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

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Sheffield April 4, 2005 2 of 31 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

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

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Sheffield April 4, 2005 4 of 31 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 SENSORS ACTUATORS IGVs VSVs MFMV A8AFMV N2T2 N25 PS14 P25 PS3T4B 01 2 14 16 25 3 4 49 5 56 689 STATIONS

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Sheffield April 4, 2005 5 of 31 MPC Applied to Jet Engine Step in power demand Constraints Fuel flow Exit nozzle area Constraint-driven controller

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Sheffield April 4, 2005 6 of 31 MPC Applied to Jet Engine Step in power demand Constraints Fuel flow Exit nozzle area Stage 3 pressure Two inputs One state

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Sheffield April 4, 2005 7 of 31 Message MPC works in handling constraints on the model With accurate state estimates — this is fine for the real plant too

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Sheffield April 4, 2005 8 of 31 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

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Sheffield April 4, 2005 9 of 31 Modifying constraints Want and we have Keep in MPC problem x+ x+ ^ x- x- ^ g t g- t t+T x ^

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Sheffield April 4, 2005 10 of 31 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

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Sheffield April 4, 2005 11 of 31 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 degree of stability model error bound

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Sheffield April 4, 2005 12 of 31 Comparison model Main lemma: For any control The controlled behavior of dominates that of Uses a control Lyapunov function for the unconstrained system

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Sheffield April 4, 2005 13 of 31 Including constraints three systems real system comparison system model system

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Sheffield April 4, 2005 14 of 31 MPC with Comparison Model Subject to If feasible at t=0 then Feasible for all t Real system is stable, constrained and

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Sheffield April 4, 2005 15 of 31 Example From Fukushima & Bitmead, Automatica, 2005, pp. 97-106

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Sheffield April 4, 2005 16 of 31 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

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Sheffield April 4, 2005 17 of 31 Approximately Normal State Manage constraints by controlling the conditional mean state Use the control independence of x n i ~N( ˆ x n i|n, n i|n ) Pr(x n i X) ˆ x n i|n ( , )

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Sheffield April 4, 2005 18 of 31 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 …

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Sheffield April 4, 2005 19 of 31 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

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Sheffield April 4, 2005 20 of 31 Collision avoidance with estimation

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Sheffield April 4, 2005 21 of 31 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

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Sheffield April 4, 2005 22 of 31 Back to the Trains Low Performance plus String Instability

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Sheffield April 4, 2005 23 of 31 Relaxed Target Schedules Low Performance but no string instability Constraints not active

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Sheffield April 4, 2005 24 of 31 Improved Communication High performance, no string instability

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Sheffield April 4, 2005 25 of 31 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

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Sheffield April 4, 2005 26 of 31 Single Node in Network Queue length q t is the state variable Constraint q t ≤Q else retransmission required Control signals are the source command data rates v i,t Propagation delays d i exist between sources and node Available bit rate t is a random process Model as an autoregressive process P(q t ≥Q)<0.05

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Sheffield April 4, 2005 27 of 31 Fair Congestion Control 50 retransmissions per 1000 samples

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Sheffield April 4, 2005 28 of 31 Simulated Source Rates — Fair! mean = 0.0012 variance = 0.0419 mean = 0.0013 variance = 0.0184 mean = 0.0013 variance = 0.0129

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Sheffield April 4, 2005 29 of 31 A Tougher Example From Yan & Bitmead, Automatica, 2005 pp.595-604

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Sheffield April 4, 2005 30 of 31 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

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Sheffield April 4, 2005 31 of 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

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Sheffield April 4, 2005 32 of 31 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

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Sheffield April 4, 2005 33 of 31 Constraints in design The appeal of MPC is that it can handle constraints Constraints provide a natural design paradigm Lane keeping potential function

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Sheffield April 4, 2005 34 of 31 A Design Bonus The MPC/KF design is much less sensitive to selection of design parameters than LQG Constraints work well in design — simplicity From Yan & Bitmead, Automatica, 2005 pp.595-604

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