Model Predictive Control Manfred Morari Institut f¨ ur Automatik ETH Z¨ urich Spring Semester 2014 Manfred Morari Model Predictive Control Spring Semester 2014
Lecturers Prof. Dr. Manfred Morari Prof. Dr. Francesco Borrelli ETH Zurich University of California, Berkeley Institut für Automatik (IfA) Model Predictive Control Lab Dr. Paul J. Goulart Dr. Alexander Domahidi ETH Zurich inspire-IfA Institut für Automatik (IfA) Manfred Morari Model Predictive Control Spring Semester 2014
Head Teaching Assistants Xiaojing (George) Zhang xiaozhan@control.ee.ethz.ch David Sturzenegger sturzenegger@control.ee.ethz.ch Manfred Morari Model Predictive Control Spring Semester 2014
Lecture Material Compilation: Xiaojing (George) Zhang, David Sturzenegger Please email suggestions and typos to xiaozhan@control.ee.ethz.ch sturzenegger@control.ee.ethz.ch Software: Beamer for LaTeX by Till Tantau & Vedran Milenti ´ c Printed material: Available in 2-page layout Sold during lecture or later at ETL I23 for CHF 30 Recordings: Entire lecture is video recorded Link will be provided on lecture homepage Homepage: http://control.ee.ethz.ch/index.cgi?page=lectures Manfred Morari Model Predictive Control Spring Semester 2014
About the Lecture Duration: Monday, 17. Feb 2014 – Friday, 28. Feb 2014 Credits: 6 credits for passing the exam Exercises: Computer excercises, ETZ D61.1/2 Exam: Fri, 14. March 2014 (written), Location: tba Week 1: Lectures Date Topic Exercises Time Location Mon, Feb 17 Linear Systems I 9.15 – 12 HG E3 13.15 – 17 Tue, Feb 18 Linear Systems II 9.15 – 12 HG E3 13.15 – 17 Wed, Feb 19 Optimization I 9.15 – 12 HG D16.2 13.15 – 17 Thu, Feb 20 Optimization II 9.15 – 12 HG D16.2 13.15 – 17 Fri, Feb 21 Introduction to MPC 9.15 – 12 HG E3 13.15 – 17 Manfred Morari Model Predictive Control Spring Semester 2014
About the Lecture Duration: Monday, 17. Feb 2014 – Friday, 28. Feb 2014 Credits: 6 credits for passing the exam Exercises: Computer excercises, ETZ D61.1/2 Exam: Fri, 14. March 2014 (written), Location: tba Week 2: Lectures Date Topic Exercises Time Location Mon, Feb 24 Numerical Methods 9.15 – 12 HG E3 13.15 – 17 Tue, Feb 25 Advanced Topics I 9.15 – 12 HG E3 13.15 – 17 Wed, Feb 26 Invited Talks 9.15 – 17 HG D16.2 — Thu, Feb 27 Design Exercise — — 10.15 – 17 Fri, Feb 28 Advanced Topics II 9.15 – 12 HG D16.2 — Manfred Morari Model Predictive Control Spring Semester 2014
Model Predictive Control Part I – Introduction C. Jones † , F. Borrelli ∗ , M. Morari Institut f¨ ur Automatik ETH Z¨ urich ∗ UC Berkley † EPFL Spring Semester 2014 C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014
3 Summary and Outlook 3.2 Literature Literature Model Predictive Control: Predictive Control for linear and hybrid systems, F. Borrelli, A. Bemporad, M. Morari, 2013 Cambridge University Press [ http://www.mpc.berkeley.edu/mpc-course-material ] Model Predictive Control: Theory and Design, James B. Rawlings and David Q. Mayne, 2009 Nob Hill Publishing Predictive Control with Constraints, Jan Maciejowski, 2000 Prentice Hall Optimization: Convex Optimization, Stephen Boyd and Lieven Vandenberghe, 2004 Cambridge University Press Numerical Optimization, Jorge Nocedal and Stephen Wright, 2006 Springer C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 3-33
1 Concepts Table of Contents 1. Concepts 1.1 Main Idea 1.2 Classical Control vs MPC 1.3 Mathematical Formulation 2. Examples 2.1 Ball on Plate 2.2 Autonomous Quadrocopter Flight 2.3 Autonomous dNaNo Race Cars 2.4 Energy E ffi cient Building Control 2.5 Kite Power 2.6 Automotive Systems 2.7 Robotic Chameleon 3. Summary and Outlook 3.1 Summary 3.2 Literature C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014
1 Concepts 1.1 Main Idea Main Idea Objective : Minimize lap time Constraints : Avoid other cars Stay on road Don’t skid Limited acceleration Intuitive approach: Look forward and plan path based on Road conditions Upcoming corners Abilities of car etc... C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-2
1 Concepts 1.1 Main Idea Optimization-Based Control Minimize (lap time) while avoid other cars stay on road ... Solve optimization problem to compute minimum-time path C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-3
1 Concepts 1.1 Main Idea Optimization-Based Control Minimize (lap time) while avoid other cars stay on road ... Solve optimization problem to compute minimum-time path What to do if something unexpected happens? We didn’t see a car around the corner! Must introduce feedback C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-3
1 Concepts 1.1 Main Idea Optimization-Based Control Minimize (lap time) while avoid other cars stay on road ... Solve optimization problem to compute minimum-time path Obtain series of planned control actions Apply first control action Repeat the planning procedure C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-3
1 Concepts 1.1 Main Idea Model Predictive Control Objectives Model Constraints Optimizer Reference Input Output Plant Measurements Do Plan Do Plan Do Plan Time Receding horizon strategy introduces feedback . C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-4
1 Concepts 1.2 Classical Control vs MPC Table of Contents 1. Concepts 1.1 Main Idea 1.2 Classical Control vs MPC 1.3 Mathematical Formulation C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014
1 Concepts 1.2 Classical Control vs MPC Two Di ff erent Perspectives Classical design: design C MPC: real-time, repeated optimiza- tion to choose u ( t ) Dominant issues addressed Dominant issues addressed Disturbance rejection ( d → y ) Control constraints (limits) Noise insensitivity ( n → y ) Process constraints (safety) Model uncertainty (usually in time domain ) (usually in frequency domain ) C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-5
1 Concepts 1.2 Classical Control vs MPC Constraints in Control All physical systems have constraints : Physical constraints, e.g. actuator limits Performance constraints, e.g. overshoot Safety constraints, e.g. temperature/pressure limits Optimal operating points are often near constraints. constraint Classical control methods: output Ad hoc constraint management Set point su ffi ciently far from constraints set point Suboptimal plant operation time constraint Predictive control: output Constraints included in the design Set point optimal set point Optimal plant operation time C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-6
1 Concepts 1.3 Mathematical Formulation Table of Contents 1. Concepts 1.1 Main Idea 1.2 Classical Control vs MPC 1.3 Mathematical Formulation C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014
1 Concepts 1.3 Mathematical Formulation MPC: Mathematical Formulation N − 1 ÿ U ∗ t ( x ( t )) := argmin q ( x t + k , u t + k ) U t k =0 subj. to x t = x ( t ) measurement x t + k +1 = Ax t + k + Bu t + k system model x t + k ∈ X state constraints u t + k ∈ U input constraints U t = { u 0 , u 1 , . . . , u N − 1 } optimization variables Problem is defined by Objective that is minimized, e.g., distance from origin, sum of squared/absolute errors, economic,... Internal system model to predict system behavior e.g., linear, nonlinear, single-/multi-variable, ... Constraints that have to be satisfied e.g., on inputs, outputs, states, linear, quadratic,... C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-7
1 Concepts 1.3 Mathematical Formulation MPC: Mathematical Formulation At each sample time: Measure / estimate current state x ( t ) Find the optimal input sequence for the entire planning window N : U ∗ t = { u ∗ t , u ∗ t +1 , . . . , u ∗ t + N − 1 } Implement only the first control action u ∗ t C. Jones † , F. Borrelli ∗ , M. Morari Model Predictive ControlPart I – Introduction Spring Semester 2014 1-8
2 Constrained Optimal Control: 2-Norm 2.1 Problem Formulation Problem Formulation Quadratic cost function N ≠ 1 ÿ J 0 ( x (0) , U 0 ) = x Õ N Px N + x Õ k Qx k + u Õ (2) k Ru k k =0 with P ≤ 0 , Q ≤ 0 , R º 0 . Constrained Finite Time Optimal Control problem (CFTOC). J ú 0 ( x (0)) = min J 0 ( x (0) , U 0 ) U 0 subj. to x k +1 = Ax k + Bu k , k = 0 , . . . , N ≠ 1 (3) x k œ X , u k œ U , k = 0 , . . . , N ≠ 1 x N œ X f x 0 = x (0) N is the time horizon and X , U , X f are polyhedral regions. F. Borrelli ∗ , M. Morari, C. Jones † Model Predictive Control Part II – Constrained Finite Time Optimal Control Spring Semester 2014 2-6
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