Robust Nonlinear Model Predictive Control using Volterra Models and the Structured Singular Value (  ) Rosendo Díaz-Mendoza and Hector Budman ADCHEM 2009.

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Robust Nonlinear Model Predictive Control using Volterra Models and the Structured Singular Value (  ) Rosendo Díaz-Mendoza and Hector Budman ADCHEM 2009 July 12–

► Chemical processes are nonlinear ► Nonlinear Model Predictive Control (NMPC) ► First principles or empirical models ► Robustness issues ► Robustness of NMPC ► Simulation studies for different parameter values ► Develop a Robust-NMPC methodology that considers parameter uncertainty Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Background and Motivation

A model is required to calculate ŷ Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionModel Predictive Control Model Predictive Control MPC Parameters: ► p, prediction horizon ► m, control horizon ► p ≥ m ► n y, number of outputs ► n u, number of inputs

Volterra Models Why Volterra Models? ► Represent a wide variety of nonlinear behavior ► Model structure: nominal model + uncertain model ► M, system memory ► n u, number of inputs ► x є [1, ,n y ]; n y, number of outputs Schetzen, M., The Volterra and Wiener theories of nonlinear systems; Robert E. Krieger, 1989 Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionVolterra Models

CSTR A→B CACA C A +C B cooling fluid cooling fluid ► Truncation error (M = 3) ► High order dynamics Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionVolterra Models

Nowak, R. D., and Van Veen, B. D. (1994). Random and pseudorandom inputs for Volterra filter identification, IEEE Transactions on Signal Processing, 42 (8), 2124–2135. ► Multilevel pseudo random binary sequence (PRBS) ► Nominal value = mean (parameters) ► Uncertainty = 2  (parameters) Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionVolterra Models Identification

Output equation with parameter uncertainty SISO System ► h n, h i,j, nominal value ►  h n,  h i,j, parameter uncertainty Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionVolterra Models

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionVolterra Models

Nonlinear Model Predictive Control SISO System How to consider parameter uncertainty? Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Introduction Nonlinear Model Predictive Control

Structured Singular Value (  ) Calculation of the worst ŷ(k) to ŷ(k+p) when parameter uncertainty is taken in consideration, i. e., for ŷ(k) Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control Doyle, J., (1982). Analysis of feedback systems with structured uncertainties, IEE Proceedings D Control Theory & Applications, 129 (6), 242–250

Structured Singular Value  (SSV) SSV Theorem Skew  problem (convex) Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control Braatz, R. D., Young, P. M., Doyle, J. C., and Morari, M. (1994). Computational complexity of  calculation, IEEE Transactions on Automatic Control, 39 (5), 1000–10002.

M, interconnection matrix Δ, uncertainty block structure M  Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

0 0 Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control Interconnection Matrix Example Nominal Uncertain Feedback

NMPC Cost Function Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

Additional terms Manipulated variables movement penalization Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

Additional terms Manipulated variables constraints Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

Additional terms Terminal Condition Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

NMPC Cost Function NMPC Algorithm at each sampling instant Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV IntroductionNonlinear Model Predictive Control

CSTR A→B Control Specifications ► CV: x 1 (dimensionless reactant concentration) ► MV: x c (cooling jacket dimensionless temperature) ► β: process disturbance Parameter Calculation ► Multilevel PRBS ► Parameter uncertainty Doyle III, F. J., Packard, A., and Morari, M. (1989). Robust controller design of a nonlinear CSTR, Chemical Engineering Science, 44 (9), 1929–1947. CACA C A +C B cooling fluid cooling fluid Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesSISO Case Study SISO System CSTR with first order exothermic reaction

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesSISO Disturbance Characteristics

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesSISO

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesSISO

Sum absolute error Robust = 1.46 Sum absolute error Non-Robust = % improvement Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesSISO

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesSISO WiΔuWiΔu Robust Controller is better than Non-Robust 1.537% 141% % % 25 different disturbances for each weight

► X, biomass concentration ► S, substrate concentration ► P, product concentration ► D, dilution rate ► S f, feed substrate concentration Control Specifications ► CV: X and P ► MV: D and S f ► Y X/S : process disturbance Parameter calculation ► Multilevel PRBS ► Parameter uncertainty Fermenter XSPXSP DSfDSf Saha, P., Hu, Q., and Rangaiah, G., P. (1999). Multi-input multi-output control of a continuous fermenter using nonlinear model based controllers, Bioprocess Engineering, 21, 533–542. Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesMIMO Case Study MIMO System

Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Case StudiesMIMO

Conclusions ► A Robust-NMPC algorithm was developed ► The algorithm considers all the features of previous NMPC formulations ► In average the robust controller results in better performance as the input weight is decreased Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Preliminary Conclusions

► Computational demand ► Multivariable control Diaz-Mendoza R. and Budman HRobust NMPC using Volterra Models and the SSV Current challenges Preliminary ConclusionsChallenges