

– p. 86/129

avantages of LMI formulations ➠

very general wrt DGKF, no assumptions required

➠

singular problems

➠

admits similar discrete-time counterpart

➠

has educational value for students (shorter proofs)

➠

See http://www.cert.fr/dcsd/cdin/apkarian/ for details

➠

See M ATLAB LMI Control Toolbox for codes.

87

– p. 87/129

uncertain systems analysis

88

• Lyapunov technique • Time-invariant and time-varying parameters • Parameter-dependent Lyapunov functions.

– p. 88/129

analysis of uncertain systems - example

89

Consider the uncertain system d x(t) = A(δ) x(t); dt

x(0) = x0

➠

δ = [δ1 , . . . , δL ]′ ∈ RL uncertain and possibly time-varying real parameters

➠

A(δ) = A0 + δ1 A1 + . . . + δL AL

δ(t)

˙ δ(t)

is the system stable for all admissible δ(t) ? – p. 89/129

Affine Quadratic Stability (AQS)

90

The system is Affinely Quadratically Stable, if ∃ V (x, δ) := x′ P (δ)x,

P (δ) = P0 +δ1 P1 +. . .+δL PL

s. t. V (x, δ) > 0, dV /dt < 0 along all admissible parameter trajectories. • Lyapunov theory ⇒ (exponential) stability. P (δ) := P0 + δ1 P1 + . . . + δL PL L(δ, d δ) := A(δ)′ P (δ) + P (δ)A(δ) + dt

> 0 dP (δ) dt

< 0

• turned into LMIs ⇒ multi-convexity, S-procedure ,... ! – p. 90/129

analysis - continued • cases Time-Invariant Parameters Arbitrary rate of variation (quad. stab.)

• extensions H∞ , H2 , LMI regions,...

91

• components LFT uncertainties nonlinear components (IQC theory, (Rantzer & Megretsky) µ analysis

– p. 91/129

gain-scheduling and LPV control

92

• motivations and concepts • classes of LPV system • synthesis conditions for LFT systems

– p. 92/129

motivations #1

93

➠

handle full operating range

➠

gain-scheduled controllers exploit knowledge on the plant’s dynamics in real time knowledge on plant

measurement signal

controller dynamics

control signal

controller mechanism is changed during operation

– p. 93/129

motivations #2

94

Gain-Scheduling techniques are applicable to • Linear Parameter-Varying Systems (LPV): d x = A(θ)x + B(θ)u , dt y = C(θ)x + D(θ)u. where θ := θ(t) is an exogenous variable. • “ Quasi-Linear” Systems: d x = A(y )x + B(y )u , sche sche dt y = C(ysche )x + D(ysche )u. where ysche is a sub-vector of the plant’s output y. – p. 94/129

motivations #3 ➠

to get higher performance

➠

some LPV system are not stabilizable via a fixed LTI controller

➠

bypass critical phases of pointwise interpolation and switching

➠

engineering insight is preserved (freeze scheduled variable for analysis).

➠

nonlinear models can be handled by immersion into an LPV plant.

95

– p. 95/129

LPV systems in practice ➠ Aeronautics (longitudinal motion of aircraft)           0 α˙ −Zα 0 α az −Zα V + = δ, = q˙ −mα 0 q mδ q 0

96

  α 0 , q 1

where Zα , mα and mδ are functions of speed, altitude and angle of attack. ➠ Robotics (flexible two-link manipulator) M (θ2 )¨ q (t) + Dq(t) ˙ + Kq(t) = F u(t), where θ2 is the scheduled variable (conf. of 2nd beam). ➠ and many others

– p. 96/129

example: different control principles -d 6

-d 6

-d 6

- K

- P (θ)

97

y

-

Robust control

?

θ

- K(θ)

- P (θ)

y

-

LPV control ?

- K(y)

y - P (y)

y

-

Output gain-scheduling

– p. 97/129

description of LPV systems

98

• LPV systems x˙ = A(θ)x + B(θ)u , y = C(θ)x + D(θ)u. are characterized by





A() B() ➠ the functional dependence of on θ, C() D() ➠

the operating domain Θ of the system trajectories, θ(t) ∈ Θ,

➠

the rate of variations of θ(t) (if available) in the form of bounds θ˙i (t) ∈ [θi ; θ¯i ]. – p. 98/129

LPV / LTV / LTI systems LPV

99

θ(t) ∈ Θ ∀t ≥ 0

select trajectory

freeze parameter

θ = θ0

θ(t) = θ∗ (t)

LTV

LTI freeze the time

t = t0

– p. 99/129

LPV / LTV / LTI - off-line vs. on-line

100

➠

LTI and LTV systems are off-line systems, the state-space data A, B,... and A(t), B(t),... must be known in advance.

➠

LPV systems are on-line systems since the dynamics depend on the trajectory θ(t) experienced by the plant in Θ. Θ

θ(t) – p. 100/129

LPV systems interpretations x˙ = A(θ)x + B(θ)u, y = C(θ)x + D(θ)u.

101

θ(t) ∈ Θ

• θ may be subject to various assumptions: ➠

θ(t) is uncertain → robust control problem,

θ(t) is known in real-time → Gain-scheduling problem,   θ1 (t) ➠ θ(t) := , where θ1 is known and θ2 is θ2 (t) uncertain → mixed problem ➠

– p. 101/129

LPV pathologies - LPV/LTI stabilities

102

• stability over a domain ➠

LTI Stability : Reλi (A(θ)) < 0, ∀θ ∈ Θ,

➠

LPV Stability : Φθ (t) → 0, for t → ∞, for all trajectory θ(t) in Θ.

• intuitive conjectures like ➠

LTI stability ⇒ LPV stability,

➠

LPV stability ⇒ LTI stability,

are FALSE !

– p. 102/129

LPV vs. LTI Stability • Conjecture #1    −1 + aθ12 x˙ 1 = x˙ 2 −1 + aθ1 θ2

1 + aθ1 θ2 −1 + aθ22

103



x1 x2



,

with trajectories θ1 := cos(t) and θ2 (t) := sin(t) is LTI stable (for a < 2) but LPV unstable. • Conjecture #2      2 −1 − 5θ1 θ2 1 − 5θ1 x˙ 1 x1 = , 2 x˙ 2 x2 −1 + 5θ2 −1 + 5θ1 θ2 with trajectories θ1 := cos(t) and θ2 (t) := sin(t) is LTI unstable (poles +1 and −3) but LPV stable. – p. 103/129

creating an LPV stability

104

consider the autonomous LPV system: x¨ + ω 2 (t)x = 0 , where we are allowed to switch between two values ω1 and ω2 . x2

system trajectories

switch

ω1 ω2

x1

unstable behavior

– p. 104/129

Slowly Varying LPV Systems x˙ = A(θ)x

105

• Sufficient stability cond. • Sufficient instability cond. (1) Reλi (A(θ)) < 0, (1) Reλi (A(θ)) < 0, ˙ < α, with α sufficiently i = 1, . . . , k (2) kθk small, (2) Reλi (A(θ)) > 0, ⇒ LPV stability (Rosen. 63) i = k + 1, . . . , n (3) stable and unstable eigenvalues do not mix ˙ < α, with α sufficiently (4) kθk small, ⇒ LPV instability (Skoog 72)

– p. 105/129

slowly varying parameters

106

LPV stability can be inferred from LTI stability for slowly varying parameters (but not constructive conditions).

– p. 106/129

LPV systems in the LFT class

107

diag(θi Iri )



 A Bθ B  Cθ Dθθ Dθ•  C D•θ D

y

»

A(θ) C(θ)

B(θ) D(θ)

–

:=

»

A C

B D

–

»

Bθ + D•θ

–

u

Θ(I − Dθθ Θ)−1 [ Cθ

Dθ• ] , – p. 107/129

LPV systems in the polytopic class θ ∈ Θ , Θ Polytope

»

A(θ) C(θ)

B(θ) D(θ)

–

∈ Cov

»

Ai Ci

Bi Di

–

108

, i = 1, 2, . . . , r

ff

θ(t)

»

y

A1 B1 C1 D1

»

–

A2 B2 C2 D»2

– Ai Bi Ci Di

u –

– p. 108/129

general LPV systems

109

A(θ), B(θ), C(θ), D(θ) are arbitrary but continuous matrix-valued function of θ. • far more difficult to handle but of great practical interest since they capture arbitrary nonlinearities

– p. 109/129

formulation of synthesis problem: LFT

110

Θ

w

z

P (s) y

u

K(s)

Θ

gain−scheduled controller – p. 110/129

formulation of synthesis problem: LFT

111

find LPV controller Fl (K(s), Θ(t)) s.t. ➠

closed-loop stability,

➠

the L2 -induced norm of the operator Tw→z satisfies kTw→z (Θ)k < γ

for all admissible trajectory θ(t).

– p. 111/129

LPV-LFT systems: notations

112

• 

Dθθ P (s) =  D1θ D2θ

Dθ1 D11 D21

 



Dθ2 Cθ D12 + C1  (sI−A)−1 [ Bθ D22 C2

B1

B2 ] ,

Assumptions: (A, B2 , C2 ) stabilizable  and  detectable,  D22 = 0. Cθ b Dθθ Dθ1 b b Notations: B1 = [ Bθ B1 ], C1 = , D11 = , C1 D1θ D11 T T 0 ], D12 NY := Ker [ B2T Dθ2 NX := Ker [ C2 D2θ D21 0 ].

– p. 112/129

LPV-LFT systems - proof scheme synthesis structure

113

synthesis structure with parameter augmentation

Θ(t) zθ

wθ

z

w

P(s)

y

w ˜θ

Θ(t)

0

0

Θ(t)

zθ

u z

z˜θ

wθ w

P(s)

K(s)

Pa (s) z˜θ

w ˜θ Θ(t)

y

y˜

u K(s)

u ˜

– p. 113/129

LPV-LFT systems - proof scheme

114

➠

redraw the control configuration into a robust control problem with repeated uncertainty,

➠

formulate the Bounded Real Lemma with scalings for the closed-loop system,

➠

apply the Projection Lemma to derive the LMI characterization.

– p. 114/129

LMI characterization

NYT

2 AY + Y AT 6 Cθ Y + Γ3 BθT 6 6 C1 Y 6 4 Σ BT B1T

⋆ T −D Γ −Σ3 + Γ3 Dθθ θθ 3 −D1θ Γ3 T Σ3 Dθθ T Dθ1

⋆ ⋆ −γI T Σ3 D1θ T D11

⋆ ⋆ ⋆ −Σ3 0

⋆ 3 ⋆ 7 7 ⋆ 7 7 NY ≺ 0, ⋆ 5 −γI

2 AT X + XA 6 BθT X + T3 Cθ 6 6 B1T X 6 4 S3 Cθ C1

⋆ T T −S3 + T3 Dθθ − Dθθ 3 T T −Dθ1 3 S3 Dθθ D1θ

⋆ ⋆ −γI T S3 Dθ1 D11

⋆ ⋆ ⋆ −S3 0

⋆ 3 ⋆ 7 7 ⋆ 7 7 NX ≺ 0, ⋆ 5 −γI

3

T NX

115

θ

»

Y I

I X

–

0

S3 ≻ 0, Σ3 > 0; T3 , Γ3 skew − symmetric .

– p. 115/129

scaling sets asso. with structure Θ ⊕ Θ

116

• symmetric SΘ := {S : S > 0, SΘ = ΘS} • symmetric augmented   S1 S2 SΘ⊕Θ = { T : S1 , S2 ∈ SΘ and S2 Θ = ΘS2 , ∀Θ ∈ Θ}. S2 S3 • skew-symmetric   T1 T2 TΘ⊕Θ = { : T1 , T2 ∈ TΘ and T2 Θ = ΘT2 , ∀Θ ∈ Θ}. T −T2 T3

– p. 116/129

robust synthesis condition 

ATcℓ Xcℓ + Xcℓ Acℓ  B T Xcℓ + T Ccℓ cℓ Ccℓ

T T Xcℓ Bcℓ + Ccℓ T T T −S + T Dcℓ + Dcℓ T Dcℓ

where

117



T Ccℓ T ≺ 0 Dcℓ −S −1

➠

Acℓ, Bcℓ , . . . closed-loop data

➠

S, T scalings for Θ ⊗ Θ ⊗ ∆, and ∆ fictitious performance block.

– p. 117/129

cast in Projection Lemma form

118

Can be rewritten Ψ + QTX ΩP + P T ΩT QX ≺ 0, where 

T

A Xcℓ + Xcℓ A Ψ =  B1T Xcℓ + T C1 C1

P = [ C2

D21

0],

C1T T T



C1T T  D11 −1

Xcℓ B1 + , −S + T D11 + D11 T T D11 −S   T T T T QX = B2 Xcℓ D12 T D12 .

– p. 118/129

cast in Projection Lemma form continued 2

2

A 4 C1 C2

B1 D11 D21

6 6 6 6 3 6 6 B2 6 6 5 D12 = 6 6 6 ΩT 6 6 6 6 4

A

0

0

Bθ

B1

0

B2

0

0

0

0

0

0

I

0

0

0

0

0

0

0

0

0

I

Cθ

0

0

Dθθ

Dθ1

0

Dθ2

0

C1

0

0

D1θ

D11

0

D12

0

0

I

0

0

0

AT K

T CK1

T CKθ

C2

0

0

D2θ

D21

T BK1

T DK11

T DKθ1

0

0

I

0

0

T BKθ

T DK1θ

T DKθθ

and b1 = [ Bθ B

B1 ] ,

b1 = C

»

– Cθ , C1

b 11 = D

»

Dθθ D1θ

Dθ1 D11

–

119

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

.

• LMI characterization follows from explicit computation of projections and using matrix completion Lemmas.

– p. 119/129

LPV-LFT systems - controller Construction120 ➠

Testing solvability falls within the scope of convex semi-definite programming

➠

A gain-scheduled controller is easily constructed from the quadruple (Y, X, L3 , J3 ) by solving a scaled Bounded Real Lemma LMI condition. u

y 3 AK BK1 BKθ 4 CK1 DK11 DK1θ 5 CKθ DKθ1 DKθθ 2

Θ(t) – p. 120/129

other variants of this technique ➠

polytopic LPV systems

➠

general LPV systems (capture slow variations of parameters)

➠

LFT systems ang generalized scalings

➠

multi-objective/channel LPV synthesis

121

see webpage: http://www.cert.fr/dcsd/cdin/apkarian/

– p. 121/129

hard non-LMI problems

122

• most analysis problems reduce to LMIs • some synthesis problems reduce to LMIs but • many practical problems do not reduce to LMI/SDP (synthesis) ➠

reduced- and fixed-order synthesis (PID H∞ , etc.)

➠

structured and decentralized synthesis problems

➠

general robust control with uncertain and/or nonlinear components

➠

simultaneous model/controller design, multimodel control

➠

unrelaxed LTI and LPV multi-objective

➠

combinations of the above – p. 122/129

new algorithms for hard problems

123

new algorithms needed ! good research direction

– p. 123/129

example: synthesis of static controller stabilize x˙ = Ax + Bu y = Cx with u = Ky (K static )

124

has characterization NC′ (A′ X + XA)NC < 0 ′ ′ ′ (Y A + AY )N < 0 N ′ B B   X I > 0 I Y XY − I = 0

constraints XY − I = 0 leads to hard problems LMI + nonlinear equality constraints

– p. 124/129

augmented Lagrangian method

125

with g(x) = 0 equ. constraints and A(x) ≺ 0 LMI, replace the difficult program by the more convenient (Pλ,µ )

minimize c′ x + λ′ g(x) + µ1 kg(x)k2 subject to A(x)  0

➠

µ is penalty, xµ → x∗ when µ → 0

➠

for good estimates λ (Lagrange multiplier), solution of (Pλ,µ ) is close to solution of original problem

➠

use first-order update rule to improve estimate λ

➠

solve (Pλ,µ ) by a succession of SDPs – p. 125/129

augmented Lagrangian method

➠

B. Fares and P. Apkarian and D. Noll, IJC, 2001

➠

B. Fares and D. Noll and P. Apkarian , SIAM Cont. Optim. 2002

➠

P. Apkarian and D. Noll and H. D. Tuan, 2002, IJRNC to appear.

➠

D. Noll and M. Torki and P. Apkarian, working paper, 2002

126

– p. 126/129

conclusions, perspectives

127

➠

A single framework for a great variety of methods

➠

LMI techniques extend the scope of classical techniques

➠

LPV control is a very successful example (industrial)

➠

Analysis meth. immediately applicable for validation

➠

Have educational merits see http://www.cert.fr/dcsd/cdin/apkarian/ for course plan

➠

not discussed: robust filtering and estimation, combinatorial optimization, graphs, etc. – p. 127/129

recent concrete control applications ➠

Analysis robustness evaluation of controllers for: ➟ ARIANE Launcher ➟ satellites ➟ long flexible civil aircraft (structural modes)

➠

Synthesis Preliminary tests show that LPV controllers are competitive for launcher control in atmospheric flight

➠

Synthesis control of the landing phase for civil aircraft under study with multiobjective LMI methods

➠

Synthesis Missiles ? still on paper

128

– p. 128/129

The End

129

GRAZIE MILLE !

– p. 129/129