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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. x. γ. wall- target. g. x=l.

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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=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

### Outline

• Introduction

• Multi-parametric Dynamic Optimization

• Explicit Control Law

• Results

• Concluding Remarks

### 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

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

### 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

Parametric Programming Developments

Theory, Algorithms and Software Tools

for Multi-parametric Optimization Problems

• 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)

Objective

Discrete Model

Current States

Constraints

### Parco / Explicit MPC Solution

• 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*

### mp-DO Solution

Three methods

mp- (MI)DO (1)

Complete discretization

Discrete state space model

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

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*)

### 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

### Control Law

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

### mp-DO Result

Results for constrained region:

### mp-DO Result

Results for constrained region:

### mp-DO Result

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=xtanθ+h

y

### Brachistrone Problem

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

### Brachistrone Problem - Results

Absence of disturbance: open=closed-loop profile

### Brachistrone Problem - Results

Presence of disturbance

### Concluding Remarks

• 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