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

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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.

x

γ

wall- target

g

x=l

plane-obstacle

y=xtanθ+h

y

Find closed-loop trajectory γ(x,y) of a gravity driven ball such that it will reach the opposite wall in minimum time

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

- Solve an optimization problem at each time interval

Accounts for

- Optimality

- Constraints

- Logical Decisions

Shortcomings

-Demanding Computations

-Applies to slow processes

-Uncertainty handling

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

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

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

Formulate mp-QP (mp-LP)

Obtain piecewise affine control law

Pistikopoulos et al., (2002)

Bemporad et al.,(2002)

Objective

Discrete Model

Current States

Constraints

- Complex
- Approximate

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 Functionf(x*), x*X*
- Optimizer, statesv*(x*,t), x(x*,t),x*X*

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

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Unconstrained problem

(No inequality constraints)

Two point boundary value problem

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Unconstrained problem

Constraint bound

g(x,v)

tf

to

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

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Constrained problem

Complementarily Conditions

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Constrained problem

States - Continuity

Costates - Adjoints

Hamiltonian – Switching points

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*)

- 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

Applied for

t* t t*+Δt

OR

Implement continuously

Continuous Control Law viamp-DO

- Property 1:
- Property 2:
- Property 3:
- Property 4:

Feasible region: X* convex but each critical region non-convex

mp-DO Result

Region

Results for constrained region:

Results for constrained region:

Complexity

mp-QP:

Max number of regions

mp-DO:

Max number of regions

Reduced space of optimization variables and constraints

mp-DO Result - Simulations

Constrained

Unconstrained

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

mp-DO Result - Suboptimal

x

γ

wall- target

g

x=l

plane-obstacle

y=xtanθ+h

y

Find trajectory of a gravity driven ball such that it will reach the opposite wall in minimum time

Absence of disturbance: open=closed-loop profile

Presence of disturbance

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