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Packetized Predictive Control for Rate-Limited Networks via Sparse Representation

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Presented at IEEE CDC 2012, Maui, USA.
http://control.disp.uniroma2.it/cdc2012/

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Packetized Predictive Control for Rate-Limited Networks via Sparse Representation

  1. 1. Packetized Predictive Control for Rate-Limited Networks via Sparse Representation Masaaki Nagahara1 Daniel E. Quevedo2 Jan Østergaard3 1 Kyoto University 2 The University of Newcastle 3 Aalborg University 2012 Dec. 10 IEEE CDC 2012 1 / 33
  2. 2. MotivationNetworked Control Packet dropouts Bit-rate limitation Plant unreliable and rate-limited network Controller 2 / 33
  3. 3. MotivationNetworked Control Packet dropouts → Packetized Predictive Control Bit-rate limitation Plant unreliable and rate-limited network Controller 3 / 33
  4. 4. MotivationNetworked Control Packet dropouts → Packetized Predictive Control Bit-rate limitation → Sparse Representation Plant unreliable and rate-limited network Controller 4 / 33
  5. 5. Table of Contents 1 Packetized predictive control (PPC) 2 Sparsity optimization in PPC 3 Stability theorem 4 Optimization via Orthogonal Matching Pursuit (OMP) 5 Simulation results 6 Conclusion 5 / 33
  6. 6. Table of Contents 1 Packetized predictive control (PPC) 2 Sparsity optimization in PPC 3 Stability theorem 4 Optimization via Orthogonal Matching Pursuit (OMP) 5 Simulation results 6 Conclusion 6 / 33
  7. 7. Packetized Predictive Control Suppose the plant is modeled by x(k + 1) = Ax(k) + Bu(k), k = 0, 1, . . . , x(0) = x0 ∈ Rn . Compute a tentative control sequence u0 , u1 , . . . , uN −1 for a finite horizon (length N ) of future time instants based on state observation x(k) and state prediction x0|k := x(k), x1|k , . . . , xN −1|k . Transmit the control sequence as a packet u = [u0 , u1 , . . . , uN −1 ] to a buffer. If a packet is dropped out, use a ”future” control ui (i ≥ 2) stemming from a previously received packet stored in the buffer. x(k) u(x(k)) u(k) x(k) Controller Buffer Plant 7 / 33
  8. 8. Packetized Predictive Control At time k=0, compute the control packet u(x(0)) using the observation x(0). u(x(0)) u0 u2 u1 u3 x(0) u(x(0)) u(0) x(0) Controller Buffer Plant 8 / 33
  9. 9. Packetized Predictive Control The control packet u(x(0)) is successfully transmitted, and then the packet is stored in the buffer. u(x(0)) u0 u2 u1 u3 x(0) u(x(0)) u(0) x(0) Controller Buffer Plant 9 / 33
  10. 10. Packetized Predictive Control Use the first element u0 in the buffer as the control input u(0). u(x(0)) u0 u2 u1 u3 x(0) u(x(0)) u(0) x(0) Controller Buffer Plant 10 / 33
  11. 11. Packetized Predictive Control At time k=1, compute the control packet u(x(1)) using the observation x(1). u(x(1)) u0 u2 u1 u3 x(1) u(x(1)) x(1) Controller Buffer Plant 11 / 33
  12. 12. Packetized Predictive Control The control packet u(x(1)) is transmitted to the buffer. u(x(1)) u0 u2 u1 u3 x(1) u(x(1)) x(1) Controller Buffer Plant 12 / 33
  13. 13. Packetized Predictive Control The control packet u(x(1)) is transmitted to the buffer. u(x(1)) u0 u2 Packet dropout occurs! u1 u3 x(1) u(x(1)) x(1) Controller Buffer Plant 13 / 33
  14. 14. Packetized Predictive Control Use the second element of u(x(0)) stored in the buffer as the control u(1) at k=1. u(x(1)) u(x(0)) u0 u2 u0 u2 u1 u3 u1 u3 x(1) u(x(1)) u(1) x(1) Controller Buffer Plant 14 / 33
  15. 15. Quadratic Packetized Predictive Control In a standard PPC, we minimize the following quadratic (or 2) cost function for the control packet u(x(k)), k = 0, 1, 2, . . . : N 2 2 J(u) = xN |k P + xi|k Q + λ u 2, 2 i=0 where u = [u0 , . . . , uN −1 ] ∈ RN and x0|k = x(k), xi+1|k = Axi|k + Bui , i = 0, 1, . . . , N − 1. 15 / 33
  16. 16. Quadratic Packetized Predictive Control In a standard PPC, we minimize the following quadratic (or 2) cost function for the control packet u(x(k)), k = 0, 1, 2, . . . : N 2 2 J(u) = xN |k P + xi|k Q + λ u 2, 2 i=0 where u = [u0 , . . . , uN −1 ] ∈ RN and x0|k = x(k), xi+1|k = Axi|k + Bui , i = 0, 1, . . . , N − 1. A large horizon length N leads to robustness against packet dropouts, but it increases the size of the packet, which should be avoided for rate-limited networks. 16 / 33
  17. 17. Quadratic Packetized Predictive Control In a standard PPC, we minimize the following quadratic (or 2) cost function for the control packet u(x(k)), k = 0, 1, 2, . . . : N 2 2 J(u) = xN |k P + xi|k Q + λ u 2, 2 i=0 where u = [u0 , . . . , uN −1 ] ∈ RN and x0|k = x(k), xi+1|k = Axi|k + Bui , i = 0, 1, . . . , N − 1. A large horizon length N leads to robustness against packet dropouts, but it increases the size of the packet, which should be avoided for rate-limited networks. Can we reduce the data size of the packet without reducing the horizon length N ? 17 / 33
  18. 18. Quadratic Packetized Predictive Control In a standard PPC, we minimize the following quadratic (or 2) cost function for the control packet u(x(k)), k = 0, 1, 2, . . . : N 2 2 J(u) = xN |k P + xi|k Q + λ u 2, 2 i=0 where u = [u0 , . . . , uN −1 ] ∈ RN and x0|k = x(k), xi+1|k = Axi|k + Bui , i = 0, 1, . . . , N − 1. A large horizon length N leads to robustness against packet dropouts, but it increases the size of the packet, which should be avoided for rate-limited networks. Can we reduce the data size of the packet without reducing the horizon length N ? Use sparse representation of the packet. 18 / 33
  19. 19. Sparse Control Packet DesignIdeaSparsify the control packet (vector) with the sparsity-promotingoptimization: u(x(k)) arg min u 0 u∈RNsubject to N −1 xN |k 2 P + xi|k 2 Q ≤ x(k) W x(k). i=0Sparsity index u 0 u 0 is the number of nonzero elements in u = [u0 , u1 , . . . , uN −1 ] .Trade-off parameter WW is a positive semi-definite matrix specifying the trade-off between thesparsity and control performance. 19 / 33
  20. 20. Sparse Control Packet Design Sparse vectors can be effectively encoded by simple means. Assume memoryless uniform scalar quantizer for encoding u. For dense vector u0 u1 u2 u3 u4 u5 u6 u7 Q û0 û1 û2 û3 û4 û5 û6 û7 8 ⇥ 8 = 64 bit 8bit 8bit 8bit 8bit 8bit 8bit 8bit 8bit For sparse vector u0 0 0 u3 0 u5 u6 0 Q û0 û3 û5 û6 8 ⇥ 4 = 32 bit 8bit 8bit 8bit 8bit 40 bit + location data 1 0 0 1 0 1 1 0 8 bit 20 / 33
  21. 21. Sparse Control Packet Design In general, assume Sampling frequency : fs > 0 [Hz] Horizon length: N ≥ 1 Packet sparsity: S = u 0 < N Quantizer precision: b ≥ 1 [bit] For dense vectors (obtained by e.g., 2 optimization), one needs N · b · fs [bit/sec] For sparse vectors, quantizing the values and the location of the nonzero elements requires S · b · fs + N · fs = (Sb + N )fs [bit/sec] for values for location If N b > Sb + N , or S < 1 − b−1 N then sparse vectors can reduce the bit rate for transmission by (1 − b−1 )N − S bfs [bit/sec]. 21 / 33
  22. 22. Stability resultAssumptionThe number of consecutive packet-dropouts is uniformly bounded by thehorizon length N . In other words, a packet will never be dropped outwhen the buffer is empty. x(k) u(x(k)) u(k) x(k) Controller Buffer Plant 22 / 33
  23. 23. Stability resultTheoremFor every Q > 0, there exist matrices P > 0 and W > 0 in theoptimization N −1 2 2 u(x(k)) = arg min u 0 , s.t. xN |k P + xi|k Q ≤ x(k) W x(k) u∈RN i=0such that the networked control system is asymptotically stable, i.e.,limk→∞ x(k) = 0. The procedure to obtain such P and W is shown inthe article. x(k) u(x(k)) u(k) x(k) Controller Buffer Plant 23 / 33
  24. 24. How to solve it? The optimization N −1 u(x(k)) arg min u 0 , s.t. xN |k 2 + P xi|k 2 Q ≤ x(k) W x(k) u∈RN i=0 can be rewritten as 2 u(x(k)) = arg min u 0 , s.t. Gu − Hx(k) 2 ≤ x(k) W x(k) u∈RN for some matrices G and H. The optimization is combinatorial, and hence hard to solve. 24 / 33
  25. 25. How to solve it? The optimization N −1 u(x(k)) arg min u 0 , s.t. xN |k 2 + P xi|k 2 Q ≤ x(k) W x(k) u∈RN i=0 can be rewritten as 2 u(x(k)) = arg min u 0 , s.t. Gu − Hx(k) 2 ≤ x(k) W x(k) u∈RN for some matrices G and H. The optimization is combinatorial, and hence hard to solve. A greedy algorithm called Orthogonal Matching Pursuit (OMP) can be used. 25 / 33
  26. 26. How to solve it? The optimization N −1 u(x(k)) arg min u 0 , s.t. xN |k 2 + P xi|k 2 Q ≤ x(k) W x(k) u∈RN i=0 can be rewritten as 2 u(x(k)) = arg min u 0 , s.t. Gu − Hx(k) 2 ≤ x(k) W x(k) u∈RN for some matrices G and H. The optimization is combinatorial, and hence hard to solve. A greedy algorithm called Orthogonal Matching Pursuit (OMP) can be used. OMP may give a local minimum, but it always gives a feasible solution, and hence leads to asymptotic stability. 26 / 33
  27. 27. Simulation Results Controlled plant (unstable): a linearized model of an aircraft [Maciejowski, Predictive Control with Constraints] ˙ xc = Ac xc + Bc u,     −1.2822 0 0.98 0 −0.3  0 0 1 0   0  Ac =   −5.4293  , Bc =   −17  .  0 −1.8366 0  −128.2 128.2 0 0 0 poles: 0, 0, −1.5594 ± j2.2900 Discrete-time model is obtained via zero-order-hold discretization with sampling time 0.5 (sec). Horizon length (= packet size): N = 10 Packet-dropout probability: 50% if there have been N − 1 = 9 consecutive dropouts, we set the next dropout probability to be 0. 27 / 33
  28. 28. Simulation Results Comparison: 1 OMP for the optimization (proposed) 2 u(x(k)) = arg min u 0 s.t. Gu − Hx(k) 2 ≤ x(k) W x(k) u∈RN 1 2 2 - optimization [Nagahara & Quevedo, IFAC, 2011], [Gallieri & Maciejovski, ACC, 2012] 1 2 u(x(k)) = arg min µ u 1 + Gu − Hx(k) 2 u∈RN 2 2 3 optimization (conventional) 2 1 2 u(x(k)) = arg min µ u 2 + Gu − Hx(k) 2 u∈RN 2 28 / 33
  29. 29. Simulation Results Sparsity ||u||0 14 OMP Ideal L2 12 L1/L2 (i) L1/L2 (ii) 10 8 ||u||0 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100 k (OMP): arg minu u 0 s.t. Gu − Hx(k) 2 ≤ x(k) W x(k). 2 (L1/L2): arg minu µ u 1 + (1/2) Gu − Hx(k) 2 2 with (i) µ = 5.3 × 103 and (ii) µ = 5.3 (L2): arg minu µ u 2 + (1/2) Gu − Hx(k) 2 2 2 with µ = 3.1 × 102 (reg) and µ = 0 (ideal). 29 / 33
  30. 30. Simulation Results [log plot] 2−norm of the state x(k) 0 10 log10 ||x(k)||2 −20 OMP 10 Ideal L2 L1/L2 (i) L1/L2 (ii) −40 10 0 10 20 30 40 50 60 70 80 90 100 k [linear plot] 2−norm of the state x(k) 5 OMP 4 Ideal L2 3 L1/L2 (i) ||x(k)||2 L1/L2 (ii) 2 1 0 0 10 20 30 40 50 60 70 80 90 100 k (OMP): arg minu u 0 s.t. Gu − Hx(k) 2 ≤ x(k) W x(k). 2 (L1/L2): arg minu µ u 1 + (1/2) Gu − Hx(k) 2 2 with (i) µ = 5.3 × 103 and (ii) µ = 5.3 (L2): arg minu µ u 2 + (1/2) Gu − Hx(k) 2 2 2 with µ = 3.1 × 102 (reg) and µ = 0 (ideal). 30 / 33
  31. 31. Simulation Results Computational time OMP Ideal L2 −2 10 L1/L2 (i) L1/L2 (ii) −3 10 Time (sec) −4 10 −5 10 −6 10 0 10 20 30 40 50 60 70 80 90 100 k (OMP): arg minu u 0 s.t. Gu − Hx(k) 2 ≤ x(k) W x(k). 2 (L1/L2): arg minu µ u 1 + (1/2) Gu − Hx(k) 2 2 with (i) µ = 5.3 × 103 and (ii) µ = 5.3 (L2): arg minu µ u 2 + (1/2) Gu − Hx(k) 2 2 2 with µ = 3.1 × 102 (reg) and µ = 0 (ideal). 31 / 33
  32. 32. Conclusion We have proposed Packetized predictive control for packet dropouts with sparse representation for rate-limited networks The control system is asymptotically stable. The optimization can be effectively solved via Orthogonal Matching Pursuit (OMP). Simulation results show effectiveness of the proposed method. 32 / 33
  33. 33. Conclusion We have proposed Packetized predictive control for packet dropouts with sparse representation for rate-limited networks The control system is asymptotically stable. The optimization can be effectively solved via Orthogonal Matching Pursuit (OMP). Simulation results show effectiveness of the proposed method. Mahalo! 33 / 33

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