Part II: Model Predictive Control

Transcription

1 Part II: Model Predictive Control Manfred Morari Alberto Bemporad Francesco Borrelli

2 Unconstrained Optimal Control Unconstrained infinite time optimal control V (x 0 )= inf U k=0 [ x k Qx k + u k Ru k] subj. to x k+1 = Ax k + Bu k, k 0, x 0 = x 0(1) where U = {u k } k=o Unconstrained finite time optimal control V (x 0 )= inf U N N 1 k=0 [ x k Qx k + u k Ru ] k + x N Px N subj. to x k+1 = Ax k + Bu k, k 0, x 0 = x 0(2) where U N = {u k } N 1 k=0 The solution is known!

3 Constrained Optimal Control Constrained infinite time optimal control V (x 0 )= inf U k=0 [ x k Qx k + u k Ru k] subj. to x k+1 = Ax k + Bu k, k 0, x 0 = x 0 U U x X (3) where U = {u k } k=o, x = {x k} k=0, U = {U u k U, k 0}, X = {x x k X, k 0} Constrained finite time optimal control V (x 0 )= N 1 [ inf x k Qx k + u U N k Ru ] k + x N Px N 1 k=0 subj. to x k+1 = Ax k + Bu t, k 0, x 0 = x 0 U U N x X N (4) where U = {u k } N 1 k=0, x = {x k} N k=0, U N = {U u k U, 0 k N 1}, X N = {x x k X, 0 k N} How to solve it???

4 Model Predictive Control r(t) Optimizer Input u(t) Plant Output y(t) Measurements MODEL : a model of the plant is needed to predict the future behaviour of the plant PREDICTIVE : optimization is based on the predicted future evolution of the plant CONTROL : control complex constrained multivariable plants

5 Optimization Problem past future Predicted outputs y(t+k t) Manipulated u(t+k) Inputs t t+1 t+m t+p input horizon output horizon Compute the optimal sequence of manipulated inputs which minimizes tracking error = output - reference subject to constraints on inputs and outputs On-line optimization Receding Horizon

6 Receding Horizon past future Predicted outputs Manipulated u(t+k) Inputs t t+1 t+m t+p t+1 t+2 t+1+m t+1+p Optimize at time t (new measurements) Only apply the first optimal move u(t) Repeat the whole optimization at time t +1 Advantage of on-line optimization: FEEDBACK!

7 Receding Horizon - An Example Chess Game

8 MPC Based on QP min U {u t,...,u t+n u 1} N y 1 k=0 [ x t+k t Qx t+k t + u t+k Ru ] t+k + +x t+n y t Px t+n y t subj. to y min y t+k t y max, k =1,...,N c u min u t+k u max, k =0, 1,...,N u x t t = x(t) x t+k+1 t = Ax t+k t + Bu t+k, k 0 y t+k t = Cx t+k t, k 0 u t+k = Kx t+k t, N u k<n y (5) x t+k t : predicted state vector at time t + k, obtained by applying the input sequence u t,...,u t+k 1 to system model starting from x(t). Q = Q 0, R = R 0, P 0, Q = C C with (C, A) detectable N y, N u, N c are the output, input, and constraint horizons, (N u N y and N c N y 1) The optimization problem (5) can be translated into a QP.

9 By substituting MPC Based on QP x t+k t = A k x(t)+ k 1 j=0 A j Bu t+k 1 j (6) in (5), this can be rewritten in the form V (x(t)) = min U subj. to 1 2 U HU + x (t)fu GU W + Ex(t) (7) where U [u t,...,u t+n u 1 ] R s, s mn u,isthe optimization vector, H = H 0, and H, F, Y, G, W, E obtained from Q, R, and (5) (6) Control policy: At time t compute U (t) ={u t,...,u t+n u 1 } Apply u(t) =u t (8) repeat the at time t + 1, based on the new state x(t +1).

10 MPC Based on QP - Tracking At time t: Measure (or estimate) current state x(t) Find the input sequence U {δu (t),δu (t +1),...,δu (t + N u )} solving min U J( U, t) N y k=0 w 2 y y(t + k t) r(t + k t) 2 + w 2 δu δu(t) 2 [δu(t) u(t) u(t 1)] subj. to u min u(t + k) u max δu min δu(t + k) δu max y min y(t + k t) y max Apply only u(t) =δu (t)+u(t 1), and discard δu (t +1),δu (t +1),... Repeat the same procedure at time t +1

11 Linear MPC - Example Amplitude Plant: G(s) = 1 s s Step Response T s =0.5 sec Model: x(t +1) = Time (sec.) [ ] x(t) [ ] 1 u(t) 0 y(t) = [ 0 1 ] x(t) Performance index: J(U, t) 10 k=0 [y(t + k t) r(t + k t)] δu 2 (t)

12 Linear MPC - Example Constraint 0.8 u(t) 1.2 (amplitude) Constraint 0.2 δu(t) 0.2 (slew-rate)

13 Anticipating/Causal MPC Reference not known in advance (causal): r(t + k t) r(t), k =0, 1,...,N y Future reference samples (partially) known in advance (anticipating): r(t + k t) = { r(t + k) if k =0,...,Nr r(t + N r ) if k = N r +1,...,N y Example:

14 MPC Based on LP min U {u t,...,u t+n u 1} subj. to N y 1 k=0 Qx t+k t + Ru t+k + Px t+ny t y min y t+k t y max, k =1,...,N c u min u t+k u max, k =0, 1,...,N u x t t = x(t) x t+k+1 t = Ax t+k t + Bu t+k, k 0 y t+k t = Cx t+k t, k 0 u t+k = Kx t+k t, N u k<n y (9) x t+k t : predicted state vector at time t + k, obtained by applying the input sequence u t,...,u t+k 1 to system model starting from x(t). Q, R, P nonsingular The optimization problem (9) can be translated into a LP.

15 MPC Based on LP Introduce slack variables {ε x 1,...,εx N y,ε u 1,...,εu N u } 1 n ε x k Qx t+k t k =1, 2,...,N y 1 1 n ε x k Qx t+k t k =1, 2,...,N y 1 1 r ε x N y Px t+ny t 1 r ε x N y Px t+ny t 1 m ε u k+1 Ru t+k k =0, 1,...,N u 1 1 m ε u k+1 Ru t+k k =0, 1,...,N u 1 where 1 k [1... 1] T, By substituting } {{ } k (10) x t+k t = A k x(t)+ k 1 j=0 A j Bu t+k 1 j in (9), this can be rewritten in the form V (x(t)) = min z subj. to c 1 z Gz W + Ex(t) (11) where z {ε x 1,...,εx N y,ε u 1,...,εu N u,u t,...,u t+nu 1} The implementation of MPC requires the on-line solution of a LP at each time step Same policy

16 Nonlinear MPC min J(x t, u N u N t ) t J(x t, u N N 1 t )= L(x t+i t,u t+i )+T(x t+n t ) subject to: i=0 x k+1 = f(x k,u k ) x t given u t+l U x t+l+1 t X system dynamics initial condition input constraints state contraints l =0,...,N 1 with: u N t = {u t,u t+1,...,u t+n 1 } N horizon length usually U := {u k R m u min u k u max } X := {x k t R n x min x k t x max }

17 Model Represent the plant Simple for computations Capture most significant dynamics Linear step/impulse response state space/transfer function Nonlinear first principle models multiple linear models artificial neural networks NARMAX models Robust Linear models + uncertainty description Hybrid Mixed logical dynamical systems

18 Features Multivariable non-square systems (i.e. #inputs #outputs) Optimal delay compensation Anticipating action for future reference changes Integral action, i.e. no offset for step-like inputs

19 Features Any model linear (state space, finite impulse response,... ) nonlinear (state space, neural network,... ) single variable/multivariable time delays constraints on inputs, outputs, states Any objective function sum of squared errors sum of absolute errors worst error over time horizon economic objective function BUT...

20 Features... BUT depending on model/objective function different computational complexity different (theoretical) performance guarantees, e.g. stability

21 Present Industrial Practice linear impulse/step response models sum of squared errors objective function executed in supervisory mode Particularly suited for problems with many inputs and outputs constraints varying objectives (e.g. because of fault) Tariq Samad, Honeywell (1997): For us multivariable control is predictive control

22 Present Industrial Practice MPC is capable of handling single variable loops with deadtime inverse response... BUT equivalent performance can be obtained with other simpler techniques HOWEVER MPC allows (in principle) UNIFORMITY (i.e. same technique for wide range of problems) reduce training reduce cost

23 Historical Goal : MPC - Theory Explain the success of DMC Present Goal : Improve, simplify, and extend industrial algorithms Areas : Linear MPC linear model Nonlinear MPC nonlinear model Robust MPC uncertain (linear) model MPC+Logic model integrating logic, NEW! dynamics, and constraints Issues : Feasibility Stability Open loop vs closed loop Computation

24 Feasibility Issues - An Example Consider the system { x(t +1) = and the MPC problem [ 1 x 1 0 t+k+1 t 0 1 k=0 [ ] 1 1 x(t)+ 0 1 subject to the input constraints ] [ ] 0 u(t) 1 x t+k+1 t +0.1u 2 t+k 1 u t+k 1, k =0, 1 and the state constraints 5 x t+k t 5, k =1, 2 Step 1 (feasible) x(0) = [ 4; 4] U =[ 1; 1], x pred (1) = [0; 3]; x pred (2) = [3; 2] Step 2 (feasible) x(0) = [0; 3] U =[ 1; 1], x pred (1) = [3; 2] x pred (2) = [5; 1] Step 3 x(0) = [3, 2]: problem infeasible!

25 MPC Feasibility When N c < there is no guarantee that the optimization problem (5) will remain feasible at all future time steps t. Setting N c =, optimization problem with an infinite number of constraints = impossible to handle. If the set of feasible state+input vectors is bounded and contains the origin in its interior, N y can be chosen ensuring feasibility of the MPC problem at all time instants. (theory of maximal output admissible set) References: E.C. Kerrigan and J.M. Maciejowski, Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control, Proc. 39th IEEE Conf. on Decision and Control, 2000.

26 Predicted and Actual Trajectories in Finite Horizon MPC Even assuming perfect model, no disturbances: predicted open-loop trajectories closed-loop trajectories Performance Goal: min i=0 L(x k+i,u k+i ) What is achieved by repeatedly minimizing N 1 i=0 L(x k+i,u k+i )? Stability Why should the closed-loop be stable?

27 MPC Stability Stability is a complex function of the tuning parameters N u, N y, N c, P, Q, andr. Impose conditions on N y, N c and P so that stability is guaranteed for all Q 0, R 0. Q and R can be freely chosen as tuning parameters to affect performance. Problem (5) is augmented with a so called stability constraint imposing over the prediction horizon explicitly forces the state vector either to shrink in some norm or to reach an invariant set at the end of the prediction horizon.

28 MPC Stability Additional constraints to ensure stability: Infinite Output/Prediction Horizon N y End constraint: (Rawlings and Muske, 1993) x(t + N y t) =0 Relaxed terminal constraint x(t + N y t) Ω (Kwon and Pearson, 1977) (Keerthi and Gilbert 88) (Scokaert and Rawlings, 1996) Contraction Constraint constraint x(t +1 t) α x(t),α<1 (Polak and Yang, 1993) (Bemporad, 1998) All the proofs use the value function V (t) = min U J(U, t) as a Lyapunov function

29 Stability Theorem Theorem 1 Consider the system { x(t +1) = φ(x(t),δu(t)), 0=φ(0, 0) y(t) = η(x(t)) 0 = η(0, 0) and let for simplicity r(t) 0. Then the MPC law based on min U J(U, t) = N y k=0 w 2 y y(t + k t) 2 + w 2 δu δu(t) 2 subj. to constraints for either N y or with the extra constraint x(t+n t) = 0 is such that lim t = 0 lim t = 0 for all w y,w u > 0. (Keerthi and Gilbert, 1988), (Bemporad et al., 1994)

30 Stability - Proof IDEA: Use V (t) =J(t, Ut ) as a Lyapunov function. Recall: J(U, t) = N y k=0 w 2 y y(t + k t) 2 + w 2 δu δu(t) 2 At time t + 1, extend the previous sequence and consider U shift {δu t (t),δu t (t +1),...,δu t (t + N u), 0} By construction, U shift is feasible at time t +1, and V (t +1) = J(t +1,U t+1 ) J(t +1,U shift) = = J(t, U t ) w y y(t) 2 w u δu(t) 2 = = V (t) w y y(t) 2 w u δu(t) 2 = V (t) is nonnegative and decreasing lim t V (t) lim t V (t) V (t +1)=0 w y y(t) 2 + w u δu(t) 2 V (t) V (t +1) 0 y(t), δu(t) 0

31 Example: AFTI F16 Linearized model: ẋ = x u 0 0 y = [ ] x, Inputs: elevator and flaperon angle Outputs: attack and pitch angle Sampling time: T s =.05 s (+ zero-order hold) Constraints: max 25 o on both angles Open-loop response: unstable (open-loop poles: , ± j, )

32 AFTI F16 - I N y = 10, N u =3,w y = {10, 10}, w δu = {.01,.01}, u min = 25 o, u max =25 o N y = 10, N u =3,w y = {100, 10}, w δu = {.01,.01}, u min = 25 o, u max =25 o

33 AFTI F16 - II N y = 10, N u =3,w y = {10, 10}, w δu = {.01,.01}, u min = 25 o, u max = 25 o, y 1,min = 0.5 o, y 1,max = 0.5 o N y = 10, N u =3,w y = {10, 10}, w δu = {.01,.01}, unconstrained MPC ( linear controller) + actuator saturation ±25 o UNSTABLE!

34 Tuning Parameters Guidelines Control more aggressive when N u : input horizon increased N y : output horizon decreased w y : output weight increased w δu : input weight decreased

35 MPC Extensions δu-formulation introduces m new states (the last input u(t 1)) Reference Tracking provides offfset-free fracking for constant reference the translated problem is min U subj. to 1 2 U HU + [ x (t) u (t 1) r (t) ] FU GU W + E Measured Disturbances x(t) u(t 1) r(t)

36 MPC Extensions Soft Constraints A slack variable ɛ is introduced into the constraints Ax b + ɛm, where Mis a matrix of suitable dimensions, and the term ɛ Eɛ is added to the objective to penalize constraint violations Variable Constraints the bounds y min, y max, δu min, δu max, u min, u max may change depending on the operating conditions the bounds can be treated as parameters in the mp-qp and added to the vector x Output Feedback state-feedback MPC controller and a linear state observer ˆx(t +1) =Aˆx(t) + L[y(t) Cˆx(t)] + Bu(t) can be combined together

37 MPC - Computations The on-line optimization problem is a Quadratic Program (QP) No need to reach global optimum (see proof of the theorem) Algorithms: Simplex method (small/medium size) Interior point methods (large size) Commercial and public domain software available Alternative: Linear Programming (with 1, norms), but worse performance

38 QP Computation Time Non-optimized Dantzig s routine Interior-point methods might be better for large scale Tests in Matlab 5.3 on Pentium II 300 MHz (MEX file from C code)

39 FHCOC and IHCOC FHCOC: Finite-Horizon Constrained Optimal Control IHCOC: Infinite-Horizon Constrained Optimal Control Problem P V (x 0 )= inf U [ xk Qx k + u k Ru k] k=0 subj. to x k+1 = Ax k + Bu k, k 0, x 0 = x 0 U U x X where U = {u k } k=o, x = {x k} k=0, U = {U u k U, k 0}, X = {x x k X, k 0} (12) Problem P N V (x 0 )= inf U N + k=0 [ xk Qx k + u k Ru k] subj. to x k+1 = Ax k + Bu k, k 0, x 0 = x 0 U U N x X N where U = {u k } N 1 k=0, x = {x k} N k=0, U N = {U u k U, 0 k N 1}, X N = {x x k X, 0 k N} (13) D. Chmielewski, V. Manousiouthakis, On constrained infinite-time linear quadratic optimal control, System and Control Letters, pp , 1996

40 FHCOC and IHCOC Problem P N can be equivalently rewritten as V (x 0 )= inf U N N 1 k=0 [ x k Qx k + u kru k ] + x N Px N subj. to x k+1 = Ax k + Bu k, k 0, x 0 = x 0 U U N x X N where U = {u k } N 1 k=0, x = {x k} N k=0, U N = {U u k U, 0 k N 1}, X N = {x x k X, 0 k N} (14) where P is the positive-definite solution to the algebraic Riccati equation Assuptions: 1. Q>0, R>0 2. X and U are closed,bounded and convex 3. 0 intx, 0 intx 4. x 0 X 0 = {x 0 R n u U with x X and V (x 0 ) < }

41 FHCOC and IHCOC Result: For a given x 0 exists N such that P N = P For all x 0 X 0 exists such N The proof is constructive. Uses the sets O = {x R n (A BK LQ ) k x X; k 0} X = {x R n x XK LQ x U} An upper bound on N can be found for every given polyhedral set of initial conditions

42 Example: Servo Motor R V θ M J M β M ρ T J L θ L βl Linear Model: k θ J ẋ = L β L k θ J L ρj L k θ ρj M 0 k θ ρ 2 J M β M+k 2/R T J M θ L = [ ] x [ ] T = k θ 0 k θ 0 ρ x x k T RJ M V Symbol Value (MKS) Meaning L S 1.0 shaft length d S 0.02 shaft diameter J S negligible shaft inertia J M 0.5 motor inertia β M 0.1 motor viscous friction coefficient R 20 resistance of armature K T 10 motor constant ρ 20 gear ratio k θ torsional rigidity Ĵ L 20J M nominal load inertia β L 25 load viscous friction coefficient

43 Amplitude To: Y(1) To: Y(2) Servo Motor - I R V θ M J M β M ρ T J L θ L βl Finite shear strength of steel shaft (τ adm = 50 N/mm 2 ) T Nm DC voltage limits V 220 V Sampling time: T s =.1 s (+ zero-order hold) 0.08 Step Response From: U(1) Time (sec.)

44 Servo Motor - II N y =10 N u =3 w y = {10, 0} w δu =.05 u min u max y min y max = 220 V = 220 V = {, } Nm = {, } Nm

45 Servo Motor - III N y =10 N u =3 w y = {10, 0} w δu =.05 u min u max y min y max = 220 V = 220 V = {, } Nm = {, } Nm

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47 Model Predictive Control Toolbox for Matlab/Simulink Manfred Morari N. Lawrence Ricker Alberto Bemporad

48 MPC Toolbox On the market MPC Toolbox v1.0 Under beta-testing MPC Graphical User Interface (GUI) MPC Simulink Library Under development MPC Toolbox v2.0

49 MPC Graphical User Interface Load and configure models (LTI, Simulink, Identification, MPC formats) Open-loop model testing MPC Controller design from model Closed-loop simulation Export controller to MPC object and Simulink Easy Design/Simulation of MPC (no M-script)

50 MPC Graphical User Interface Software and documentation:

51 MPC Simulink Library Software and documentation: