Conversion of transfer function to

canonical state variable models

Presented by:

JYOTI SINGH

ME (I &C)

REGULAR (142511)

Contents

• Introduction

• First Companion Form

• Second Companion Form

• Jordan Canonical Form

• Computing Standard Forms in MATLAB

Introduction

• Realization of transfer function into state variable models is needed even if the

control system design based on frequency-domain design method.

• In these cases the need arises for the purpose of transient response simulation.

• But there is not much software for the numerical inversion of Laplace transform.

• So one ways is to convert transfer function of the system to state variable

description and numerically integrating the resulting differential equations rather

than attempting to compute the inverse Laplace transform by numerical method.

There are three problems involved in realization of a given transfer function into state

variable models.

1. Is it possible to obtain state variable description from the given transfer function?

2. If yes , is the state variable description unique for a given transfer function?

3. How do we obtain the state variable description from the given transfer function?

Answer 1: Yes it is possible if and only if G(s) is a proper rational function.

A proper rational transfer function will have state

ẋ (t) = A x(t) + B u(t)

y (t) = C x(t) + D u(t)

A strictly proper rational function will have state model of the form

ẋ (t) = A x(t) + B u(t)

y (t) = C x(t)

(1)

(2)

Answer 2: There are numerable system that have same transfer function so the

representation of a transfer function in state variable form is obviously not unique.

Answer 3: There are three standard or canonical representation of transfer functions.

• A LTI-SISO system is described by transfer function of the form

where αi and βi are real constant scalar .

• We now drive the result for m=n ; then use it for m<n by setting appropriate βi

coefficients equal to zero.

• So we have to obtain state variable model corresponding to the transfer function

nm

nsnsnmsmsm

sG

;

......11

........110)(

nsnsnnsnsn

sG

......11

........110)( (3)

First Companion Form

• Let us consider a transfer function in the form

)()()......11( sUsZnsnsn

nsnsnsU

sZ

......11

1

)(

)(

)()(......)(11)( tutzntzpntzpn

where

tkd

tzd ktzpk )()(

• Now solving for highest derivative of z(t) we get the equation

(4)

which can be written as

• The corresponding differential equation is

α2α1 αn-1 αn

+

+

+

+

+

+

+

upnz pn-1z pn-2z pz z

-

)()(......)(11)( tutzntzpntzpn

After realizing the system let us consider the transfer function in two parts.

)........11( 0)( nsnsnsY

nsnsnsU

sZ

......11

1

)(

)(

(6a)

(6b)

(5)

Realization of equation (5)

nsnsn ......11

1

nsnsn

........

110

Z(s)U(s) Y(s)

Decomposition of transfer function (3)

• Now the realization of transfer (5a) is straightforward. The output

)(........)(11)(0)( tzntzpntzpnty

is nothing but the sum of the scaled versions of the input to the n integrators.

• To get the state variable model of the system , identify the output of each integrator

with a state variable starting a the right & proceeding to the left.

• The corresponding differential equation using this identification of state variable are

ẋ1 = x2

ẋ2 = x3

:

ẋn = -αnx1 - αn-1x2 - .......... α1xn+u

(7a)

α2α1αn-1 αn

+

+

+

+

+

+

+

u xnxn-1 x1

-

β0 β1 βnβn-1

x2

y

z

+

+ +

+ +

+

Realization of the system (3)

β2

+

+

The output equation can be written as

y = ( βn - β0αn ) x1 + ( βn-1 - β0αn-1 ) x2 +……..+ ( β1 - β0α1 ) xn + β0 u (7b)

The state & output equations are organized in vector matrix form

ẋ (t) = A x(t) + B u(t)

y (t) = C x(t) + D u(t)with

0 1 0 … 0

0 0 1 … 0

: : : :

0 0 0 … 1

-αn -αn-1 -αn-2 … -α1

A =;

0

0

:

0

1

B =

C = [ βn - β0αn , βn-1 - β0αn-1 , . . . . . . . , β1 - β0α1] ; D = β0

(8)

• The matrix A has a very special structure : the coefficients of the denominator of the

transfer function preceded by minus sign form a string along the bottom of the

matrix.

• The rest of matrix is zero except for “superdiagonal” terms which all are unity.

• In matrix theory this structure is said to be in Companion form. For this reason the

realization of transfer function is identified as Companion form realization.

• This state-space realization is also called controllable canonical form because the

resulting model is guaranteed to be controllable (i.e., because the control enters a

chain of integrators, it has the ability to move every state).

Second Companion Form

• In this form the coefficient appear in a column of the A matrix.

• This can be obtain by writing equation (3) as

)()........110()()........1

1( sUnsnsnsYnsnsn

or

0)](0)([..........)](1)(1[1)](0)([ sUsYnsUsYsnsUsYsn

• On dividing by and solving for Y(s), we obtainsn

)]()([1.......)](1)(1[1)(0)( sYnsUnsn

sYsUs

sUsY

Note that is the transfer function of a chain of n integrators.sn1

(9)

• The signal passes through n integrators ; the signal

passes through n-1 integrators and so forth to complete the realization of equation

βn βn-1 βn-2 β1β0

αnαn-1 αn-2 α1

u

+

-

x1x2

xn-1 xn y

Realization of equation (9)

ynun ynun 11

•.• To write the differential equation for the realization identify the output of each

integrator with a state variable starting at the left and proceeding to the right

ẋn = xn-1 -α1 ( xn + β0 u ) + β1 u

ẋn-1 = xn-2 - α2 ( xn + β0 u ) + β2 u

:

ẋ2 = x1 - αn-1 ( xn + β0 u ) + βn-1 u

ẋ1 = - αn ( xn + β0 u ) + βn u

and the output equation is

y = xn + β0 u

• The state and output equation organized in vector matrix form are given below

ẋ (t) = A x(t) + B u(t)

y (t) = C x(t) + D u(t)(10)

0 0 … 0 -αn

1 0 … 0 -αn-1

0 1 … 0 -αn-2

: : : : :

0 0 0 1 -α1

A = ; B =

βn – αn β0

βn-1 – αn-1 β0

βn -2 – αn-2 β0

:

- β1 – α1 β0

C = [ 0 0 … 0 1 ]

;

D = β0

A , B , C or D matrix of second companion form correspond ot the transpose of

the A , B , C or D respectively to the first one.

• This state-space realization is also called observable canonical form because the

resulting model is guaranteed to be observable (i.e., because the output exits from

a chain of integrators, every state has an effect on the output).

• These form also play an important role in pole placement design through state

feedback.

Jordan Canonical Form

• In this form the poles of the transfer function form a string along the main diagonal of the

matrix.

nsnnsn

nsnsnsG

......1

......110)(

• By long division , G(s) can be written as

)('0.....1

1

'.....1'1

0)( sG

nsnsnnsn

sG

or

ns

rn

s

r

s

r

sU

sYsG

.....

2

2

1

10)(

)()(

• The coefficient (i = 1,2,…….,n ) are the residue of the transfer function G’(s)

at the poles at s = ( i = 1,2,…..,n).

rii

(11)

• The transfer function consists of a direct path with gain , and first order transfer

function in parallel.

0

λ1

λ2

λn

β0

r1

r2

rn

+

u y

Realization of G(s) in equation (11)

x1

x2

xn

• Identifying the outputs of integrator with the state variables results in following state

and output equations:

λ1 0 … 0

0 λ2 … 0

0 0 … 0

: : : :

0 0 0 λn

ẋ (t) = x(t) + B u(t)

y (t) = C x(t) + D u(t)

ʌ = ;

1

1

1

:

1

B =

C = [ r1 r2 ….. rn ] ; D = β0

• It is observed that for this canonical state variable model , the matrix A is a diagonal

with the poles of G(s) as its diagonal elements.

• The unique decoupled nature of the canonical model is obvious from eqn (12); the

n first order differential equation are independent of each other.

ẋ (t) = λi xi(t) + u(t) ; i = 1 , 2 , 3 …….,n

(12)

(13)

• Assume that G(s) has m distinct poles at s = λ1 , λ2 , ……… , λm of multiplicity

n1 , n2 , ……… , nm respectively: s = n1 + n2 + ……… + nm i.e. G(s) is of the form

)(. . . . . . . . .)2( 2)1( 1

'. . . . . . . . .2'2

1'1

0)(

ms nms ns nnsnsn

sG

• The partial fraction expansion of G(s) is of the form

)(

)()(.......)(10)(

sU

sYsH msHsG

where

)(

)(

)(.........

)( 12

)(

1)(sU

sY i

is

rini

is ni

ri

is ni

risH i

• The first term in Hi(s) can be synthesized as a chain of ni identical, first order

systems , each having transfer function 1/(s-λi).

• The second order term can be synthesized by a chain of (ni-1) first order system ,

and so forth.

(14)

(15)

• The entire Hi(s) can be synthesized by the system having block diagram shown in

figure.

rin1 ri2r i1

λi λi λi

u +

+

yi

xini xi2 xi1

Realization of Hi(s) in equation (15)

• Now to get state variable we identify the output of each integrator with a state variable

starting at the right and proceeding to the left.

• The corresponding differential equation are

ẋi1 = λi xi1 + xi2

ẋi2 = λi xi2 + xi3

:uxiniixini

.

And the output is given by

xiniin

yi x rrxr iiii

i

2211 .........

• If the state vector for the subsystem is defined by

rinixixiT

xi 21

• Then equation can be written in standard form

ẋi = ʌi xi + Bi u

yi = Ci xi

where

i

i

i

i

000

1000

010

001

1

0

0

0

Bi; ;

rrrC ini

iii

21

(16a)

(16b)

(17)

Note that the matrix has two diagonals- the principle diagonal has the

corresponding characteristic root (pole) and the super diagonal has all 1’s.

i

• In matrix theory , a matrix having this structure is said to be in Jordan form. That’s

why this realization is identified as Jordan Canonical Form.

• The state vector of the overall system consists of the concatenation of state vector

of each of the Jordan blocks:

xm

x

x

x

2

1

• Since there is no coupling between any of the subsystem , the matrix of the

overall system is ‘block diagonal’: where each of the sub matrices is in the

Jordan canonical form.

i

ẋ1=ʌ1x1+B1u

y1=C1x1

ẋ1=ʌ2x2+B2u

y2=C2x2

ẋm=ʌmxm+Bmu

ym=Cmxm

0

y1

y2

ym

yu

m

00

020

001

;

BM

B

B

B

2

1

C = [ C1 C2 … Cm] ; D = β0

Subsystems in Jordan canonical form combined into overall system

Computing Standard Forms in

MATLAB

• MATLAB contains a function for automatically transforming a state space equation

into a companion (e.g., controllable or observable canonical form) form.

[Ap, Bp, Cp, Dp, P] = canon(A, B, C, D, 'companion');

• Moving from one companion form to the other usually involves elementary

operations on matrices and vectors (e.g., transposes or interchanging rows).

• compan(P): P is the vector with the coefficients of a characteristic

polynomial

[Ap, Bp, Cp, Dp, P] = canon(tf(Pnum,Pden), 'companion');

P= vector with the coefficients of a transfer function's numerator polynomial.

• To transform a state space equation into a modal (e.g., diagonal) form, the same

command can be used.

[Ap, Bp, Cp, Dp, P] = canon(A, B, C, D, 'modal');

• However, MATLAB also includes a command to compute the Jordan form of a

matrix, which is a modified modal form suited for matrices that have repeated

eigenvalues. jordan(A)

• There are some more function which can be used to convert transfer function to

canonical state variable form.

csys=ss(A,B,C,D) Controllable form

osys=ss(A’,C’,B’,D) Observable form

Problem: A feedback system has a closed loop transfer function

Construct three different state models for this system:

(a) One where the system matrix A is a diagonal matrix.

(b) One where A is in first companion form.

(c) One where A is in second companion form.

)3)(1(

)4(10

)(

)(

sss

s

sR

sY

THANKS