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EEM 462 DIGITAL CONTROL SYSTEMS

Chapters

1 – z-Plane Analysis2 – State-Space Analysis3 – Conventional Design4 – State-Space Design

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z-PLANE ANALYSIS

z-Transform by the Convolution IntegralThe Pulse Transfer Function (PTF)PTF of Various ConfigurationsDigital PIDRealization of Digital ControllersDigital Filters

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DEFINITIONS

Digital: Refers to those control systems where some signals arenot CT but DT

In practical usage, “discrete-time” and “digital” are interchanged:“Discrete-time” is frequently used in theoretical study,“Digital” is used in connection with hardware or softwarerealizations

DLTI: Discrete-time linear, time-invariant systems

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x(t)

t0

(a) CT analog signal

x(t)

0 t

(b) CT quantized signal

x(t)

t0 0 t

(c) Sampled-data signal (d) Digital signal

x(t)

5

Sample-and-Hold (S/H) samples an analog signal and holdsit at a constant value for a specified period of time

Analog-to-Digital Converter (A/D) also called an encoder, isa device that converts an analog signal into a digital signal,usually a numerically coded signal

Digital-to Analog Converter (D/A) also called a decoder, isa device that converts a digital signal (numerically coded signal) into an analog signal

Transducer is a device that converts a physical signal intoa different type e.g. a pressure signal into a voltage signalThere are analog or digital transducers

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7

z-PLANE ANALYSIS OF DT CONTROL SYSTEMS

Sampler converts a CT signal into a train of pulses occurring at the sampling instants; 0, T, 2T, 3T, …

ZOH (Zero-order Hold) keeps the o/p at a constant levelduring the sampling

ZOHx(t) T x(kT) or x*(t) h(t)

Sampler

x(t)

0 t

x(kT)

0 T 2T kT

h(t)

0 T 2T t

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If we consider the sampler o/p to be a train of weighted impulses, then we can have

0 0

* )()()()()(k k

kTtkTxkTttxtx

where a train of unit impulses is considered as

0

)()(k

T kTtt dT(t)

0 T 2T tThe Laplace transformation yields

0

** )()]([)(k

kTsekTxtxLsX

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Define eTs =z or s=(1/T)ln z

where z is a complex variable. Then,

0

** )(ln1)()(k

kzkTxzT

XsXzX

where X(z) is called the z-transform of x*(t);

)()( * txZzX and

0

)()(k

kzkTxzX

As in the case of Fourier and Laplace transforms, there is a table of z-transforms.

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At the o/p of ZOH, we have

h(kT+t)=x(kT)

The transfer fn. Of a ZOH is

sesG

Ts

h

1)(

[See, Ogata, p.78-79]

Note that the impulse sampler is a fictitous sampler introduced purely for the mathematical analysis. It is not possible to physically implement a sampler that generates impulses.

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z-TRANSFORM BY THE CONVOLUTION INTEGRAL

If x(t) is expressed in its L as X(s)=L[x(t)], then it is possible to obtain Z of x(t) directly from X(s) utilizing the convolutionintegral method (Ogata, p. 83-90);

)(_____)(__)( sXofpolestheat

ezzsXofresiduezX Ts

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For a simple pole at s=sj the corresponding residue Kj is given by

Tsjssj ez

zsXssKj

)()(lim

If a pole at s=si has a multiplicity ni then the residue Ki is

Tsn

in

n

ssi

i ezzsXss

dsd

nK i

i

i

i

)()(lim)!1(

11

1

13

X(s) with ZOH

Let X(s) be formed as the product of the process TF and ZOH;

where G1(s)=G(s)/s.The convolution integral yields,

)(1)(1)(1)( 1 sGessGesG

sesX TsTs

Ts

t

Ts TtgdgTtsGeL0

1111 )()()()(

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and the z-transform gives,

Hence, X(z) with ZOH is obtained as

or

)()( 11

1 zGzTtgZ

)()1()()()( 11

11

1 zGzzGzzGzX

ssGZzzX )()1()( 1

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THE PULSE TRANSFER FUNCTION (PTF)

The pulse TF relates the z-transform of the o/p at the sampling instants to that of the sampled i/p.Consider a CT system G(s) driven by an impulse-sampled signal;

We also consider a fictitious (synchronized) sampler at the o/p and observe that the sequence of values taken by y(t) only atinstants t=kT.

G(s)x(t) x*(t)

dT

dT

y*(t)

y(t)

16

The convolution sum gives the o/p as

For a causal system, the sum can be taken from 0 to 8 as

where g(k-h)=0 for h>k.

Thus, the z-transform of y(k) becomes

k

h

hxhkgkgkxky0

)()()(*)()(

0

)()()(h

hxhkgky k=0, 1, 2..

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0 0 0

)(

0

)()()()(h m h

hmhm

m

zhxzmgzhxmg

0 0 0

)()()()(k k h

kk zhxhkgzkyzY

= G(z)X(z) with m=k-h

The PTF is given by

)]([)()()( sGZzG

zXzY

G(z)X(z) Y(z)

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The presence or absence of the i/p sampler is crucial in determining the PTF.Consider a CT system without an i/p sampler,

Y(s)=G(s)X(s)

In terms of the z-transform,

Y(z) = Z[Y(s)] = Z[G(s)X(s)] = Z[GX(s)] = GX(z) ? G(z)X(z)

Note that the presence or absence of a sampler at the o/p does not affect the PTF.

G(s)x(t)

X(s)

y(t)

Y(s)

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PTF OF VARIOUS CONFIGURATIONS

(a) Cascaded Elements:

G(s) H(s)x(t)

dTdT dT

x*(t) u(t) u*(t) y(t) y*(t)

)()()()()(

)()(

zXzYzGHzHzG

zXzY

G(s) H(s)x(t)

dT dT

x*(t) y(t) y*(t)

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(b-1) Closed-Loop Systems:

G(s)

H(s)

R(s) C(s)E(s)

dT- )(1)(

)()(

zGHzG

zRzC

(b-2) Closed-Loop Systems:

G(s)

H(s)

dT dT

R(s) C(s)

- )()(1)(

)()(

zHzGzG

zRzC

?

(c) Digital Control System

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“*” indicates L of impulse-sampled signals

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From the block diagram we have,

and C(s) = G(s) GD*(s) E*(s) or C*(s) = G*(s) GD

*(s) E*(s)[See Ogata p.103-104 for the use of Starred L]In terms of the z-transform notation,

C(z) = G(z) GD(z) E(z)

Since E(z) = R(z) – C(z) we have,

C(z) = GD(z)G(z)[R(z) – C(z)]and

)()(1 sGsGse

p

Ts

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where GD(z) represents any digital controller of the form;

)()(1)()(

)()(

zGzGzGzG

zRzC

D

D

nn

nn

D zazazbzbb

zEzMzG

...1...

)()()( 1

1

110

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(d-1) Digital PID Controller (Positional Form):

whereKP = K - KT/2Ti = K – KI/2 = proportional gainKI = KT/Ti = integral gainKD = KTd/T = derivative gain

and K, Ti, Td are the proportional gain, integral (reset) time andderivative (rate) time respectively of the analog PID as

(d-2) Digital PID Controller (Velocity Form):

)1(1

)( 11

zK

zKKzG D

IpD

t

di dt

tdeTdtteT

teKtm0

])()(1)([)(

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The integral term can not be excluded when velocity form isused because it involves the i/p R(z).

The velocity form PID exhibits better response characteristics than the positional form PID and does not require initializationwhen the operation is switched from manual to automatic.

The hardware restrictions of analog PID are ignored for the digital PID because control laws can be implemented by software.

)()1()1(

)()()()( 11 zCzK

zzCzRKzCKzM DIP

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REALIZATION OF DIGITAL CONTROLLERS

The general form of the PTF is

where n=m and ai, bi are real quantities. This form is applicableto many digital controllers. For example, the PID controller has a 2nd order form as,

nn

mm

zazazbzbb

zXzYzG

...1...

)()()( 1

1

110

G(z)x(k)

X(z)

y(k)

Y(z)

1

21

1)2()()(

z

zKzKKKKKzG DDPDIPD

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where b0 = KP + KI +KD, a1 = -1b1 = -(KP + 2KD), a2 = 0b2 = KD

The realizations can be implemented by using the direct programming and the standard programming where the coefficients will appear as multipliers and z-1 will correspondto unit delays.

22

11

22

110

1

zazazbzbb

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(a) Direct Programming

From the general form, we get, after carrying out cross multiplication;

Y(z) = - a1z-1Y(z) - a2z-2Y(z) - … - anz-nY(z)

+ b0X(z) + b1z-1X(z) + … + bmz-mX(z)

In this realization (m+n) no. of delay elements are used. A minimum possible no. of delay elements are provided by thestandard programming, which is also referred to as the direct decomposition.

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The PTF is rewritten as

where

nn

mm zaza

zbzbzbbzXzH

zHzY

zXzY

...11)...(

)()(

)()(

)()(

11

22

110

)...()()( 1

10m

m zbzbbzHzY

? )(...)()()( 110 zHzbzHzbzHbzY m

m

and

nn zazazX

zH

...1

1)()(

11

? )(...)()()( 11 zHzazHzazXzH n

n

32

33

To avoid the accumulation of errors in the coefficients ai, bi asthe order of the realization gets higher, three decomposition schemes may be utilized.

As a result of these implementations, the system becomes lesssensitive to coefficient inaccuracies:

* Series Programming* Parallel Programming* Ladder Programming

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(b) Series Programming

G(z) is factorized into 1st and/or 2nd TF as,

G(z) = G1(z) G2(z) … Gp(z)

j

i

p

ji ii

ii

i

i

zdzczfze

zazb

1 121

21

1

1

11

11

G1(z) G2(z) … Gp(z)x(k)

X(z)

y(k)

Y(z)

35

H(z)

36

H(z)

37

(c) Parallel Programming

G(z) is expanded into partial fractions as

G(z) = A + G1(z) + G2(z) + … + Gq(z)

Note that the Ladder Programming is implemented by expanding G(z) into the continued-fraction form (See, Ogata p. 128-135)

j

i

q

Ji ii

ii

i

i

zdzczfe

zabA

1 121

1

1 11

38

39

H(z)

40

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DIGITAL FILTERS

(a) IIR (Infinite-Impulse Response) Filter

Consider PTF,

or in terms of the difference eq.,

y(k) = - a1y(k-1) – a2y(k-2) - … - any(k-n)

+ b0x(k) + b1x(k-1) + … + bmx(k-m)

nn

mm

zazazbzbb

zXzY

...1...

)()(

11

110 n=m

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Observe that the previous values of the o/p together with the present and past values of i/p are used to obtain the currento/p y(k).

Hence, this type of a digital filter is called recursive filter,where errors in previous o/p may accumulate.

A recursive filter is recognized by the presence of both ai ans biin the block diagram realization.

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(b) FIR (Finite-Impulse Response) Filter

Consider a digital filter where all ai’s are zero;filter

or in terms of the difference eq.,

y(k) = b0x(k) + b1x(k-1) + … + bmx(k-m)

The impulse response is now limited to a finite no. of samplesdefined over a finite range of time intervals, i.e. the impulseresponse is finite. This type is also called nonrecursive or a moving-average filter.

mm zbzbb

zXzY ...

)()( 1

10

44

Observe that;(i) FIR is non-recursive, so that no accumulation of errorsthrough the feedback,

(ii) Implementation does not require feedback, so that directand standard programming are identical,

(iii) The poles are at the origin and therefore it is always stable,

(iv) A disadvantage of FIR over IIR is that when i/p has highfrequency components, then the no. of delay elements in thefilter increases.

mm

mm

zbzbzb

zXzY ...

)()( 1

10

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