Working with Uncertainty in Model Predictive Control

1. Working with Uncertainty in Model Predictive Control Bob Bitmead University of California, San Diego Nonlinear MPC Workshop 4 April, 2005, Sheffield UK

2. Outline • Model Predictive Control • Constrained receding horizon optimal control • Based on full-state information or certainty equivalence • How do we include estimated states? • Accommodate estimate error — tighten the constraints • Coordinated vehicles example • Vehicles solve local MPC problems • Interaction managed via constraints • Estimation error affects the constraints — back-off • Communication bandwidth affects state error • Control, Performance, Communication tied-in Sheffield April 4, 2005

3. Model Predictive Control Works Full Authority Digital Engine Controller (FADEC) Commercial jet engine

4. AFMV A8 IGVs VSVs MFMV ACTUATORS PS14 T2 N2 PS3 T4B N25 SENSORS P25 25 49 4 3 5 56 6 8 9 16 14 2 0 1 STATIONS MPC with Constraints — Jet Engine • Full-Authority Digital Engine Controller (FADEC) • Multi-input/multi-output control 5x6 • Constrained in • Inputs - max fuel flow, rates of change • States - differential pressures, speeds • Outputs - turbine temperature • Control problem solved via Quadratic Programming (every 10 msec) • State estimator - Extended Kalman Filter • State estimate used as if exact — Certainty Equivalence Sheffield April 4, 2005

5. MPC Applied to Jet Engine • Step in power demand • Constraints • Fuel flow • Exit nozzle area • Constraint-driven controller Sheffield April 4, 2005

6. MPC Applied to Jet Engine • Step in power demand • Constraints • Fuel flow • Exit nozzle area • Stage 3 pressure • Two inputs • One state Sheffield April 4, 2005

7. Message • MPC works in handling constraints on the model • With accurate state estimates • — this is fine for the real plant too Sheffield April 4, 2005

8. What if the estimates are not accurate? • Tighten the constraints imposed on the model • — to ensure their satisfaction on the plant Remember. The MPC problem works on the estimate only Sheffield April 4, 2005

9. g g-g ^ x ^ x+g t t t+T ^ x-g Modifying constraints • Want and we have • Keep in MPC problem Sheffield April 4, 2005

10. Handling uncertainty • Two kinds of uncertainty • Modeling errors • State model is an inaccurate description of the real system • State estimation errors • Remember the MPC constrained control calculation works with the model and not the real system • Constraints must be asserted on the real system Sheffield April 4, 2005

11. degree of stability model error bound Working with model error • Total Stability Theorem (Hahn, Yoshizawa) • Uniform convergence rate of nominal system • + bounds on model error  bounds on state error • MPC formulation of Total Stability • Robust Control Lyapunov Function idea Sheffield April 4, 2005

12. Comparison model Main lemma: For any control The controlled behavior of dominates that of Uses a control Lyapunov function for the unconstrained system Sheffield April 4, 2005

13. real system model system comparison system  Including constraints three systems Sheffield April 4, 2005

14. MPC with Comparison Model • If feasible at t=0 then • Feasible for all t • Real system is stable, constrained and Subject to Sheffield April 4, 2005

15. Example From Fukushima & Bitmead, Automatica, 2005, pp. 97-106 Sheffield April 4, 2005

16. Working with state estimates • Kalman filtering framework • Gaussian state estimate errors • Probabilistic constraints are needed • State estimate error • Rework this as • The constrained controller will need to be cognizant of • This is a non-(certainty-equivalence) controller Information quality is of importance • Same concept of tightening constraints Sheffield April 4, 2005

17. ˆ x ~ N ( x , ) S n i n i | n n i | n + + + Pr( x X ) > < e n i + ˆ x ( , )  < b e S n i | n + Approximately Normal State • Manage constraints by controlling the conditional mean state • Use the control independence of  Sheffield April 4, 2005

18. Pause for breath • Our formulation so far • Model errors • Tighten constraints on the nominal system • State estimate errors • Tighten constraints to accommodate the estimate covariance • Preserves the MPC structure and properties • Original constraints inherited by real system • Perhaps with probabilistic measures • Feasibility and stability properties • Via terminal constraint as usual • Some examples … Sheffield April 4, 2005

19. The Shinkansen Example • One dimensional problem • Three Shinkansen [Bullet Trains] on one track • Uncertainty in knowledge of other trains’ positions • Uniformly distributed with known width • Follow the same reference with each train • Constraint — no crash with preceding train • Leader-follower strategy • Each solves an MPC problem with state estimation Sheffield April 4, 2005

20. Collision avoidance with estimation Sheffield April 4, 2005

21. Train coordination • All trains have the same schedule • Osaka to Tokyo in three hours • Depart at 09:00, arrive at 12:00 • Each solves their own MPC problem • Minimize departure from schedule • No-collision constraint • Estimates of other trains’ positions • Trains separate early • Separation reflects quality of position knowledge Sheffield April 4, 2005

22. Back to the Trains • Low Performance plus String Instability Sheffield April 4, 2005

23. Relaxed Target Schedules • Low Performance but no string instability • Constraints not active Sheffield April 4, 2005

24. Improved Communication • High performance, no string instability Sheffield April 4, 2005

25. Big Issues • Constraints • Quality of Information • Communication • Network and Control Architecture • Tools for systematic design of complex interacting dynamical systems • Model Predictive Control and State Estimation Sheffield April 4, 2005

26. Single Node in Network • Queue length qt is the state variable • Constraint qt≤Q else retransmission required • Control signals are the source command data rates vi,t • Propagation delays di exist between sources and node • Available bit rate mt is a random process • Model as an autoregressive process P(qt≥Q)<0.05 Sheffield April 4, 2005

27. Fair Congestion Control 50 retransmissions per 1000 samples Sheffield April 4, 2005

28. mean = 0.0012 variance = 0.0419 mean = 0.0013 variance = 0.0184 mean = 0.0013 variance = 0.0129 Simulated Source Rates — Fair! Sheffield April 4, 2005

29. A Tougher Example Sheffield April 4, 2005 From Yan & Bitmead, Automatica, 2005 pp.595-604

30. Network Control • A variant of the train control problem • Much greater degree of connectivity — higher dimension • Improved performance is achievable by sending more frequent or more accurate state information upstream to control data flows • This consumes network resources and must be managed • MPC and State Estimation (Kalman Filtering) tools prove of value Sheffield April 4, 2005

31. Conclusions • MPC plus State Estimation • Tools for coordinated control performance • with managed communication complexity • Information architecture • Resource/bandwidth assignment • … as a function of system task Sheffield April 4, 2005

32. Acknowledgements • Hiroaki Fukushima, Jun Yan, Tamer Basar, Soura Dasgupta, Jon Kuhl, Keunmo Kang • NSF, Cymer Inc • GE Global Research Labs, Pratt & Whitney, • United Technologies Research Center • My gracious UK and Irish hosts, IEEE Sheffield April 4, 2005

33. Constraints in design • The appeal of MPC is that it can handle constraints • Constraints provide a natural design paradigm • Lane keeping potential function Sheffield April 4, 2005

34. A Design Bonus • The MPC/KF design is much less sensitive to selection of design parameters than LQG • Constraints work well in design — simplicity Sheffield April 4, 2005 From Yan & Bitmead, Automatica, 2005 pp.595-604

35. Sheffield April 4, 2005

36. Sheffield April 4, 2005