Rosendo D í az-Mendoza and Hector Budman ADCHEM 2009 July 12–15 2009

1. 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

2. Background and Motivation 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

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

4. Introduction Volterra Models 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

5. Introduction Volterra Models 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

6. Introduction Volterra Models Volterra Models Identification • Multilevel pseudo random binary sequence (PRBS) • Nominal 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

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

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

9. Introduction Nonlinear Model Predictive Control 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

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

11. Introduction Nonlinear Model Predictive Control 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

12. Introduction Nonlinear Model Predictive Control 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

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

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

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

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

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

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

19. Introduction Nonlinear Model Predictive Control 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

20. Case Studies SISO Case Study SISO System CSTR with first order exothermic reaction CA • Control Specifications • CV: x1 (dimensionless reactant concentration) • MV: xc (cooling jacket di-mensionless temperature) • β: process disturbance Parameter Calculation • Multilevel PRBS • Parameter uncertainty CA+CB cooling fluid CSTR A→B cooling fluid 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

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

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

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

24. 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

25. Case Studies SISO 25 different disturbances for each weight Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

26. Case Studies MIMO D Sf X S P Fermenter Case Study MIMO System • X, biomass concentration • S, substrate concentration • P, product concentration • D, dilution rate • Sf, feed substrate concentration • Control Specifications • CV: X and P • MV: Dand 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

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

28. Preliminary Conclusions Preliminary Conclusions 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

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