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Robust LFR-based technique for stability analysis of periodic solutions Dimitri PEAUCELLE & Christophe FARGES & Denis ARZELIER LAAS-CNRS - Universit e de Toulouse, FRANCE LAPS - Universit e de Bordeaux I, FRANCE

  1. Robust LFR-based technique for stability analysis of periodic solutions Dimitri PEAUCELLE † & Christophe FARGES ‡ & Denis ARZELIER † † LAAS-CNRS - Universit´ e de Toulouse, FRANCE ‡ LAPS - Universit´ e de Bordeaux I, FRANCE

  2. Introduction Robustness and Non-Linear problems ✪ Robustness: prove stability whatever uncertainties in bounded sets ✪ Lyapunov theory: prove stability whatever initial conditions in bounded sets ➙ Model non-linearities on the states as uncertainties on an LTI model Stability of periodic solutions and periodic systems ✪ Dynamics of relative motion: stability of the origin of periodic system ✪ Numerical methods for periodic systems: sampling ➙ Modeling of sampled uncertainties Outline ① Discrete-time Periodic Uncertain models for stability of periodic solutions ② Linear Matrix Inequality (LMI) results ③ Illustrative example 1 IFAC PSYCO’07, August 2007, St. Petersburg

  3. Linear-Fractional Representation (LFR) η ( t ) = f ( η ( t ) , t, δ ) ˙ Non-linear, uncertain, system with periodic solution: η s ( t ) = f ( η s ( t ) , t, δ ) , ∀ δ ∈ [ δ, δ ] : η s ( t + T ) = η s ( t ) ˙ Quasi-linearization around periodic solution ( x c = η s − η ) x c ( t ) = A c ( t ) x c ( t ) + B c δ ( t ) w δ ( t ) + B c ˙ Ω ( t ) w Ω ( t ) z c δ ( t ) = C c δ ( t ) x c ( t ) + D c δδ ( t ) w δ ( t ) + D c δ Ω ( t ) w Ω ( t ) z c Ω ( t ) = C c Ω ( t ) x c ( t ) + D c Ω δ ( t ) w δ ( t ) + D c ΩΩ ( t ) w Ω ( t ) where uncertainties and non-linearities are feedback connected δ w c δ ( t ) = δz c δ ( t ) Ω (x(t)) w c Ω ( t ) = Ω( x c ( t )) z c Ω ( t ) x(t) = A(t)x(t)... 2 IFAC PSYCO’07, August 2007, St. Petersburg

  4. LFR Hypothesis LMI results for robust stability need quadratic separators T      1  1 ∃ Θ : Υ Θ ( δ 1 ) = Θ  ≤ 0 , ∀ δ ∈ [ δ, δ ]      δ 1 δ 1 ➙ Let Θ the set of all Θ such that this holds. Bounded Ω( x ) for x bounded ✪ HYP 0: Exists a positive definite matrix Q and a matrix Ξ such that T     1 1 x T Qx ≤ γ ⇒ Υ Ξ (Ω( x )) = Ξ  ≤ 0        Ω( x ) Ω( x ) ➙ For given Q and γ , let Ξ γ the set of all Ξ such that HYP 0 holds. ➙ Based on HYP 0, non-linearities may be treated as if uncertainties. 3 IFAC PSYCO’07, August 2007, St. Petersburg

  5. Illustrative Example [Jordan & Smith 1987] η 1 + ( η 2 η 2 ¨ 1 + ˙ 1 − 1) ˙ η 1 + η 1 = u Non-linear system with uncertain control input u ( t ) = ( κ + δ )(cos t − η 1 ( t )) ⇒ η s 1 ( t ) = cos t x c 1 = η s 1 − η 1 , x c x c 2 = ˙ LFR around periodic solution 1     0 1 0  B c A c ( t ) = δ =    − 1 − κ + sin 2 t cos 2 t − 1 − 1   0 0 0 0 B c Ω ( t ) =   − sin t 2 cos t − 3 sin t − 1   � � C c D c D c x c δ = δδ = 0 δ Ω = 0 0 1 0 1    x c  0   2 Ω( x c ) =       x c 0  1 0   2 C c D c D c Ω = Ω δ = 0 ΩΩ = 0      x c 2 1 + x c 2 0 0 1 2 4 IFAC PSYCO’07, August 2007, St. Petersburg

  6. Sampling of the LFT model Periodic sampling strategy ✪ Sampling sequence { T s ( k ) } k ≥ 0 : T s ( N ) = T , T s ( k + N ) = T s ( k ) x ( k ) = x c ( T s ( k )) , w δ ( k ) = w c Ω( k ) = Ω( x c ( T s ( k ))) δ ( T s ( k )) . . . ✪ HYP 1: median approximation of LTI model A ( k ) = A c (1 ∀ t ∈ [ T s ( k ) , T s ( k +1)] : A c ( t ) ≃ ˜ 2( T s ( k )+ T s ( k +1))) . . . ✪ HYP 2: first-order hold approximation of exogenous signals [Imbert 2001] t − T s ( k ) w c δ ( t ) ≃ w δ ( k ) + T s ( k + 1) − T s ( k )( w δ ( k + 1) − w δ ( k )) . . . ✪ HYP 3: β -bounded growth of the state x over sampling period ( β ≥ 1 ) x T ( k ) Qx ( k ) ≤ q ⇒ x T ( k + 1) Qx ( k + 1) ≤ βq 5 IFAC PSYCO’07, August 2007, St. Petersburg

  7. Discrete-time LFR-model HYP 0-3 : T -periodic continuous ⇒ N -periodic discrete δ δ (t) Ω (k) Ω (k+1) Ω T (k) s x(t) = A(t)x(t)... x(k+1)=A(k)x(k)... x ( k + 1) = A ( k ) x ( k ) + B δ ( k ) ˜ w δ ( k ) + B Ω ( k ) ˜ w Ω ( k ) w δ ( k ) = δ ˜ ˜ z δ ( k ) z δ ( k ) = C δ ( k ) x ( k ) + D δδ ˜ ˜ w δ ( k ) + D δ Ω ( k ) ˜ w Ω ( k ) w ω ( k ) = ˜ ˜ Ω( k )˜ z Ω ( k ) z Ω ( k ) = C Ω ( k ) x ( k ) + D Ω δ ˜ ˜ w δ ( k ) + D ΩΩ ( k ) ˜ w Ω ( k )    with Υ Ξ ∈ Ξ q (Ω( k )) ≤ 0  Ω( k ) 0 ˜ if x T ( k ) Qx ( k ) ≤ q . Ω = 0 Ω( k + 1) Υ ˆ Ξ ∈ Ξ βq (Ω( k + 1)) ≤ 0 6 IFAC PSYCO’07, August 2007, St. Petersburg

  8. Invariant set - LMI results Invariant sets x T Qx ≤ q : If there exist Ξ( k ) ∈ Ξ q , ˆ Ξ( k ) ∈ Ξ βq , Θ( k ) ∈ Θ that satisfy for all k = 1 . . . N the LMIs N T δ ( k )Θ( k ) N δ ( k )    Q 0 N T x ( k )  N x ( k ) < + N T Ω ( k )Ξ( k ) N Ω ( k )   − Q 0 + ˆ Ω ( k )ˆ Ξ( k ) ˆ N T N Ω ( k ) where N x , N δ , N Ω and ˆ N Ω matrices are functions of model matrices A, B δ . . . then for any bounded initial conditions such that x T (0) Qx (0) ≤ q the trajectory remains bounded such that x T ( k ) Qx ( k ) ≤ q . 7 IFAC PSYCO’07, August 2007, St. Petersburg

  9. Asymptotic stability - LMI results Convergence to periodic solution for initial conditions x T ˆ Qx ≤ q : If there exist P ( k ) ≥ 0 , Ξ( k ) ∈ Ξ q , ˆ Ξ( k ) ∈ Ξ βq , Θ( k ) ∈ Θ that satisfy for all k = 1 . . . N the LMIs ˆ Q ( k ) = Q + P ( k ) > 0 N T δ ( k )Θ( k ) N δ ( k )   ˆ Q ( k ) 0 N T x ( k )  N x ( k ) < + N T Ω ( k )Ξ( k ) N Ω ( k )   − ˆ  Q ( k ) 0 + ˆ Ω ( k )ˆ Ξ( k ) ˆ N T N Ω ( k ) then for any bounded initial conditions such that x T (0) ˆ Q (0) x (0) ≤ q the trajectory satisfies invariance x T ( k ) Qx ( k ) ≤ x T ( k ) ˆ Q ( k ) x ( k ) ≤ q decreasing Lyapunov function x T ( k +1) ˆ Q ( k +1) x ( k +1) ≤ x T ( k ) ˆ Q ( k ) x ( k ) and convergence to the periodic solution x ( k ) = η s ( k ) − η ( k ) → 0 . 8 IFAC PSYCO’07, August 2007, St. Petersburg

  10. LMI conservative representations of separators Need for tools to choose Θ ∈ Θ and Ξ ∈ Ξ γ If Θ satisfies the LMIs Vertex separator: T      0  0  ≥ 0 , Υ Θ ( δ 1 ) ≤ 0 , Υ Θ ( δ 1 ) ≤ 0 Θ  1 1 then Θ ∈ Θ ( i.e. Υ Θ ( δ 1 ) ≤ 0 ∀ δ ∈ [ δ, δ ] ) LMIs conditions of Ξ for Υ Ξ (Ω( x )) ≤ 0 whatever x T Qx ≤ γ ?   x 1 0    0  x 2   with x T Qx = x T x = x 2 1 + x 2 Example: Ω( x ) = 2 .     0 x 2       x 2 1 + x 2 0 2 9 IFAC PSYCO’07, August 2007, St. Petersburg

  11. LMI conservative representations of separators Take Ξ structured as   − γ ( α 1 + α 4 ) − γ ( α 2 + α 5 ) 0 0 0 α 2    − γ ( α 2 + α 5 ) − γ (2 α 3 + α 6 + γα 7 ) 0 0 0  α 3       0 0 0 0 0 α 1   Ξ =     0 0 0 α 1 + α 4 α 5 0       0 0 0 α 5 α 6 0       0 0 0 α 2 α 3 α 7        0 0  α 1 α 2  α 4 α 5  ≥ 0 , Ξ 2 =  ≥ 0 , Ξ 3 =  ≥ 0 Ξ 1 = 2 α 3 0 α 2 α 5 α 6 α 7 then 2 ) 2 − γ 2 )Ξ 3 ≤ 0 Υ Ξ (Ω( x )) = ( x 2 1 + x 2 2 − γ )Ξ 1 + ( x 2 2 − γ )Ξ 2 + (( x 2 1 + x 2 for all x T Qx = x T x = x 2 1 + x 2 2 ≤ γ . 10 IFAC PSYCO’07, August 2007, St. Petersburg

  12. Results for the example η 1 + ( η 2 η 2 ¨ 1 + ˙ 1 − 1) ˙ η 1 + η 1 = u Non-linear system with uncertain control input u ( t ) = ( κ + δ )(cos t − η 1 ( t )) ⇒ η s 1 ( t ) = cos t Tests made for control gain κ = 4 . 5 and δ = − δ = 0 . 5 uncertainty. LMI tests LMIs solved for N = 20 samples per period (uniformly spaced), β = 1 . 5 (assumed bound on state growth over one sample) and various values of q : feasible for all q ≤ q max = 0 . 056 . When taking β = 1 . 1 then q max = 0 . 061 . Computation time (YALMIP+SeDuMi, 3GHz+1GB) ≃ 1 . 4 sec. 11 IFAC PSYCO’07, August 2007, St. Petersburg

  13. Simulations for η 1 (0) = 0 . 78 , 0 . 91 and 1 . 09 ! $"# ! % $ ! $"# ! ! ! !"# ! ! ! $"# $ $"# ! ! ! 12 IFAC PSYCO’07, August 2007, St. Petersburg

  14. Conclusions ➙ Methodology to analyze robust stability of periodic trajectories ➙ HYP 3 on bounded growth of the state embeds errors due to sampling ➚ LMI type results provide access to efficient numerical tools ➘ Results are conservative ➚ State-feedback design and others may be considered as well ➘ Difficulties to describe the sets Ξ via LMIs. 13 IFAC PSYCO’07, August 2007, St. Petersburg

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