A Global Optimization Approach to Structured Regulation Design under - PowerPoint PPT Presentation

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion A Global Optimization Approach to Structured Regulation Design under H ∞ Constraints Dominique Monnet, Jordan Ninin, Benoˆ ıt Cl´ ement LAB-STICC, UMR 6285 / ENSTA-Bretagne December 12, 2016 1 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Plan 1 Introduction 2 Global optimization approach: Branch and Bound 3 H ∞ norm of MISO systems 4 Example 5 Conclusion 2 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Structured H ∞ synthesis problem w z y u P K P interconnected with K : z = T w → z w Structured H ∞ synthesis problem � Find K ∈ K s such that || T w → z || ∞ ≤ γ and K stabilizes T w → z or � K ∈ K s || T w → z || ∞ min subject to K stabilizes T w → z 3 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Problem solved? || T w → z || ∞ K Convex approaches (LMI). Non convex approaches based on local optimization. Non convex approaches based on global optimization → only for SISO (polynomial formulation). Global optimization for MIMO? 4 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Almost possible d P W 2 u ˜ ˜ W 1 ˜ W 3 y e y + + e u r K G − n + Constraints: || T w → z || ∞ is an over approximation: || T w → ˜ e || ∞ ≤ 1. max( || T w → ˜ e || ∞ , || T w → ˜ u || ∞ , || T w → ˜ y || ∞ ) ≤ || T w → z || ∞ || T w → ˜ u || ∞ ≤ 1. || T w → ˜ y || ∞ ≤ 1. 5 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Almost possible K ( k, s ) is a parametric rational function. PID example: K ( k, s ) = k p + k i s + k d s, k = ( k p , k i , k d ) Problem � Find k ∈ k such that || T w → z j ( k ) || ∞ ≤ γ, j ∈ { 1 , ..., p } and K stabilizes T w → z or � min k ∈ k { max j || T w → z j ( k ) || ∞ } subject to K stabilizes T w → z The global optimization algorithm provides: A guaranteed enclosure [ l b , u b ] of the minimum. A good controller K . And a certificate of unfeasibility if l b > γ . 6 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Plan 1 Introduction 2 Global optimization approach: Branch and Bound 3 H ∞ norm of MISO systems 4 Example 5 Conclusion 7 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Branch and Bound Algorithm Minimization problem � � min max j || T w → z j ( k ) || ∞ k ∈ k max j || T w → z j || ∞ k 8 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Branch and Bound Algorithm max j || T w → z j || ∞ k 8 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Branch and Bound Algorithm max j || T w → z j || ∞ u b l b k 8 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Branch and Bound Algorithm max j || T w → z j || ∞ u b l b k 8 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Branch and Bound Algorithm max j || T w → z j || ∞ u b l b k 8 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Enclosure of infinity norm of MISO Problem Find b inf and b sup such that: b inf ≤ max j || T w → z i ( k ) || ∞ ≤ b sup , ∀ k ∈ k b sup b inf k k 9 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Plan 1 Introduction 2 Global optimization approach: Branch and Bound 3 H ∞ norm of MISO systems 4 Example 5 Conclusion 10 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Expression of || . || ∞ of MISO systems T w → z ( k, s ) is a matrix of rational functions which maps w to z :     T w 1 → z 1 ( k, s ) T w n → z 1 ( k, s ) T w → z 1 ( k, s ) ... . . . . . . T w → z ( k, s ) =  =     . . .    T w 1 → z p ( k, s ) ... T w n → z p ( k, s ) T w → z p ( k, s ) � � T w → z j ( k, s ) = T w 1 → z j ( k, s ) ... T w n → z j ( k, s ) Proposition �� n l =1 | T w l → z j ( k, iω ) | 2 || T w → z j ( k ) || ∞ = sup ω ≥ 0 11 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Demonstration || T w → z j ( k, s ) || ∞ = sup { σ max ( T w → z j ( k, iω )) } ω ≥ 0 � = sup λ max ( T w → z j ( k, iω ) T w → z j ( k, iω ) ∗ ) ω ≥ 0 �     T w 1 → z j ( k, iω ) � � . � � � . = sup � λ max T w 1 → z j ( k, iω ) , ...T w n → z j ( k, iω )     . �     ω ≥ 0 T w n → z j ( k, iω ) �� n l =1 | T w l → z j ( k, iω ) | 2 = sup ω ≥ 0 12 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Maximization problem Problem Find b inf and b sup such that: b inf ≤ max j || T w → z i ( k ) || ∞ ≤ b sup , ∀ k ∈ k   � n �   � � max j || T w → z j ( k ) || ∞ = max  sup | T w l → z j ( k , iω ) | 2 � j ω  l =1   � n �   � � | T w l → z j ( k , iω ) | 2 = sup  max � j ω  l =1 Use another branch and bound! → Finite frequency range 13 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion 2 nd Branch and Bound �� n l =1 | T w l → z j ( k , iω ) | 2 T w → z 1 T w → z 2 ω 14 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion 2 nd Branch and Bound �� n max l =1 | T w l → z j ( k , iω ) | 2 j ω 14 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion 2 nd Branch and Bound �� n max l =1 | T w l → z j ( k , iω ) | 2 j b sup b inf ω 14 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Min-Max formulation The problem has a Min-Max formulation: Min-Max Problem � � �� �� n l =1 | T w l → z j ( k, iω ) | 2 min sup max k ∈ k j ω s.t. K stabilizes the system: P ( k ) ≤ 0 Complexity of a branch and bound: exponential. → Branch and bound inside a branch and bound: Tricky problem. 15 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Plan 1 Introduction 2 Global optimization approach: Branch and Bound 3 H ∞ norm of MISO systems 4 Example 5 Conclusion 16 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Example 10 s +100 10 s +1 z 1 z 2 1000 s +1 s +10 + y e u k p + k i s + k d s 1 100 s +1 z 3 w s 2 +1 . 4 s +1 1+ s s +10 − k p ∈ [ − 10 , 10], k i ∈ [ − 10 , 10], k d ∈ [ − 10 , 10] Method Cpu (s) || T w → z || ∞ max j ( || T w → z j || ∞ ) H ∞ full 2 1.0258 1.0161 H ∞ struct 73 (500 rand start) 1.0411 1.0411 H ∞ hifoo 88 (30 rand start) 1.0349 1.0348 H ∞ systune 73 (2.1e6 rand start) 1.0986 0.9912 GO struct 80 1.0811 [0.9496,0.9947] 17 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Plan 1 Introduction 2 Global optimization approach: Branch and Bound 3 H ∞ norm of MISO systems 4 Example 5 Conclusion 18 / 19

Introduction Global optimization approach: Branch and Bound H ∞ norm of MISO systems Example Conclusion Conclusion Analytic expression of || . || ∞ of MISO system (square root of a rational function). Lower bound on the minimum = ⇒ we can prove that the CSP is not feasible. Another step towards structured robust synthesis? (yes). � n � � � | T w l → z j ( k, ∆ , iω ) | 2 } G ( ∆ , s ) → sup { max � j ω, ∆ l =1 19 / 19