Optimal H Control Design under Model Information Limitations and State Measurement Constraints

Transcription

1 Optimal H Control Design under Model Information Limitations and State Measurement Constraints F. Farokhi, H. Sandberg, and K. H. Johansson ACCESS Linnaeus Center, School of Electrical Engineering, KTH-Royal Institute of Technology, Stockholm, Sweden. s: {farokhi,hsan,kallej}@ee.kth.se The 5nd IEEE Conference on Decision and Control (CDC 03) Friday December 3, 03 Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

2 Decentralized Control P C 4 P 7 P 5 C P C 3 P 4 C 6 C 5 C 7 C P 3 P 6 Decentralized control extensively studied, e.g., [Witsenhausen 68 and 7; Ho and Chu 7; Sandell and Athans 74; Anderson and Moore 8; Šiljak 89; Davison and Chang 90; Rotkowitz and Lall 06]; Full model information {P i } required. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

3 Control Design with Limited Plant Model Information P C 4 P 7 P 5 C P C 3 P 4 C 6 C 5 C 7 C P 3 P 6 Design of C i only depends on the model of surrounding P j ; Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

4 Control Design with Limited Plant Model Information P C 4 P 7 P 5 C P C 3 P 4 C 6 C 5 C 7 C P 3 P 6 Design of C i only depends on the model of surrounding P j ; Recent studies: P i LTI and C i static/dynamic/adaptive [Langbort & Delvenne; Farokhi, et al]; P i stochastically-varying parameters and C i static [Farokhi & Johansson]. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

5 Control Design with Limited Plant Model Information P C 4 P 7 P 5 C P C 3 P 4 C 6 C 5 C 7 C P 3 P 6 Design of C i only depends on the model of surrounding P j ; Recent studies: P i LTI and C i static/dynamic/adaptive [Langbort & Delvenne; Farokhi, et al]; P i stochastically-varying parameters and C i static [Farokhi & Johansson]. Here: Numerical method for optimal control design over general plant model model constraints Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

6 Networked Control Systems P P x u x u x K K G K G C P P } {{ } Architectural Constraints }{{} Model Information Constraints Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

7 Networked Control Systems P P x u x u x K K G K G C P P } {{ } Architectural Constraints }{{} Model Information Constraints Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

8 Networked Control Systems P P x u x u x K K G K G C } {{ } Architectural Constraints }{{} Model Information Constraints Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

9 Networked Control Systems P P x u x u x K K G K G C } {{ } Architectural Constraints }{{} Model Information Constraints Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

10 Networked Control Systems P P x u x u x K K G K G C } {{ } Architectural Constraints }{{} Model Information Constraints Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

11 Plant Model Subsystem i is a parameter-dependent linear system N [ ẋ i (t) = Aij (α i )x j (t) + (B w ) ij (α i )w i (t) + (B u ) ij (α i )u i (t) ], j= where x i (t) R n i, w i (t) R m w,i, u i (t) R m u,i input, and control input; α i R p i is its parameter vector. are its state, exogenous Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 5 / 5

12 Plant Model Subsystem i is a parameter-dependent linear system N [ ẋ i (t) = Aij (α i )x j (t) + (B w ) ij (α i )w i (t) + (B u ) ij (α i )u i (t) ], j= where x i (t) R n i, w i (t) R m w,i, u i (t) R m u,i input, and control input; α i R p i is its parameter vector. Augmenting these subsystems results in are its state, exogenous ẋ(t) = A(α)x(t) + B w (α)w(t) + B u (α)u(t), where α = (α i ) N i= belongs to the set of feasible parameters A. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 5 / 5

13 Output Feedback Controller Subsystem i observes N [ y i (t) = (Cy ) ij (α i )x j (t) + (D yw ) ij (α i )w j (t) ] R o y,i, j= and uses static control law u i (t) = K ii y i (t). G K [ ] (Cy ) C y = (C y ) 0 (C y ) [ (Dyw ) D yw = (D yw ) 0 (D yw ) ] Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 6 / 5

14 Output Feedback Controller Subsystem i observes N [ y i (t) = (Cy ) ij (α i )x j (t) + (D yw ) ij (α i )w j (t) ] R o y,i, j= and uses static control law u i (t) = K ii y i (t). G K [ ] (Cy ) C y = (C y ) 0 (C y ) [ (Dyw ) D yw = (D yw ) 0 (D yw ) ] Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 6 / 5

15 Output Feedback Controller Subsystem i observes N [ y i (t) = (Cy ) ij (α i )x j (t) + (D yw ) ij (α i )w j (t) ] R o y,i, j= and uses static control law u i (t) = K ii y i (t). G K [ ] (Cy ) C y = (C y ) 0 (C y ) [ (Dyw ) D yw = (D yw ) 0 (D yw ) ] Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 6 / 5

16 Control Design Strategy Subsystem i uses control design strategy K ii = Γ ii ({α j j i in G C }). G C K = Γ (α ) K = Γ (α ) The set of all such control design strategies Γ is denoted by C. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 7 / 5

17 Control Design Strategy Subsystem i uses control design strategy K ii = Γ ii ({α j j i in G C }). G C K = Γ (α ) K = Γ (α ) The set of all such control design strategies Γ is denoted by C. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 7 / 5

18 Performance Metric Let us introduce the performance measure output vector z(t) = C z x(t) + D zw w(t) + D zu u(t) R oz, and define the closed-loop performance measure J(Γ, α) = T zw (s; Γ, α), where T zw (s; Γ, α): Closed-loop transfer function from w(t) to z(t). Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 8 / 5

19 Objective Find the best control design strategy with limited information: inf sup J(Γ, α) = inf sup T zw (s; Γ, α) Γ C α A Γ C α A Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 9 / 5

20 Objective Find the best control design strategy with limited information: inf sup J(Γ, α) = inf sup T zw (s; Γ, α) Γ C α A Γ C α A For general G K, the outer problem is not necessarily convex and even if convex, calculating the optimal controller might be difficult; The outer problem is an infinite-dimensional optimization problem; See (Ho and Chu, 97; Sandell and Athans, 974; Rotkowitz and Lall, 006) for conditions to guarantee convexity and (Swigart and Lall, 00, 0; Shah and Parrilo, 00) for explicitly calculating the optimal controller. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 9 / 5

21 Objective Find the best control design strategy with limited information: inf sup J(Γ, α) = inf sup T zw (s; Γ, α) Γ C α A Γ C α A For general G K, the outer problem is not necessarily convex and even if convex, calculating the optimal controller might be difficult; The outer problem is an infinite-dimensional optimization problem; Find saddle point solutions of J using a numerical method See (Ho and Chu, 97; Sandell and Athans, 974; Rotkowitz and Lall, 006) for conditions to guarantee convexity and (Swigart and Lall, 00, 0; Shah and Parrilo, 00) for explicitly calculating the optimal controller. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 9 / 5

22 Assumptions A is a compact set; There exists a basis set (ξ l ) L l= such that A span((ξ l ) L l=) n n, B w span((ξ l ) L l=) n mw, B u span((ξ l ) L l=) n mu, C y span((ξ l ) L l=) oy n, D yw span((ξ l ) L l=) oy mw ; There exists a basis set (η l ) L l = such that D zud zu = I and D yw D yw = I. Γ span((η l ) L l =) mu oy ; Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 0 / 5

23 Saddle Point Solution Find a saddle point of J(Γ, α) for Γ C span((η l ) L l = ) and α A. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

24 Saddle Point Solution Find a saddle point of J(Γ, α) for Γ C span((η l ) L l = ) and α A. Saddle Point A pair (Γ, α ) [C span((η l ) L exist constants ɛ, ɛ > 0 such that l = )] A is a saddle point of J(, ) if there J(Γ, α) J(Γ, α ) J(Γ, α ), for any (Γ, α) [C span((η l ) L l = )] A where Γ Γ ɛ and α α ɛ. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

25 Saddle Point Solution Find a saddle point of J(Γ, α) for Γ C span((η l ) L l = ) and α A. Saddle Point A pair (Γ, α ) [C span((η l ) L exist constants ɛ, ɛ > 0 such that l = )] A is a saddle point of J(, ) if there J(Γ, α) J(Γ, α ) J(Γ, α ), for any (Γ, α) [C span((η l ) L l = )] A where Γ Γ ɛ and α α ɛ. If G K is appropriately chosen (e.g., partially nested property) and (η l ) L l = is not degenerate (i.e., α s.t. all basis functions become zero), then ɛ can be arbitrarily large (when considering dynamic controllers of high enough order); For special (ξ l ) L l= (e.g., affine functions), ɛ may also be arbitrarily large. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

26 Numerical Procedure for Finding Saddle Points of J : Calculate Γ J(Γ, α) and α J(Γ, α); [Apkarian and Noll, 06] : For each Γ (k), numerically extract α (Γ (k) ); 3: Then, update Γ (k+) = Γ (k) + µ k g k where g k Γ J(Γ (k), α (Γ (k) )). Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

27 Numerical Procedure for Finding Saddle Points of J : Calculate Γ J(Γ, α) and α J(Γ, α); [Apkarian and Noll, 06] : For each Γ (k), numerically extract α (Γ (k) ); 3: Then, update Γ (k+) = Γ (k) + µ k g k where g k Γ J(Γ (k), α (Γ (k) )). Theorem Let {µ k } k=0 be chosen such that lim k k z= µ z = and lim k k z= µ z <. Assume that the subgradients are uniformly bounded for all iterations. If the numerical procedure converges, it gives a saddle point of J. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 / 5

28 Numerical Example: Vehicle Platooning Regulating inter-vehicle distances d and d 3 v (t) ϱ /m v (t) b /m 0 0 d (t) d (t) v (t) d 3 (t) = 0 0 ϱ /m 0 0 v (t) d 3 (t) + 0 b /m 0 u w (t) (t) w (t) u (t) + w 3 (t) u 3 (t) w 4 (t) v 3 (t) ϱ 3 /m 3 v 3 (t) 0 0 b 3 /m 3 w 5 (t) ϱ = ϱ = ϱ 3 = 0. and b = b = b 3 =.0 3 G K Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

29 Numerical Example: Vehicle Platooning Regulating inter-vehicle distances d and d 3 v (t) ϱ /m v (t) b /m 0 0 d (t) d (t) v (t) d 3 (t) = 0 0 ϱ /m 0 0 v (t) d 3 (t) + 0 b /m 0 u w (t) (t) w (t) u (t) + w 3 (t) u 3 (t) w 4 (t) v 3 (t) ϱ 3 /m 3 v 3 (t) 0 0 b 3 /m 3 w 5 (t) ϱ = ϱ = ϱ 3 = 0. and b = b = b 3 =.0 3 G K Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

30 Numerical Example: Vehicle Platooning Regulating inter-vehicle distances d and d 3 v (t) ϱ /m v (t) b /m 0 0 d (t) d (t) v (t) d 3 (t) = 0 0 ϱ /m 0 0 v (t) d 3 (t) + 0 b /m 0 u w (t) (t) w (t) u (t) + w 3 (t) u 3 (t) w 4 (t) v 3 (t) ϱ 3 /m 3 v 3 (t) 0 0 b 3 /m 3 w 5 (t) ϱ = ϱ = ϱ 3 = 0. and b = b = b 3 =.0 3 G K Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

31 Numerical Example: Vehicle Platooning Regulating inter-vehicle distances d and d 3 v (t) ϱ /m v (t) b /m 0 0 d (t) d (t) v (t) d 3 (t) = 0 0 ϱ /m 0 0 v (t) d 3 (t) + 0 b /m 0 u w (t) (t) w (t) u (t) + w 3 (t) u 3 (t) w 4 (t) v 3 (t) ϱ 3 /m 3 v 3 (t) 0 0 b 3 /m 3 w 5 (t) z(t) = [ d (t) d 3 (t) u (t) u (t) u 3 (t) ] Find a saddle point of J(Γ, α) = T zw (s; Γ, α) when α = [m m m 3 ] [0.5,.0] 3 and Γ belongs to the set of polynomials of α i, i =,, 3, up to the second order. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

32 Numerical Example: Vehicle Platooning Regulating inter-vehicle distances d and d 3 v (t) ϱ /m v (t) b /m 0 0 d (t) d (t) v (t) d 3 (t) = 0 0 ϱ /m 0 0 v (t) d 3 (t) + 0 b /m 0 u w (t) (t) w (t) u (t) + w 3 (t) u 3 (t) w 4 (t) v 3 (t) ϱ 3 /m 3 v 3 (t) 0 0 b 3 /m 3 w 5 (t) z(t) = [ d (t) d 3 (t) u (t) u (t) u 3 (t) ] Find a saddle point of J(Γ, α) = T zw (s; Γ, α) when α = [m m m 3 ] [0.5,.0] 3 and Γ belongs to the set of polynomials of α i, i =,, 3, up to the second order. G K 3 Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 3 / 5

33 Numerical Example: Vehicle Platooning Control Design with Local Model Information 3 max α A Tzw ( s; Γ local, α ) = GC Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

34 Numerical Example: Vehicle Platooning Control Design with Local Model Information 3 max α A Tzw ( s; Γ local, α ) = GC Control Design with Limited Model Information 5.8% 3 max α A Tzw ( s; Γ limited, α ) = GC Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

35 Numerical Example: Vehicle Platooning Control Design with Local Model Information 3 max α A Tzw ( s; Γ local, α ) = GC Control Design with Limited Model Information 5.8% 3 max α A Tzw ( s; Γ limited, α ) = GC Control Design with Full Model Information 5.4% 3 max α A T zw ( s; Γ full, α ) = GC Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 4 / 5

36 Conclusions and Future Work Conclusions Optimal control design for continuous-time linear parameterdependent systems under limited model information and partial state measurements; Expanded the control design strategy using basis functions; A numerical optimization method using the subgradients of the closed-loop performance measure. Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 5 / 5

37 Conclusions and Future Work Conclusions Optimal control design for continuous-time linear parameterdependent systems under limited model information and partial state measurements; Expanded the control design strategy using basis functions; A numerical optimization method using the subgradients of the closed-loop performance measure. Future Work Finding a good basis functions for expanding the control design strategies. For more information, please visit Farokhi, Sandberg, and Johansson (KTH) Optimal H Control Design... Friday December 3, 03 5 / 5