st 1 HYCON PhD School on Hybrid Systems www.ist-hycon.org www.unisi.it Model Predictive Control of Hybrid Systems Alberto Bemporad University of Siena, Italy bemporad@dii.unisi.it scimanyd suounitnoc enibmoc smetsys dirbyH lacipyt (snoitauqe ecnereffid ro laitnereffid) scimanyd etercsid dna stnalp lacisyhp fo fo lacipyt (snoitidnoc lacigol dna atamotua) fo senilpicsid gninibmoc yB .cigol lortnoc ,yroeht lortnoc dna smetsys dna ecneics retupmoc dilos a edivorp smetsys dirbyh no hcraeser ,sisylana eht rof sloot lanoitatupmoc dna yroeht fo ngised lortnoc dna ,noitacifirev ,noitalumis egral a ni desu era dna ,''smetsys deddebme`` ria ,smetsys evitomotua) snoitacilppa fo yteirav ssecorp ,smetsys lacigoloib ,tnemeganam ciffart .(srehto ynam dna ,seirtsudni HYSCOM IEEE CSS Technical Committee on Hybrid Systems 9 Siena, July 1 9-22, 2005 - Rectorate of the University of Siena
Outline Outline Model Predictive Control Model Predictive Control Model Predictive Control Model Predictive Control of Hybrid Systems of Hybrid Systems of Hybrid Systems of Hybrid Systems • Model Predictive Control (MPC) concepts • Hybrid models for MPC Alberto Bemporad Alberto Bemporad Dept. Information Engineering Dept. Information Engineering - - Universi University of Universi ty of Siena, Ital Siena, Italy Dept. Information Engineering Dept. Information Engineering University of ty of Siena, Ital Siena, Italy • MPC of hybrid systems http://www.dii.unisi.it/~bemporad www.dii.unisi.it/~bemporad http:// • Explicit MPC (multiparametric programming) • Optimization-based reachability analysis • Examples University of Siena Universit y of Siena (founded in 1240) (founded in 1240) EIHS EIHS – EIHS EIHS – Eu Europ Europ Eu ropean In ropean In an Instit an Instit itut itut ute ute of H of Hybrid id Sy Systems of of H Hybrid id Sy Systems st HYCON 1 st HYCON PhD School on Hybrid Systems (Siena, 20/7/05) hD School on Hybrid Systems (Siena, 20/7/05) 2 Model Predictive Control Model Predictive Control Optimizer Plant Reference Input Output Model Predictive Control Model Predictive Control u ( t ) r ( t ) y ( t ) Measurements • MPC concepts • MODEL: a model of the plant is needed to predict • Linear MPC the future behavior of the plant • Matlab tools for linear MPC • PREDICTIVE: optimization is based on the predicted future evolution of the plant • CONTROL: control complex constrained multivariable plants 3 4 Receding Horizon Philosophy Receding Horizon - - Example Example Receding Horizon Philosophy Receding Horizon • At time t : y ( t + k ) r ( t ) Solve an optimal Predicted outputs • MPC is like playing chess ! control problem over a u ( t + k ) finite future horizon N : Manipulated Inputs t + N t t +1 – minimize – subject to constraints t +1 t +2 t + N +1 • Only apply the first optimal move • Get new measurements, and repeat the optimization at time t +1 Advantage of on-line optimization: FEEDBACK! 5 6
Constrained Optimal Control Constrained Optimal Control Constrained Optimal Control Constrained Optimal Control • Optimization problem: • Linear Model: By substituting , we get (quadratic) • Constraints: (linear) • Constrained optimal control problem (quadratic performance index): Convex QUADRATIC Convex Q ADRATIC PROGRAM (QP) PROGRAM (QP) 7 8 MPC of Linear Systems of Linear Systems Model Predictive Control Toolbox Model Predictive Control Toolbox MPC past future • MP MPC Toolbox 2.0 C Toolbox 2.0 (Bemporad, Ricker, Morari, 1998-2004): Predicted y(t+k|t) outputs – Object-oriented implementation (MPC object) Manipulated Inputs u(t+k) – MPC Simulink Library t t+1 t+N – MPC Graphical User Interface At time t : – RTW extension (code generation) • Get/estimate the current state x ( t ) • Solve the QP problem and let U = { u * (0),..., u * ( N -1) } be the solution (=finite-horizon constrained open-loop optimal control) • Apply only and discard the remaining optimal inputs Only linear models are handled Only linear models are handled • Go to time t +1 http://www.mathworks.com 9 10 Example: AFTI- -16 16 Example: AFTI- -16 16 Example: AFTI Example: AFTI • Linearized model: see demo afti16.m (MPC-Tbx) 11 12
Convergence Convergence Convergence Proof Convergence Proof Lyapunov function (Keerthi and Gilbert, 1988)(Bemporad et al ., 1994) Proof: Use value function as Lyapunov function Global optimum is not needed to prove convergence ! 13 14 Convergence Proof Convergence Proof Convergence Proof Convergence Proof 15 16 Outline MPC and LQR Outline MPC and LQR • Consider the MPC control law: � Model Predictive Control (MPC) concepts Jacopo Francesco Riccati (1676 - 1754) • Hybrid models for MPC • MPC of hybrid systems • Explicit MPC (multiparametric programming) • Optimization-based reachability analysis • In a polyhedral region around the origin the MPC control • Examples law is equivalent to the constrained LQR controller with weights Q,R. (Chmielewski, Manousiouthakis, 1996) (Scokaert and Rawlings, 1998) MPC ≡ constrained LQR • The larger the horizon, the larger the region where MPC=LQR 17 18
Hybrid Systems Hybrid Systems Control Computer Theory Science Continuous Finite Hybrid Models for MPC Hybrid Models for MPC dynamical state systems machines B • Discrete Hybrid Automata (DHA) Hybrid systems A • Mixed Logical Dynamical (MLD) Systems B C u ( t ) y ( t ) system • Piecewise Affine (PWA) Systems B C C A 19 20 “Intrinsically Hybrid” Systems Intrinsically Hybrid” Systems Cruise Control Problem “ Cruise Control Problem continuous inputs continuous inputs GOAL: GOAL: command gear ratio, gas pedal, and brakes to track track a desired speed and minimize consumptions discrete input discrete input Continuous Discrete input Continuous inputs + + dynamical states (1,N,2,3,4) (brakes, gas, clutch) CHALLENGE CHALLE GES: S: (velocities, torques, air-flows, fuel level) • continuous and discrete inputs • dynamics depends on gear • nonlinear torque/speed maps 21 22 Key Requirements for Hybrid Models Key Requirements for Hybrid Models Piecewise Affine Systems Piecewise Affine Systems state+input space • Descriptive Descriptive enough to capture the behavior of the system – continuous dynamics (physical laws) – logic components (switches, automata, software code) – interconnection between logic and dynamics • Simple Simple enough for solving analysis and synthesis problems ? (Sontag 1981) linear systems nonlinear systems linear hybrid systems • Can approximate nonlinear/discontinuous dynamics arbitrarily well “Make everything as simple as possible, but not simpler.” — Albert Einstein Albert Einstein 24
Switched Affine System Switched Affine System Discrete Hybrid Automaton Discrete Hybrid Automaton (Torrisi, Bemporad, 2004) Event Generator Switched Affine System 1 Finite State 2 Machine time or event s counter The affine dynamics depend on the current mode i ( k ): mode Mode Selector 25 26 Mode Selector Mode Selector Event Generator Event Generator Event variables are generated by linear threshold conditions over The active mode i ( k ) is selected by a Boolean function of the continuous states, continuous inputs, and time: current binary states, binary inputs, and event variables: Example: 0 1 0 Example: [ δ =1] ↔ [ x c ( k ) ≥ 0] the system has 3 modes 1 27 28 Finite State Machine Finite State Machine Logic and Inequalities Logic and Inequalities Glover 1975, Williams 1977 The binary state of the finite state machine evolves according to a Boolean state update function: Switched Affine System Event Finite State Mode Selector Generator Machine 1 2 Example: s 29 30
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