1. Explicit Non-linear Optimal Control Law for Continuous Time Systems via Parametric Programming Vassilis Sakizlis, Vivek Dua, Stratos Pistikopoulos Centre for Process Systems Engineering Department of Chemical Engineering Imperial College, London.

2. Brachistrone Problem wall- target plane-obstacle x y g x=l y=x  tanθ+h γ Find closed-loop trajectory γ(x,y) of a gravity driven ball such that it will reach the opposite wall in minimum time

3. Outline Introduction Multi-parametric Dynamic Optimization Explicit Control Law Results Concluding Remarks

4. Introduction Model Predictive Control Accounts for - Optimality - Constraints - Logical Decisions Shortcomings -Demanding Computations -Applies to slow processes -Uncertainty handling  Solve an optimization problem at each time interval

5. Application - Parametric Controllers (Parcos) Explicit Control law Eliminate expensive, on-line computations Optimization Problem Parametric Solution Parametric Controller v(t)=g(x * ) PLANT Process Outputs y Input Disturbances w Plant State x * Control v

6. Theory of PARCOS Complete mapping of optimal conditions in parameter space Function  c (x),v c (x),  c (x) Critical regions CR c (x)  0 c=1,N c What is Parametric Programming? Features Region CR 1

7. Theory, Algorithms and Software Tools for Multi-parametric Optimization Problems Quadratic and convex nonlinear Mixed integer linear, quadratic and nonlinear Bilinear Applications Process synthesis and planning Design under Uncertainty Reactive scheduling / Bilevel Programming Stochastic Programming Model based and hybrid control Parametric Programming Developments

8. Model – based Control via Parametric Programming Formulate mp-QP (mp-LP) Obtain piecewise affine control law Pistikopoulos et al., (2002) Bemporad et al.,(2002) Objective Discrete Model Current States Constraints

9. Parco / Explicit MPC Solution Complex Approximate

10. Multi-parametric Dynamic Optimization mp-DO Feasible Set X * For each x *  X * there exists an optimizer v * (x *,t) such that the constraints g(v *,x * ) are satisfied. Value Function  ( x * ), x *  X * Optimizer, states v * ( x *,t), x(x*,t), x *  X *

11. mp-DO Solution Three methods Complete discretization Discrete state space model (Bemporad and Morari, 1999) mp-(MI)QP (LP) (Dua et al., 2000,2001) Lagrange Polynomials for Parameterizing the Controls (Vassiliadis et al., 1994) semi-infinite program - two stage decomposition. (similar to Grossmann et al., 1983) mp- (MI)DO (1) mp- (MI)DO (2) Euler – Lagrange conditions of Optimality No state or control discretization No state or control discretization

12. Multi-parametric Dynamic Optimization mp-DO Optimality Conditions - Unconstrained problem (No inequality constraints) Two point boundary value problem

13. Multi-parametric Dynamic Optimization mp-DO Unconstrained Optimality Conditions - Unconstrained problem tftf toto g(x,v) Constraint bound

14. Multi-parametric Dynamic Optimization mp-DO Constrained Optimality Conditions - Constrained problem tftf toto g(x,v) - constraint t1t1 t2t2 Boundary constrained arc Unconstrained arc Unknowns Switching points

15. Multi-parametric Dynamic Optimization mp-DO Optimality Conditions - Constrained problem Complementarily Conditions

16. Multi-parametric Dynamic Optimization mp-DO Optimality Conditions - Constrained problem States - Continuity Costates - Adjoints Hamiltonian – Switching points

17. Multi-parametric Dynamic Optimization mp-DO Optimality Conditions - Constrained problem Solve analytically the dynamics, get time profiles of variables Substitute into Boundary Conditions  Eliminate time Linear in  Non Linear in t 1,2 Solve for ξ (sole unknown) and back-substitute into dynamics Get profiles of x(t,x*), v(t,x*), λ(t,x*), μ(t,x*)

18. Solution of mp-DO 1.Fix a point in x-space 2.Solve DO and determine active constraints and boundary arcs 3.Determine optimal profiles for μ(t,x * ),λ(t,x * ),v(t,x * ),t 1 (x*),t 2 (x * ) 4.Determine region where profiles are valid: Optimality condition Feasibility condition

19. Control Law Applied for t *  t  t * +Δt OR Implement continuously

20. Continuous Control Law via mp-DO Property 1: Property 2: Property 3: Property 4: Feasible region: X * convex but each critical region non-convex

21. 2 - state Example open-loop unstable system

22. mp-DO Result Region

23. mp-DO Result Results for constrained region:

24. mp-DO Result Results for constrained region:

25. mp-DO Result Complexity mp-QP:Max number of regions mp-DO:Max number of regions Reduced space of optimization variables and constraints

26. Constrained Unconstrained mp-DO Result - Simulations

27. mp-DO Result - Suboptimal Feature: 25 regions correspond to the same active constraint over different time elements Merge and get convex Hull Compute feasible Control law In Hull v = -6.92x 1 -2.9x 2 -1.59 v = -6.58x 1 -3.02x 2

28. mp-DO Result - Suboptimal

29. Brachistrone Problem wall- target plane-obstacle x y g x=l y=x  tanθ+h γ Find trajectory of a gravity driven ball such that it will reach the opposite wall in minimum time

30. Brachistrone Problem

31. Brachistrone Problem - Results

32. Absence of disturbance: open=closed-loop profile

33. Brachistrone Problem - Results Presence of disturbance

34. Concluding Remarks Issues Unexplored area of research Non-linearity in path constraints even if dynamics are linear Complexity of solution Advantages Improved accuracy and feasibility over discrete time case Suitable for the case of model – based control Reduction in number of polyhedral regions Relate switching points to current state