Robust Multi-Objective Control for Linear Systems Elements of theory and ROMULOC toolbox Dimitri PEAUCELLE & Denis ARZELIER LAAS-CNRS, Toulouse, FRANCE with the participation of Master’s Students Maud SEVIN & Alberto BORTOTT
General Features of RoMulOC ➞ Matlab Toolbox - freely distributed www/laas.fr/OLOCEP/romuloc I - Modeling features of uncertain LTI systems ➞ State-space systems with performance input/output channel ➞ Both polytopic and LFT uncertain systems - large variety of uncertainty models ➞ Basic model manipulations II - Robust performances in Lyapunov framework ➞ Robust control objectives: stability, transient response (pole location), perturbation rejection ( H ∞ , H 2 and impulse-to-peak) ➞ All performances are recast in a Lyapunov framework ➞ Robustness is achieved with either Unique Lyapunov function or 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 WCFCL ’08, September 26th, 2008, Sienna
Demo example - problem formulation 6-th order mechanical system ( x ∈ R 3 ) x ( t ) + M − 1 (∆ M ) D (∆ D ) ˙ x ( t ) + M − 1 (∆ M ) Kx ( t ) = M − 1 (∆ M ) E (∆ E ) w ( t ) ¨ z ( t ) = C (∆ C ) x ( t ) + Fw ( t ) where 0.35 0.3 0.25 0.2 0.15 M (∆ M ) = M 0 + M 1 ∆ M M 2 with ∆ M ∈ R 1 × 2 in ellipsoid 0.1 0.05 0 � 0.05 � 0.1 � 0.15 � 0.35 � 0.3 � 0.25 � 0.2 � 0.15 � 0.1 � 0.05 0 0.05 0.1 0.15 D (∆ D ) = D 0 + D 1 ∆ D D 2 with ∆ D ∈ R 2 × 2 norm-bounded ∆ T D ∆ D ≤ 0 . 25 2 I E (∆ E ) = E 0 + E 1 ∆ E E 2 with ∆ E ∈ [ − 0 . 25 0 . 25] scalar in interval � � ∆ [1] C , ∆ [2] C , ∆ [3] C (∆ C ) = C 0 + C 1 ∆ C C 2 with ∆ C ∈ R 2 × 2 in polytope co C Robust analysis ➞ Robust pole location in a sector (robust bound on damping of all modes) ➞ Robust H ∞ norm of w → z transfer : � T z/w ( s, ∆) � ∞ ≤ γ . RoMulOC toolbox 2 WCFCL ’08, September 26th, 2008, Sienna
Demo example - solved with RoMulOC >> sys=ssmodel(’mechanical system’); >> sys.A = [ zeros(n) , eye(n) ; -iM0*D0 , -iM0*K ]; ... >> sys.Bw = [ zeros(n) ; iM0*E0 ]; >> 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 = dstability( usys, r1 ); >> pb1 = pb1 + ctrpb( ’analysis’, ’Lyap unique’ ); >> IsDstable = solvesdp( pb1 ); >> pb2 = hinfty( pb2, usinf ); >> pb2unique = pb2 + ctrpb( ’analysis’, ’Lyap unique’ ); >> HinfLyapUnique = solvesdp( pb2unique ); >> pb2PDLF = pb2 + ctrpb( ’analysis’ , ’PDLF’ ); >> HinfPDLF = solvesdp( pb2PDLF ); RoMulOC toolbox 3 WCFCL ’08, September 26th, 2008, Sienna
I - Uncertain LTI systems and performances General Robust Multi-Objective Control Problem ∆ : errors in modeling, operating conditions, mass-production... ∆ : uncertainty 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 . F F 2 Σ ( Δ ) Σ (0) Σ 2 ( Δ ) 1 1 1 1 2 K K K RoMulOC today ➞ Modeling tools ready for the global design problem (uncertainties restricted to be constant) ➞ Analysis : Unique Lyapunov function and a PDLF method ➞ Control design : only for unique Lyapunov function RoMulOC toolbox 4 WCFCL ’08, September 26th, 2008, Sienna
I - Uncertain LTI systems and performances [2] Σ Σ ( Δ ) ➞ Polytopic models z w [1] [N] Σ < γ Σ u y K ✪ Affine polytopic models : convex hull of N vertices A (∆) = � ζ i A [ i ] , B w (∆) = � ζ i B [ i ] : ζ i ≥ 0 , � ζ i = 1 . . . w RoMulOC toolbox 5 WCFCL ’08, September 26th, 2008, Sienna
I - Uncertain LTI systems and performances [2] Σ Σ ( Δ ) ➞ Polytopic models z w [1] [N] Σ < γ Σ u y K ✪ Affine polytopic models : convex hull of N vertices A (∆) = � ζ i A [ i ] , B w (∆) = � ζ i B [ i ] : ζ i ≥ 0 , � ζ i = 1 . . . w ➥ Parallelotopic models with N P axes A (∆) = A [0] + � ξ i A [ i ] , B w (∆) = B [0] w + � ξ i B [ i ] . . . : | ξ i | ≤ 1 w ➾ 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 RoMulOC toolbox 6 WCFCL ’08, September 26th, 2008, Sienna
I - Uncertain LTI systems and performances Δ w z Δ Δ w z ➞ LFT models Σ < γ u y K sx ( t ) = Ax ( t ) + B ∆ w ∆ ( t ) + B w w ( t ) + B u u ( t ) w ∆ ∈ C q ∆ z ∆ ( t ) = C ∆ x ( t ) + D ∆∆ w ∆ ( t ) + D ∆ w w ( t ) + D ∆ u u ( t ) : z ∆ ∈ C p ∆ 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 ) 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 WCFCL ’08, September 26th, 2008, Sienna
I - Uncertain LTI systems and performances ➞ Uncertainty sets ✪ { X, Y, Z }− dissipative matrices : X + Y ∆ + ∆ ∗ Y ∗ + ∆ ∗ Z ∆ ≤ O , X ≤ O , Z ≥ O ∆ ∈ C q w × p z � � ➾ {− ρ 2 I , O , I }− dissipative ➥ Norm-bounded uncertainties : � ∆ � ≤ ρ 1 ➥ Positive real uncertainties : ∆ + ∆ ∗ ≥ O (eg. s ) ➾ { O , − I , O }− dissipative RoMulOC toolbox 8 WCFCL ’08, September 26th, 2008, Sienna
I - Uncertain LTI systems and performances ➞ Uncertainty sets ✪ { X, Y, Z }− dissipative matrices : X + Y ∆ + ∆ ∗ Y ∗ + ∆ ∗ Z ∆ ≤ O , X ≤ O , Z ≥ O ∆ ∈ C q w × p z � � ➾ {− ρ 2 I , O , I }− dissipative ➥ Norm-bounded uncertainties : � ∆ � ≤ ρ 1 ➥ Positive real uncertainties : ∆ + ∆ ∗ ≥ O (eg. s ) ➾ { O , − I , O }− dissipative ✪ Polytopic uncertainties ∆ = � ζ i ∆ [ i ] : ζ i ≥ 0 , � ζ i = 1 � � polytope N vertices ➥ Parallelotopic uncertainties ∆ = ∆ [0] + � ξ i ∆ [ i ] : | ξ i | ≤ 1 � � polytope N = 2 N P N P axes ➾ ➥ Interval uncertainties � � ∆ [1] � ∆ � ∆ [2] : δ [1] ij ≤ δ ij ≤ δ [2] ij polytope N = 2 N I N I coef. � = ➾ RoMulOC toolbox 9 WCFCL ’08, September 26th, 2008, Sienna
Demo example - solved with RoMulOC >> sys=ssmodel(’mechanical system’); >> sys.A = [ zeros(n) , eye(n) ; -iM0*D0 , -iM0*K ]; ... >> sys.Bw = [ zeros(n) ; iM0*E0 ]; >> 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 = dstability( usys, r1 ); >> pb1 = pb1 + ctrpb( ’analysis’, ’Lyap unique’ ); >> IsDstable = solvesdp( pb1 ); >> pb2 = hinfty( pb2, usinf ); >> pb2unique = pb2 + ctrpb( ’analysis’, ’Lyap unique’ ); >> HinfLyapUnique = solvesdp( pb2unique ); >> pb2PDLF = pb2 + ctrpb( ’analysis’ , ’PDLF’ ); >> HinfPDLF = solvesdp( pb2PDLF ); RoMulOC toolbox 10 WCFCL ’08, September 26th, 2008, Sienna
II - Lyapunov based analysis Nominal performance analysis V ( x ) = x T Px Lyapunov function ( P > O ) A T P + PA < O A T PA − P < O | ✪ Stability r 11 P r 12 P I � � < O ✪ D-Stability A ∗ I r ∗ 12 P r 22 P A A T P + PA + C T PB w + C T z C z z D zw < O ✪ H ∞ norm B T w P + D T − γ 2 I + D T zw C z zw D zw A T P + PA + C T z C z < O ✪ H 2 norm trace ( B T w PB w ) < γ 2 A T P + PA < O B T w PB w < γ 2 I ✪ Impulsion-to-peak C T D T zw D zw < γ 2 I z C z < P RoMulOC toolbox 11 WCFCL ’08, September 26th, 2008, Sienna
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 WCFCL ’08, September 26th, 2008, Sienna
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