MPC of Nonlinear Systems B. Wayne Bequette Motivation Challenging behavior Model Predictive Control Various Options EKF-based NMPC Multiple Model Predictive Control Summary Theory and Applications

Challenging Behavior Input Multiplicity Connection with RHP zeros: Sistu & Bequette, Chem. Eng. Sci. (1996) Output Multiplicity Achievable performance is a strong function of the operating condition Region of instability Russo & Bequette, AIChE J. (1995)

Global Bifurcation Diagram Design parameter Space III, IV, V: Infeasible operating regions Russo and Bequette, AIChE J. (1995)

Model Predictive Control (MPC) Constraints Multivariable Time-delays Objective function? Optimization technique? Model type? Disturbances? Initial cond./state est.?

Many Publications/Researchers Will not attempt a reasonable overview Every plenary speaker has worked on the topic! Reviews  Bequette (1991)  Henson (1998) Will focus on work of my graduate students

Our Approaches Quadratic Objective Function Models  Fundamental: numerical integration or collocation  Fundamental with linearization at each time step  Multiple model  Artificial neural network State Estimates/Initial Conditions  Additive output disturbance (e.g. DMC)  Estimation horizon (optimization)  Extended Kalman Filter Importance of stochastic states

Non-Convex Problem Sistu and Bequette, 1992 ACC

Input Multiplicity Example Sistu and Bequette, 1992 ACC

Additive Disturbance Assumption Bequette, ADCHEM (1991)

Stability Infinite Horizon  Meadows and Rawlings (1993) Terminal State Constraints  Mayne and Michalska (1990) Dual Model (Region, State Feedback)  Michalska and Mayne (1993) Quasi-Infinite Horizon  Chen and Allgower (1998) Numerical Lyapunov - Regions of Attraction  Sistu and Bequette (1995)

State Estimation Output Disturbance (DMC, not a good idea)  Garcia (1984) Extended Kalman Filter  Gattu and Zafiriou (1992)  Lee and Ricker (1994) Estimation Horizon, Optimization  Ramamurthi et al. (1993)

EKF-based NMPC (Lee & Ricker, 1994) Nonlinear Model State Estimation: Extended Kalman Filter Prediction  One integration of NL ODEs based on set of control moves  Perturbation (linear) model - effect of changes in control moves Optimization  SQP

Multi-rate EKF Implementation Frequent temperature Infrequent concentration and/or MWD Schley et al. NL-MPC, Ascona (1998), Prasad et al. J. Proc. Cont. (2002)

Multiple Model Predictive Control Fundamental Model  Time consuming, often impractical (biomedical, etc.) ANN, other NL Empirical Model  Much data required, large validation effort, “overfitting” Multiple Model Predictive Control  Extension of multiple model adaptive control (MMAC)  MMAC developed for aircraft Many flight conditions  Bank of possible linear models Controller-model pairing Switching vs. weighting

Multiple Model Predictive Control Optimization Prediction r(k) Reference Model Plant 1 2 m Model Bank  u(k) y(k)  i (k) y(k) ^ w i (k) Weight Computation Constrained MPC X X X y i (k) ^ y(k+1:P) ^ Rao et al. IEEE Eng. Med. Biol. Mag (2001)

Multiple Models and Weighting Probabilities Weights

Example Comparison of MMPC with EKF-based NMPC ABC A+AD Cain F Cb F Constant V,T,  Aufderheide et al., 2001 ACC Aufderheide and Bequette, Comp. Chem. Eng. (2003)

Feed Concentration Disturbance Aufderheide et al., 2001 ACC

Feed Concentration Disturbance w/noise

Biomedical Control Anesthesia  Adaptation Multiple models  Constraints  Recovery time Diabetes  Blood glucose  s.c. measurement  Sensor recalibration  Meal disturbances controller sensor pumppatient glucose setpoint drugs infused blood pressure cardiac output

Current Status of NMPC Modeling: the biggest challenge  Fundamental: much effort, many parameters  Empirical: much data, range of conditions? Estimation  Biased estimates Adaptation  Parameter, operating condition changes  Failure detection and compensation Cost-Benefit  Nonlinear vs. Better Performing Linear (e.g. not DMC)

Potential Techniques Multiobjective Optimization-based MPC Distributed: Multiple MPC  Individual optimization  Communicate solution Birds Bugs

Summary Motivation: nonlinear behavior  Multiplicities Nonlinear model predictive control  Various, including full NMPC  EKF-based NMPC  MMPC Current and Future Work

El Dorado’s (Troy, NY, 1994) Ravi Gopinath Kevin Schott Lou Russo Phani Sistu Wayne Bequette

Troy Pub and Brewery (1998) Matt Schley Wayne Bequette Manoel Carvalho Deepak Nagrath Brian Aufderheide Vinay Prasad Venkatesh Natarajan Ramesh Rao