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.

Brachistrone Problem

wall- target

plane-obstacle

x

y

g

x=l

y=xtanθ+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

Outline

• Introduction

• Multi-parametric Dynamic Optimization

• Explicit Control Law

• Results

• Concluding Remarks

Introduction Model Predictive Control

Accounts for- Optimality- Constraints- Logical Decisions

Shortcomings-Demanding Computations-Demanding Computations-Applies to slow processes-Applies to slow processes-Uncertainty handling

Solve an optimization problem at each time interval

Application - Parametric Controllers (Parcos)

• Explicit Control law• Eliminate expensive, on-line computations

Optimization Problem

Parametric Solution

)( *xv

*xParametric Controller

v(t)=g(x*)

PLANT Process Outputs yInputDisturbances

w

Plant State x*Control v

Theory of PARCOS

•Complete mapping of optimal conditions in parameter space•Function c(x),vc (x),c

(x) •Critical regions CRc(x)0 c=1,Nc

What is Parametric Programming?

FeaturesFeatures

binary

continuous

parameters

)( s.t.

)(

::

δvx

xvg

xvfxv

:0,,

,,min)(,

)()(

xxv

x

Region CR1

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

Model – based Control via Parametric Programming

0.15t ,10

,0

5.1),(0

""

01221.0992.0170.0

0.16954107.0269.1 ..

)100198.0

0084.00116.0(min)

,2,1||

*

,2,11,2

,2,11,1

24,2,1

1

0

2,2

2,1||

*

N

Nk

xxvxg

xx

vxxx

vxxxts

vxx

xxPxxφ(x

ktkttktk

t

ktktktkt

ktktktkt

ktktkt

N

kktkttN

TtN

vN

Formulate mp-QP (mp-LP)Obtain piecewise affine control lawPistikopoulos et al., (2002)Bemporad et al.,(2002)

c

o

cNc

Nc

XxCR

(x),φ(xv

,1

,0)(

),*

**

Objective

Discrete Model

Current States

Constraints

Parco / Explicit MPC Solution

c

cc

Nc

CRxCR

bxav

cc

,...1

0 if

law Control

2*1

*0

• Complex• Approximate

Multi-parametric Dynamic Optimizationmp-DO

),[

0)( ,0)()(

v(t)]dtv(t))()([min)(

*

221

T*

*

fo

o

f

t

t

Tf

Tf

v

ttt

xx

txGbtvDtxDg

vBxAx

RtxQtxPxxxf

• 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*

mp-DO Solution

Three methods

Complete discretization

Discrete state space model(Bemporad and Morari, 1999)

mp-(MI)QP (LP)mp-(MI)QP (LP) (Dua (Dua et al., et al., 2000,2001)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)mp- (MI)DO (2)• Euler – Lagrange conditions of OptimalityEuler – Lagrange conditions of Optimality• No state or control discretizationNo state or control discretization

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Unconstrained problem(No inequality constraints)

)()(

)(

ConditionsBoundary

],[

Equations of System alDifferenti Augmented

*

1

ff

o

fo

T

T

tPxt

xtx

ttt

BRv

AQx

BvAxx

Two point boundary value problem

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - UnconstrainedUnconstrained problem

tfto

g(x,

v)

Constraint bound

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - ConstrainedConstrained problem

tfto

g(x,

v) -

co

nstr

aint

t1 t2

Boundary constrained arc

Unconstrained arc

UnknownsSwitching points

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Constrained problem

],[0 ,0

)(

Equations of System alDifferenti Augmented

1

fo

iii

TT

TT

tttg

x

gBRv

x

gAQx

BvAxx

Complementarily Conditions

Multi-parametric Dynamic Optimizationmp-DO

Optimality Conditions - Constrained problem

TTTT

T

ff

o

gRvvQxxxH

tHtH

tHtH

tt

x

gtt

tPxt

txtx

txtx

xtx

)()(

)()(

)()(

)()(

)()(

)()(

)()(

)(

ConditionsBoundary

22

11

22

11

22

11

*

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

1111

2,1*

2,12,1

*2,12,1

,:

0)(,0))()()((

)()(

ofxwhere

ttbxtFtN

xtStM

Linear in 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-DO1. 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*),t1(x*),t2(x*)

4. Determine region where profiles are valid:

0)},(~

{min)(0),(~

0)},({max)(0),(

*

],[

*2

*

*

],[

*1

*

21

txxGtx

txgxGtxg

ttt

tttfo

Optimality condition

Feasibility condition

Control Law

c

cc

Nc

xCRxxCR

xtbxxtAv

,,1

),()(0

if

),(),(ˆ

*2**1

***

Applied fort* t t*+Δt

OR

))(,())(,(lim)(ˆ **

0

* xttbxxttAxv kco

kc

t

Implement continuously

Continuous Control Law viamp-DO

* offunction nonlinear are , ,continuous xv

• Property 1:

• Property 2:

• Property 3:

• Property 4:

convex ,continuous is )( *x

NL continuous )(x t),(xt *kx

*kt

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

2 - state Exampleopen-loop unstable system

]5.1,0[

1001000]15.1[

0

1

01

63.063.1

]dtv100084.00099.0

0099.00116.0[min)(

*

24*

*

txx

vxg

vxx

xxPxxx

o

t

t

Tf

Tf

v

f

mp-DO Result

);43.113.0(0.21)93.009.1(4.12)( 2186.0

2176.10 xxexxetv tt

Region

)43.113.0

94.009.1ln(1.0 :where

04.2)43.113.0(22.0)94.009.1(4.1

21

21

2186.0

2177.10

xx

xxt

xxexxe tt

mp-DO Result

Results for constrained region:

mp-DO Result

Results for constrained region:

mp-DO Result

Complexity

mp-QP:

10

00

)21(10

i

Nnv

i ii

NqMax number of regions

mp-DO: 4211

00

i

nv

i ii

qMax number of regions

Reduced space of optimization variables and constraints

Constrained

Unconstrained

mp-DO Result - Simulations

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 lawIn Hull

v = -6.92x1-2.9x2-1.59

v = -6.58x1-3.02x2

mp-DO Result - Suboptimal

Brachistrone Problem

wall- target

plane-obstacle

x

y

g

x=l

y=xtanθ+h

γ

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

Brachistrone Problem

),0[

)(

)(

1 ,)(

1,5.0)( tan,)tan(

)sin(2

)cos(2

min),( 00

f

oo

oo

f

f

tt

yty

xtx

lltx

hhxy

gyy

gyx

txy

Brachistrone Problem - Results

Brachistrone Problem - Results

Absence of disturbance: open=closed-loop profile

Brachistrone Problem - ResultsPresence of disturbance

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