CNRS - RAS cooperation Seminar - 18th June, 2004 - IPME, St. Petersburg Quadratic Separation for Robustness and Design Dimitri Peaucelle
What is LAAS-CNRS? 1 French National Center for Scientific Research. ➥ Public basic-research org. producing knowledge and making it available to society. ➥ 26,000 employees (11,600 researchers). ➥ 1,260 units, spread throughout the country, cover all fields of research. Analysis and Architecture of Systems ➥ Part of CNRS - STIC department (Science and Technology for Information and Communication) ➥ 500 employees (200 researchers) ➥ 12 research groups ➥ Control Theory, Robotics, Micro and Nano-Systems, Computer science ➥ In Toulouse, France. Quadratic Separation for Robustness and Design
Methods and Algorithms in Control group 2 MAC group http://www.laas.fr/MAC ❏ Research fields : Robust control & Non-linear control ❏ Application fields : Aeronautics & Space industry & Environment Research in robust control ❏ MIMO LTI systems with parametric uncertainty ❏ State-space modeling and Lyapunov theory ❏ Stability and performance ( H ∞ , H 2 , pole location, impulse to peak) ❏ Analysis & Control design (state-feedback, full-order and static output-feedback) ❏ LMI based results & Semi-definite programming ❏ Robust MULti-Objective Control toolbox (V1 in September) http://www.laas.fr/OLOCEP Quadratic Separation for Robustness and Design
Outline 3 Quadratic separation for LTI systems Examples of results for robustness and design ➙ Preliminaries and notations ➙ Robust analysis and losslessness of quadratic separators ➙ Quadratic separation and control design Quadratic Separation for Robustness and Design
✞ ✁ ☎ ✆ ✁ ✄ ✄ ☎ ✂ ✁ ✄ ✁ ✆ ✁ ✄ ✡ � ✝ ✁ ✞ ✄ Methodology and notations 4 Uncertain model ❏ Engineering problem modeled as uncertain differential equations with constraints ❏ State-space LTI systems / parametric uncertainty / pole and induced norm constraints Optimization problem ❏ At best: lossless formulation with a global polynomial-time algorithm ❏ Conveniently: Conservative formulation with a global polynomial-time algorithm ❏ Usually: Conservative with sub-optimal heuristic algorithm n 6 5 ❏ LMI formulated results ➾ convex SDP & algorithms ❏ YALMIP interface in Matlab & Solvers: SeDuMi, SDPT3, CSDP,... g ∆ ∆ v x ˙ A x B v min γ Θ γ : P ∆ ∆ v C x D v ✄✠✟ ∞ Quadratic Separation for Robustness and Design
✘ ✘ ✄ ☎ ☛ ☞ ✌ ✄ ☎ ✙ ✁ ✁ ✁ ✁ ✞ ✑ ✄ ☎ ✄ ✍ ✎ ✏ ✁ ☎ ✒ ✄ ✁ ✌ ✁ ☞ ✄ ☎ ☛ ☞ ✌ ☛ ☎ ✄ ✏ ✔ ✁ ✁ ✞ ✓ ✄ ☎ ✄ ✍ ✎ ✁ Topological Separation 5 Graph of Σ 1 and inverse graph of Σ 2 : z z Σ 1 : Σ 1 Σ 2 : Σ 2 I x z w 0 x w z 0 w w Stability result [Safonov]: Σ 1 z The interconnected system is stable w Σ 2 Σ 1 Σ 2 I ❏ iff 0 Σ 1 d x 0 x ❏ iff ✞✗✖ : ✄✠✕ d Σ 2 I d x 0 x ✞✗✖ d : topological separator (see also “supply rate” in dissipative theory [Willems]) Quadratic Separation for Robustness and Design
✭ ✄ ✧ ✱ ✰ ✥ ✦ ✯ ✧ ★ ✚ ✮ ✚ ✌ ✭ ✬ ✦ ☎ ✭ ✭ ✔ ✭ ✭ ☛ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ★ ✥ ✭ ✛ ☎ ✬ ✁ ✙ ✄ ☎ ✳ ✚ ✲ ★ ☎ ✱ ✚ ✘ ✜ ✲ ✰ ✢ ✯ ✚ ✜ ★ ✢ ✮ ✣ ✚ ☎ ✬ ✕ ✳ ☞ Quadratic Separation 6 Quadratic function for the topological separator: Θ x Θ Θ m p m p d x x Lossless for linear systems ✣✠✤ E.g. for matrix gains Σ 1 p m w z The interconnected system is well-posed Σ 2 m p Σ 1 Σ 2 ❏ iff det 0 ✁✩★✫✪ Σ 1 Θ Σ w w 0 w 0 1 ✞✗✖ Θ ❏ iff : Θ Σ 0 0 z z z 2 Σ 2 ✞✗✖ Quadratic Separation for Robustness and Design
✭ ✭ ✴ ✳ ✔ ✲ ★ ☛ ✭ ✭ ✭ ✭ ✭ ✭ ✕ ☞ ✱ ✭ ✭ ✭ ✭ ✭ ✭ ✌ ✮ ✰ ✚ ✘ ✡ ★ ✘ ✴ ✘ ✖ ✞ ✥ ✦ ✡ ✧ ✙ ✳ ✲ ✴ ✮ ✵ ✛ ★ ✤ ✱ ✰ ✯ ✚ ★ ✄ ✬ ☎ ✯ One interconnected operator is uncertain 7 Robust analysis Σ 1 p m is robustly well-posed for all Σ 2 w m p z Σ 2 Σ 1 Σ 2 Σ 2 ❏ iff det 0 ✁✩★✫✪ ✞✗✖ Σ 1 Θ Σ 1 Θ ❏ iff : Θ Σ 2 Σ 2 Σ 2 ➥ Quadratic separation results are LMI-based. ➥ Is it possible to handle the infinite-dimensional constraint without conservatism? Quadratic Separation for Robustness and Design
★ ✲ ✷ ✡ ★ ✮ ✡ ✕ ✳ ★ ✪ ✲ ✱ ✰ ✳ ✯ ★ ✕ ✮ ✚ ✯ ✭ ✙ ✛ ✘ ☎ ✷ ✰ ✖ ✞ ✡ ✳ ✪ ✲ ★ ✱ ✷ ✡ ★ ✱ ✰ ✌ ✭ ✔ ✪ ✱ ✸ ★ ✲ ✳ ✸ ☎ ★ ✷ ✷ ✪ ✕ ✤ ✡ ✶ ✘ ✛ ✢ ✰ ✯ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✘ ☛ ✡ ✞ ✔ ✮ ★ ✢ ✛ ✢ One interconnected operator is uncertain 8 Example: Lyapunov matrix w x z x ˙ n n A 1 is stable (interconnection well-posed for all s ) 1 s A Θ A T Θ ❏ iff : Θ 1 s s 1 s P A Θ Θ A T P ❏ iff A T : P PA P ➥ Lossless quadratic separator. ➥ Other sets than ➾ pole location. Quadratic Separation for Robustness and Design
✡ ✡ ✡ ✡ ★ ✱ ✪ ✡ ★ ✰ ✪ ✡ ✡ ✱ ✹ ✹ ✹ ✪ ✡ ✹ ✺ ✕ ✮ ✚ ✳ ✺ ✺ ✺ ✺ ✯ ✲ ★ ★ ✡ ✡ ✡ ✡ ✹ ✰ ✕ ✪ ✆ ✷ ✆ ✱ ✰ ✳ ✲ ★ ✡ ✆ ✲ ✱ ✰ ✳ ✕ ✡ ★ ✡ ☎ ✞ ✲ ✳ ☎ ✪ ★ ✙ ✰ ✢ ✆ ✛ ✘ ✱ ✷ ✆ ✳ ✲ ✡ One interconnected operator is uncertain 9 Example: bounded real lemma ∆ T ∆ 1 is robustly stable ( s ) A B ❏ iff there exists a separator such as: C D P w z P τ 1 Θ s , P τ ∆ 0 τ that satisfies the LMI constraint: Σ 1 τ C T C τ C T D A T P PA PB Θ Σ 1 τ D T C τ τ D T D B T P ➥ Lossless quadratic separator. Quadratic Separation for Robustness and Design
✚ ✡ ☎ ✻ ✚ ✙ ✲ ✳ ✞ ✆ ✕ ✆ ★ ☎ ☎ ★ ✱ ✕ ✆ ✞ ✆ ✳ ✚ ✲ ✚ ✆ ✚ ✱ ✰ ✙ ✰ ✰ ✻ ✲ ✚ ✪ ☎ ✮ ★ ✡ ✚ ✕ ✯ ✰ ✱ ✞ ✳ ✚ ✱ ✚ ✲ ✳ ✰ ✱ ★ ✆ ✲ ✳ ✙ ✡ ✻ ✆ ☎ ☎ Conservative and lossless separators 10 Lossless quadratic separators ❏ Full-block dissipative ( -procedure) X Y X Y Θ τ τ ∆ 0 D ∆ D Y Z Y Z ❏ Disk located, repeated, complex valued scalar ∆ δ c α P β P Θ P β γ P α δ c β δ c β δ c δ c γ P 0 ❏ Bounded, repeated, real valued scalar ∆ δ r α P β P P Q Θ β P γ P α 2 δ r β δ 2 r γ Q 0 Q Q Quadratic Separation for Robustness and Design
✞ ✞ ✞ ✽ ✿ ✽ ✾ ✽ ☎ ✄ ✄ ✁ ✽ ✁ ✽ ✲ ✽ ✁ ✳ ☎ ✁ ✽ ✡ ✕ ☎ ✞ ✳ ✰ ✡ ✙ ✳ ✲ ✿ ✞ ✾ ✽ ★ ✱ ❀ ✄ ✯ ✿ ✽ ✾ ✡ ★ ✮ ✝ ✽ ❀ ✆ ✲ ✄ ✚ ✲ ✽ ✚ ✞ ✽ ✽ ✱ ✰ ✯ ✽ ✄ ✰ ★ ✮ ✝ ✼ ✽ ★ ☎ ☎ ✰ ✱ ✳ ✱ ✽ ✼ ✼ ✄ ✚ ✆ ✼ ✞ ✙ ✼ ✁ ✄ ☎ ★ ✱ ✰ ☎ ✽ ✝ ✡ ✙ ✳ ✲ ✁ ✁ Conservative and lossless separators 11 Conservative quadratic separators for block diagonal uncertainty ❏ Repeated full-block dissipative ∆ ∆ D T X T Y r Θ X Y T Y T Z ∆ D ∆ D Y Z T ❏ Block diagonal polytopic ∆ ∆ 1 ∆ r Θ ∆ i diag ∆ ✞✠✽ i ∆ ∑ζ i ∆ ∑ζ i ζ i i : 1 0 Θ 22 ii ❏ Any block diagonal structure of previous types (lossless if 2 m r m c m D 3) Θ 11 Θ 12 diag diag ∆ ∆ 1 Θ 1 1 diag Θ 21 Θ 22 diag diag 1 1 Quadratic Separation for Robustness and Design
✌ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✳ ✭ ✲ ✮ ★ ✚ ✄ ✁ ✁ ✄ ★ ✯ ✱ ✁ ☛ ✄ ✄ ✰ ✴ ✘ ✡ ✙ ❁ ✳ ❂ ✲ ★ ✖ ✱ ✘ ✴ ✄ ✕ ✁ ✘ ✴ ✯ ✚ ✖ ★ ✘ ✴ ✮ ✔ ✡ ✁ ✰ Both interconnected operators are uncertain 12 Robust analysis: parameter-dependent separators Σ 1 ∆ 1 ∆ 1 ∆ 2 w is robustly well-posed for all z 1 2 ✞✗✖ ∆ 2 Σ 1 ∆ 1 Θ ∆ 1 Σ ∆ 1 1 ∆ 1 Θ ∆ 1 ❏ iff : 1 Θ ∆ 1 ∆ 2 ∆ 2 2 ∆ 2 ✞✗✖ ➥ Infinite number of LMI variables & infinite number of constraints Quadratic Separation for Robustness and Design
★ ✡ ✯ ✳ ✄ ✙ ✁ ✚ ✮ ✁ ✌ ✭ ✭ ✭ ✭ ✭ ✭ ✲ ★ ☞ ✳ ✚ ✄ ★ ✮ ✡ ✕ ✲ ✄ ★ ✄ ✰ ✁ ✱ ✱ ✰ ✭ ✭ ✁ ✽ ✘ ✽ ✞ ✢ ✶ ✘ ❂ ✞ ✽ ❁ ✞ ✓ ✵ ✶ ✢ ✴ ✒ ✭ ✄ ✭ ✭ ✭ ✭ ✭ ☛ ✞ ✖ ✖ ✁ ✘ ✔ ✢ ✶ ✘ ✴ ✯ Both interconnected operators are uncertain 13 Example: µ -analysis Σ j ω is robustly stable ( ω ∆ w ) z ∆ Σ j ω Θ j ω Σ j ω ω Θ j ω ❏ iff : Θ j ω ∆ ∆ ∆ ω 1 ω N ➥ An optimistic bound on µ can then be obtained by gridding . ➥ For each ω i , build finite dimensional LMIs. Quadratic Separation for Robustness and Design
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