Explicit Non-linear Optimal Control Law for Continuous Time Systems via Parametric Programming

1. ExplicitNon-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. x γ wall- target g x=l plane-obstacle y=xtanθ+h y Brachistrone Problem 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 • Solve an optimization problem at each time interval Accounts for - Optimality - Constraints - Logical Decisions Shortcomings -Demanding Computations -Applies to slow processes -Uncertainty handling

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

6. Theory of PARCOS What is Parametric Programming? Region CR1 Features • Complete mapping of optimal conditions in parameter space • Function fc(x),vc (x),dc(x) • Critical regionsCRc(x)0 c=1,Nc

7. Parametric Programming Developments 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

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

9. Parco / Explicit MPC Solution • Complex • Approximate

10. Multi-parametric Dynamic Optimizationmp-DO • Feasible SetX*For each x*X* there exists an optimizer v*(x*,t) such that the constraints g(v*,x*) are satisfied. • Value Function f(x*), x*X* • Optimizer, statesv*(x*,t), x(x*,t),x*X*

11. mp-DO Solution Three methods mp- (MI)DO (1) 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 (2) • Euler – Lagrange conditions of Optimality • No state or control discretization

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

13. Multi-parametric Dynamic Optimizationmp-DO Optimality Conditions - Unconstrained problem Constraint bound g(x,v) tf to

14. Multi-parametric Dynamic Optimizationmp-DO Unknowns Switching points Optimality Conditions - Constrained problem Boundary constrained arc g(x,v) - constraint Unconstrained arc tf to t1 t2

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

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

17. Multi-parametric Dynamic Optimizationmp-DO Optimality Conditions - Constrained problem • Solve analytically the dynamics, get time profiles of variables • Substitute into Boundary Conditions  Eliminate time Linear in x Non Linear in t1,2 • 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 • Fix a point in x-space • Solve DO and determine active constraints and boundary arcs • Determine optimal profiles for μ(t,x*),λ(t,x*),v(t,x*),t1(x*),t2(x*) • Determine region where profiles are valid: Feasibility condition Optimality condition

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

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

21. 2 - state Exampleopen-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. mp-DO Result - Simulations Constrained Unconstrained

27. mp-DO Result - Suboptimal Compute feasible Control law In Hull v = -6.58x1-3.02x2 v = -6.92x1-2.9x2-1.59 Feature: 25 regions correspond to the same active constraint over different time elements Merge and get convex Hull

28. mp-DO Result - Suboptimal

29. x γ wall- target g x=l plane-obstacle y=xtanθ+h y Brachistrone Problem 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. Brachistrone Problem - Results Absence of disturbance: open=closed-loop profile

33. Brachistrone Problem - Results Presence of disturbance

34. Concluding Remarks 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 Issues • Unexplored area of research • Non-linearity in path constraints even if dynamics are linear • Complexity of solution