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An Over View of Digital

Control System

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Digital Control System

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Linear Difference Equation

y((k+n)T)+a1 y((k+n-1)T)+ a2 y((k+n-2)T)++an y(kT) =

b0 r((k+m)T) +b1 r((k+m-1)T)+bm r(kT)

or can be written as

y(k+n)+a1 y(k+n-1)+ a2 y(k+n-2)++an y(k) =

b0 r(k+m) +b1 r(k+m-1)+bm r(k)

n m

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Consider a continuous time system

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The sampled data form of this system is

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Output of continuous time system can be written as

With t0 = kT and t = (k+1)T

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Solution of this equation can be written as

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The zero input solution of this equation is

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Basic Components

Sample and Hold (S/H)

Analog to Digital Converter (A/D)Digital to Analog Convertor (D/A)

Plant or Process

Transducers

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Mathematical Model of S/H

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)(te can be expressed as

....)3()2()2(

)2()()()()()0()(

TtuTtuTe

TtuTtuTeTtutuete

Taking Laplace of this

0

2

322

)(1

......)2()()0(1

....11

)2(11

)(11

)0()(

k

skTsT

sTsTsT

sTsTsTsTsT

ekTes

e

eTeeTees

e

e

s

e

s

Tee

s

e

s

Tee

ss

esE

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Second term on the RHS can be treated as input to ZOH and iswritten as

0

*)()(

k

skTekTesE

Therefore the Transfer Function of the ZOH

s

e

sE

sETF

sT

ZOH

1

)(

)(*

Therefore S/H can be represented as

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Starred Transform

Taking inverse Laplace ofE*(s) to get e*(t)

0

*)()(.......)2()2()()()()0()(

k

kTtkTeTtTeTtTetete

where )(t is the unit impulse

Determination of starred Laplace Transform

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TsEofPoles

sT

k

eEofresidues

esE

OPERATIONCONVOLUTIOtheiswheretILteLteL

kTttIwhere

tIteL

TtTeTtTeteLteL

)(

)(

*

0

*

1

1*)(

1

1)(

))(())(())((

)()(

))().((

.......))2()2()()()()0(())((

This expression is useful in generating tables for Starred Transforms

Starred Transform

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Starred Transform

)2)(1(

1)(

sssE

TsTs

TsTs

TsEofPoles

ee

ee

eEofresiduessE

)2()1(

2)(

1)(

)()(

*

1

1

1

1

)1)(1(

1

)1)(2(

1

1

1*)()(

ExampleDetermineE*(s) given that

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Hold Devices

T. F. of First Order Hold Device

2

2

*

)1)(1(

)(

)(

Ts

eTs

sE

sETF

sT

FOH

T. F. of Fractional Order Hold Device

2

)1()1)(1(

se

Tk

sekeTF

sTsTsT

Hk

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Z-Transform

We Know that

0

*)()()(

k

kTtkTete

0

*)()(

k

skTekTesE

e-kTs

is called Transcendental Equation

The reason for using Z-Transform is to convert Transcendental

Equations of s into an algebraic function of z.

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Z-Transform

Define z = esT where js

)ln(1

z

T

s

0

*)()ln

1())(()(

k

kzkTez

TEteZzE

z is an operatoroperator z is an advance operator representing advance

one sampling period.

operatorz-1 = e-sTis an delay operator representing a delay of

one sampling period.

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Z-Transform

Consider the transfer function

The corresponding impulse transfer function in the s domain is

and in the z domain,

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Z-TRANSFORM THEOREMS

Theorem 1Translation in Time (Time Shift) Shifting to the right (delay) yields

Shifting to the left (advance) yields

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Z-TRANSFORM THEOREMS

)(zfLimz

Theorem 2

Final Value: IfF(z) converges for mod(z) >1 and all

poles of(1-z)F(z) are inside the unit circle, then

Theorem 3

Initial Value: If exists, then

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Z-Transform

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Inverse Z-Transform

The partial fraction method

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Inverse Z-Transform

The power series method (Direct division method)

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Inverse Z-Transform

The power series method (Direct division method)

By direct comparison with equation 1

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Inverse Z-TransformThe inversion integral method

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Mapping of s-Plane to z-Plane

The transformation of the s plane into the z plane can be investigatedby inserting js into

Lines of constant the s plane map into circles of radius equal to

eT in the z plane

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Plots of loci of constant and loci of constant n

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SYNTHESIS IN THE Z DOMAIN

(DIRECT METHOD)Consider the system given below

For the block diagram shown, the system equations are

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(DIRECT METHOD)

Solving for E*(s) and substituting into (1)

The starred transform C*(s)

The Z transform of this equation is obtained by replacing each

starred transform by the corresponding function of z:

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(DIRECT METHOD)

Although the output c (t) is continuous, the inverse of C(z) yields

only the set of values C(kT), k = 0, 1, 2,. . ., corresponding to

the values at the sampling instants.

Thus, C(kT) is the set of impulses from an ideal fictitious sampler

located at the output, as shown in figure, which operates in synchronism

with the sampler of the actuating signal.

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(DIRECT METHOD)Example:

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Note that the input R(z) is incorporated as part of G1R(z); thus this

equation cannot be manipulated to obtain the desired control ratio.

A pseudo control ratio can be obtained as follows:

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The Frequency Response

For a given pulse TF G(z) replace z by ejT

where T is the sampling period

Example

Consider the system defined byX(kT)=u(kT)+ax((k-1)T) 0 < a < 1

U(kT) is the sampled sinusoid defined as u(kT) = A sin kT

Z Transform of the system is

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The Frequency Response

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The Bode Plot

In the z-plane frequency appearing as z = ejT

if it is treated in the same fashion the simplicity of Bode Plot is lost

Thus the direct application of frequency response method is not

worthy

Solution to this is

Bilinear Transformation and w-plane

wT

wTz

)2/(1

)2/(1

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The Bode Plot

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The Bode Plot

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The Bode Plot

By converting a given pulse TF in the z-plane into a rational function

of w, the frequency response methods can be extended to discrete

time control systems.

Inverse relationship

1

12

z

z

Tw

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State-Space Representation of the System

The state equation may be written asx(k+1) = f(x(k),u(k),K)

y(k+1)=g(x(k),u(k),K)

For a linear system

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Solution of Discrete Time State Equation

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