1. Multiple Model approach to Multi-Parametric Model Predictive Control of a Nonlinear Process a simulation case study Boštjan Pregelj, Samo Gerkšič Jožef Stefan Institute, Ljubljana, Slovenia bostjan.pregelj@ijs.si, samo.grerksic@ijs.si 10 th PhD Workshop on Systems and Control September 2009, Hluboka nad Vltavou, Czech Republic

2. Introduction with explicit solution the MPC is expanding its application area to low-level control disturbance rejection offset-free tracking output feedback (states usually not measurable) »controller – estimator interplay complexity (significant offline computation burden) hybrid mp-MPC methods control of hybrid or nonlinear systems hybrid estimator required controller and estimator model stitching/switching extremly demanding computation & complex partition multiple-model approach simplified, suboptimal solution

3. Outline multi-parametric MPC tracking controller and offset removal case study plant pressure control in wire annealer nonlinear simulation model controller design PWA process model controller & Kalman filter tuning results remarks & conclusions

4. Model predictive controller, an MPC linear system defined by a SS model state and input constraints MPC optimisation problem = CFTOC s.t.:

5. Explicit solution of MPC u(k) = function of current state! PWA on polyhedra control law where describes i -th region (polyhedron) properties: regions have affine boundaries value function J * k is convex, continuous, piece- wise quadratic function of x(k), optimizer: x * k is affine function of x(k), possibly discontinuous (at some types of boundaries)

6. State controller -> Tracking contrl. offset-free reference tracking »velocity form augmentation elimination of offset due to disturbance »tracking error integration »disturbance estimation output feedback »Kalman filter observer »additional integrating disturbance state d(k) »additional KF tuning possibilities > responce tuning with disturb. on states, inputs > input/output step disturbance model

7. Process: pressure control in annealer nonlinear high-order process, disturbances actuators: pump – slow response, large operating range valve – fast response, small operating range two input single output constrained system additional DOF constraints 0 < u 1 < 50 [s -1 ], 0 < u 2 < 100 [%], -5 < Δu 1 < 5 [s -2 ], -50 < Δu 2 < 50 [%/s]. 0 < p < 133 [mbar]

8. Process: nonlinear simulation model 2 nd order linear dynamics static input nonlinearities u 1 : polynomial function y = f(u 1 ) u 2 : affine function > y = k i u 2 + n i > i = f (u 1 ) u 2 nonlinearity »narrow the input constraint limit to linear range f(u 1 ) f(u 2 )

9. Control design: hybrid PWA model augment the original linear model with data from other operating points model switching »f(x 2 ) »f(x 2, x 4 ) boundary lines: OPu 1 [HZ]u 2 [%]u 1 gainu 2 gain 1 (low extreme)1530-0.3203-1.0057 2 (high extreme)1030-1.0010-2.4136 3 (intermediate)12.530-0.7007-1.7096

10. Control design: PWA process model gains for each local dynamical model defined in output equation (Wiener model) continuous transitions between models desired controller implementation active controller takes current state and computes control action PWA dynamic (i) OUTPUT ( GAIN ) MATRIX C I offset (g i ) 1[ 0 -1.0010 0 -2.4136 ]4.2408 2[ 0 -0.7007 0 -1.7096 ]-1.2497 3[ 0 -0.3203 0 -1.0057 ]-8.5920

11. Control design: tuning controller parameter tuning guide: reasonable computation time of controller tuning using LLA (Local Linear Analysis) » root loci of dominant controller poles » parameters: N = 6, N u = 2, R du = diag([0.1 0.05]), R u = diag([10 -6 0.02]) KF tuning extended LLA of closed loop system parameters: Q K = diag([10 -6 10 -6 10 -6 10 -6 1]) R K = 10 -3

12. Results: simulation studies MM mp-MPC (N=6,Nu=2) vs linear mp-MPC (N=6, Nu=2) tracking reference signal steps along three local dynamical models) linear model (black) from intermediate OP controller partition composed of 3x100 reg. (hybrid mp-MPC 200k)

13. Results: simulation studies MM mp-MPC (N=27,Nu=2) vs linear mp-MPC (N=27, Nu=2) improved performance due to longer horizons. controller resuling in ~3x300 regions hybrid mp-MPC not really feasible

14. Conclusions improved performance due do reduced plant- to-model mismatch low computation demand & complexity emphasis to nonlinear PWA plane matching suboptimal solution controller does not anticipate switch in prediction controller sellection via scheduling variable better results achievable other suboptimal approaches (current & future work) »simplified hybrid mp-MPC »restrict switching among dynamics in prediction »keeps higher level of optimality

15. Thank you!

16. Multiple Model approach to Multi-Parametric Model Predictive Control of a Nonlinear Process a simulation case study Boštjan Pregelj, Samo Gerkšič Jožef Stefan Institute, Ljubljana, Slovenia bostjan.pregelj@ijs.si, samo.grerksic@ijs.si 10 th PhD Workshop on Systems and Control September 2009, Hluboka nad Vltavou, Czech Republic