robust multi-objective control for linear systems elements

Robust Multi-Objective Control for Linear Systems

Elements of theory and ROMULOC toolbox

OLOCEP project

Denis Arzelier & Didier Henrion & Jean Lasserre & Dimitri Peaucelle

LAAS-CNRS, Toulouse, FRANCE

CNRS-RAS cooperation

Outline

I - Uncertain LTI systems and performances

➞ Control objectives: stability, transient response, perturbation rejection...

➞ Structured parametric uncertainties: extremal values, bounded sets...

II - LMIs and convex polynomial-time optimization

➞ Semi-Definite Programming and LMIs

➞ SDP solvers and parsers

III - Conservative LMI results

➞ Methods: Lyapunov, S-procedure, Finsler lemma, Topologic Separation...

➞ The ROMULOC toolbox

Seminar at IPME 1 March 21st, 2005, St. Petersburg


Ia - Control objectives

➞ Stability & D-stability

sx(t) = Ax(t) + Buu(t)

y(t) = Cyx(t) + Dyuu(t):

x ∈ Cn u ∈ Cqu

y ∈ Cpy

Design an LTI control u = Ky s.t. poles belong to a region of the complex plane

(or for given K analyse pole location):

DR = { s ∈ C : r11 + sr12 + s∗r12 + ss∗r22 ≤ 0 } , R = (rij)

Such regions are half-planes and discs

R =

0 1

1 0

R =

−1 0

0 1

R =

−2α cosψ cosψ − i sinψ

cosψ + i sinψ 0

ψ

α



Ia - Control objectives

➞ H∞ & H2 & Impulse-to-peak performance

sx(t) = Ax(t) + Bww(t) + Buu(t)

z(t) = Czx(t) + Dzww(t) + Dzuu(t)

y(t) = Cyx(t) + Dyww(t) + Dyuu(t)

:w ∈ Cqw

z ∈ Cpz

H∞ performance : ‖z‖ ≤ γ∞‖w‖ ; bounded-real lemma

H2 performance : ‖z‖ ≤ γ2 for w g.w.n. ; max |z| ≤ γ2‖w‖

Impulse-to-peak performance : max |z| ≤ γi2p for w = δ.

Design an LTI control u = Ky s.t. given specification γ is fulfilled

or that minimizes γ

(or for given K analyse performance level)



Ib - Robust Multi-Objective Control

∆ : errors in modeling, operating conditions, mass-production...

∆ : parametric uncertainty, assumed constant, belongs to a set ∆∆.

sx(t) = A(∆)x(t) + Bw(∆)w(t) + Bu(∆)u(t)

z(t) = Cz(∆)x(t) + Dzw(∆)w(t) + Dzu(∆)u(t)

y(t) = Cy(∆)x(t) + Dyw(∆)w(t) + Dyu(∆)u(t)

Find controllerK that fulfills robust specifications Πi defined for models Σi(∆i) with ∆i ∈ ∆∆i.

K

Σ2 F2(∆ )2

K

1(∆ )1

K

1Σ (0)1FΣ



Ic - Structured uncertainties

➞ Polytopic models zw

K

u y< γ

[2]Σ

Σ[1]Σ[N]

Σ(∆)

✪ Affine polytopic models : convex hull of N vertices

A(∆) =∑ζiA

[i] , Bw(∆) =∑ζiB

[i]w . . . : ζi ≥ 0 ,

∑ζi = 1




➞ Polytopic models zw

K

u y< γ

[2]Σ

Σ[1]Σ[N]

Σ(∆)

✪ Affine polytopic models : convex hull of N vertices

A(∆) =∑ζiA

[i] , Bw(∆) =∑ζiB

[i]w . . . : ζi ≥ 0 ,

∑ζi = 1

➥ Parallelotopic models with NP axes

A(∆) = A[0] +∑ξiA

[i] , Bw(∆) = B[0]w +

∑ξiB

[i]w . . . : |ξi| ≤ 1

➾ polytope with N = 2NP vertices

➥ Interval models with NI non equal coefficients

A[1] � A(∆) � A[2] : a[1]ij ≤ aij(∆) ≤ a

[2]ij

➾ parallelotope with axes in the euclidian basis of matrices

➾ polytope with N = 2NI vertices




➞ LFT modelszΣ

∆w z

K

yu< γ

w

∆ ∆

sx(t) = Ax(t) + B∆w∆(t) + Bww(t) + Buu(t)

z∆(t) = C∆x(t) + D∆∆w∆(t) + D∆ww(t) + D∆uu(t)

z(t) = Czx(t) + Dz∆w∆(t) + Dzww(t) + Dzuu(t)

y(t) = Cyx(t) + Dy∆w∆(t) + Dyww(t) + Dyuu(t)

:w∆ ∈ Cq∆

z∆ ∈ Cp∆

Linear - Fractional Transformation:

A(∆) = A+B∆∆(I−D∆∆∆)−1C∆ , Bw(∆) = Bw+B∆∆(I−D∆∆∆)−1D∆w . . .

Any model rational in δi parameters ➾ LFT (not unique) with diagonal ∆ =diag(δ1, δ1, ..., δ2, ...).




➞ Uncertainty sets

✪ {X,Y, Z}−dissipative matrices{∆ ∈ Cqw×pz : X + Y∆ + ∆∗Y ∗ + ∆∗Z∆ ≤ O , X ≤ O , Z ≥ O

}➥ Norm-bounded uncertainties : ‖∆‖ ≤ ρ1 ➾ {−ρ2I,O, I}−dissipative

➥ Positive real uncertainties : ∆ + ∆∗ ≥ O (eg. s) ➾ {O,−I,O}−dissipative




➞ Uncertainty sets

✪ {X,Y, Z}−dissipative matrices{∆ ∈ Cqw×pz : X + Y∆ + ∆∗Y ∗ + ∆∗Z∆ ≤ O , X ≤ O , Z ≥ O

}➥ Norm-bounded uncertainties : ‖∆‖ ≤ ρ1 ➾ {−ρ2I,O, I}−dissipative

➥ Positive real uncertainties : ∆ + ∆∗ ≥ O (eg. s) ➾ {O,−I,O}−dissipative

✪ Polytopic uncertainties

polytope N vertices

{∆ =

∑ζi∆[i] : ζi ≥ 0 ,

∑ζi = 1

}➥ Parallelotopic uncertainties

NP axes ➾ polytope N = 2NP

{∆ = ∆[0] +

∑ξi∆[i] : |ξi| ≤ 1

}➥ Interval uncertainties

NI coef. 6= ➾ polytope N = 2NI

{∆[1] � ∆ � ∆[2] : δ

[1]ij ≤ δij ≤ δ

[2]ij

}


Outline




zw

K

u y< γ

[2]Σ

Σ[1]Σ[N]

Σ(∆)zΣ

∆w z

K

yu< γ

w

∆ ∆



➞ SDP solvers and parsers






IIa - Semi-Definite Programming and LMIs

✪ Extension of LP to semi-definite matrices

min cx : Ax = b , xi ≥ 0 (LP ) | mat(x) ≥ O (SDP )

➥ Convexity, duality, polynomial-time algorithms (O(n6.5 log(1/ε))).

max bT y : AT y − cT = z , mat(z) ≥ O

✪ 1st developments and 1st results : LMI formalism & Control Theory

min∑

giyi : F0 +∑

Fiyi ≥ O

➥ The H∞ norm computation example for G(s) ∼ (A,B,C,D) :

‖G(s)‖2∞ = min γ : P > O ,

ATP + PA+ CTz Cz BwP + CT

z Dzw

PBTw +DT

zwCz −γI +DTzwDzw

≤ O



IIb - SDP solvers and parsers

➞ LMI Control Toolbox ➾ Control Toolbox

1st solver, dedicated to LMIs issued from Control Theory, Matlab, owner.

➞ SDP solvers: SP, SeDuMi, SDPT3, CSDP, DSDP, SDPA...

Active field, mathematical programing, C/C++, free.

➞ Parsers: tklmitool, sdpsol, SeDuMiInterface, YALMIP

Convert LMIs to SDP solver format, Matlab (Scilab), free.



IIc - SDP-LMI issues and prospectives

✪ Any SDP representable problem is ”solved” (numerical problems due to size and structure)

➥ Find ”SDP-ables” problems

(linear systems, performances, robustness, LPV, saturations, delays, singular systems...)

➥ Equivalent SDP formulations ➾ distinguish which are numerically efficient

➥ New SDP solvers: faster, precise, robust (need for benchmark examples)









✪ Any ”SDP-able” problem has a dual interpretation

➥ New theoretical results, new proofs (Lyapunov functions = Lagrange multipliers)

➥ SDP formulas numerically stable (KYP-lemma)












✪ Non ”SDP-able” : Robustesse & Multi-objective & Relaxation of NP-hard problems

➥ Optimistic / Pessimistic (conservative) results

➥ Reduce the gap (upper/lower bounds) while handling numerical complexity growth.












✪ Non ”SDP-able” : Robustesse & Multi-objective & Relaxation of NP-hard problems

➥ Optimistic / Pessimistic (conservative) results

➥ Reduce the gap (upper/lower bounds) while handling numerical complexity growth.

✪ Develop software for ”industrial” application / adapted to the application field

➾ ROMULOC toolbox


Outline




zw

K

u y< γ

[2]Σ

Σ[1]Σ[N]

Σ(∆)zΣ

∆w z

K

yu< γ

w

∆ ∆



➞ SDP solvers and parsers ➾ YALMIP and all solvers






IIIa - Nominal performance analysis V (x) = xTPx Lyapunov function (P > O)

✪ Stability ATP + PA < O | ATPA− P < O

✪ D-Stability[

I A∗] r11P r12P

r∗12P r22P

I

A

< O

✪H∞ norm

ATP + PA+ CTz Cz PBw + CT

z Dzw

BTwP +DT

zwCz −γ2I +DTzwDzw

< O

✪H2 normATP + PA+ CT

z Cz < O

trace(BTwPBw) < γ2

✪ Impulsion-to-peakATP + PA < O BT

wPBw < γ2I

CTz Cz < P DT

zwDzw < γ2I



IIIb - Robust performance analysis V (x,∆) parameter-dependent Lyapunov function.

✪ Nominal analysis (LMI)→ Robust analysis (NP-hard)

∃ P : LΣ(P ) < O → ∀∆ ∈ ∆∆ , ∃ P (∆) : LΣ(∆)(P (∆)) < O

Test over sample values in ∆∆ gives optimistic results.





∃ P : LΣ(P ) < O → ∀∆ ∈ ∆∆ , ∃ P (∆) : LΣ(∆)(P (∆)) < O


➥ Choice of P (∆) for having a finite number of decision variables :

➙ “Quadratic Stability”: P (∆) = P

➙ Polytopic PDLF: P (∆) =∑ζiP

[i]

➙ P (∆) polynomial w.r.t. ζi

➙ Quadratic-LFT PDLF: P (∆) =[

I ∆TC

]P

[I

∆C

]: ∆C = (I−∆D∆∆)−1∆C∆

➙ P (∆) polynomial w.r.t. A(∆)





∃ P : LΣ(P ) < O → ∀∆ ∈ ∆∆ , ∃ P (∆) : LΣ(∆)(P (∆)) < O


➥ Choice of P (∆) for having a finite number of decision variables :



[i]

➙ P (∆) polynomial w.r.t. ∆ coefficients

➙ Quadratic-LFT PDLF: P (∆) =[

I ∆TC

]P

[I

∆C

]: ∆C = (I−∆D∆∆)−1∆C∆

➙ P (∆) polynomial w.r.t. A(∆)

➥ LMIs over infinite number of variables

∀∆ ∈ ∆∆ , ∃ P (∆) : LΣ(∆)(P (∆)) < O

⇐ ∃ P [i] : ∀∆ ∈ ∆∆ , LΣ(∆)(P (∆)) < O



IIIc - Conservative LMIs for polytopic models

Example : stability of x = A(∆)x with A(∆) =∑ζiA

[i] : ζi ≥ 0 ,∑ζi = 1





[i] : ζi ≥ 0 ,∑ζi = 1


V (x) < 0 ⇔ AT (∆)P + PA(∆) < O ⇔ A[i]TP + PA[i] < O





[i] : ζi ≥ 0 ,∑ζi = 1


V (x) < 0 ⇔ AT (∆)P + PA(∆) < O ⇔ A[i]TP + PA[i] < O


[i] x

x

T O P (∆)

P (∆) O

x

x

< 0 :[A(∆) −I

] x

x

= 0

⇔ Finsler Lemma O P (∆)

P (∆) O

+G(∆)[A(∆) −I

]+

AT (∆)

−I

GT (∆) < O

⇐ G(∆) = G & convexity O P [i]

P [i] O

+G[A[i] −I

]+

A[i]T

−I

GT < O



IIId - Conservative LMIs for LFT models

Example : stability of x = Ax+B∆w∆ with w∆ = ∆z∆ = ∆C∆x+ ∆D∆∆w∆






V (x) =

x

w∆

∗ I O

A B∆

∗ O P

P O

I O

A B∆

x

w∆

< 0

:[

∆ −I] C∆ D∆∆

O I

x

w∆

= 0

⇔ Finsler Lemma

M∗A

O P

P O

MA < τM∗C

∆∗

−I

[∆ −I

]MC






V (x) =

x

w∆

∗ I O

A B∆

∗ O P

P O

I O

A B∆

x

w∆

< 0

:[

∆ −I] C∆ D∆∆

O I

x

w∆

= 0

⇔ Finsler Lemma

M∗A

O P

P O

MA < M∗CΘMC ≤ τM∗

C

∆∗

−I

[∆ −I

]MC

with[

I ∆∗]Θ

I

∆

≤ O






V (x) =

x

w∆

∗ I O

A B∆

∗ O P

P O

I O

A B∆

x

w∆

< 0

:[

∆ −I] C∆ D∆∆

O I

x

w∆

= 0

⇔ Finsler Lemma

M∗A

O P

P O

MA < M∗CΘMC ≤ τM∗

C

∆∗

−I

[∆ −I

]MC

with[

I ∆∗]Θ

I

∆

≤ O

➙ Quadratic-LFT PDLF - same methodology.



IIId -Conservative LMIs for LFT models

➞ LMI conservative choices of Quadratic separators Θ

✪ {X,Y, Z}−dissipative matrices ∆∆ = {∆ : X + Y∆ + ∆∗Y ∗ + ∆∗Z∆ ≤ O }

[I ∆∗

]Θ

I

∆

≤ O : ∀∆ ∈ ∆∆ ⇔ Θ ≤ τ

X Y

Y ∗ Z

, τ ≥ 0






[I ∆∗

]Θ

I

∆

≤ O : ∀∆ ∈ ∆∆ ⇔ Θ ≤ τ

X Y

Y ∗ Z

, τ ≥ 0

✪ {X,Y, Z}−dissipative repeated scalars ∆∆ = {∆ = δI : x+ yδ + δ∗y∗ + δ∗zδ ≤ 0 }

[I δ∗I

]Θ

I

δI

≤ O : ∀δ ∈ ∆∆ ⇐ Θ ≤

xQ yQ

y∗Q zQ

, Q ≥ O






[I ∆∗

]Θ

I

∆

≤ O : ∀∆ ∈ ∆∆ ⇔ Θ ≤ τ

X Y

Y ∗ Z

, τ ≥ 0

✪ {X,Y, Z}−dissipative repeated scalars ∆∆ = {∆ = δI : x+ yδ + δ∗y∗ + δ∗zδ ≤ 0 }

[I δ∗I

]Θ

I

δI

≤ O : ∀δ ∈ ∆∆ ⇐ Θ ≤

xQ yQ

y∗Q zQ

, Q ≥ O

✪ Polytopic uncertainties ∆∆ ={

∆ =∑ζi∆[i] : ζi ≥ 0 ,

∑ζi = 1

}[

I ∆[i]∗]Θ

I

∆[i]

≤ O ,[

O I]Θ

O

I

≥ O


Outline




zw

K

u y< γ

[2]Σ

Σ[1]Σ[N]

Σ(∆)zΣ

∆w z

K

yu< γ

w

∆ ∆



➞ SDP solvers and parsers ➾ YALMIP and all solvers





robust multi-objective control for linear systems elements

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