SLIDE 1
Robust Multi-Objective Control for Linear Systems Elements of theory - - PowerPoint PPT Presentation
Robust Multi-Objective Control for Linear Systems Elements of theory - - PowerPoint PPT Presentation
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
SLIDE 2
SLIDE 3
I - Uncertain LTI systems and performances
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 R = −1 1 R = −2α cos ψ cos ψ − i sin ψ cos ψ + i sin ψ
ψ α
Seminar at IPME 2 March 21st, 2005, St. Petersburg
SLIDE 4
I - Uncertain LTI systems and performances
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| ≤ γ2w
Impulse-to-peak performance : max |z| ≤ γi2p for w = δ. Design an LTI control u = Ky s.t. given specification γ is fulfilled
- r that minimizes γ
(or for given K analyse performance level) Seminar at IPME 3 March 21st, 2005, St. Petersburg
SLIDE 5
I - Uncertain LTI systems and performances
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 controller K that fulfills robust specifications Πi defined for models Σi(∆i) with ∆i ∈ ∆
∆i.
K Σ2 F2 (∆ )
2
K
1
(∆ )
1
K
1
Σ (0)
1
F Σ
Seminar at IPME 4 March 21st, 2005, St. Petersburg
SLIDE 6
I - Uncertain LTI systems and performances
Ic - Structured uncertainties
➞ Polytopic models
z w 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
Seminar at IPME 5 March 21st, 2005, St. Petersburg
SLIDE 7
I - Uncertain LTI systems and performances
Ic - Structured uncertainties
➞ Polytopic models
z w 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
Seminar at IPME 6 March 21st, 2005, St. Petersburg
SLIDE 8
I - Uncertain LTI systems and performances
Ic - Structured uncertainties
➞ LFT models
z
Σ ∆
w z K y u < 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, ...). Seminar at IPME 7 March 21st, 2005, St. Petersburg
SLIDE 9
I - Uncertain LTI systems and performances
Ic - Structured uncertainties
➞ 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
Seminar at IPME 8 March 21st, 2005, St. Petersburg
SLIDE 10
I - Uncertain LTI systems and performances
Ic - Structured uncertainties
➞ 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. = ➾
polytope N = 2NI
- ∆[1] ∆ ∆[2] : δ[1]
ij ≤ δij ≤ δ[2] ij
- Seminar at IPME
9 March 21st, 2005, St. Petersburg
SLIDE 11
Outline
I - Uncertain LTI systems and performances
➞ Control objectives: stability, transient response, perturbation rejection... ➞ Structured parametric uncertainties: extremal values, bounded sets...
z w K u y < γ
[2]
Σ Σ
[1]
Σ
[N]
Σ(∆) z
Σ ∆
w z K y u < γ w
∆ ∆
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 10 March 21st, 2005, St. Petersburg
SLIDE 12
II - LMIs and convex polynomial-time optimization
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/ǫ))). 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 , AT P + PA + CT
z Cz
BwP + CT
z Dzw
PBT
w + DT zwCz
−γI + DT
zwDzw
≤ O
Seminar at IPME 11 March 21st, 2005, St. Petersburg
SLIDE 13
II - LMIs and convex polynomial-time optimization
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. Seminar at IPME 12 March 21st, 2005, St. Petersburg
SLIDE 14
II - LMIs and convex polynomial-time optimization
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)
Seminar at IPME 13 March 21st, 2005, St. Petersburg
SLIDE 15
II - LMIs and convex polynomial-time optimization
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)
Seminar at IPME 14 March 21st, 2005, St. Petersburg
SLIDE 16
II - LMIs and convex polynomial-time optimization
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.
Seminar at IPME 15 March 21st, 2005, St. Petersburg
SLIDE 17
II - LMIs and convex polynomial-time optimization
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. ✪ Develop software for ”industrial” application / adapted to the application field ➾ ROMULOC toolbox
Seminar at IPME 16 March 21st, 2005, St. Petersburg
SLIDE 18
Outline
I - Uncertain LTI systems and performances
➞ Control objectives: stability, transient response, perturbation rejection... ➞ Structured parametric uncertainties: extremal values, bounded sets...
z w K u y < γ
[2]
Σ Σ
[1]
Σ
[N]
Σ(∆) z
Σ ∆
w z K y u < γ w
∆ ∆
II - LMIs and convex polynomial-time optimization
➞ Semi-Definite Programming and LMIs ➞ SDP solvers and parsers ➾ YALMIP and all solvers
III - Conservative LMI results
➞ Methods: Lyapunov, S-procedure, Finsler lemma, Topologic Separation... ➞ The ROMULOC toolbox
Seminar at IPME 17 March 21st, 2005, St. Petersburg
SLIDE 19
III - Conservative LMI results
IIIa - Nominal performance analysis V (x) = xT Px Lyapunov function (P > O)
✪ Stability AT P + PA < O | AT PA − P < O ✪ D-Stability
- I
A∗
-
r11P r12P r∗
12P
r22P I A < O ✪ H∞ norm AT P + PA + CT
z Cz
PBw + CT
z Dzw
BT
wP + DT zwCz
−γ2I + DT
zwDzw
< O ✪ H2 norm AT P + PA + CT
z Cz < O
trace(BT
wPBw) < γ2
✪ Impulsion-to-peak AT P + PA < O BT
wPBw < γ2I
CT
z Cz < P
DT
zwDzw < γ2I
Seminar at IPME 18 March 21st, 2005, St. Petersburg
SLIDE 20
III - Conservative LMI results
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.
Seminar at IPME 19 March 21st, 2005, St. Petersburg
SLIDE 21
III - Conservative LMI results
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. ➥ 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
∆T
C
- P
- I
∆C
- : ∆C = (I − ∆D∆∆)−1∆C∆
➙ P(∆) polynomial w.r.t. A(∆)
Seminar at IPME 20 March 21st, 2005, St. Petersburg
SLIDE 22
III - Conservative LMI results
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. ➥ 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. ∆ coefficients ➙ Quadratic-LFT PDLF: P(∆) =
- I
∆T
C
- 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
Seminar at IPME 21 March 21st, 2005, St. Petersburg
SLIDE 23
III - Conservative LMI results
IIIc - Conservative LMIs for polytopic models Example : stability of ˙
x = A(∆)x with A(∆) = ζiA[i] : ζi ≥ 0 , ζi = 1
Seminar at IPME 22 March 21st, 2005, St. Petersburg
SLIDE 24
III - Conservative LMI results
IIIc - Conservative LMIs for polytopic models Example : stability of ˙
x = A(∆)x with A(∆) = ζiA[i] : ζi ≥ 0 , ζi = 1 ➙ “Quadratic Stability”: P(∆) = P ˙ V (x) < 0 ⇔ AT (∆)P + PA(∆) < O ⇔ A[i]T P + PA[i] < O
Seminar at IPME 23 March 21st, 2005, St. Petersburg
SLIDE 25
III - Conservative LMI results
IIIc - Conservative LMIs for polytopic models Example : stability of ˙
x = A(∆)x with A(∆) = ζiA[i] : ζi ≥ 0 , ζi = 1 ➙ “Quadratic Stability”: P(∆) = P ˙ V (x) < 0 ⇔ AT (∆)P + PA(∆) < O ⇔ A[i]T P + PA[i] < O ➙ Polytopic PDLF: P(∆) = ζiP [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
Seminar at IPME 24 March 21st, 2005, St. Petersburg
SLIDE 26
III - Conservative LMI results
IIId - Conservative LMIs for LFT models Example : stability of ˙
x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆
Seminar at IPME 25 March 21st, 2005, St. Petersburg
SLIDE 27
III - Conservative LMI results
IIId - Conservative LMIs for LFT models Example : stability of ˙
x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆ ➙ “Quadratic Stability”: P(∆) = P ˙ 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
Seminar at IPME 26 March 21st, 2005, St. Petersburg
SLIDE 28
III - Conservative LMI results
IIId - Conservative LMIs for LFT models Example : stability of ˙
x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆ ➙ “Quadratic Stability”: P(∆) = P ˙ 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
Seminar at IPME 27 March 21st, 2005, St. Petersburg
SLIDE 29
III - Conservative LMI results
IIId - Conservative LMIs for LFT models Example : stability of ˙
x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆ ➙ “Quadratic Stability”: P(∆) = P ˙ 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.
Seminar at IPME 28 March 21st, 2005, St. Petersburg
SLIDE 30
III - Conservative LMI results
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
Seminar at IPME 29 March 21st, 2005, St. Petersburg
SLIDE 31
III - Conservative LMI results
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 ✪ {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
Seminar at IPME 30 March 21st, 2005, St. Petersburg
SLIDE 32
III - Conservative LMI results
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 ✪ {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
Seminar at IPME 31 March 21st, 2005, St. Petersburg
SLIDE 33
Outline
I - Uncertain LTI systems and performances
➞ Control objectives: stability, transient response, perturbation rejection... ➞ Structured parametric uncertainties: extremal values, bounded sets...
z w K u y < γ
[2]
Σ Σ
[1]
Σ
[N]
Σ(∆) z
Σ ∆
w z K y u < γ w
∆ ∆