Robust Nonlinear Model Predictive Control using Volterra Models

Robust Nonlinear Model Predictive Control using Volterra Models and the Structured Singular Value ( ) Rosendo Díaz-Mendoza and Hector Budman ADCHEM 2009 July 12– 15 2009

Background and Motivation ► 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 H Robust NMPC using Volterra Models and the SSV

Introduction Model Predictive Control MPC Parameters: ► p, prediction horizon ► m, control horizon ► p≥m ► ny, number of outputs ► nu, number of inputs Diaz-Mendoza R. and Budman H A model is required to calculate ŷ Robust NMPC using Volterra Models and the SSV

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

Introduction Volterra Models CA CA+CB cooling fluid CSTR A→B cooling fluid ►Truncation error (M = 3) ►High order dynamics Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Volterra Models Identification ►Multilevel ►Nominal pseudo random binary sequence (PRBS) value = mean (parameters) ►Uncertainty = 2 (parameters) 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. Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Volterra Models Output equation with parameter uncertainty SISO System ► h n, hi, j, nominal value ► hn, hi, j, parameter uncertainty Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Volterra Models Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

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

Introduction Nonlinear Model Predictive Control Diaz-Mendoza R. and Budman H Robust 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) Doyle, J. , (1982). Analysis of feedback systems with structured uncertainties, IEE Proceedings D Control Theory & Applications, 129 (6), 242– 250 Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control Structured Singular Value (SSV) SSV Theorem Skew problem (convex) 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. Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control D M M, interconnection matrix Δ, uncertainty block structure Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control Interconnection Matrix Example 0 0 Feedback Nominal Diaz-Mendoza R. and Budman H Uncertain Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control NMPC Cost Function Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control Additional terms Manipulated variables movement penalization Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control Additional terms Manipulated variables constraints Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control Additional terms Terminal Condition Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Introduction Nonlinear Model Predictive Control NMPC Cost Function NMPC Algorithm at each sampling instant Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Case Studies SISO Case Study SISO System CSTR with first order exothermic reaction Control Specifications CA CA+CB cooling fluid CSTR A→B ► ► cooling fluid ► CV: x 1 (dimensionless reactant concentration) MV: xc (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. Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Case Studies SISO Disturbance Characteristics Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Case Studies Diaz-Mendoza R. and Budman H SISO Robust NMPC using Volterra Models and the SSV

Case Studies Diaz-Mendoza R. and Budman H SISO Robust NMPC using Volterra Models and the SSV

Case Studies SISO Sum absolute error Robust = 1. 46 Sum absolute error Non-Robust = 1. 55 6% improvement Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Case Studies WiΔu SISO 1. 5 Robust Controller is better than Non-Robust 37% 1 41% 0. 75 50% 0. 50 66% 25 different disturbances for each weight Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Case Studies MIMO Case Study MIMO System D Sf ► Fermenter X S P ► ► X, biomass concentration S, substrate concentration P, product concentration D, dilution rate Sf, feed substrate concentration Control Specifications ► CV: X and P ► MV: D and Sf ► YX/S: process disturbance Parameter calculation ► Multilevel PRBS ► Parameter uncertainty 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 H Robust NMPC using Volterra Models and the SSV

Case Studies Diaz-Mendoza R. and Budman H MIMO Robust NMPC using Volterra Models and the SSV

Preliminary 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 H Robust NMPC using Volterra Models and the SSV

Preliminary Conclusions Challenges Current challenges ► Computational demand ► Multivariable control Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV