Multiple Model Approach to Multi-Parametric Model Predictive Control of a Nonlinear Process: A Simulation Case Study

1. Multiple Model approach toMulti-Parametric Model PredictiveControl 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 10th 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 – estimatorinterplay • complexity (significant offline computation burden) • hybrid mp-MPC methods • control of hybrid or nonlinear systems • hybrid estimator required • controllerandestimator model stitching/switching • extremlydemandingcomputation & complexpartition • multiple-modelapproach • simplified, suboptimalsolution

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 < u1 < 50 [s-1], 0 < u2 < 100 [%], -5 < Δu1 < 5 [s-2], -50 < Δu2 < 50 [%/s]. 0 < p < 133 [mbar]

8. Process: nonlinear simulation model • 2nd order linear dynamics • static input nonlinearities • u1: polynomial function y = f(u1) • u2: affine function • y = kiu2 + ni • i = f (u1) • u2 nonlinearity • narrow the input constraint limit to linear range f(u1) f(u2)

9. Control design: hybrid PWA model • augment the original linear model with data from other operating points • model switching • f(x2) • f(x2, x4) boundary lines:

10. 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 Control design: PWA process model

11. Control design: tuning • controller parameter tuning • guide: reasonable computation timeof controller • tuning using LLA (Local Linear Analysis) • root loci of dominant controller poles • parameters: N = 6, Nu = 2, Rdu = diag([0.1 0.05]), Ru = diag([10-6 0.02]) • KF tuning • extended LLA of closed loop system • parameters: QK = diag([10-6 10-6 10-6 10-6 1]) RK = 10-3

12. MMmp-MPC (N=6,Nu=2) vslinearmp-MPC (N=6, Nu=2) tracking reference signal steps along three local dynamical models) linear model (black) from intermediate OP controllerpartitioncomposedof 3x100 reg. (hybridmp-MPC 200k) Results:simulation studies

13. MMmp-MPC (N=27,Nu=2) vslinearmp-MPC (N=27, Nu=2) improvedperformancedue to longerhorizons. controllerresuling in ~3x300 regions hybridmp-MPC not reallyfeasible Results:simulation studies

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 toMulti-Parametric Model PredictiveControl 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 10th PhD Workshop on Systems and Control September 2009, Hluboka nad Vltavou, Czech Republic