Design of structured optimal feedback gains for interconnected systems

Transcription

1 Design of structured optimal feedback gains for interconnected systems Mihailo Jovanović mihailo joint work with: Makan Fardad Fu Lin Technische Universiteit Delft; Sept 6, 2010

2 M. JOVANOVIĆ, U OF M 1 Distributed systems stimation Error sensor networks variance estimate APPLICATIONS: arrays of micro-cantilevers UAV formations satellite constellations OF INCREASING IMPORTANCE IN MODERN TECHNOLOGY to the reference INTERACTIONS CAUSE COMPLEX BEHAVIOR cannot be predicted by analyzing isolated subsystems SPECIAL STRUCTURE every unit has sensors and actuators

3 M. JOVANOVIĆ, U OF M 2 Array of micro-cantilevers POTENTIAL APPLICATION: MASSIVELY PARALLEL DATA STORAGE ELECTROSTATICALLY ACTUATED MICRO-CANTILEVERS problem: solution: slow scans low throughput go massively parallel ISSUES: tightly coupled dynamics spatio-temporal instabilities large number of devices localized control imperative

4 FORMATION FLIGHT FOR AERODYNAMIC ADVANTAGE e.g. additional lift in V-formations MICRO-SATELLITE FORMATIONS e.g. for synthetic aperture precise control needed M. JOVANOVIĆ, U OF M 3 Coordinated control of formations MAKE VEHICLES SMALLER AND CHEAPER USE MANY cooperative control becomes a major issue

5 G0 G1 G2 K FULLY DECENTRALIZED G0 G1 G2 best performance excessive communication worst performance no communication M. JOVANOVIĆ, U OF M 4 Controller architectures CENTRALIZED K 0 K 1 K 2 LOCALIZED G0 G1 G2 K0 K1 K2 many possible architectures

6 M. JOVANOVIĆ, U OF M 5 Outline ❶ STRUCTURED OPTIMAL DESIGN sparsity constraints on feedback gains ❷ OPTIMAL LOCALIZED CONTROL OF VEHICULAR PLATOONS design of spatially-varying feedback gains ❸ PERFORMANCE VS. SIZE coherence of formation ❹ PARTING THOUGHTS

7 STRUCTURED OPTIMAL DESIGN M. JOVANOVIĆ, U OF M 6

8 M. JOVANOVIĆ, U OF M 7 Structured H 2 problem STRUCTURAL CONSTRAINTS K S ẋ = A x + B 1 d + B 2 u [ ] [ ] Q 1/2 0 z = x + 0 R 1/2 u u = K x K S centralized fully decentralized localized OBJECTIVE: design stabilizing K S that minimizes d z 2 2

9 ẋ = (A B 2 K) x + B 1 d [ ] Q z = 1/2 R 1/2 x, K S K STRUCTURED H 2 PROBLEM: minimize J(K) := trace ( ) P (K)B 1 B1 T subject to { (A B2 K) T P + P (A B 2 K) = ( Q + K T RK ) K S J(K) nonconvex function of K M. JOVANOVIĆ, U OF M 8 CLOSED-LOOP SYSTEM RELATED PROBLEMS: static output feedback: Levine & Athans, IEEE TAC 70 structured dynamic controller: Wenk & Knapp, IEEE TAC 80

10 M. JOVANOVIĆ, U OF M 9 Necessary conditions for optimality (A B 2 K) T P + P (A B 2 K) = ( Q + K T RK ) (A B 2 K) L + L (A B 2 K) T = B 1 B T 1 [( RK B T 2 P ) L ] I S = 0 K = I S - structural identity I S = SPECIAL CASES: no constraints A T P + P A P B 2 R 1 B T 2 P + Q = 0 K c = R 1 B T 2 P expensive control of stable open-loop systems perturbation analysis: Fardad, Lin, Jovanović, CDC 09

11 M. JOVANOVIĆ, U OF M 10 Perturbation analysis EXPENSIVE CONTROL: R = (1/ε) I, 0 < ε 1 P = n = 0 O(1) : K 0 = 0 O(ε) : O(ε 2 ) : ε n P n, L = A T P 0 + P 0 A = Q n = 0 AL 0 + L 0 A T = B 1 B T 1 ε n L n, K = [K 1 L 0 ] I S = [ B T 2 P 0 L 0 ] IS n = 0 ε n K n A T P 1 + P 1 A = (matrix function of K 1 and P 0 ) AL 1 + L 1 A T = (matrix function of K 1 and L 0 ) [K 2 L 0 ] I S = [ B T 2 (P 0 L 1 + P 1 L 0 ) K 1 L 1 ] IS followed by homotopy

12 M. JOVANOVIĆ, U OF M 11 Numerical computation (A B 2 K) T P + P (A B 2 K) = ( Q + K T RK ) NEWTON S METHOD FEATURES: (A B 2 K) L + L (A B 2 K) T = B 1 B T 1 [( RK B T 2 P ) L ] I S = 0 K i+1 = K i + s i Ki until J(K i ) < tolerance Newton direction K i : conjugate-gradient method sparsity utilized step size s i : backtracking line search stability guaranteed initial condition: truncated centralized gain K c I S exponential decay Bamieh, Paganini, Dahleh, IEEE TAC 02; Motee & Jadbabaie, IEEE TAC 08

13 M. JOVANOVIĆ, U OF M 12 Augmented Lagrangian K = 0 = k 12 = [ 1 0 ] [ ] [ ] k 11 k 12 0 k 21 k 22 1 K = [ ] k k 22 ([ 0 0 = trace 1 0 K S trace ( Λ T K ) = 0, Λ S c [ ] Λ = [ 0 λ1 λ 2 0 ] [ 0 1, IS c = 1 0 ] [ ]) k11 k 12 k 21 k 22 ] AUGMENTED LAGRANGIAN: L γ (K, Λ) = J(K) + trace ( Λ T K ) + γ K I c S 2 F

14 M. JOVANOVIĆ, U OF M 13 Augmented Lagrangian minimization min L γ(k, Λ) K OPTIMALITY OF L γ : NECESSARY CONDITIONS (A B 2 K) T P + P (A B 2 K) = ( Q + K T RK ) (A B 2 K) L + L (A B 2 K) T = B 1 B T 1 2 ( RK B T 2 P ) L + Λ + γ (K I c S ) = 0 NEWTON S METHOD WITH LINE SEARCH Stability guaranteed Deals with ill-conditioning UPDATE RULES Λ i+1 = Λ i + γ i (K i I c S ), γ i+1 = c γ i, c > 1 STOPPING CRITERION K i I c S F < tolerance

15 M. JOVANOVIĆ, U OF M 14 Part 1: summary STRUCTURED H 2 PROBLEM Primal formulation (perturbation analysis & homotopy) Augmented Lagrangian COMPUTATION OF OPTIMAL STRUCTURED GAINS Possible CHALLENGES Understand limitations of proposed methods Understand structure of augmented Lagrangian formulation Perturbation analysis/homotopy for unstable open-loop systems Hidden convexity?

16 CONTROL OF VEHICULAR FORMATIONS M. JOVANOVIĆ, U OF M 15

17 AUTOMATED CONTROL OF EACH VEHICLE tight spacing at highway speeds KEY ISSUES (also in: control of swarms, flocks, formation flight) M. JOVANOVIĆ, U OF M 16 Vehicular platoons Is it enough to only look at neighbors? How does performance scale with size? Are there any fundamental limitations? FUNDAMENTALLY DIFFICULT PROBLEM (scales badly) Jovanović & Bamieh, IEEE TAC 05 Bamieh, Jovanović, Mitra, Patterson, IEEE TAC 10 (conditionally accepted)

18 M. JOVANOVIĆ, U OF M 17 Problem formulation ẋ n = u n + d n control disturbance DESIRED TRAJECTORY: DEVIATIONS: { xn := vt + n constant velocity x n := x n x n ũ n := u n v } x n = ũ n + d n CONTROL: ũ = K x, K: feedback gain

19 RELATIVE POSITION FEEDBACK: ũ n = f n ( x n x n 1 ) b n ( x n x n+1 ) ũ = [ ] [ ] C F f F f b x C b M. JOVANOVIĆ, U OF M 18 DESIGN K TO USE NEAREST NEIGHBOR FEEDBACK K f f 2 0 } 0 0 {{ f 3 } F f } 0 1 {{ 1 } C f + b b 2 0 } 0 0 {{ b 3 } F b } 0 0 {{ 1 } C b

20 M. JOVANOVIĆ, U OF M 19 Structured feedback design PERFORMANCE MEASURES x = d + ũ [ ] [ ] Q 1/2 0 z = x + ũ 0 I ũ = [ ] [ ] C F f F f b x, F C f, F b b spatially uniform: F f = F b = I MICROSCOPIC: local position deviation ( x n x n 1 ) Q l diagonal MACROSCOPIC: global position deviation x n Q g = I

21 M. JOVANOVIĆ, U OF M 20 Performance vs. size (1/N) H 2 2 MICROSCOPIC: MACROSCOPIC: SPATIALLY UNIFORM CONTROL: (1/N) H 2 2

22 M. JOVANOVIĆ, U OF M 21 Most energetic spatial profiles sin (πn/(n + 1)): cos (0.5π(n/(N + 1) 1)):

23 STRUCTURED FEEDBACK DESIGN M. JOVANOVIĆ, U OF M 22

24 M. JOVANOVIĆ, U OF M 23 Convex structured design SYMMETRIC GAIN: ũ n = k n ( x n x n 1 ) k n+1 ( x n x n+1 ) ũ = K x, K = K T > 0 CONVEX PROBLEM: minimize trace ( K + Q K 1) subject to 0 < K T = K S

25 M. JOVANOVIĆ, U OF M 24 Performance vs. size SYMMETRIC STRUCTURED OPTIMAL (WITH Q g = I) (1/N) H 2 2 MICROSCOPIC: MACROSCOPIC: CONTROL: (1/N) H 2 2

26 M. JOVANOVIĆ, U OF M 25 Optimal design of non-symmetric gains SPATIALLY UNIFORM (K 0 = C f + C b ) inversely optimal wrt Q 0 = K 2 0 O(1) : O(ε) : P Q 0 = 0 K 0 P = = P 0 S n = 0 } ε n P n, L = P Q 0 = 0 K 0 = P 0 K 0 L 0 L 0 K 0 = I n = 0 do design with: Q = Q 0 + ε (Q d Q 0 ) ε n L n, K = K 0 P 1 P 1 K 0 = (Q d Q 0 ) [( [ ] [ ] ) C Ff1 F f [ b1 P C 1 L 0 C T f Cb T b followed by homotopy n = 0 ε n K n ] ] [ I I ] = 0

27 HOMOTOPY FORWARD/BACKWARD GAINS: FORWARD GAINS: M. JOVANOVIĆ, U OF M 26 PERTURBATION ANALYSIS (WITH Q g = I)

28 M. JOVANOVIĆ, U OF M 27 Performance vs. size STRUCTURED OPTIMAL (WITH Q d = I) (1/N) H 2 2 MICROSCOPIC: MACROSCOPIC: CONTROL: (1/N) H 2 2

29 (1/N) H 2 2 MACROSCOPIC: M. JOVANOVIĆ, U OF M 28 spatial uniform vs. symmetric structured optimal vs. structured optimal vs. centralized optimal

30 M. JOVANOVIĆ, U OF M 29 Part 2: summary Optimal gains are spatially non-uniform PLATOONS: Limitations due to chaining of open-loop integrators Jovanović & Bamieh, IEEE TAC 05 Must have global interactions to address tightness problem Even then, convergence of Merge & Split Maneuvers scales badly with N Jovanović, Fowler, Bamieh, D Andrea, SCL 08 STRUCTURED OPTIMAL CONTROL OF VEHICULAR FORMATIONS FUNDAMENTAL LIMITATIONS FOR SPATIALLY INVARIANT LATTICES Bassam s talk Bamieh, Jovanović, Mitra, Patterson, IEEE TAC 10 (accepted)

31 M. JOVANOVIĆ, U OF M 30 Acknowledgments TEAM: SUPPORT: Makan Fardad Fu Lin (Syracuse University) (U of M) NSF CAREER Award CMMI NSF Award CMMI