Model Predictive Control of Hybrid Systems. Model Predictive Control of Hybrid Systems. Receding Horizon - Example. Receding Horizon Philosophy

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1 Model Predictive Control of Hybrid Systems Alberto Bemporad Dip. di Ingegneria dell Informazione Università degli Studi di Siena Università degli Studi di Siena Facoltà di Ingegneria Reference r(t) Model Predictive Control of Hybrid Systems Controller Input u(t) Measurements Hybrid System MODEL: a model of the plant is needed to predict the future behavior of the plant PREDICTIVE: optimization is based on the predicted future evolution of the plant CONTROL: control complex constrained multivariable plants Output y(t) At time t : Receding Horizon Philosophy Solve an optimal control problem over a finite future horizon p : minimize y à r + ú u t t+ r(t) Predicted outputs Manipulated Inputs u(t+k) t+p Receding Horizon - Example MPC is like playing chess! subject to constraints u min ô u ô u max y min ô y ô y max Only apply the first optimal move t+ t+ t+p+ u ã (t) Get new measurements, and repeat the optimization at time t + Advantage of on-line optimization: FEEDBACK!

2 MPC for Hybrid Systems Closed-Loop Stability past future t t+ Predicted outputs y(t+k t) Manipulated Inputs u(t+k) t+t Model Predictive (MPC) Control At time t solve with respect to the finite-horizon open-loop, optimal control problem: (Bemporad, Morari 999) Apply only u(t)=u ã (t) (discard the remaining optimal inputs) Repeat the whole optimization at time t+ Proof: Easily follows from standard Lyapunov arguments Stability Proof Hybrid MPC - Example Switching System: " # ô cosë(t) à sinë(t) x(t +)=.8 x(t) + õ u(t) sinë(t) cosë(t) Lyapunov function y(t) = [ ] x(t) ù if [ ]x(t) õ ë(t) = 3 ù à if [ ]x(t) < 3 Closed loop: y(t), r(t) u(t) Constraint: à ô u(t) ô Open loop: time t time t Note: Global optimum not needed for convergence!

3 MIQP Formulation of MPC (Bemporad, Morari, 999) Optimal Control of Hybrid Systems: Computational Aspects Mixed Integer Quadratic Program (MIQP) MILP Formulation of MPC Introduce slack variables: ï x k ï u k õ kqy(t + k t)k õ kru(t + k)k min x ï x k ï x k ï u k ï u k Set ø,[ï x,...,ïx N y,ï u,...,ï Tà,U,î,z] Mixed Integer Linear Program (MILP) (Bemporad, Borrelli, Morari, ) min ï s.t. ï õ x ï õàx õ [Qy(t + k t)] i i =,...,p, k =,..., T à õ à[qy(t + k t)] i i =,...,p, k =,...,Tà õ [Ru(t + k))] i i =,...,m, k=,...,tà õ à[ru(t + k))] i i =,...,m, k=,...,tà Tà min J(ø, x(t)) = X ï x + i ïu i ø k= s.t.gø ô W + Sx(t) Mixed-Integer Program Solvers Mixed-Integer Programming is NP-hard Phase transitions have been found in computationally hard problems. BUT Ratio of Constraints to Variables (Monasson et al., Nature, 999) General purpose Branch & Bound/Branch & Cut solvers available for MILP and MIQP (CPLEX, Xpress-MP, BARON, GLPK,...) More solvers and benchmarks: No need to reach global optimum (see proof of the theorem), although performance deteriorates but not for fast sampling (e.g. ms) / cheap hardware! Cost of Computation var 4 var var Good for large sampling times (e.g., h) / expensive hardware

4 On-Line vs. Off-Line Optimization Explicit Form of Model Predictive Control via Multiparametric Programming min J(U, x(t)), P U k= MLD model subj. to x(t t) =x(t) x(t + T t) = Tà kqy(t + k + t)k + kru(t + k)k On-line optimization: given x(t), solve the problem at each time step t Mixed-Integer Linear Program (MILP) Off-line optimization: solve the MILP for all x(t) min J(ø, x(t)),f ø ø s.t.gø ô W + Fx(t) multi-parametric Mixed Integer Linear Program (mp-milp) Example of Multiparametric Solution Multiparametric LP ( ø R ) 6 CR{,4} CR{,,3} Linear Model: Linear MPC 4 x - -4 CR{,3} CR{,3} Constraints: Optimal control problem (quadratic performance index): x

5 Substitution: Linear MPC Multiparametric Quadratic Programming (Bemporad et al., ) Optimization problem: Convex QUADRATIC PROGRAM (QP) (quadratic) (linear) Objective: solve the QP for all Assumption: (always satisfied if QP problem originates from optimal control problem) x X Linearity of the Solution solve QP to find identify active constraints at form matrices by collecting active constraints Determining a Critical Region Impose primal and dual feasibility: linear inequalities in x! Remove redundant constraints KKT optimality conditions: From () : critical region CR x CR = {Ax ôb} CR x-space From () : In some neighborhood of x, λ and U are explicit affine functions of x! X CR is the set of all and only parameters x for which Gà,Wà, Sà is the optimal combination of active constraints at the optimizer

6 X X Multiparametric QP Multiparametric QP R R x CR R 3 CR = {Ax ôb} R i = {x X : A i x>b i, A j z ôb j, j<i} R R x CR R 3 CR = {Ax ôb} R i = {x X : A i x>b i, A j z ôb j, j<i} x-space R N R 4 X Theorem: {CR,R,...,R N} is a partition of X ò R n Proceed iteratively: for each region R i repeat the whole procedure with X R i x-space R N R 4 X Theorem: {CR,R,...,R N} is a partition of X ò R n Note: while CR is characterizing a set of active constraints, R i is not The recursive algorithm terminates after a finite number of steps, because the number of combinations of active constraints is finite Keep track of the CR already explored, don t split CRs Mp-QP More efficient method (Tøndel, Johansen, Bemporad, 3) Mp-QP Properties x x x CR CR CR continuous, piecewise affine x The active set of a neighboring region is found by using the active set of the current region + knowledge of the type of hyperplane we are crossing: x The corresponding constraint is added to the active set The corresponding constraint is withdrawn from the active set x convex, continuous, piecewise quadratic, C (if no degeneracy) Corollary: The linear MPC controller is a continuous piecewise affine function of the state

7 .5.5 Complexity Reduction CR CR CR3 CR4 CR5 CR6 CR7 CR8 CR9.5.5 CR CR CR3 CR4 CR5 CR6 CR7 System: Double Integrator Example ô y(t) = s u(t) x(t +)= õ ô x(t)+ õ u(t) sampling + ZOH T s = s y(t) = [ ] x(t) Constraints: à ô u(t) ô Control objective: minimize P y (t)y(t)+ t= u (t) u t+k = K LQ x(t + k t) k õ N u Optimization problem: for N u = Regions where the first component of the solution is the same can be joined (when their union is convex). (Bemporad, Fukuda, Torrisi, Computational Geometry, ) (cost function is normalized by max λ(h)) mp-qp solution Complexity N u =3 N u =4 N u =5 N u = N u =6 CPU time # Regions # free moves

8 Worst- M, P`= q Complexity case complexity analysis: q ` ( ) = q N r ô PMà k!q k k= combinations of active constraints upper- bound to the number of regions Extensions Tracking of reference r(t) : îu(t) =F(x(t),u(t à ),r(t)) Rejection of measured disturbance v(t) : îu(t) =F(x(t),u(t à ),v(t)) Numerical Tests: Number of regions in the state-space partition Soft constraints: y min à ï ô y(t + k t) ô y max + ï u(t) =F(x(t)) Variable constraints: u(t) =F(x(t),u min (t),...,y max (t)) Computation time (s) [Matlab 5.3, Pentium II 3MHz] u min (t) ô u(t + k) ô u max (t) y min (t) ô y(t + k t) ô y max (t) Linear norms: min J(U, x(t)), P p kqy(t + k t)k U k= + kru(t + k)k (Bemporad, Borrelli, Morari, IEEE TAC, ) Closed-Loop MPC and Hybrid Systems MPC Regulation of a Ball on a Plate Task: Tune an MPC controller by simulation, using the MPC Simulink Toolbox DHA Get the explicit solution of the MPC controller. Validate the controller on experiments. Motivation: Use hybrid techniques to analyze closed-loop MPC systems! (Bemporad, Heemels, De Schutter IEEE TAC, )

9 Ball&Plate Experiment General Philosophy: () MPC Design y Step : Tune the MPC controller (in simulation) δ γ x Specifications: Angle: -7 deg +7deg Plate: -3 cm +3 cm Input Voltage: - V + V Computer: PENTIUM66 Sampling Time:3 ms α α' β' β Model: LTI 4 states Constraints on inputs and states E.g: MPC Toolbox for Matlab (Bemporad, Morari, Ricker, 3) General Philosophy: () Implementation MPC Tuning Step : Solve mp- MPC QP and implement Explicit Sampling time: Prediction horizon: T s = 3 ms y p = 5 u Free control moves: m = m p Output constraint horizon: Input constraint horizon: Weight on position error: (soft constraint) (hard constraint) 5 Weight on input voltage changes: E.g: Real- Time Workshop + xpc Toolbox

10 Controller: Explicit MPC Solution o x-mpc: sections at α x =, α x =, u x =, r x =8, r α = 6 x: Regions, y: 3 Regions MPC Regulation of a Ball on a Plate Design Steps: Tune an MPC controller by simulation, using the MPC Simulink Toolbox. Get the explicit solution of the MPC controller. Region : Region 6: Region 6: LQR Controller (near Equilibrium) Saturation at - Saturation at + Validate the controller on experiments. Comments on Explicit MPC MILP Formulation of MPC (Bemporad, Borrelli, Morari, ) Multiparametric Quadratic Programs (mp-qp) can be solved efficiently Model Predictive Control (MPC) can be solved off-line via mp-qp Explicit solution of MPC controlleru = f(x) is Piecewise Affine Introduce slack variables: min x min ï s.t. ï õ x ï õàx ï x k ï u k õ õ kqy(t + k t)k kru(t + k)k ï x k ï x k ï u k ï u k õ [Qy(t + k t)] i i =,...,p, k =,..., T à õ à[qy(t + k t)] i i =,...,p, k =,...,Tà õ [Ru(t + k))] i i =,...,m, k=,...,tà õ à[ru(t + k))] i i =,...,m, k=,...,tà Eliminate heavy on-line computation for MPC Make MPC suitable for fast/small/cheap processes Set ø,[ï x,...,ïx N y,ï u,...,ï Tà,U,î,z] Mixed Integer Linear Program (MILP) Tà min J(ø, x(t)) = X ï x + i ïu i ø k= s.t.gø ô W + Sx(t)

11 Multiparametric MILP min f ø c + d ø d ø={øc,ø d } s.t. Gø c + Eø d ô W + Fx ø c R n ø d {, } m mp-milp can be solved (by alternating MILPs and mp-lps) (Dua, Pistikopoulos, 999) Theorem: The multiparametric solution ø ã (x) is piecewise affine Corollary: The MPC controller is piecewise affine in x Solutions via Dynamic Programming (Borrelli, Bemporad, Baotic, Morari, 3) (Mayne, ECC ) Explicit solutions to finite- time optimal control problems for PWA systems can be obtained using a combination of Dynamic Programming Multiparametric Linear (- norm, - norm), or or Quadratic (squared - norm) programming Note: in the -norm case, the partition may not be polyhedral Hybrid Control - Example Switching System: Hybrid Control Example (Revisited) " # ô cos ë(t) à sinë(t) x(t +)=.8 x(t) + õ u(t) sinë(t) cos ë(t) y(t) = [ ] x(t) ù if [ ]x(t) õ ë(t) = 3 ù à if [ ]x(t) < 3 Constraint: à ô u(t) ô Closed loop: y(t), r(t) u(t) Open loop: time t time t

12 Hybrid MPC - Example mp-milp Solution MLD system State x(t) Input u(t) Aux. binary vector δ(t) Aux. continuous vector z(t) variables variables variables 4variables Prediction Horizon T= mp- MILP optimization problem minn oj(v,x(t)),p kq k= (v(k) à u e)k + kq (î(k t) à î e)k + kq 3(z(k t) à z e)k + kq 4(x(k t) à x e)k v subject to constraints to be solved in the region à 5 ô x ô 5 à 5 ô x ô 5 Computational complexity of mp- Linear constraints Continuous variables Binary variables Parameters Time to solve mp-milp Number of regions MILP 84 3 min 7 PWA law ñ MPC law Linear constraints Continuous variables Binary variables Parameters Time to solve mp-milp Number of regions 84 3 min 7 mp-milp Solution X X Hybrid Control Example: Traction Control System X Prediction Horizon T=3 Prediction Horizon T=4 X

13 Vehicle Traction Control Improve driver s ability to control a vehicle under adverse external conditions (wet or icy roads) Simple Traction Model Mechanical system ò ó ü t ωç e = ü c à b eω e à Je gr ü t vç v = mv r t Model nonlinear, uncertain, constraints Controller suitable for real-time implementation Manifold/fueling dynamics ü c = b i ü d (t à ü f ) Tire torque τ t is a function of slip ω and road surface adhesion coefficient µ ω e v v ω = à gr rt Wheel slip MLD hybrid framework + optimization-based control strategy ω Tire Force Characteristics Hybrid Model Maximum Braking Lateral Force Tire Forces Maximum Cornering Slip Target Zone Longitudinal Force Maximum Acceleration Steer Angle Tire Slip Longitudinal Force Lateral Force Torque Torque µ Slip µ Slip Nonlinear tire torque τ t =f( ω, µ) PWA Approximation HYSDEL (Hybrid Systems Description Language) (PWL Toolbox, Julian, 999) Mixed-Logical Dynamical (MLD) Hybrid Model (discrete time)

14 MLD Model Performance and Constraints Control objective: min P N k= ω(k t) à ω des subj. to. MLD Dynamics State x(t) 9 variables Constraints: Limits on the engine torque: à Nm ô ü d ô 76Nm Input u(t) variable Aux. Binary vars δ(t) 3 variables Aux. Continuous vars z(t) 4 variables Logic Constraint: Hysteresis The MLD matrices are automatically generated in Matlab format by HYSDEL Experimental Apparatus Experimental Apparatus

15 Experiment Hybrid Control Example: Cruise Control System >5 regions ms sampling time Pentium 66Mhz + Labview Hybrid Control Problem Hybrid Model Renault Clio.9 DTI RXE Vehicle dynamics mẍ =F e à F b à ìxç xç = vehicle speed F e = traction force F b = brake force discretized with sampling time T s =.5 s GOAL: command gear ratio, gas pedal, and brakes to track a desired speed and minimize consumption Transmission kinematics R ω = g(i) ks xç F e = R g(i) ks M ω M i = engine speed = engine torque = gear

16 Hybrid Engine torque Max engine torque Piecewise-linearization: (PWL Toolbox, Julián, 999) Min engine torque Hybrid Model à C à e (ω) ô M ô C+ e (ω) C + e (ω) requires: 4 binary aux variables 4 continuous aux variables C à e (ω) =ë + ì ω Gear selection (traction force): R F e = g(i) ks M ω = ks Hybrid Model Gear selection: for each gear #i, define a binary input g i {, } depends on gear #i define auxiliary continuous variables: R IF g i = THEN F ei = g(i) M ELSE ks F e = F er + F e + F e + F e3 + F e4 + F e5 Gear selection (engine/vehicle speed): R g(i) xç similarly, also requires 6 auxiliary continuous variables MLD model Hybrid Model Hysdel Model x(t +)=Ax(t)+B u(t) + B î(t)+b 3 z(t) y(t) =Cx(t)+D u(t) + D î(t)+d 3 z(t) E î(t)+e 3 z(t) ô E 4 x(t)+e u(t)+e 5 continuous states: continuous inputs: 6 binary inputs: continuous output: x, v M, F b g R,g,g,g 3,g 4,g 5 v (vehicle position and speed) (engine torque, brake force) (gears) (vehicle speed) 6 auxiliary continuous vars: 4 auxiliary binary vars: (6 traction force, 6 engine speed, 4 PWL max engine torque) (PWL max engine torque breakpoints) 96 mixed-integer inequalities

17 Max-speed controller Hybrid Controller Max-speed controller Hybrid Controller max J(u t,x(t)),v(t + t) ut ú MLD model subj. to x(t t) =x(t) 5 5 Velocity (km/h) Gear Fraction of Max Torque (Nm).5.5 Brakes (Nm) MILP optimization problem Linear constraints Continuous variables Binary variables Parameters Time to solve mp-milp (Sun Ultra ) Number of regions s v(t) 5 5 x(t) ( x(t) is irrelevant) Road Slope (deg) Engine speed (rpm) Engine Torque (Nm) Power (kw) Hybrid Controller Hybrid Controller Tracking controller Tracking controller min ut J(u t,x(t)), v(t + t) à v d(t) + ú ω min J(u t,x(t)), v(t + t) à v d (t) + ú ω ut ú MLD model subj. to x(t t) =x(t) 5 Velocity (km/h), Desired velocity (km/h) Gear.5 Fraction of Max Torque (Nm) Brakes (Nm) -.5 MILP optimization problem Linear constraints Continuous variables Binary variables Parameters Time to solve mp-milp (Sun Ultra ) Number of regions m 49 v d (t) v(t) ú =. Road Slope (deg) Engine speed (rpm) Engine Torque (Nm) - Power (kw)

18 Smoother tracking controller Hybrid Controller Smoother tracking controller Hybrid Controller min J(u t,x(t)), v(t + t) à v d (t) + ú ω ut v(t + t) à v(t) <T s a max subj. to MLD model x(t t) =x(t) MILP optimization problem Linear constraints Continuous variables Binary variables Parameters Time to solve mp-milp (Sun Ultra ) Number of regions 9 8 m 54 v d (t) v(t) Velocity (km/h), Desired velocity (km/h) Gear Fraction of Max Torque (Nm) Brakes (Nm) Road Slope (deg) Engine speed (rpm) Engine Torque (Nm) Power (kw)