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OPTIMAL CONTROL SYSTEMS

AIM

To provide an understanding of the principles of optimization techniques in the static and dynamic contexts.

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LEARNING OBJECTIVES

On completion of the module the student should be able to demonstrate:

 - an understanding of the basic principles of optimization and the ability to apply them to linear and non-linear unconstrained and constrained static problems,

 - an understanding of the fundamentals of optimal control.

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LABORATORY WORK

The module will be illustrated by laboratory exercises and demonstrations on the use of MATLAB and the associated Optimization and Control Tool Boxes for solving unconstrained and constrained static optimization problems and for solving linear quadratic regulator problems.

 

ASSESSMENT Via written examination.

 MSc only - also laboratory session and report

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PRINCIPLES OF OPTIMIZATION Typical engineering problem: You have a process that can be represented by a mathematical model. You also have a performance criterion such as minimum cost. The goal of optimization is to find the values of the variables in the process that yield the best value of the performance criterion.

  Two ingredients of an optimization problem:

(i) process or model

(ii) performance criterion

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Some typical performance criteria:        maximum profit

        minimum cost

        minimum effort

        minimum error

        minimum waste

        maximum throughput

        best product quality

 Note the need to express the performance criterion in mathematical form.

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Static optimization: variables have numerical values, fixed with respect to time.

Dynamic optimization: variables are functions of time.

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Essential Features

Every optimization problem contains three essential categories:

1.        At least one objective function to be optimized

2.        Equality constraints

3.        Inequality constraints

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By a feasible solution we mean a set of variables which satisfy categories 2 and 3. The region of feasible solutions is called the feasible region.

feas

ible

regi

on

linearinequalityconstraint

linearequalityconstraint

nonlinearinequalityconstraint

nonlinearinequalityconstraints

x2

x1linear inequality constraint

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An optimal solution is a set of values of the variables that are contained in the feasible region and also provide the best value of the objective function in category 1.

For a meaningful optimization problem the model needs to be underdetermined.

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Mathematical Description

1 2

Minimize : ( ) objective function

( ) equality constraintsSubject to: ( ) inequality constraints

where , is a vector of n variables ( , , , )

( ) is a ve

nn

f

x x x

x

h x 0g x 0

x

h x

1

2

ctor of equalities of dimension

( ) is a vector of inequalities of dimension

m

mg x

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Steps Used To Solve Optimization Problems

1. Analyse the process in order to make a list of all the variables.

2. Determine the optimization criterion and specify the objective function.

3. Develop the mathematical model of the process to define the equality and inequality constraints. Identify the independent and dependent variables to obtain the number of degrees of freedom.

4. If the problem formulation is too large or complex simplify it if possible.

5. Apply a suitable optimization technique.

6. Check the result and examine it’s sensitivity to changes in model parameters and assumptions.

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Classification of optimization Problems

Properties of f(x)

    single variable or multivariable

    linear or nonlinear

    sum of squares

    quadratic

    smooth or non-smooth

    sparsity

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Properties of h(x) and g(x)

        simple bounds

        smooth or non-smooth

        sparsity

        linear or nonlinear

        no constraints

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Properties of variables x

•time variant or invariant

•continuous or discrete

•take only integer values

•mixed

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Obstacles and Difficulties

• Objective function and/or the constraint functions may have finite discontinuities in the continuous parameter values.

• Objective function and/or the constraint functions may be non-linear functions of the variables.

• Objective function and/or the constraint functions may be defined in terms of complicated interactions of the variables. This may prevent calculation of unique values of the variables at the optimum.

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• Objective function and/or the constraint functions may exhibit nearly “flat” behaviour for some ranges of variables or exponential behaviour for other ranges. This causes the problem to be insensitive, or too sensitive.

• The problem may exhibit many local optima whereas the global optimum is sought. A solution may be obtained that is less satisfactory than another solution elsewhere.

• Absence of a feasible region.

• Model-reality differences.

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Typical Examples of Application static optimization • Plant design (sizing and layout).

• Operation (best steady-state operating condition).

• Parameter estimation (model fitting).

• Allocation of resources.

• Choice of controller parameters (e.g. gains, time constants) to minimise a given performance index (e.g. overshoot, settling time, integral of error squared).

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dynamic optimization

• Determination of a control signal u(t) to transfer a dynamic system from an initial state to a desired final state to satisfy a given performance index.

• Optimal plant start-up and/or shut down.

• Minimum time problems

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BASIC PRINCIPLES OF STATIC optimization THEORY

Continuity of Functions

Functions containing discontinuities can cause difficulty in solving optimization problems.

Definition: A function of a single variable x is continuous at a point xo if:

(a) exists

(b) exists

(c)

f x

f x

f x f x

o

x x

x xo

o

o

( )

lim ( )

lim ( ) ( )

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If f(x) is continuous at every point in a region R, then f(x) is said to be continuous throughout R.

x

f(x)

x

f(x)

f(x) is discontinuous.

f(x) is continuous, but( ) ( )dff x x

dx is not.

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Unimodal and Multimodal Functions

A unimodal function f(x) (in the range specified for x) has a single extremum (minimum or maximum).A multimodal function f(x) has two or more extrema.

There is a distinction between the global extremum (the biggest or smallest between a set of extrema) and local extrema (any extremum). Note: many numericalprocedures terminate at a local extremum.

If ( ) 0f x at the extremum, the point iscalled a stationary point.

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f(x)

x

local max (stationary)

local min (stationary)

global max (not stationary)

stationary point(saddle point)

global min (stationary)

A multimodal function

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Multivariate Functions - Surface and Contour Plots

We shall be concerned with basic properties of a scalar function f(x) of n variables (x1,...,xn).If n = 1, f(x) is a univariate functionIf n > 1, f(x) is a multivariate function.

n 1

For any multivariate function, the equationz = f(x) defines a surface in n+1 dimensional space .

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In the case n = 2, the points z = f(x1,x2) represent a three dimensional surface. Let c be a particular value of f(x1,x2). Then f(x1,x2) = c defines a curve in x1 and x2 on the plane z = c.

If we consider a selection of different values of c, we obtain a family of curves which provide a contour map of the function z = f(x1,x2).

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1 2 21 2 1 2 2(4 2 4 2 1)xz e x x x x x contour map of

-3 -2 -1 0 1 2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x2

x1

0.20.4

0.71.0

1.7

1.8 2

3 4 56

z = 20

1.8

2

1.7

3

4

56

local minimum

saddle point

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27

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Example: Surface and Contour Plots of “Peaks”

Function

2 2 21 1 2

3 5 2 21 1 2 1 2

2 21 2

3(1 ) exp ( 1)

10(0.2 )exp( )

1 3exp ( 1)

z x x x

x x x x x

x x

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0 5

10 15

20

0 5

10 15

20 -10

-5

0

5

10

x2

z

x1

multimodal!

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90

100

x1

x2

global max

global min

local min

local max

local max

saddle

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Gradient VectorThe slope of f(x) at a pointx x in the directionof the ith co-ordinate axis is

f

xi

( )x

x x

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The gradient vector at a point x xis normal to the the contour through that point in the direction of increasing f.

x

f ( )xincreasing f

At a stationary point:

f ( )x 0 (a null vector)

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e.g.

f T( )x c x x c f( )=

Note: If f ( )x is a constant vector,

f(x) is then linear.

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Convex and Concave Functions

A function is called concave over a given region R if:

f f f

Ra b a b

a b

( ( ) ) ( ) ( ) ( )

, , .

x x x x

x x

1 1

0 1where: and

The function is strictly concave if is replaced by >.

A function is called convex (strictly convex) if is replaced by (<).

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concave function

convex function

xxa xb

f(x)

f x( ) 0

xxa xb

f(x)

f x( ) 0

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I f then is concave.

I f then is convex.

f xf

xf x

f xf

xf x

( ) ( )

( ) ( )

2

2

2

2

0

0

For a multivariate function f(x) the conditions are:-

Strictly convex +ve defconvex +ve semi defconcave -ve semi defstrictly concave -ve def

f(x) H(x) Hessian matrix

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Tests for Convexity and Concavity

H is +ve def (+ve semi def) iff

x Hx x 0T 0 0 ( ), .

H is -ve def (-ve semi def) iff

x Hx x 0T 0 0 ( ), .

Convenient tests: H(x) is strictly convex (+ve def) (convex) (+ve semi def)) if:

1. all eigenvalues of H(x) are or 2. all principal determinants of H(x) are

0 0 ( )

0 0 ( )

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H(x) is strictly concave (-ve def)(concave (- ve semi def)) if:

1. all eigenvalues of H(x) are or 2. the principal determinants of H(x) are alternating in sign:

0 0 ( )

1 2 3

1 2 3

0 0 0

0 0 0

, , ,

( , , , )

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Example f x x x x x( ) 2 3 212

1 2 22

2 2

1 2 21 1 21

2

1 2 22 2

( ) ( ) ( )4 3 4 3

( ) ( )3 4 4

f f fx x

x x xx

f fx x

x x

x x x

x x

1 2

4 3 4 3( ) , 4, 7

3 4 3 4

H x

22

1 2

4 3eigenvalues: | | 8 7 0

3 4

1 Hence, ( ) is strictly conv, 7. e .xf

I H

x

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Convex Regionxa

xb

non convexregion

A convex set of points exist if for any two points, xa

and xb, in a region, all points:x x x a b( ) ,1 0 1

on the straight line joining xa and xb are in the set.If a region is completely bounded by concave functions then the functions form a convex region.

xa

xb

convexregion

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Necessary and Sufficient Conditions for an Extremum of

an Unconstrained FunctionA condition N is necessary for a result R if R can be true only if N is true.

R N

A condition S is sufficient for a result R if R istrue if S is true.

S R

A condition T is necessary and sufficient for a result R iff T is true.

T R

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There are two necessary and a single sufficientconditions to guarantee that x* is an extremum of afunction f(x) at x = x*:1. f(x) is twice continuously differentiable at x*.

2. , i.e. a stationary point exists at x*.

3. is +ve def for a minimum to exist at x*, or -ve def for a maximum to exist at x*

f ( )*x 0

2 f ( ) ( )* *x H x

1 and 2 are necessary conditions; 3 is a sufficient condition.Note: an extremum may exist at x* even though it is not possible to demonstrate the fact using the three conditions.

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optimization with Equality Constraints

min ( );

( ) ;x

x x

h x 0

nf

subject to: m constraints (m n)

Elimination of variables:

example: min ( ),x xf x x

x x1 2

4 5

2 3 6

12

22

1 2

x

(a)

s.t. (b)

and substituting into (a) :- f x x x( ) ( )2 22

226 3 5

Using (b) to eliminate x1 gives:xx

126 3

2

(c)

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At a stationary point

f x

xx x

x x

( )( )

.*

2

22 2

2 2

0 6 6 3 10 0

28 36 1286

Then using (c): xx

126 3

21071*

*

.

Hence, the stationary point (min) is: (1.071, 1.286)