How To Solve A Plant Control Problem

Transcription

1 Model Predictive Control Lecture 1 (Palle Andersen) pa@es.aau.dk Automation & Control Aalborg University Denmark mpc1 p. 1/32

2 Book Predictive Control with Constraints by J.M. Maciejowski 1. Introduction 2. A Basic Formulation of Predictive Control 3. Solving Predictive Control Problems (4. Step Response and Transfer Function Formulations) 5. Other Formulations of Predictive Control 6. Stability 7. Tuning 8. Robust Predictive Control (9. Two Case Studies) 10. Perspectives mpc1 p. 2/32

3 Software Suggested: Multi Parametric Toolbox (MPT) MPT is very flexible, and can be used for PWA systems. mpt/downloads/mptmanual.pdf You can also use MatLab s own MPC toolbox, or even set things up manually for quadprog.m. mpc1 p. 3/32

4 Course plan Lecture 1: motivation, constraints, quadratic problems, MPT Literature: Maciejowski, Chapters 1-3 Lecture 2: soft constraints, stability, feasibility Literature: Maciejowski, 3.4, Chapters 5-6 Lecture 3: robustness, recursive feasibility, linear programs Literature: Maciejowski, Chapter 8, papers Lecture 4: MH estimation, time-varing dynamics Literature: Maciejowski, 10, papers Lecture 5: explicit MPC, nonlinear systems mpc1 p. 4/32

5 Basic idea of MPC Predict future plant response using a model over a prediction horizon of H p samples. Use a prediction of the reference and disturbances. Set up cost function over the horizon. Define constraints on inputs and states/outputs. Determine control inputs over a control horizon H u H p that minimise the cost function. Apply first sample of determined inputs. Shift the horizon and repeat the procedure (receding horizon). mpc1 p. 5/32

6 Basic idea of MPC Known Future Reference Predicted output Depend of control Predicted Free response k k+hp Predicted control u k k+hu mpc1 p. 6/32

7 Motivation The achievable performance of control problems is often dominated by constraints rather than plant dynamics. Constraints can be a more natural way to specify requirements than through penalties on variations from set-point. Motivating example: Buffer tank. We want to reduce the strain on the producer. We do not care about the tank level, only that it does not run empty or overflow. Nonlinear behaviour of MPC can improve performance. mpc1 p. 7/32

8 Pros and cons Advantages: Constraints are respected. Natural requirement specification. Anti-windup follows automatically. Standard tools for solving optimisation. Fairly natural extensions to nonlinear/hybrid systems. Disadvantages: A finite (and sometimes short) horizon is necessary problems with stability and feasibility. Essentially state feedback uncertainty (disturbances, model errors) is problematic. Complexity. mpc1 p. 8/32

9 MPC in the hierarchy Local loops should handle fast dynamics. Set-point optimisation is often done by static considerations. mpc1 p. 9/32

10 Prediction model In principle, any dynamic model can be used. We will focus on linear state space models. x(k +1) = Ax(k)+Bu(k) y(k) = C y x(k) z(k) = C z x(k) y: measurements z: performance outputs, constrained signals mpc1 p. 10/32

11 Delta-formulation We are usually interested in minimising/constraining u rather than u. [ ] x(k +1) u(k) = [ A B 0 I ][ ] x(k) u(k 1) + [ ] B I u(k) Alternatively, we can integrate the outputs instead: [ ] [ ][ ] [ ] x(k +1) A 0 x(k) B = + u(k) z(k +1) C z A I y(k) C z B The main difference arises when formulating constraints. mpc1 p. 11/32

12 Cost function Most commonly we use a quadratic cost function, penalising the control signal derivative: V(k) = H p i=h w ẑ(k +i k) r(k +i k) 2 Q(i) + H u 1 i=0 û(k +i k) 2 R(i) ẑ(k + i k) are predictions, assuming inputs û(k + i k). Note that Q and R may vary over the horizon. mpc1 p. 12/32

13 Constraints in MPC The main difference between LQ and MPC is that MPC handles constraints in the optimisation. Constraints in control moves are introduced as matrix inequalities in all u s with one row per constraint. For example, if a plant has two inputs with constraints in u 1 and constraints in u 2 2 u 1 2 { 1 2 u u In the simple case with a control horizon H u = 1 this gives [ ] 1 û 1 (k k) [ ] [ ] [ U(k) 1 û 2 (k k) E } 0 0 ] mpc1 p. 13/32

14 Constraints in MPC Similarly constraints in u may be rewritten like { } u u u With a control horizon H u = 1 this gives [ ] û 1 (k k) [ ] û 2 (k k) F [ U(k) 1 ] [ 0 0 ] For every constraint in a control move u j and every constraint in a u j we introduce H u inequalities. With lower and upper bounds u and u of l controls we potentially have dimensions 2lH u H u +1 of both E and F mpc1 p. 14/32

15 Constraints in MPC It is possible also to formulate constraints in outputs z z 1 0 z 2 0 3z 1 +5z 2 15 z 1 0 z z z Horizon H p (= 2) introduce up to m z H p inequalities ẑ 1 (k +1 k) ẑ 2 (k +1 k) [ ] ẑ 1 (k +2 k) G 0 ẑ 2 (k +2 k) [ Z(k) 1 ] mpc1 p. 15/32

16 Predictive Control Problem The control problem is now to minimise V(k) = where H p i=h w ẑ(k +i k) r(k +i k) 2 Q(i) + H u 1 i=0 û(k +i k) 2 R(i) ˆx(k +i+1 k) = Aˆx(k +i k)+bû(k +i k) ẑ(k +i k) = C zˆx(k +i k) subject to the constraints [ ] [ ] [ U(k) 0 E, F 1 0 U(k) 1 ] [ 0 0 ], G [ Z(k) 1 ] [ 0 0 ] mpc1 p. 16/32

17 Solving Predictive Control Problems Without constraints we have seen earlier that the problem can be solved using dynamic programming The problem with constraints has usually been solved by solving a minimisation problem which is quadratic in control moves U To do this we first have to express U and Z in terms of U mpc1 p. 17/32

18 Prediction of future states ˆx(k +1) = Aˆx(k)+Bû(k) ˆx(k +2) = Aˆx(k +1)+Bû(k +1) = A 2ˆx(k)+ABû(k)+Bû(k +1). ˆx(k +Hp) = A Hpˆx(k)+A Hp 1 Bû(k)+ +Bû(k +Hp 1) In model predictive control we want to express future outputs in terms of control moves u(k) = u(k) u(k 1) and furthermore we allow only H u future control moves. mpc1 p. 18/32

19 Prediction of future states Predictions of future in terms of predicted control moves. ˆx(k +1) = Aˆx(k)+B( û(k)+u(k 1)) ˆx(k +2) = A 2ˆx(k)+(AB +B)u(k 1) + AB û(k)+b û(k +1). ˆx(k +Hp) = A Hpˆx(k)+ + H p 1 i=0 H p 1 i=1 A i Bu(k 1) A i B u(k)+ + H p H u i=0 A i B û(k +H u 1 k) mpc1 p. 19/32

20 Prediction of future states Defining X(k) = ˆx(k +1 k). ˆx(k +H p k), U(k) = û(k k). û(k +H u 1 k), We may find matrices A, B u and B u such that X(k) = Aˆx(k k)+b u u(k 1)+B u U(k) mpc1 p. 20/32

21 Prediction of future outputs We define vectors with future outputs and references Z(k) = ẑ(k +H w k). ẑ(k +H p k) = ˆr(k +H w k). ˆr(k +H p k) Further define C = diag(c z ), Ψ = CA, Υ = CB u and Θ = CB u and find Z(k) = Ψˆx(k k)+υu(k 1)+Θ U(k) we also define the vectors of future predicted errors with U(k) = 0 E(k) = T (k) Ψˆx(k k) Υu(k 1) mpc1 p. 21/32

22 Rewriting performance To rewrite the performance we further define the extended weighting matrices Q = R = Q(H w ) Q(H w +1) Q(H p ) R(0) R(1) R(H u ) mpc1 p. 22/32

23 Rewriting performance With vectorized signals performance may be rewritten V(k) = Z(k) T (k) 2 Q + U(k) 2 R or if we insert the expression for Z(k) and subsequently E(k) we may express the performance in terms of signals which are known (if the state vector is known) at time k and U(k) V(k) = Θ U(k) E(k) 2 Q + U(k) 2 R mpc1 p. 23/32

24 Solving the unconstrained case We can now easily find the solution in the case where we have no constraints (which may also be the solution if no constraints are active). V(k) = Θ U(k) E(k) 2 Q + U(k) 2 R V(k) = E(k) T QE(k) 2 U(k)Θ T QE(k)+ U(k) T [ΘQΘ+R] U(k) with V(k) = const U(k)G + U(k) T H U(k) G = 2Θ T QE(k), H = ΘQΘ+R mpc1 p. 24/32

25 Solving the unconstrained case The unconstrained control law can be solved by finding the gradient and putting this to zero U(k) = G +2H U(k) = 0 U(k) opt = 1 2 H\G Now we pick out the control move at time k as the first l rows of U(k) opt u(k) opt = [I l 0 l 0 l ] U(k) opt This control law is equivalent to a finite horizon LQ controller with feedback from the current state vector (of which u(k 1) should be an element in this case). The control law also has feed forward from the reference trajectory T (k) which is equal to the term we found using LQ mpc1 p. 25/32

26 Reformulation of the constrained case In the case where constraints are present we want to express these in terms of U(k) and variables known at time k. Since U(k) can be found from u(k 1) and U(k) we may find F,F 1,f such that F [ U(k) 1 ] [ 0 0 ] F U(k) F 1 u(k 1) f mpc1 p. 26/32

27 Reformulation of the constrained case We have also found Z(k) = Ψˆx(k k)+υu(k 1)+Θ U(k) and if we put G = [Γ g] the constraints in z may be rewritten like [ ] [ ] Z(k) 0 G ΓΘ U(k) Γ[Ψˆx(k k)+υu(k 1)] g 1 0 Finally putting E = [W w] we may rewrite [ ] [ ] U(k) 0 E, W U(k) w 1 0 mpc1 p. 27/32

28 Quadratic program The optimisation problem: minimise U(k)G + U(k) T H U(k) subject to F ΓΘ W U(k) F 1 u(k 1) f Γ[Ψˆx(k k)+υu(k 1)] g w This is a quadratic programming problem (QP): 1 min θ 2 θt Φθ +φ T θ, s.t. Ωθ ω mpc1 p. 28/32

29 Solving quadratic programs 1 min θ 2 θt Φθ+φ T θ s.t. Ωθ ω QP s are convex and we have efficient solvers, for instance quadprog.m in MatLab. MPT (or the MPC Toolbox) converts the plant and problem description into a quadratic program, which can then be solved by e.g. quadprog. mpc1 p. 29/32

30 Observers The prediction requires an estimate ˆx(k k) of the state. If the entire state vector is not measured then an observer must be used. The optimisation problem is the same as before, but sometimes uncertainty must be taken into account. State/output constraints can lead to feasibility problems when the state is uncertain. mpc1 p. 30/32

31 Recap An MPC design requires A prediction model A choice of prediction and control horizon lengths A definition of constraints A cost function Minimizing the cost over input signals mpc1 p. 31/32

32 Exercises 1) Design an MPC for a two-tank system. 2) Consider MPC in your project. mpc1 p. 32/32