Multiplexed Model Predictive Control. K.V. Ling, J.M. Maciejowski, B. Wu bingfang

Transcription

1 Multiplexed Model Predictive Control KV Ling, JM Maciejowski, B Wu ekvling@ntuedusg jmm@engcamacuk bingfang wu@pmailntuedusg Cambridge University NTU Singapore 1

2 Model Predictive Control (MPC) Control by using on-line optimization (QP or LP) Increasingly used for fast systems MPC on a Chip or FPGA size limits Reduce complexity of optimization problem Various decentralised schemes proposed but all with synchronous control updates 2

3 Multiplexed MPC multi-input systems Sequential updates of control inputs Plant inputs (MV s) Plant inputs (MV s) T time Left: Conventional MPC, time Right: Multiplexed MPC 3

4 Assumptions Only one input updated at each time step (at time kt/m), in sequence Measurements of state vector are made at intervals of T/m Current state x k is known when deciding the update of each input x k is known to each controller Scheme 1: Optimise all inputs over future horizon at each step Can be thought of as 1 controller Scheme 2: Optimise only one input over future horizon at each step Can be thought of as m controllers 4

5 Update one input at a time Plant: If only one input is updated at each k then x k+1 = Ax k + m B j u j,k j=1 = Ax k + B σ(k) ũ k where σ(k) = (k mod m) + 1 is a periodic switching function: σ(k + m) = σ(k) So multi-input LTI plant looks like periodic single-input plant 5

6 Stability of Scheme 1 Cost function: J k = ( x k+i+1 2 q + u k+i 2 r) = i=0 N 1 i=0 for suitable P k+n+1 ( xk+i+1 2 q + u k+i 2 ) r + x T k+n+1 P k+n+1 x k+n+1 Infinite horizon closed-loop stable (if feasible) Constraints can be imposed during first N steps of horizon 6

7 Periodic Riccati Equation P k = A T P k+1 A A T P k+1 B σ(k) (Bσ(k) T P k+1b σ(k) + r) 1 Bσ(k) T P k+1a + q This converges to a periodic solution (given suitable final condition) State feedback after end of prediction horizon: K k = (Bσ(k) T P k+1b σ(k) + r) 1 Bσ(k) T P k+1a Stability of Scheme 1 follows easily if feasible 7

8 Scheme 2 Controller j decides future sequence of j th input only Other inputs are treated as known disturbances Assume that controller j knows the future plans of the other controllers, and assumes u σ(k),k+i = K σ(k) x k+i beyond the planning horizon Stability proof idea: J k is a Lyapunov function, if problem is feasible 8

9 U k = u 1,k u 2,k+1 u 1,k+2 u 1,k+2N 4 u 2,k+2N 3 u 1,k+2(N 1) K 2 x k+2n 1 K 1 x k+2n K 2 x k+2n+1 K 1 x k+2n+2, U k+1 = u 2,k+1 u 1,k+2 u 2,k+3 u 2,k+2N 3 u 1,k+2(N 1) u 2,k+2N 1 K 1 x k+2n K 2 x k+2n+1 K 1 x k+2n+2 Pattern of control updates in Scheme 2 Entries in bold get updated 9

10 Scheme 2: How to compute the cost? Assume that, after end of prediction horizon (of length N): Let Monodromy matrices: u σ(k),k+i = K σ(k) x k+i Φ j = A + B j K j Ψ 1 = Φ m Φ m 1 Φ 2 Φ 1 Ψ 2 = Φ 1 Φ m Φ 3 Φ 2 Ψ m = Φ m 1 Φ m 2 Φ 1 Φ m Stability condition: λ j (Ψ i ) < 1 for all j, for any i 10

11 Scheme 2: Cost function J k Paper has errors here! Details wrong, Idea OK At time k, controller σ(k) evaluates J k as: J k = m(n 1) i=0 ( xk+i+1 2 q + u k+i 2 ) r + x T k+m(n 1)+1 P σ(k)x k+m(n 1)+1 What is P σ(k)? The tail of J k is J k+m(n 1)+1 = ( xk+i+1 2 q + u k+i 2 r) i=m(n 1)+1 m = x k+m(n 1)+i 2 q + i=2 i=n 1 11 ( Xk+m(i+1)+1 2 Q + U k+mi+1 2 R)

12 where X k = x k x k+1, U k = u σ(k),k u σ(k+1),k+1 x k+m 1 u σ(k+m 1),k+m 1 and Q = diag[q,,q], R = diag[r,,r] State transition equation if i > m(n 1): X k+1+m(n+i+1) = Ψ σ(k+1) Ψ σ(k+2) 0 X k+1+m(n+i) 0 0 Ψ σ(k) 12

13 Recall: If x i+1 = Φx i and u i = Kx i and J = i=0 ( x i+1 2 Q + u i 2 R ) then J = x T 0 Px 0 where P = Φ T PΦ + Φ T QΦ + K T RK Applying this to our problem: J k+m(n 1)+1 = Xk+m(N 1)+1 T diag[π σ(k+1),,π σ(k+m) ]X k+m(n 1)+1 where Π l = Ψ T l Π l Ψ l + Ψ T l qψ l + K T l rk l for l = 1, 2,,m 13

14 But x k+m(n 1)+2 = Φ σ(k+1) x k+m(n 1)+1 x k+m(n 1)+3 = Φ σ(k+2) Φ σ(k+1) x k+m(n 1)+1 etc so we obtain J k+m(n 1)+1 = x T k+m(n 1)+1 P σ(k)x k+m(n 1)+1 where P σ(k) = Π σ(k+1) + Φ T σ(k+1)π σ(k+2) Φ σ(k+1) + + [Φ T σ(k+1) Φ T σ(k+m 1)Π σ(k+m) Φ σ(k+m 1) Φ σ(k+1) ] 14

15 Scheme 2 stability proof Standard monotonically decreasing cost argument: If, at step k + 1, controller σ(k + 1) leaves the moves u σ(k+1),k+1,, u σ(k+1),k+m(n 1)+1 unchanged, then J k+1 = J o k x k+1 2 q u k 2 r < J o k But J o k+1 J k+1 by optimality Hence J o k+1 Jo k, (equality only if x k = 0) Hence J o k is a Lyapunov function Scheme 2 is stable if feasible 15

16 Example y 1(s) y 2 (s) = 1 7s+1 2 8s+1 1 3s+1 1 4s+1 u 1(s) u 2 (s) m = 2, T = 1 sec, T/m = 05 sec No constraints Step disturbance on y 1 at t = 701 sec Step disturbance on y 2 at t = 1401 sec 16

17 15 y1 15 y multiplex synchronize u1 u Time (seconds) Time (seconds) 17

18 Conclusions Multiplexed MPC updates one input at a time Do something sooner can be better than Do optimal thing later Extension of Chmielewski-Manousiouthakis approach using periodic systems theory Constraints, feasibility etc not addressed yet Complexity reduction requires constraint decoupling too Generalisations: Unequal intervals; Groups of inputs 18