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- 1. ME 190M Introduction to Model Predictive Control Francesco Borrelli Fall 2009 Department of Mechanical Engineering University of California Berkeley, USA
- 2. Instruction • Instructor: Francesco Borrelli, Room 5139 EH, 643-3871, fborrelli@me.berkeley.edu Office Hours: Tu and Th 9.30-11 • Teaching Assistant: None • Lectures: Friday 11-12 in Room 1165, Etcheverry Hall • Class Notes: Slides distributed before (sometime after) the class • Class Web Site: bSpace
- 3. Grading • Homework assignments: 100% • Every 1 week – Includes matlab programming and simulation assignments • Only selected ones (~5) will be graded (will announce it beforehand)
- 4. Matlab • Matlab running the computers in 2109 Etcheverry Hall • Card key access required • I will submit class list to so that everyone in the class has access to that room – Please enroll in the class ASAP • Need additional toolbox/Software distributed through bSpace
- 5. Recommended software purchase Student Price: ~$ 100.00 – The Scholar’s Workstation • Contains: – Matlab, Simulink, Symbolic Math, and two books/manuals • Professional price is about $5,000. – Some minor limitations, but you would be hard pressed to notice them
- 7. Modelpast future Predictive Control 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 • Optimization using current measurements Feedback
- 8. MPC Algorithm At time t: • Measure (or estimate) the current state x(t) • Find the optimal input sequence U* ={u*t , u*t+1, u*t+2, … , u*t+N-1} • Apply only u(t)=u*t , and discard u*t+1, u*t+2, … Repeat the same procedure at time t +1 Multivariable, Model Based Nonlinear, Constraints Satisfaction, Prediction
- 9. Important Issues in Model Predictive Control Even assuming perfect model, no disturbances: predicted open-loop trajectories ≠ closed-loop trajectories • Feasibility Optimization problem may become infeasible at some future time step. • Stability Closed-loop stability is not guaranteed. • Performance Goal: What is achieved by repeatedly minimizing • Real-Time Implementation
- 10. Feasibility Issues Infeasible Terminal region t+3 t+2 t+1 t
- 11. Stability Issues Terminal region t+3 t+2 t+1 t
- 12. Feasibility and Stability Constraints Modified Problem (Large Body of Literature) Xf (Robust) Invariant Set p(x) Control Lyapunov Function
- 13. Feasibility and Stability Issues t+2 Terminal region t+1 t
- 15. Class Goals • Design and Implement a “simple” MPC Controller in Matlab (for linear and nonlinear systems) • Tune it for achieving Desired Performance • Understand main issues of Stability and Feasibility
- 16. Example 1 Data from PeMS • California Freeway Performance Measurement System • Collects real-time data on CA freeways via loop detectors • Able to communicate average traffic speed at loop location every 5 minutes • Loops typically positioned every 0.3- 3 miles
- 17. Example 1 Audi SmartEngine Vehicle ACC Desired Speed Actual Speed • Design and MPC Controller regulating the desired speed (through an Automatic Cruise Control) in order to reach the destination in the most fuel-efficient way • Prediction: Max and Min Speed of traffic, Grade • Constraints: Max and Min Speed (of traffic and of vehicle)
- 18. Example 2 Ball and Plate Experiment y δ γ x β' α' α β • Specification of Experiment: Angle: -17°… +17° , Plate:-30 cm…+30 cm Input Voltage: -10 V… +10 V Computer: PENTIUM166 Sampling Time: 30 ms
- 19. Example 2 Ball and Plate Experiment
- 20. Summarizing… Need: • A discrete-time model of the system (Matlab, Simulink) • A state observer • Set up an Optimization Problem (Matlab, MPT toolbox/Yalmip) • Solve an optimization problem (Matlab/Optimization Toolbox, NPSOL) • Verify that the closed-loop system performs as desired (avoid infeasibility/stability) • Make sure it runs in real-time and code/download for the embedded platform
- 21. Summarizing… Need: • A discrete-time model of the system (Matlab, Simulink) • A state observer • Set up an Optimization Problem (Matlab, MPT toolbox/Yalmip) • Solve an optimization problem (Matlab/Optimization Toolbox, NPSOL) • Verify that the closed-loop system performs as desired (avoid infeasibility/stability) • Make sure it runs in real-time and code/download for the embedded platform
- 22. Class Topics (Subject to changes) Week 1/2: Modeling • Cont. time vs Discrete time , Transfer function vs State Space, Linear vs Nonlinear Week 3/4/5: Fundamentals of Optimization • Basis Concept of Optimizations • Linear Program, Quadratic Program, Nonlinear program. • Polyhedral and their manipulation • Piecewise-linear Optimization. Week 6/7/8: Constrained Optimal Control • General Formulation of constrained control problems • Linear 2-norm, Linear 1-norm, nonlinear • Solution via batch approach and dynamic programming Week 9/10/11: Predictive Control • General formulation, Fundamental Properties • Invariant set and Feasibility • Soft constraints and tracking Week 12/13/14/15: Examples and Review Matlab Oriented
- 23. Initial Remarks • Continuous-Time versus Discrete-Time • MPC Name • s-functions
- 24. Movie