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
I - Uncertain LTI systems and performances Ia - Control objectives ➞ Stability & D-stability x ∈ C n u ∈ C q u sx ( t ) = Ax ( t ) + B u u ( t ) : y ∈ C p y y ( t ) = C y x ( t ) + D yu u ( t ) Design an LTI control u = Ky s.t. poles belong to a region of the complex plane (or for given K analyse pole location): D R = { s ∈ C : r 11 + sr 12 + s ∗ r 12 + ss ∗ r 22 ≤ 0 } , R = ( r ij ) Such regions are half-planes and discs 0 1 − 1 0 − 2 α cos ψ cos ψ − i sin ψ R = R = R = 1 0 0 1 cos ψ + i sin ψ 0 ψ α Seminar at IPME 2 March 21st, 2005, St. Petersburg
I - Uncertain LTI systems and performances Ia - Control objectives ➞ H ∞ & H 2 & Impulse-to-peak performance sx ( t ) = Ax ( t ) + B w w ( t ) + B u u ( t ) w ∈ C q w : z ( t ) = C z x ( t ) + D zw w ( t ) + D zu u ( t ) z ∈ C p z y ( t ) = C y x ( t ) + D yw w ( t ) + D yu u ( t ) H ∞ performance : � z � ≤ γ ∞ � w � ; bounded-real lemma H 2 performance : � z � ≤ γ 2 for w g.w.n. ; max | z | ≤ γ 2 � w � Impulse-to-peak performance : max | z | ≤ γ i2p 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) Seminar at IPME 3 March 21st, 2005, St. Petersburg
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 ) + 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 . (∆ ) Σ (0) Σ 2 (∆ ) F 2 F Σ 1 1 1 1 2 K K K Seminar at IPME 4 March 21st, 2005, St. Petersburg
I - Uncertain LTI systems and performances Ic - Structured uncertainties [2] Σ Σ(∆) ➞ Polytopic models z w [1] Σ [N] Σ < γ 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 Seminar at IPME 5 March 21st, 2005, St. Petersburg
I - Uncertain LTI systems and performances Ic - Structured uncertainties [2] Σ Σ(∆) ➞ Polytopic models z w [1] Σ [N] Σ < γ 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 Seminar at IPME 6 March 21st, 2005, St. Petersburg
I - Uncertain LTI systems and performances Ic - Structured uncertainties ∆ w z ∆ ∆ w z Σ ➞ LFT models < 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 , ... ) . Seminar at IPME 7 March 21st, 2005, St. Petersburg
I - Uncertain LTI systems and performances Ic - Structured uncertainties ➞ 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 Seminar at IPME 8 March 21st, 2005, St. Petersburg
I - Uncertain LTI systems and performances Ic - Structured uncertainties ➞ 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. � = ➾ Seminar at IPME 9 March 21st, 2005, St. Petersburg
Outline I - Uncertain LTI systems and performances ➞ Control objectives: stability, transient response, perturbation rejection... ➞ Structured parametric uncertainties: extremal values, bounded sets... ∆ [2] Σ w z ∆ ∆ Σ(∆) w z Σ z w < γ [1] Σ [N] Σ < γ u y u y K K 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
II - LMIs and convex polynomial-time optimization IIa - Semi-Definite Programming and LMIs ✪ Extension of LP to semi-definite matrices min cx : Ax = b , x i ≥ 0 ( LP ) | mat ( x ) ≥ O ( SDP ) ➥ Convexity, duality, polynomial-time algorithms ( O ( n 6 . 5 log(1 /ǫ )) ). max b T y A T y − c T = z , : mat ( z ) ≥ O ✪ 1st developments and 1st results : LMI formalism & Control Theory � � min g i y i : F 0 + F i y i ≥ O ➥ The H ∞ norm computation example for G ( s ) ∼ ( A, B, C, D ) : A T P + PA + C T B w P + C T z C z z D zw � G ( s ) � 2 ≤ O = min γ : P > O , ∞ PB T w + D T − γ I + D T zw C z zw D zw Seminar at IPME 11 March 21st, 2005, St. Petersburg
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
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
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
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