# Model Predictive Control Lecture 5

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1 Model Predictive Control Lecture 5 Klaus Trangbæk Automation & Control Aalborg University Denmark. mpc5 p. 1

2 Exercise from last time 1) Assume that you know linear constraints on the states and process noise. Show that the moving horizon estimator is a QP. Assume that we choose w (on the finite horizon) and x a as decision variables. The first step is to set up a vectorised predictor in terms of these: X = Γx a + BW. Secondly express the performance function in terms of the decision variables. This is done by inserting V = Y m ΛX. The file MHEQP.m sets up the predictor and performance weights. 2) Have a look at the three tank simulation. Replace the saturating observer with an MHE using quadprog to solve. MHE3tanks.m A simpler example is found in MHEsimple.m mpc5 p. 2

3 Today Explicit MPC Nonlinear MPC Summary of the course mpc5 p. 3

4 Explicit MPC It can be shown (Bemporad et al. 2002) that LQ MPC can be formulated as a piece-wise affine state feedback. We then get the explicit formulation: u k = K i x k + d i, x k X i, which is much faster than online optimisation. The number regions can be gigantic. Locating the current region can then be time consuming in itself. (mpt_demo4.m) mpc5 p. 4

5 Nonlinear MPC Nonlinear models are usually formulated in continuous time: dx dt = f(x,u) The control signal must be parameterised by a finite number of decision variables, e.g. by discretising in time or using splines. Closing inner loops increases chance of success. mpc5 p. 5

6 Nonlinear MPC The simplest method is probably sequential QP: Linearise along trajectory -> gradient and Hessian Solve QP -> optimal control sequence Recompute trajectory Repeat until convergence A faster alternative: Linearise along trajectory (using tail) -> gradient and Hessian Solve QP -> optimal control sequence Apply first sample of control sequence mpc5 p. 6

7 Simple nonlinear example Discrete time x k+1 = tanh(0.9x 3 k + u k) Nonlex.m mpc5 p. 7

8 Nonlinear MPC example Continuous time dx dt = 2tanh((x + 1)u) x 3 Linearising by simulation of small changes to input. (Linalong.m) Exc5.m, Exc5c.m mpc5 p. 8

9 Nonlinear MPC Direct collocation: Instead of approximating the performance function, we can approximate the model by linear constraints -> sparse NLP. Multiple shooting: A mixture of sequential QP and direct collocation. The performance function is approximated on intervals. The intervals must be connected -> constraints. mpc5 p. 9

10 Literature 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 mpc5 p. 10

11 Literature Predictive Control with Constraints by J.M. Maciejowski Supplementary: The MPT manual Rao 99 on moving horizon estimation. Kothare 96 on robust MPC Bemporad 02 on explicit MPC Boyd and Vandenberghe: Convex Optimization mpc5 p. 11

12 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). mpc5 p. 12

13 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. Nonlinear behaviour of MPC can improve performance. mpc5 p. 13

14 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. mpc5 p. 14

15 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 mpc5 p. 15

16 Soft constraints Infeasibility causes the solver to yield useless results. For some constraints we can accept a violation for a short period. Soft constraints are added through the performance index: 1 min θ 2 θ T Φθ + φ T θ +ρ ǫ s.t. Ωθ ω +ǫ ǫ 0 Soft constraints increase the number of decision variables. mpc5 p. 16

17 Linear programs For some systems, a 1-norm cost: V (k) = H p m q j ẑ j (k +i k) r j (k +i) + H u p r j û j (k +1 k) i=1 j=1 i=1 j=1 is more natural. For instance, the actual cost of fuel consumption is linear. An -norm is also possible: V (k) = max i max j q j ẑ j (k + i k) r j (k + i) 1- and -norms result in linear programs. mpc5 p. 17

18 Inner loop state feedback Adding a state feedback in an inner loop has several purposes: Stabilising predictions. Dual mode schemes. Constraint handling as perturbations to optimal control. Make behaviour after control horizon more realistic. mpc5 p. 18

19 Stability Stability problems can occur because We must use a finite horizon to get a finite dimensional problem. The finite horizon causes prediction mismatch. The optimisation is solved online analysis is difficult. In summary, the main problem is that we ignore what happens after the horizon. Stability can be ensured by ensuring that the terminal state is forced into a sufficiently small set and penalised. mpc5 p. 19

20 Terminal constraints A terminal constraint specifies that the state must be in a given set by the end of the prediction horizon. The the terminal constraint set should be chosen so that the state trajectories, after entering, stay inside, i.e an invariant set. mpc5 p. 20

21 Infinite horizons Optimising over an infinite horizon yields stability. On an infinite horizon, the tail will be the first part of the optimal solution the cost function serves as a Lyapunov function. We can write the infinite horizon cost as V (k) = = i=1 H u i=1 + H u x(k + i k) 2 Q + i=1 u(k + i 1 k) 2 R x(k + i k) 2 Q + x(k + H u + 1) T Qx(k + Hu + 1) H u i=1 u(k + i 1 k) 2 R mpc5 p. 21

22 Robustness Robust control can be used for dealing with uncertainty and nonlinearities. Assume that the system matrices will belong to a convex set [A(k),B(k)] Ω. We would like to minimise the worst case performance: min U(k) max V (k) A(k+j),B(k+j) We approximate this by using the performance function as a Lyapunov function and minimising at each sample: V (x(k + j + 1)) V (x(k + j)) x(k + j) 2 Q 1 u(k + j) 2 R and V (x) = x T Px, P > 0. mpc5 p. 22

23 Robustness We now assume u(k + j) = K k x(k + j) and want to find K k that minimises V while respecting the constraints. With standard manipulations, this can be as an LMI in the auxiliary variables Q = γp 1 and Y = K k Q. The LMIs must be fulfilled for the infinitely many [A(k),B(k)] Ω. The LMIs are affine in A and B, so if Ω is a convex polytope, we only need to satisfy the LMIs at the vertices of the polytope, which is a finite-dimensional problem. mpc5 p. 23

24 Adding constraints The above method makes it fairly easy to add constraints. To ensure feasibility on an infinite horizon, we often employ the concept of invariant sets: For an autonomous system x k+1 = Fx k, S is an invariant set if x k S x k+1 = Fx k S. So if the constraints are satisfied for any x S, and S is invariant, then the constraints are satisfied forever. With some conservatism, constraints can be cast as LMIs. mpc5 p. 24

25 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. mpc5 p. 25

26 Observers Because of the constraints, we no longer have perfect separation of observer and state feedback. Methods: Separation, e.g. by high gain observers. Direct: considering the state uncertainty in the control. I/O-models. mpc5 p. 26

27 Constrained estimation One type of observer is a constrained estimator: Past values of states and inputs (disturbances) are known to respect constraints. Constraints provide extra information. Often, noise distributions are very uncertain but some limits are known. A finite horizon is necessary. Linear constrained estimation is a QP. mpc5 p. 27

28 Constrained estimation x k+1 = Ax k + Gw k, y k = Cx k + v k where w, v have covariances Q, R. We can formulate the problem min x N M,{w k } N 1 k=n M x N M ˆx N M N M 1 2 Π 1 N M + N 1 k=n M ( v k 2 R 1 + s.t. the system equations and x X, w W. This is a QP, i.e. constrained estimation is a dual of constrained control. Online optimisation -> extension to nonlinear systems. mpc5 p. 28

29 Online optimisation Online optimisation gives a lot of flexibility: the model can be changed online. This allows for adaptive/fault tolerant control. The model does not have to be constant over the horizon. This allows for time varying and even nonlinear models. mpc5 p. 29

30 Nonlinear MPC Nonlinear models are usually formulated in continuous time: dx dt = f(x,u) The control signal must be parameterised by a finite number of decision variables, e.g. by discretising in time or using splines. Closing inner loops increases chance of success. mpc5 p. 30

31 Nonlinear MPC The simplest method is probably sequential QP: Linearise along trajectory -> gradient and Hessian Solve QP -> optimal control sequence Recompute trajectory Repeat until convergence A faster alternative: Linearise along trajectory (using tail) -> gradient and Hessian Solve QP -> optimal control sequence Apply first sample of control sequence mpc5 p. 31

32 Nonlinear MPC Direct collocation: Instead of approximating the performance function, we can approximate the model by linear constraints -> sparse NLP. Multiple shooting: A mixture of sequential QP and direct collocation. The performance function is approximated on intervals. The intervals must be connected -> constraints. mpc5 p. 32

33 Using MPT MPT (or the MPC Toolbox) converts the plant and problem description into a quadratic program, which can then be solved by e.g. quadprog. MPT can handle many things beyond linear MPC, e.g. PWA, MILP, nonlinear (polynomials). The your MPC feature adds even more flexibility. However, setting up the QP by hand gives more control over details. mpc5 p. 33

34 Conclusions The main feature of MPC is constraint handling. Uncertainty (noise, model) cause problems (just as with most nonlinear methods). State constraints cause feasibility problems. It is possible to use standard solvers. Closing inner loops is usually a good idea. mpc5 p. 34

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