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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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Explicit non linear optimal control law for continuous time systems via parametric programming

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.


Brachistrone problem

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

Outline

  • Introduction

  • Multi-parametric Dynamic Optimization

  • Explicit Control Law

  • Results

  • Concluding Remarks


Introduction model predictive control

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


Explicit non linear optimal control law for continuous time systems via parametric programming

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

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


Explicit non linear optimal control law for continuous time systems via parametric programming

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


Model based control via parametric programming

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


Parco explicit mpc solution

Parco / Explicit MPC Solution

  • Complex

  • Approximate


Explicit non linear optimal control law for continuous time systems via parametric programming

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

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


Explicit non linear optimal control law for continuous time systems via parametric programming

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Unconstrained problem

(No inequality constraints)

Two point boundary value problem


Explicit non linear optimal control law for continuous time systems via parametric programming

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Unconstrained problem

Constraint bound

g(x,v)

tf

to


Explicit non linear optimal control law for continuous time systems via parametric programming

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


Explicit non linear optimal control law for continuous time systems via parametric programming

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Constrained problem

Complementarily Conditions


Explicit non linear optimal control law for continuous time systems via parametric programming

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Constrained problem

States - Continuity

Costates - Adjoints

Hamiltonian – Switching points


Explicit non linear optimal control law for continuous time systems via parametric programming

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

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

Control Law

Applied for

t* t t*+Δt

OR

Implement continuously


Explicit non linear optimal control law for continuous time systems via parametric programming

Continuous Control Law viamp-DO

  • Property 1:

  • Property 2:

  • Property 3:

  • Property 4:

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


2 state example open loop unstable system

2 - state Exampleopen-loop unstable system


Explicit non linear optimal control law for continuous time systems via parametric programming

mp-DO Result

Region


Mp do result

mp-DO Result

Results for constrained region:


Mp do result1

mp-DO Result

Results for constrained region:


Mp do result2

mp-DO Result

Complexity

mp-QP:

Max number of regions

mp-DO:

Max number of regions

Reduced space of optimization variables and constraints


Explicit non linear optimal control law for continuous time systems via parametric programming

mp-DO Result - Simulations

Constrained

Unconstrained


Explicit non linear optimal control law for continuous time systems via parametric programming

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


Explicit non linear optimal control law for continuous time systems via parametric programming

mp-DO Result - Suboptimal


Brachistrone problem1

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 problem2

Brachistrone Problem


Brachistrone problem results

Brachistrone Problem - Results


Brachistrone problem results1

Brachistrone Problem - Results

Absence of disturbance: open=closed-loop profile


Brachistrone problem results2

Brachistrone Problem - Results

Presence of disturbance


Concluding remarks

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


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