State description –Digital Processors,Sampled-data systems,Systems with dead time

New Roll No.: 167 (A.C.R)M. S. UniversityIndia

Manish A Tadvi

M.E Student

17 Sep 2012

Why State Description?

Limitation of Transfer Function

technique.

1.Highly cumbersome

2. It reveals only the system output for

a given input

Advantage Of State Description

o Provides a feedback proportional to

the internal variables of a system

CONTROLLABLE (FIRST) CANONICAL FORM:

Given a transfer function.

The coefficients can now be inserted directly into the state-space model by the following approach:

This state-space realization is called controllable canonical form.

.

OBSERVABLE (SECOND) CANONICAL FORM:

. This state-space realization is called observable canonical form.

State Descriptions of Digital Processors

SISO DTS:

State variables x1(k),x2(k),…,xn(k)

Input u(k), Output y(k)

Assumption- I/p switched on to the system at k=0

i.e. u(k)=0 for k<0,

Initial state is given by:

x(0)=x0 (n*1)vector

LTI system:

X(k+1)=F x(k) + g u(k); (state equation)…(1)

Y(k)=c x(k) + d u(k); (output equation)…(2)

where,

)(

)(

...

)(

)(

1

2

1

kx

kx

kx

kx

kx

n

n

u(k)=system input, d=scalar, (direct coupling between i/p & o/p.

Y(k)=defined output

ng

g

g

...

...g

2

1

4321 ccccc

nnnn

n

n

fff

fff

fff

F

..21

22221

11211

..

.........

.........

.....

......

State Variables:- The smallest set of variables which determine the state of a dynamic system is called state variable.

State variable describes the future response of a system, given the present state, the excitation i/p, and the eqn describing the dynamics.

State space:- The n dimensional state variables are elements of n dimensional space is called state space.

Basic Structure Of DCS :

20-Sep-12

Digital computer D/A Plant

Sensor

A/D

Controlled o/pDigital

set pt

The State-Space block implements a system whose behavior is defined by :

X(k+1)=F x(k) + G u(k) n = number of states.m= number of inputs.

Y(k)=C x(k) + D u(k) r= number of outputs

where x is the state vector, u is the input vector,

and y is the output vector.

F must be an n-by-n matrix, G must be an n-by-m matrix, C must be an r-by-n matrix, D must be an r-by-m matrix.

F G

C D

n m

n

r

Conversion of state variable to TF

X(k+1)=F x(k) + g u(k)

Y(k)=c x(k) + d u(k)

zX(z) – z x0 =F x(z) + g u(z) % z transform(zI - F) X(z) = zx0 + g u(z) % I is n x n identity matrixX(z) = (zI-F)-1 * z x0 + (zI-F)-1 * g u(z)

Y(z) = c x(z) + d u(z) % z transform

Y(z) = c * (zI-F)-1 * z x0 + (zI-F)-1 * g u(z)*c + d u(z)

Y(z) = c * (zI-F)-1 * z x0 + [ c * (zI-F)-1 * g + d ] u(z)

Y(z) = G(z) = c * (zI-F)-1 * g + d % In case of Initial condition x0 = 0U(z)

Y(z) = G(z) = c * adj(zI-F) * g + dU(z) | zI-F |

Matlab Simulation :

A=[0 1 0;-5 -2 -1;0 0 3]B=[0;1;1]c=[4 1 0]D=0[num,den]=ss2tf(A,B,C,D)sys=tf(num,den)

O/P : -A =

0 1 0-5 -2 -10 0 3

B = 011

20-Sep-12

Conversion of state variable to TF using MATLAB

Example :

c =4 1 0

D =0

num =

0 1.0000 0.0000 -16.0000den =

1 -1 -1 -15

Transfer function:s^2 - 16

------------------s^3 - s^2 - s - 15

csys= canon(sys,'companion')

a = x1 x2 x3

x1 0 0 15x2 1 0 1x3 0 1 1

b = u1

x1 1x2 0x3 0

c = x1 x2 x3

y1 1 1 -14

d = u1

y1 0Continuous-time model.

>>

Conversion of Transfer Function to Canonical State Variable Model :

First Companion form : % Direct Form : 1

Second Companion form : % Direct Form : 2

Jordan Canonical Form : % Parallel Form

zn+ 1zn-1+….+ n-1z+ n

Transfer Function : zn+ 1zn-1+…+ n-1z+ n

1, 2,… n as feedback element

1, 2,… n as feed forward element

G(z) =

Direct form : 1

20-Sep-12

a1 an-1 an

b0 b1 bn-1 bn

u(k) Xn(k) x2(k) x1(k)

+

+

+

+

_

+

+

+

+

+

+

+

y(k)

x(k+1)=Fx(k)+gu(k)y(k)=cx(k)+du(k)

1

..

...

0

0

g

]- ...., ,- ,0[ c 01101-n1-nnn

d = β 0

1..1 ..

1..000

.........

0..100

0...010

aaa nn

F

Direct Form : 1

This is called first companion form.

Direct form :- 2

20-Sep-12

bn bn-1b1 b0

an an-1 a1

+

_

+

_

+

+

+

_

+

+

u(k)

y(k)

xn(k)xn-1(k)x1(k)

1] 0 ... 0 [0 c

011

011

0

..

...

nn

nn

g

1..

2

1

..00

..........

....10

....01

.....00

a

an

n

n

F

d = β0

x(k+1)=Fx(k)+gu(k)y(k)=cx(k)+du(k)

This is called Second companion form.

Direct Form : 2

• F, g and c matrices of one companion form

correspond to the transpose of F, c and g matrices,

respectively, of the other.

• play an important role in pole-placement design

through state feedback.

Case 1:

20-Sep-12ln

b0

r1

rn

l1

+

+

+

+

+

+

+

x1(k) y(k)

xn(k)

u(k)

Parallel Form : 1

zn+ 1zn-1+….+ n-1z+ n

Transfer Function : zn+ 1zn-1+…+ n-1z+ n

G(z) =

r1 r2 rn

(z- 1) (z- ) ……………….. (z- n) =

If the transfer function involves distinct poles only as shown below :

Case 1:

n..

2

1

..00

..........

..........

0....0

0.....0

1

..

...

1

1

g

]r ... r [r c n21

d= β 0

x(k+1)=Fx(k)+gu(k)y(k)=cx(k)+du(k)

Case 1:

zn+ 1zn-1+….+ n-1z+ n

Transfer Function : zn+ 1zn-1+…+ n-1z+ n

G(z) =

=

Case 2:If the transfer function involves multiple poles as shown

below :

’1zn-1+ ’2z

n-2+… ’n(z- 1)

m(z- m+1)…(z- n)

G(z) = H1(z)+Hm+1(z)+….+Hn(z)

Hm+1(z) = rm+1 ,…., Hn(z) = rn

z- m+1 z- n

And, r11 r12 … r1m

(z- 1)m (z- )m-1 (z- )

H1(z) =

The realization of H1(z) is shown Here

Case : 2

r1m r12 r11

λ1 λ1 λ1

u(k)

+

+

+

+

+ +

++ y1(k)

+

+xm(k) x2(k)

x1(k)

Realization of H1(z)

z-1z-1z-1

1..

1

1

..00

..........

..........

0..10

0.....1

1n

m

..

1

1

..00

..........

..........

0....0

0.....0

]rrr ... r [r c n.. . 1m|1m1211

1

:

:

1

1

:

:

00

g

d= β 0

Case 2:

State space analysis of DCS applicable for LTI as well

as LTV system.

LTI systems are SISO.

State variable describes the future response of a

system, given the present state, the excitation

i/p, and the eqn describing the dynamics.

Application :

Sampled-Data Systems :

State Description of Sampled CT plants

A model of an A/D converter:

Samplerf(t) f(k)

k

t 0 1 2 3

A model of D/A converter

ZOH

tk0 1 2 3

f(k) f+(t)

f+(t)=f(k); kT <= t< (k+1)T

DT systemCTsystem samplerDiscrete time

systemZOH

Interconnection of DT and CT system

u(k) u+(t) y(t) y(k)

Equivalent Discrete

Time system

x(k+1)=Fx(k)+gu(k)y(k)=cx(k)+du(k)

bdθeAθ

gT

0

F = eAt

Find eigen values By equation | λI – A | = 0

Get λ1, λ2 …

e t = g( ) = 0 + 1

e t = g( ) = 0 + 1

e t = g( ) = 0 + 1

eAt = 0+ 1A

Thus we find λ1, λ2.

Example :

From the given BD find out state equation.

The state variable defined by:

x1(t)=q(t),

x2(t)=dq(t)/dt

State Eqn are given by:

dx(t)/dt=Ax(t)+bu+(t)

y(t)=cx(t)

20-Sep-12

.

20-Sep-12

Gh0(s)1

s(s+5)

u(t)

-

+

T=1s

u(k)

ZOHplant

Q(t)u+(t)

A= 0 1 b= 0 c=[1 0]

0 5 1

Find eigen values, 1 =0, 2=-5

e t=g( )= 0+ 1

eAt= 0+ 1A= 1 1/5(1-e-5T)

0 e-5T

20-Sep-12

Example :

g= 0.2(T-0.2+0.2e-5T)

0.2(1-e-5T)for T=0.1 sec

F = 1 0.07870 0.6065

g = 0.0430.0787

bdTeAg 0

Obtain Z-domain TF using MATLAB

.

20-Sep-12

ZOH1

s(s+2)

R(s) E(s) E*(s)

T=1s

C(s)

_

+

Matlab Simulation :

num=1

den=[1 2 0]

T=1

[numz,denz]=c2dm(num,den,T,'ZOH')

printsys(numz,denz,'z')sys = tf(numDz,denDz,-1)

axis([-1 1 -1 1])

zgrid

20-Sep-12

num =1

den =1 2 0

T =1

numz =0 0.2838 0.1485

denz =1.0000 -1.1353 0.1353

num/den = 0.28383 z + 0.1485

------------------------z^2 - 1.1353 z + 0.13534

numDz =1

denDz =1.0000 -0.3000 0.5000

Transfer function:1

-----------------z^2 - 0.3 z + 0.5

Sampling time: unspecified

Applications :

L2 norm of sampled-data system

Sampled-data H controller synthesis

Time response of sampled-data feedback system

State Description of Systemswith Dead-Time:

What is Dead time?

•Appears in many processes in Industry and in other fields like Economical and Biological Systems

They are Caused By Following Phenomena:

Transport Time

Accumulation of Time Lags

The required Processing time For Sensors

Effect of Dead time In System

Introduces additional lag in System Phase

A Heated Tank with A Long Pipe

The control input is the power W at the resistor.

The plant output is the temperature T at the end of the pipe.

dDtubtto e tAtx

ttA)()()0(e )0(

x(t)

dDbuTkT

e TkTAkTxAT

kT ]()[)(eT)x((kT

d X(t)/dt=Ax(t)+bu+(t- D)

X(kT+T)=Fx(kT)+g1u(kT-NT-T)+g2u(kT-NT)

bdTmT eAg1

bdmT

eAg 02

We can specify a first-order transfer function with dead time .Matlab Simulation :

• >> num = 5;•den = [1 1];•P = tf(num,den,'InputDelay',3.4)•

•Transfer function:• 5•exp(-3.4*s) * -----• s + 1•

•>> P0 = tf(num,den);•step(P0,'b',P,'r')•>>

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Step Response

Time (sec)

Am

plit

ude

• Digital Control and State Variable Methods- by M. Gopal

Reference :

.Wikipedia

.Matlab