Robust Multi-Objective Control for Linear Systems

Transcription

1 Robust Multi-Objective Control for Linear Systems Elements of theory and ROMULOC toolbox Dimitri PEAUCELLE & Denis ARZELIER LAAS-CNRS, Toulouse, FRANCE Part of the OLOCEP project (includes GloptiPoly) Denis Arzelier & Didier Henrion & Jean Lasserre & Dimitri Peaucelle

2 General Features of RoMulOC I - Uncertain LTI systems and robust performances Robust control objectives: stability, transient response (pole location), perturbation rejection (H, H 2 and impulse-to-peak) Guaranteed for structured parametric uncertainties: extremal values (polytopic and interval uncertainties), bounded sets (norm-bounded, passive and dissipative uncertainties). II - Lyapunov based analysis All performances are recast in a Lyapunov framework Robustness is achieved with either Unique Lyapunov function of PDLF LMI results derived using Quadratic Separation and Slack Variables III - LMIs and convex polynomial-time optimization Semi-Definite Programming and LMIs (black box) SDP solvers and YALMIP parser (user can tune solvers) RoMulOC toolbox 1 CACSD 06, October 5th, 2006, Munich

3 Demo example - problem formulation 6-th order mechanical system (x R 3 ) ẍ(t) + M 1 ( M )D( D )ẋ(t) + M 1 ( M )Kx(t) = M 1 ( M )E( E )w(t) z(t) = C( C )x(t) + F w(t) where M( M ) = M 0 + M 1 M M 2 affine with M R in ellipsoid, D( D ) = D 0 + D 1 D D 2 affine with D R 2 2 norm-bounded T D D I, E( E ) = E 0 + E 1 E E 2 affine with E [ ] scalar in interval, { } C( C ) = C 0 + C 1 C C 2 affine with C R 2 2 in polytope co [1] C, [2] C, [3] C. Robust analysis Robust pole location in a sector Robust H norm of weighted w z transfer : T z/w (s, ) γ RoMulOC toolbox 2 CACSD 06, October 5th, 2006, Munich

4 Demo example - solved with RoMulOC >> sys=ssmodel( mechanical system ); >> sys.a = [ zeros(n), eye(n) ; -im0*d0, -im0*k ];... >> Dm = udiss( X, Y, Z, Inertia ); >> Dd = unb( 2, 2, 0.25, Damping ); >> De = uinter(-0.25, 0.25, Input ); >> Dc = upoly( Dcv, Output ); >> usys = ussmodel( sys, diag(dm, Dd, De, Dc) ); >> r1 = region( plane,0,asin(0.35)); >> pb1 = ctrpb([1 1]); >> pb1 = dstability( pb1, usys, r1 ); >> IsDstable = solvesdp( pb1 ); >> pb2 = ctrpb([1 1]); >> pb2 = hinfty( pb2, usinf ); >> HinfLyapUnique = solvesdp( pb2 ); >> pb3 = ctrpb([1 11]); >> pb3 = hinfty( pb3, usinf ); >> HinfPDLF = solvesdp( pb3 ); RoMulOC toolbox 3 CACSD 06, October 5th, 2006, Munich

5 I - Uncertain LTI systems and performances General Robust Multi-Objective Control Problem : errors in modeling, operating conditions, mass-production... : parametric uncertainty, assumed constant, belongs to a set. sx(t) = A( )x(t) + B w ( )w(t) + B u ( )u(t) z(t) = C z ( )x(t) + D zw ( )w(t) + D zu ( )u(t) y(t) = C y ( )x(t) + D yw ( )w(t) + D yu ( )u(t) Find controller K that fulfills robust specifications Π i defined for models Σ i ( i ) with i i. Σ1( ) K 1 F 1 K Σ 1(0) Σ 2 ( 2 ) F 2 K RoMulOC V1 Modeling tools ready for the global design problem Two types of models : Polytopic and LFT RoMulOC toolbox 4 CACSD 06, October 5th, 2006, Munich

6 I - Uncertain LTI systems and performances [2] Σ Σ( ) Polytopic models w u Σ [1] Σ [N] y z < γ K Affine polytopic models : convex hull of N vertices A( ) = ζ i A [i], B w ( ) = ζ i B [i] w... : ζ i 0, ζi = 1 RoMulOC toolbox 5 CACSD 06, October 5th, 2006, Munich

7 I - Uncertain LTI systems and performances [2] Σ Σ( ) Polytopic models w u Σ [1] Σ [N] y z < γ K Affine polytopic models : convex hull of N vertices A( ) = ζ i A [i], B w ( ) = ζ i B [i] w... : ζ i 0, ζi = 1 Parallelotopic models with N P axes A( ) = A [0] + ξ i A [i], B w ( ) = B w [0] polytope with N = 2 N P vertices Interval models with N I non equal coefficients A [1] A( ) A [2] : a [1] ij a ij( ) a [2] ij parallelotope with axes in the euclidian basis of matrices polytope with N = 2 N I vertices + ξ i B [i] w... : ξ i 1 RoMulOC toolbox 6 CACSD 06, October 5th, 2006, Munich

8 I - Uncertain LTI systems and performances w z LFT models w u Σ y z < γ K sx(t) = Ax(t) + B w (t) + B w w(t) + B u u(t) z (t) = C x(t) + D w (t) + D w w(t) + D u u(t) z(t) = C z x(t) + D z w (t) + D zw w(t) + D zu u(t) y(t) = C y x(t) + D y w (t) + D yw w(t) + D yu u(t) : w C q z C p Linear - Fractional Transformation: A( ) = A+B (I D ) 1 C, B w ( ) = B w +B (I D ) 1 D w... Any model rational in δ i parameters LFT (not unique) with diagonal =diag(δ 1, δ 1,..., δ 2,...). RoMulOC toolbox 7 CACSD 06, October 5th, 2006, Munich

9 I - Uncertain LTI systems and performances Uncertainty sets {X, Y, Z} dissipative matrices { C q w p z : X + Y + Y + Z O, X O, Z O } Norm-bounded uncertainties : ρ1 Positive real uncertainties : + O (eg. s) { ρ 2 I, O, I} dissipative {O, I, O} dissipative RoMulOC toolbox 8 CACSD 06, October 5th, 2006, Munich

10 I - Uncertain LTI systems and performances Uncertainty sets {X, Y, Z} dissipative matrices { C q w p z : X + Y + Y + Z O, X O, Z O } Norm-bounded uncertainties : ρ1 Positive real uncertainties : + O (eg. s) { ρ 2 I, O, I} dissipative {O, I, O} dissipative Polytopic uncertainties polytope N vertices { = ζi [i] : ζ i 0, ζi = 1 } Parallelotopic uncertainties { = [0] + ξ i [i] : ξ i 1 } N P axes polytope N = 2 N P Interval uncertainties N I coef. polytope N = 2 N I { [1] [2] : δ [1] ij δ ij δ [2] ij } RoMulOC toolbox 9 CACSD 06, October 5th, 2006, Munich

11 Demo example - solved with RoMulOC >> sys=ssmodel( mechanical system ); >> sys.a = [ zeros(n), eye(n) ; -im0*d0, -im0*k ];... >> Dm = udiss( X, Y, Z, Inertia ); >> Dd = unb( 2, 2, 0.25, Damping ); >> De = uinter(-0.25, 0.25, Input ); >> Dc = upoly( Dcv, Output ); >> usys = ussmodel( sys, diag(dm, Dd, De, Dc) ); >> r1 = region( plane,0,asin(0.35)); >> pb1 = ctrpb([1 1]); >> pb1 = dstability( pb1, usys, r1 ); >> IsDstable = solvesdp( pb1 ); >> pb2 = ctrpb([1 1]); >> pb2 = hinfty( pb2, usinf ); >> HinfLyapUnique = solvesdp( pb2 ); >> pb3 = ctrpb([1 11]); >> pb3 = hinfty( pb3, usinf ); >> HinfPDLF = solvesdp( pb3 ); RoMulOC toolbox 10 CACSD 06, October 5th, 2006, Munich

12 II - Lyapunov based analysis Nominal performance analysis V (x) = x T P x Lyapunov function (P > O) Stability A T P + P A < O A T P A P < O [ ] D-Stability I A r 11P r 12 P I < O r12p r 22 P A H norm AT P + P A + Cz T C z P B w + Cz T D zw < O BwP T + DzwC T z γ 2 I + DzwD T zw H 2 norm A T P + P A + C T z C z < O trace(b T wp B w ) < γ 2 Impulsion-to-peak A T P + P A < O C T z C z < P B T wp B w < γ 2 I D T zwd zw < γ 2 I RoMulOC toolbox 11 CACSD 06, October 5th, 2006, Munich

13 II - Lyapunov based analysis Robust performance analysis V (x, ) parameter-dependent Lyapunov function. Nominal analysis (LMI) Robust analysis (NP-hard) P : L Σ (P ) < O, P ( ) : L Σ( ) (P ( )) < O Test over sample values in gives optimistic results. RoMulOC toolbox 12 CACSD 06, October 5th, 2006, Munich

14 II - Lyapunov based analysis Robust performance analysis V (x, ) parameter-dependent Lyapunov function. Nominal analysis (LMI) Robust analysis (NP-hard) P : L Σ (P ) < O, P ( ) : L Σ( ) (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 ( ) = ζ i P [i] Quadratic-LFT PDLF: [ P ( ) = I T C ] P I C, C = (I D ) 1 C RoMulOC toolbox 13 CACSD 06, October 5th, 2006, Munich

15 II - Lyapunov based analysis Conservative LMIs for polytopic models : vertex tests (with slack variables) Example : stability of ẋ = A( )x with A( ) = ζ i A [i] : ζ i 0, ζi = 1 Quadratic Stability : P ( ) = P A [i]t P + P A [i] < O Polytopic PDLF: P ( ) = ζ i P [i] O P [i] [ + G P [i] O A [i] [SCL00] I ] + A[i]T I G T < O RoMulOC toolbox 14 CACSD 06, October 5th, 2006, Munich

16 II - Lyapunov based analysis Conservative LMIs for LFT models : Quadratic Separation [Iwa97], [Sch97] Example : stability of ẋ = Ax + B w with w = z = C x + D w Quadratic Stability : P ( ) = P M A = I O A M A O P P O B M A < MCΘM C, M C = C O with [ D I ] I Θ I O Quadratic-LFT PDLF - same methodology with [ ] I I k Θ I I k O RoMulOC toolbox 15 CACSD 06, October 5th, 2006, Munich

17 III - Conservative LMI results Conservative quadratic separators {X, Y, Z} dissipative matrices = { : X + Y + Y + Z O } S-procedure, D-scaling and DG-scaling Θ = D X D Y + G I D Y G I D Z, D > 0, G = G T Polytopic uncertainties = Vertex separator [ { = ζ i diag( [i] k ) : ζ i 0, ] I [i] Θ I [i] O, Θ 22kk O ζi = 1 } RoMulOC toolbox 16 CACSD 06, October 5th, 2006, Munich

18 Demo example - solved with RoMulOC >> sys=ssmodel( mechanical system ); >> sys.a = [ zeros(n), eye(n) ; -im0*d0, -im0*k ];... >> Dm = udiss( X, Y, Z, Inertia ); >> Dd = unb( 2, 2, 0.25, Damping ); >> De = uinter(-0.25, 0.25, Input ); >> Dc = upoly( Dcv, Output ); >> usys = ussmodel( sys, diag(dm, Dd, De, Dc) ); >> r1 = region( plane,0,asin(0.35)); >> pb1 = ctrpb([1 1]); >> pb1 = dstability( pb1, usys, r1 ); >> IsDstable = solvesdp( pb1 ); >> pb2 = ctrpb([1 1]); >> pb2 = hinfty( pb2, usinf ); >> HinfLyapUnique = solvesdp( pb2 ); >> pb3 = ctrpb([1 11]); >> pb3 = hinfty( pb3, usinf ); >> HinfPDLF = solvesdp( pb3 ); RoMulOC toolbox 17 CACSD 06, October 5th, 2006, Munich

19 III - LMIs and convex polynomial-time optimization Semi-Definite Programming and LMIs The LMI problem is defined in YALMIP format Can be solved with any available SDP software result = solvesdp( problem, sdpsettings( solver, sdpt3 )) result is the answer to the problem, e.g. the guaranteed H norm Constraints and variables are also available problem.vars problem.lmi Results for the demo example with SeDuMi solver Nominal 200 rand. val rand. val. Robust PDLF Unique Lyapunov H ?? time 0.036s 13.79s 2 min NP 17.99s 6.772s RoMulOC toolbox 18 CACSD 06, October 5th, 2006, Munich

20 Conclusions and Prospective work Test yourself No need to know perfectly the theoretical results - no need to code the LMIs joloef/yalmip.php Future versions State-feedback design (October 2006) Output-feedback : full-order (LMI) and SOF (BMI) LFT uncertain Descriptor Systems Uncertain Time-Delay Systems (F. Gouaisbaut) Discrete-time uncertain Periodic Systems (C. Farges) Contribute PPDLF based on Polya [Oliveira & Peres, CDC, 2005] yours? RoMulOC toolbox 19 CACSD 06, October 5th, 2006, Munich