Advanced - PowerPoint Presentation

Slide1

Advanced Control Systems (ACS)

Dr. Imtiaz Hussainemail: imtiaz.hussain@faculty.muet.edu.pkURL :http://imtiazhussainkalwar.weebly.com/

Lecture-7

State Space Canonical formsSlide2

Lecture Outline

Canonical forms of State Space ModelsPhase Variable Canonical FormControllable Canonical form Observable Canonical formSimilarity TransformationsTransformation of coordinatesTransformation to CCFTransformation OCFSlide3

Canonical Forms

Canonical forms are the standard forms of state space models. Each of these canonical form has specific advantages which makes it convenient for use in particular design technique. There are several canonical forms of state space modelsPhase variable canonical formControllable Canonical formObservable Canonical form

Diagonal Canonical form

Jordan Canonical

Form

It is interesting to note that the dynamics properties of system remain unchanged whichever the type of representation is used.

Companion forms

Modal formsSlide4

Phase Variable Canonical formThe method of phase variables possess mathematical advantage over other representations.

This type of representation can be obtained directly from differential equations. Decomposition of transfer function also yields Phase variable form.Slide5

Phase Variable Canonical form

Consider an nth order linear plant model described by the differential equationWhere y(t

)

is the plant output and

u

(

t)

is the plant input. A state model for this system is not unique but depends on the choice of a set of state variables.

A useful set of state variables, referred to as phase variables

, is defined as:

 Slide6

Phase Variable Canonical form

Taking derivatives of the first n-1 state variables, we have

 Slide7

Phase Variable Canonical form

Output equation is simply

 Slide8

8

∫

∫

∫

∫

…

＋

＋

Phase Variable Canonical form

 Slide9

9

Phase Variable Canonical form

 Slide10

Obtain the state equation in phase variable form for the following differential equation, where u(t) is input and

y(t) is output.The differential equation is third order, thus there are three state variables:And their derivatives are (i.e state equations)

Phase Variable Canonical form

(Example-1)Slide11

Phase Variable Canonical form (Example-1)

In vector matrix form

Home Work:

Draw Sate diagramSlide12

Consider the transfer function of a third-order system where the numerator degree is lower than that of the denominator.

Transfer function can be decomposed into cascade formDenoting the output of the first block as W(s)

, we have the following input/output relationships:

Phase Variable Canonical form

(Example-2)

 Slide13

Re-arranging above equation yields

Taking inverse Laplace transform of above equations.Choosing the state variables in phase variable form

Phase Variable Canonical form

(Example-2)

+

+

 Slide14

State Equations are given asAnd the output equation is

Phase Variable Canonical form

(Example-1)

 Slide15

State Equations are given asAnd the output equation is

Phase Variable Canonical form

(Example-1)

 Slide16

State Equations are given asAnd the output equation is

In vector matrix form

Phase Variable Canonical form

(Example-1)

 Slide17

Companion Forms

Consider a system defined bywhere u is the input and y is the output. This equation can also be written as

We will

present state-space representations of the system defined

by above equations in

controllable canonical

form and observable canonical

form.

 Slide18

Controllable Canonical Form

The following state-space representation is called a controllable canonical form:

 Slide19

Controllable Canonical Form

 Slide20

Controllable Canonical Form

∫

∫

∫

∫

…

＋

＋

…

＋Slide21

Controllable Canonical Form (Example)

Let us Rewrite the given transfer function in following formSlide22

Controllable Canonical Form (Example)

 Slide23

Controllable Canonical Form (Example)

By

direct decomposition of

transfer

function

Equating

Y(s)

with numerator on the right hand side and

U(s)

with denominator on right hand side.Slide24

Controllable Canonical Form (Example)

Rearranging equation-2 yields

Draw a simulation diagram using equations (1) and (3)

1/s

1/s

U(s)

Y(s)

-2

-3

P(s)

3

1Slide25

Controllable Canonical Form (Example)

State equations and output equation are obtained from simulation diagram.

1/s

1/s

U(s)

Y(s)

-2

-3

P(s)

3

1Slide26

Controllable Canonical Form (Example)

In vector Matrix formSlide27

Observable Canonical Form

The following state-space representation is called an observable canonical form:

 Slide28

Observable Canonical Form

 Slide29

Observable Canonical Form (Example)

Let us Rewrite the given transfer function in following formSlide30

Observable Canonical Form (Example)

 Slide31

Similarity Transformations

It is desirable to have a means of transforming one state-space representation into another. This is achieved using so-called similarity transformations.Consider state space model Along with this, consider another state space model

of the same

plant

Here

the state

vector

, say, represents the physical state relative to some other reference, or even

a mathematical coordinate vector.

 Slide32

Similarity Transformations

When one set of coordinates are transformed into another set of coordinates of the same dimension using an algebraic coordinate transformation, such transformation is known as similarity transformation. In mathematical form the change of

variables is written as,

Where

T

is a nonsingular

nxn transformation matrix.

The transformed state

is written as

 Slide33

Similarity Transformations

The transformed state

is written as

Taking time derivative of above equation

 Slide34

Similarity Transformations

Consider transformed output equationSubstituting

in above equation

Since output of the system remain unchanged [i.e.

] therefore above equation is compared with

that yields

 Slide35

Similarity Transformations

Following relations are used to preform transformation of coordinates algebraically Slide36

Similarity Transformations

Invariance of Eigen ValuesSlide37

Transformation to CCF

Transformation to CCf is done by means of transformation matrix P. Where CM is controllability Matrix and is given as and W is coefficient matrix

Where the

a

i

’s

are coefficients of the characteristic polynomial

s+

 Slide38

Transformation to CCF

Once the transformation matrix P is computed following relations are used to calculate transformed matrices. Slide39

Transformation to CCF (Example)

Consider the state space system given below. Transform the given system in CCF.

 Slide40

Transformation to CCF (Example)

The characteristic equation of the system is

 Slide41

Transformation to CCF (Example)

Now the controllability matrix CM is calculated as Transformation matrix P is now obtained as

 Slide42

Transformation to CCF (Example)

Using the following relationships given state space representation is transformed into CCf as

 Slide43

Transformation to OCF

Transformation to CCf is done by means of transformation matrix Q. Where OM is observability Matrix and is given as and W is coefficient matrix

Where the

a

i

’s

are coefficients of the characteristic polynomial

s+

 Slide44

Transformation to OCF

Once the transformation matrix Q is computed following relations are used to calculate transformed matrices. Slide45

Transformation to OCF (Example)

Consider the state space system given below. Transform the given system in OCF.

 Slide46

Transformation to OCF (Example)

The characteristic equation of the system is

 Slide47

Transformation to OCF (Example)

Now the observability matrix OM is calculated as Transformation matrix Q is now obtained as

 Slide48

Transformation to CCF (Example)

Using the following relationships given state space representation is transformed into CCf asSlide49

Home WorkObtain state space representation of following transfer function in Phase variable canonical form, OCF and CCF by

Direct Decomposition of Transfer FunctionSimilarity TransformationDirect Approach

 Slide50

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