Infinite-dimensional nonlinear predictive controller design for open-channel hydraulic systems

Transcription

1 Infinite-dimensional nonlinear predictive controller design for open-channel hydraulic systems D. Georges, Control Systems Dept - Gipsa-lab, Grenoble INP Workshop on Irrigation Channels and Related Problems, Salerno, Italy, october 2-4, 28

2 Outline of the talk 1. Control of a single canal pool 2. Modelling of open-channel hydraulic systems: Saint Venant PDE 3. Some backgrounds on predictive control 4. Formulation of the associated optimal control problem 5. Necessary conditions for optimality: the adjoint PDEs 6. Computation of the related two-point boundary value problem 7. Description of the here-proposed predictive control scheme 8. Some simulation results 9. The multi-pool case: a decomposition approach via Lagrangian relaxation 1. Some conclusions and perspectives

3 Control problem Regulation of a single pool of an irrigation canal (pool = canal section delimited by 2 gates) using the upstream regulator gate: B L D h h S h Q P I Upstream Downstream Transversal section Longitudinal section

4 Why to use nonlinear predictive control? 1. In order to take nonlinear transportation and diffusion phenomena into account; 2. Because (finite-dimensional) nonlinear optimal control provides, under some specific assumptions, stabilizing feedbacks (infinite-horizon optimal control, predictive control): possible extension in the infinite-dimensional framework?

5 Open-channel modelling: Saint Venant PDE Rectangular canal section case: S : { B z t + Q x x ( Q2 Bz Q t + = q ) + gbz z x gbz(i J(Q, z)) = kq Q Bz (1) z = relative water level,(m); B=canal width,(m); Q = water flow rate, ( m3 s ); A = wet section, (m2 ); I = canal slope ; g = acc. of gravity, ( m ); J = friction ( m s 2 m ) ; q(x)= withdrawal / length unit ( m2 s ) ; k=, if q >, 1, if q <. { Q(, t) = u(t) (integral control : upstream) B.C. z(l, t) = v(t) (disturbance : downstream) Models of the gates may be introduced (2)

6 Some backgrounds on predictive control Consider a nonlinear system of the form: ẋ = F (x, u), x() = x (3) with F (, ) = (the origin is an equilibrium point). 1. At time t, x(t) = x t is measured or computed from a state observer. 2. Computation of an (open-loop) optimal control problem over a receding control interval [t, t + T ] starting with state x t (so-called associated optimal control problem (AOCP)): min u(.) t+t t L(x(τ), u(τ))dτ (4) s.t. ẋ = F (x, u), x(t) = x t and x(t + T ) E (5) where E denotes a domain of the state space in which x(t + T ) is assigned. 3. Application of the first control u(t) obtained from the sequence {u(t), u(t + t), u(t + 2 t),..., u(t + N t)}. 4. Go back to 1), t + t t.

7 Some backgrounds on predictive control Some sufficient conditions to ensure local asymptotic stability around the origin (Mayne & Michalska, TAC 9): L(.,.) is positive definite in both x and u (in the sense of Hessian matrices w.r.t. x and u) under assumptions (mainly, F is 2 times cont. differentiabble, F is uniformly Lipschitz in x w.r.t. u, local uniform controllability) ensuring existence and uniqueness of the AOCP E = {} V (x, t) = t+t t L(x(τ), u(τ))dτ is a control Lyapunov function. Extension to the infinite-dimensional case?

8 Associated optimal control problem s.t. with T T min m(u(t))dt + u S : B.C. B z t + Q x Q t + x ( Q2 Bz µ = u I.C. = q L l(z(x, t), Q(x, t))dxdt (6) ) + gbz z x gbz(i J) = kq Q Bz Q(x, ) = φ 1 (x), x [, L] z(x, ) = φ 2 (x) µ() = φ 1 () { Q(, t) = µ(t) (control : upstream end) z(l, t) = v(t) (disturbance : downstream end) and constraint on state at final time (terminal constraint): { Q(x, T ) = Q (x), x [, L] T.C. z(x, T ) = z (x) (7) (8) (9) (1)

9 Regulation around an equilibrium state Regularization term: m(u(t)) = r 2 u(t)2, r > (11) State cost function: l(z(x, t), Q(x, t)) = 1 2 [q 1(x)(z(x, t) z (x)) 2 (12) +q 2 (x)(q(x, t) Q (x)) 2 ], q 1 (.), q 2 (.) > where z (x), Q (x), x [, L] = equilibrium state, solution of: S e : { Q x = q x ( Q2 z Bz ) + gbz x gbz(i J) = kq Q Bz with Boundary Conditions: (13) B.C. { Q () = Q o z (L) = z L (14)

10 Some remarks Some other possible control objectives by choosing other m and l functionals: Regulation of the downstream-end level z(x = L, t), with upstream-end boundary control and a downstream-end boundary condition Q(x = L, t) = v(t), where v is the downstream-end water flow acting as a disturbance. Regulation of the downstream-end level z(x = L, t), with downstream-end boundary control and an upstream-end boundary condition Q(x =, t) = v(t), where v is the upstream-end water flow acting as a disturbance. Control problems with withdrawals may be also considered without restriction.

11 Lagrangian formulation L(Q, z, λ 1, λ 2, λ 3, u) = with S : + + Q S 1 = B z t S 2 = Q t + T T L x q x ( Q2 Bz = Q t + x ( Q2 S 3 = µ u {m(u) + λ 3 S 3 }dt (15) {l(z, Q) + λ 1 S 1 + λ 2 S 2 }dxdt ) + gbz z x gbz(i J) kq Q Bz Bz gbz2 ) gbz(i J) kq Q Bz (16)

12 Necessary conditions for optimality: some adjoint PDEs (D. Georges & M.L. Chen, ECC 1) derived from the Lagrangian formulation + variational calculus: In order that both u(t) [, T ] and the trajectory of system: B z = q with and S : are optimal, t + Q x Q t + x ( Q2 Bz µ = u ) + gbz z x gbz(i J) = kq Q Bz Q(x, ) = φ 1 (x), x [, L] I.C. z(x, ) = φ 2 (x) µ() = φ 1 () { Q(, t) = µ(t) (control) B.C. z(l, t) = v(t) (disturbance) T.C. Q(x, T ) = Q (x), x [, L] z(x, T ) = z (x) µ(t ) = Q (x = ) (17) (18) (19) (2)

13 it is necessary that there exists λ = (λ 1 (x, t), λ 2 (x, t), λ 3 (t)), solution of the adjoint system of S: l z B λ 1 t + λ 2 x ( Q2 gbz) λ Bz 2 2 (gb(i J) gbz J z kq Q Bz 2 ) = S adj : l Q λ 1 x λ 2 t 2 λ 2 Q x Bz λ 2(kq 1 J gbz Bz Q ) = Q(, t) λ 1 (, t) + 2λ 2 (, t) Bz(, t) + λ 3 = with and I.T.C. B.C. λ 1 (x, ), λ 1 (x, T ), free, x [, L] λ 2 (x, ), λ 2 (x, T ), free λ 3 (), λ 3 (T ) free (21) { λ2 (, t) =, t [, T ] λ 1 (L, t) + 2λ 2 (L, t) Q(L,t) Bz(L,t) = (22)

14 For all t [, T ], the optimal control u = u, is necessary solution of: m (u(t)) λ 3 (t) = A two-point boundary value problem

15 Computation of the two-point boundary value problem Computation in two stages: 1. Spatial and temporal discretization of the canonical equations: Preissman numerical scheme (1962): applied to both S and S adj ; 2. Solution of the nonlinear algebraic equations derived from the two-dimensional grid via a Newton-Raphson method.

16 Preissman scheme: a finite-difference scheme semi-implicit in time Approximation of fonctions f and their derivatives: f (x, t) = 1 θ 2 [f i+1 + f i ] + θ 2 [f + i+1 + f + i ] f 1 θ (x, t) = x x [f i+1 f i ] + θ x [f + i+1 f + i ] f t (x, t) = 1 2 t [f + i f i + f + i+1 f i+1] (23) where i = spatial index, + = t + t and θ 1 = relaxation coefficient. If θ, 5, we get an unconditionally stable integration scheme.

17 An algebraic set of nonlinear equations For the case when m(u) = 1 2 u2, if N is the number of spatial discretization points and M, the number of temporal discretization points: S + B.C. (2N + 1) M unknown variables (z(x, t), Q(x, t), µ(t)) with (2N + 1) (M 1) equations + 4N + 2 constraints (the states (z, Q, µ) are imposed at t = and t = T ). S adj + B.C. (2N + 1) M unknown variables (λ 1 (x, t), λ 2 (x, t), λ 3 (t)) with (2N + 1) (M 1) equations. 4N M + 2M variables (z(x, t), Q(x, t), µ(t), λ 1 (x, t), λ 2 (x, t), λ 3 (t)) for 4N M + 2M equations: an implicit problem Computation of both state and adjoint state via a Newton-Raphson scheme.

18 The predictive control scheme 1. At time t : computation of the open-loop optimal control problem, with both initial and final states fixed: min u s.t. with S : t+t t m(u(t))dt + B z t + Q x Q t + x ( Q2 Bz µ = u I.C. t+t L t = q l(z(x, t), Q(x, t))dxdt (24) ) + gbz z x gbz(i J) = kq Q Bz Q(x, ) = Q t (x), x [, L] z(x, ) = z t (x) µ() = Q t () (25) (26) Q t (x) et z t (x) = states at time t on [, L], { Q(, τ) = µ(τ), τ [t, t + T ] (control) B.C. z(l, τ) = ˆv(τ) (prediction of downstream end level) (27)

19 The predictive control scheme 2 Apply u(t) defined as the first optimal control input sequence computed on [t, t + T ]: the system reaches state (z(x, t + t), Q(x, t + t)), x [, L]. 3 Go to 1), with t + t t and get z t (x) = z(x, t + t), Q t (x) = Q(x, t + t).

20 Some simulation results Simulation based on Preissman numerical scheme; A 5 km long canal divided into 1 sections of 5 m each; Regulation around a uniform equilibrium state corresponding to a constant relative water level z of 1.5 m, along the pool; Simulation starting from a uniform equilibrium of 1 m; At each time step: complexity of the two-point boundary value problem: N = 11 (spatial discretization) and M = 6 (temporal discretization, t = 1s): 4N M + 2M = 288 equations for 4N M + 2M = 288 unknown variables: computation time < 6 s on a Pentium 1,8 Mhz, 512 Mo Laptop: real-time control is possible.

21 Simulations results 3.9 Water flow rate 3.8 in m3/s Time in mns 1.8 Water level in meters Time in mns

22 Simulations results

23 The multi-pool case A two-pool case: (easy extension to n pools) : Regulator gate x = Pool 1 x = L 1 x + = L 1 Pool 2 x = L 2 Dynamics of the 2 pools: 2 X 2 PDE coupled by a gate model: Q 1 (L 1, t) = Q 2(L + 1, t) = Q(t) and Q such that Q 2 = K 2 α 2 2g(z 1 (L 1, t) z 2(L + 1, t)) F (Q(t), z 1 (L 1, t), z 2(L + 1, t), α(t)) =

24 Two-pool modelling S pool1 : S pool2 : B z 1 t + Q 1 Q 1 t x = q 1 + x ( Q2 1 z Bz 1 ) + gbz 1 1 x gbz Q 1(I J) = kq 1 1 Bz 1 µ = u 1 { (29) Q B.C. 1 (, t) = µ(t) Q 1 (L 1, t) = Q(t) (3) B z 2 t + Q 2 Q 2 t α = u 2 x = q 2 + x ( Q2 2 z Bz 2 ) + gbz 2 2 x gbz Q 2(I J) = kq 2 2 Bz 2 { Q2 (L + B.C. 1, t) = Q(t) z 2 (L 2, t) = v(t) (31) (32) Q(t) 2 = K 2 α 2 2g(z 1 (L 1, t) z 2(L + 1, t)), (33)

25 Optimal control formulation min u 2 [ i=1 T T bi m i (u i )dt + l i (z i (x, t), Q i (x, t))dxdt] (34) a i s.t. S pool1 + B.C. + I.C.+ T.C. and S pool2 + B.C + I.C.+ T.C. and the additional algebraic constraint defined on [, T ]: Q 2 = K 2 α 2 2g(z 1 (L 1, t) z 2(L + 1, t)).

26 Main issues How to reduce the computational complexity? How to take advantage of distributed control architecture (supervisory control and data acquisition: SCADA) used in large-scale water distribution systems? Here-proposed solution: use of a decomposition-coordination algorithm based on Lagrangian relaxation (D. Georges, Workshop NMPC 6)

27 An augmented lagrangian formulation = L c (Q 1, z 1, Q 2, z 2, Q, λ 1 1, λ 1 2, λ 1 3, λ 2 1, λ 2 2, λ 2 3, p, u 1, u 2 ) T T T {[m 1 (u 1 ) + λ 1 3[ Q 1 (, t) u 1 (t)]}dt L 1 T L2 {l 1 (z 1, Q 1 ) + λ 1 1S λ 1 2S 1 2 }dxdt {[m 2 (u 2 ) + λ 2 3[ α u 2 (t)]}dt L + 1 {l 2 (z 2, Q 2 ) + λ 2 1S λ 2 2S 2 2 }dxdt (35)

28 + T [(p + c 2 F (Q(t), z 1(L 1, t), z 2(L + 1, t))) F (Q(t), z 1 (L 1, t), z 2(L + 1, t), α)]dt (36) p is the dual variable associated to the gate constraint F an additional quadratic term c 2 F 2 is introduced An augmented Lagrangian L c

29 Main improvement of augmented Lagrangians: For sufficiently large c >, the augmented Lagrangian admits at least one local saddle-point (no duality gap) even for non convex problems. Convergence of relaxation algorithms is guaranteed

30 Some backgrounds on price-decomposition Augmented Lagrangian: min J(u 1, u 2 ) sous θ(u 1, u 2 ) =. (37) u=(u 1,u 2 ) L c (u, p) = J(u 1, u 2 )+ < p, θ(u 1, u 2 ) > + c 2 θ(u 1, u 2 ) 2 Duality approach: Computation of a saddle-point, solution of: max min L c (u, p) p u A price-decomposition algorithm (UZAWA): 1) Solve min u L c (u, p k ) u k+1 2) p k+1 = p k + ρθ(u1 k+1, u2 k+1 )

31 If both J(u 1, u 2 ) = J 1 (u 1 ) + J 2 (u 2 ) and θ(u 1, u 2 ) = θ 1 (u 1 ) + θ 2 (u 2 ) (separable case) and if c = (ordinary lagrangian formulation), then price decomposition-coordination: 1. Decomposition: Solve two independant subproblems min ui J i (u i )+ < p k, θ i (u i ) > u k+1 i, i = 1, 2 2. Coordination: p k+1 = p k + ρθ(u1 k+1, u2 k+1 )

32 If θ is not separable and c, separability via linearization of c. 2 : ALGORITHM (Cohen, 84) 1. Decomposition : min J i (u i )+ < p k +cθ(u u 1 k, u2 k ), θ i i(u1 k, u2 k ).u i > + 1 2ɛ u i ui k 2, = u k+1 i, where θ i is the gradient of θ w.r.t. u i. 2. Coordination: p k+1 = p k + ρθ(u k+1 1, u k+1 2 ) with < ρ c et < ɛ < 1/2c (for convex problems).

33 Application to the 2-pool AOCP 1. AOCP of pool 1: min u 1 T T m 1 (u 1 )dt + T L 1 l 1 (z 1 (x, t), Q 1 (x, t))dxdt + [< p k + cf k, F k u 1.u 1 > + 1 2ɛ (u 1 u k 1 ) 2 ]dt s.t. S pool1 + B.C. + I.C.+ T.C., = u1 k+1 (.), 2. AOCP of pool 2: min u 2 T m 2 (u 2 )dt + T L2 L + 1 l i (z 2 (x, t), Q 2 (x, t))dxdt + T p k + cf k, F k u 2.u 2 > + 1 2ɛ (u 2 u k 2 ) 2 ]dt, s.t. S pool1 + B.C. + I.C.+ T.C., = u2 k+1 (.), 3. Coordination: T min [< p k + cf k, F Q k.q > + 1 Q 2ɛ (Q Qk ) 2 ]dt, = Q k+1 (.), p k+1 = p k + ρf (Q k+1, z 1 (u k+1 1 ), z 2 (u k+1 2 ), α(u k+1 2 )) [<

34 Some potential advantages Complexity reduction thanks to the decomposition in sub-problems and parallel computation: In our case study, 2 problems of complexity 4(N + 1) M versus one problem of complexity 8(N + 1) M ; Well suited for distributed control application (Networked Control Systems).

35 A distributed predictive control scheme At each instant dt, parallel computation of the two-point boundary value subproblems POOL 1 Computation of a two-point boundary value problem (similar to a single-pool problem) The local variables z1 and u1 at iteration k are sent through the network POOL 2 Computation of a two-point boundary value problem (similar to a single-pool problem) The local variables z2 and u2 at iteration k are sent through the network NETWORK Coordination: The coordination variables are the Lagrangian multiplier p associated to the gate constraint and the flow rate Q Update of p and Q ; p and Q are sent through the network k+1 -> k Broadcast of variables through the communication network until convergence

36 Conclusions and perspectives 1. Practical extension of finite-dimensional predictive control techniques to an infinite-dimensional problem; 2. Computation of the related two-point boundary value problem using a 2D discretization method based on Preissman integration scheme; 3. Theoretical analysis still to be performed (see M. Herty s presentation); 4. Extension to the multi-pool case via a Lagrangian relaxation technique (Decomposition-Coordination) is possible, but has to be validated; 5. For practical implementation: need of a state observer (can be derived by using variational calculus as a dual control problem ).