iscrete SlSO control systems via a unctions

A. Deb G . Sa r ka r M. Bhattacharjee S.K.Sen

Indexing terms: Set of delta functions, Operational transfer functions, Operational matrices, Discrete control systems

Abstract: The paper presents a computational technique through operational matrices using a set of mutually disjoint delta functions (DF) for the analysis of linear discrete control systems. Following a brief review of the well known block pulse functions (BPF), a new set of delta functions is viewed in the same light. This set is used to develop operational transfer functions in the delta function domain (DOTF) and employed for discrete system analysis which results in the same accuracy as the conventional z-transform method. The presented technique uses simple matrix manipulations and is able to do away with laborious and involved algebraic steps, including inverse transformation, associated with the z- transform analysis without losing accuracy. Also, the accuracy of sample values of the output does not depend upon m (or the sampling interval h). A few linear discrete SISO control systems, open loop as well as closed loop, having different typical plant transfer functions, are analysed as illustrative examples.

1 Introduction

Piecewise constant basis functions (PCBF) (e.g. Haar functions, Rademacher functions, Walsh functions, block pulse functions etc.) [I, 21 have been around in the literature for about eight decades. Of all PCBFs, the block pulse functions (BPFs) turned out to be the most fundamental and its qualitative as well as quanti- tative appraisal were presented by Deb et al. [2].

BPFs have been used successfully for the analysis, synthesis [3] and design of control systems, and other related problems [4, 51.

It is apparent that any set of PCBF is unsuitable for analysing discrete systems because a discrete system always deals with impulses, namely, Dirac delta func- tions, which can never be handled by the PCBFs men- tioned above. However, Chen and Wu [6] made a lone attempt to analyse a discrete control system using the

0 IEE, 1996 IEE Proceedings online no. 19960629 Paper first received 14th February 1996 and in revised form 30th May 1996 The authors are with the Department of Applied Physics, University of Calcutta, 92 A.P.C. Road, Calcutta 700 009, India

set of BPF and came up with approximate results. The main reason for such inexact results was their assump- tion that a block pulse could approximate an impulse.

In the present method, discrete system analysis is car- ried out using a set of mutually disjoint delta functions instead of the conventional z-domain. This delta func- tion set forms the &domain and is used to develop a special type of integration operational matrices of dif- ferent orders. Then an operational technique, using delta operational transfer function (DOTF), is used to analyse various linear discrete control systems via this set of delta functions and &domain operational matri- ces. The results are compared with the exact solutions.

The present analysis does not need any inverse trans- formation, or long division, like the z-transform method and algebraic manipulations associated with the z-transform analysis are avoided. Further, once the DOTF of any plant is computed, the presented tech- nique is able to analyse the plant for different input functions in one mathematical operation.

2 Brief review of block pulse functions (BPF) and the related operational matrices 171

A set of block pulse functions ~( , , ( t ) containing m component functions in the semi-open interval [0, r ) is given by:

where [ ... I T denotes transpose.

defined as:

- +( m) ( t ) 1 [ +o ( t ) .sol(t)....soz(t)...$m-l(t)IT

The ith component q,(t) of the BPF vector ~&,(t) is

(1) 1 0 otherwise

iT/m < t 5 (i + 1)T/m $i(t) = { A square integrable time function f ( t ) of Lebesgue measure may be expanded into an m-term BPF series in t E [0, r ) as:

m-1

f ( t ) - ci$z(t) , 2 = 0 ) 1 , 2 , .") m - 1 (2) z=o

= [CO c1 c2..-ci...cm-1].so( ( t ) - m)

The constant coefficients cis in eqn. 2 are given by: 1 /-(z+l)k

(3)

(4)

where h = Tlm is duration of each component BPF along the time scale.

IEE P~oc.-Control Theory Appl., Vol 143, No. 6, November 1996 514

In the m-term BPF domain, an operational matrix for integration P(m) has been derived as the following triangular matrix:

where P(m) = hi4 1 l... ... 1 1 I n m x m ( 5 )

The matrix P(m) performs as an integrator in the BPF domain and it may be inverted to compute the opera- tional matrix for differentiation.

The operational matrix for integration is pivotal in any BPF domain analysis. Though it performs approx- imate integration, it may be improved [XI if necessary.

Performing repeated integrations in the BPF domain [9], integration operational matrices of different orders may be obtained. These matrices provide better accu- racy for analysis in the BPF domain.

3 Set of mutually disjoint impulse functions

Delta function has a long as well as rich history and heritage in the area of mathematical physics. It was introduced by Paul A.M. Dirac [lo] and, from the very moment of its appearence in the literature, it never stopped fascinating scientists with its enigmatic appeal. 6 function is not a function in the ordinary sense because, although 6(t) = 0 if t z 0, 6(0) = CQ. Also, the function does not have a definite value for each point in its infinitesimal domain. Thus, 6 function is consid- ered to be a generalised function [Ill, and can be regarded as a distribution in the physical sense.

Now we form a new set of basis functions using delta functions (DF). This set is composed of m (say) com- ponent functions of which the ith member is defined as:

It is easy to see that h is the time delay between any two consecutive members of the set and may be regarded as the sampling period. In line with the defini- tions given in [lo] and [ll], we can write:

&(t) = S ( t - ih) where i = 0, 1 ,2 , ..., m - 1

( 6 ) 0 f o r t # i h 00 f o r t = i h U t ) = {

The value of 6,(t) at t = ih has no meaning but it attains numerical character if we use &(t) only within an integral.

&(t) satisfies the condition:

&,(t)dt = 1

With these basic ideas, the set of delta functions is formed in line with the set of block pulse functions dis- cussed in Section 2. But it should be kept in mind that, unlike block pulse functions, the set of delta functions does not form an orthogonal basis because the inner product of delta function is undefined.

From the nature of block pulse functions as defined by eqn. 1, it is easy to note that the component delta functions are actually samples of block pulse functions with a sampling interval equal to the width of each component block pulse.

This set of impulse functions is used to represent a function Y(t) which is the sampled output of a square integrable function At). IEE Proc-Control Theory Appb, Vol. 143, No. 6, November 1996

Thus, in the interval [0, T), the sampled functionf(t) is given by:

m-l

f * ( t ) = f ( t ) S ( t - ih) (7) 2=0

Considering the nature of D F and comparing eqns. 1 and 6, we observe that any two members of the set of D F is mutually disjoint in t E [0, T), like the block pulse functions. But, other properties of orthogonal/ orthonormal basis functions like completeness, approx- imations etc. are not defined for this set of DF.

Iff(t) is fed to a sampling device, the output of the device modulatesflt) as in eqn. 7. Thus,

m-1

j * ( t ) = C f , ~ , ( t ) , i = 0 , 1 , 2 , ..., m - 1 2=0

n = [ fo f l ...fm-lILJ(,)N = F(,).LJ,,)(t> ( 8 )

a,,, ( t ) = [bo(t) 61 (t)...62 (t)...6( m-I) (OIT

where we have chosen an m-set delta functions given by:

Unlike the coefficients c,s of block pulse functions given by eqn. 2, the coefficients J1 s of eqn. 8 are given by:

T f , = 1 f ( t )&( t - ih)dt, ih < T (9)

0

4 integration

In this Section we develop the operational matrix for integration in delta function domain.

It is noted that, if we integrate an m-set delta func- tion, the result is a set of delayed unit step functions [2]. This integrated version may be represented by means of the same set of delta functions (i.e. the output step functions are sampled) to obtain their DF equiva- lent.

It is easy to see the following relationship involving integration of an m-set delta function:

Delta function operational matrices for

1 A(,, ( W t = F(m) (t)

and FTm) ( t ) = PI(,) .A(,) ( t ) (10) Eqn. 10 needs some explaining. Integration of A(,,(t) produces a set of delayed unit step functions F,)(t). We sample the vector F(m,(t) to obtain a set involving impulse functions only. This is represented by the vec- tor F*tm)(t). This sampled function set may be repre- sented by the product Pl(m),A(m)(t) as shown in eqn. 10. We call the upper triangular matrix P1(,) the opera- tional matrix for integration of order m in the 6- domain. Thus,

PI(,) = ti 1 i...i 1 inmxm

P1(,) = P ( m ) - Q(m)Ip1

(11)

(12)

Pl(,) can also be expressed as:

where I(m, is a unit matrix and Q(m) is the delay matrix both of orders m [12]. It is noted that P1(,) has an elgenvalue 1 repeated m times.

The character of PlCm). is slightly different from P,,,, the conventional operational matrix for integration in the BPF domain. While the matrix P(m) operates upon a BPF vector to obtain the BPF equivalent of the inte- grated time function, the matrix Pltm) operates upon a

515

DF vector not only to integrate it, but also to sample the integrated output. Thus, the operational matrix

l(m) may be called an integrator-modulator which relates two impulse trains as input-output with the inte- gration operation in the sense of Euler.

Following eqn. 10, we can integrate F*(,)(t) to:

1 FT,) (W = G(m) ( t )

and GT,, ( t ) = .A,,, ( t ) (13) It is obvious that, at each stage of integration, E'+,) introduces some error. If we repeat the integration n times in this manner, the resulting E'l(,)n is obtained, using eqn. 12, as:

py,, = [I(,) - &(m)I-", n < m (14) If we work with Pl(,)", n = 1, 2, 3, ..., m - 1, for dis- crete system analysis, introduction of error at each of the n stages will cripple the usefulness of the DF domain analysis.

To avoid such a situation, we integrate the D F set directly n times and then expand the result of integra- tion in terms of the set of delta functions. The method is similar to the one shot operational matrices for repeated integration (QSOMRI) introduced by Rao and Palanisamy [9, 131 for Walsh function analysis.

When n = 2, the resulting one shot operational matrix P2(,) is obtained as:

~ 2 ( , ) = IO 1 2 3 4... ...(m - i)imxm (15) Upon a detailed investigation, the operational matrix for n times repeated integration Pn(,) is obtained as:

We cannot find any operational matrix for differentia- tion for DF like that for block pulse or Walsh function analysis. However, the operational matrices for integra- tion of all orders may be helpful in analysing discrete control systems as discussed in the next Section.

ear discrete §!SO control delta operational transfer

In this Section we present the principle of the opera- tional transfer function technique for linear discrete system analysis and develop delta operational transfer functions (DQTF) for several standard plants. These DQTFs are the equivalents of the Laplace domain transfer functions. It will be evident from the following that any system having any type of complex transfer function can be analysed using the results obtained for the simple plants with the help of the partial fraction technique.

For a linear SISQ control system, with the usual notation

C ( S ) = G ( s ) R ( s ) (17) We can express the sampled versions of the input r( t ) and output c(t) in terms of an m-set delta function as:

A ?( t ) = RA(,)

c*( t ) = CA(,, (18) A

Now the Laplace transfer function G(s) of eqn. 17 is expanded in a power series containing negative powers

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of the Laplace operator s. Since the terms of this series involve integrators of different orders, it is logical that these can be replaced by one shot operational matrices of respective orders to obtain an equivalent operational transfer function in the discrete &domain. Hence, eqns. 11, 15 and 16 can now be used to determine the delta domain operational transfer function DOTF(,). It should be noted that DOTF(,) is always a square matrix and is operative upon delta functions only.

Using DOTF(,) and eqn. 18, we can write an equa- tion similar to eqn. 17 to obtain the relation:

C = R.DOTF(,) (19) which is the key equation.

We can test this system for n number of inputs by making R a rectangular matrix of dimension (n x m) and the simple multiplication operation on the RHS of eqn. 19 will yield giving n number of outputs. That is, input-output correspondence exists between each row of R(nx,l and each row of C(nxm).

5. I First-order plant Consider a first-order plant having a transfer function given by:

Gl(s) = ( S + U)-'

where a is real and greater than zero. Expressing Gl(s) as a power series involving negative powers of the Laplace operator s, we have:

Gl(s) = (S + U)-'

= a-l[as-l-a2s-2+a3s-3-U4s-4+ ... to 001

= a-1[aP1(,)-a2P2(,)+a3P3(,)-a4P4(,)+ ... to 001

Following the principle outlined above, we can write: DOTFl(,)

where m basis functions have been considered. Upon simplification, we have

DOTF1(,) = 0[1 exp(-ah) exp(-2ah) exp(-3ah) . .. exp[- (m - 1)ahIm (20)

Eqn. 20 will now be used to analyse the sampled-data control systems shown in Figs. 1-3 for step inputs con- sidering m = 10, T = 10s and h = 1s.

r--------- 1 I

Rl(S), - / I S I I I L _ _ _ _ _ _ _ _ _ J

sampler plant

Fig. 1 function

Open loop discrete control system having $ut-order transfer

r-------- 1 R

I L - - - - _ - _ _ J

sampler ' pl2 Fig. 2 Simple feedback control system with sampler at input

IEE Proc.-Control Theory Appl., Vol. 143, No. 6, November 1996

r - - ------- R 3 k ) '

plant L _ _ _ _ _ _ _ - _ J sampler

Fig. 3 Error sampledfeedback control system

Following eqn. 8, the step input in the &domain is

u*(t) = R l . ~ l ( ~ ~ ) ( t ) = [l 1 1...1 1].4(10)(t) For the system shown in Fig. 1, following eqn. 19, the output C1 in the delta function domain is given by

where DOTFl is obtained from eqn. 20 putting a = 1.

D F domain is

where DOTF1' is obtained considering a = 2 in eqn. 20. In a similar fashion the D F domain output C3 for

the system shown in Fig. 3 is:

Using the conventional z-transform approach, the exact solutions with step input, for the systems shown in Figs. 1-3 are, respectively, given by:

given by n

C1= R l . D O T F 1 (21)

For the system shown in Fig. 2, the output C2 in the

C2 = Rl .DOTF1 ' (22)

C 3 = R l . D O T F l [ I + DOTFII-1 (23)

(24) 2'

2' - 1.36787942 + 0.3678794 Cl(2) =

(25)

(26)

2'

z2 - 1.13533522 + 0.1353352

0.52'

C2(2) =

C3(2) = 2' - 1.18393972 + 0.1839397

Eqns. 21, 22 and 23 are solved and the results are com- pared with the exact solutions given by eqns. 24, 25 and 26. It is observed that the results of the DF analy- sis match exactly with the results obtained from z- transform analysis.

5.2 nth order plant with a single pole of multiplicity n Consider a plant having a transfer function G2(s) = (s + a)-.. As before, G2(s) can be expanded in a power series given by:

The equivalent DOTF is given by:

DOTF2(,) = [O uiz ui3 u14... uil)... Ulmlmxm

where (27)

( j - 1)n-1 U l j = exp[-(j - l ) a ] , for 1 5 j 5 m

(n - l)! Consider an open loop control system having a plant

transfer function G2(s) with a = 1 and n = 2, and sam- pler at the input. For step input, the output responses of the system are determined by both the conventional z-transform analysis and the b-domain analysis for m = 10, T = lOs, and h = 1s. The results are the same as those given by the z-transform analysis.

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5.3 Second-order plant with imaginary roots Consider a system having transfer function G3(s) = (s2 + a2)-l. This can be decomposed as:

where j = - 1) G3(s) = - 1 [- 1 - -1 1

23a s - j a s + j a Using eqn. 20, the DOTF is given by:

DOTF3(,) = 10 ~ 1 2 'U13 'u14... ~ j . . . ~ l m ] m x m (28) where vl1 = (l/a) sin[(j - l)a], for 1 5 j I m. Consider an open loop sampled-data control system having a plant transfer function G3(s) with a = d2. For step input, the output responses of the system are determined by both the conventional z-transform analysis and the 6- domain approach for m = 10, T = lOs, and h = 1s. The results are the same as those obtained by the z-trans- form analysis.

5.4 Second-order plant having complex roots Let us consider a system with complex conjugate poles having a transfer function:

G4(s) = ( s2 + QS + @)-I, a,p > 0 This can be decomposed as:

- s + a + j b

where a = 1/2u and b = l/z d(48 - u2). Again, using eqn. 20, the DOTF is given by:

DOTF4(,) = U0 w12 w13 w14..* w3... wlmllmxm (29) where wlJ = (l/b)exp[-(j - l)a] sin[(j - l)b] for 1 I j I m.

Consider an open loop sampled-data system having a plant transfer function G4(s) with U = fi = 1. For a step input, the output responses of the system are computed via both the conventional z-transform method and the b-domain technique for m = 10, T = 1Os, and h = 1s. The results obtained are the same as those given by the z-transform method.

6 Conclusion

The present work uses a set of mutually disjoint Dirac delta functions for discrete control system analysis. The principle of the delta domain operational technique is presented and using this set, a delta operational trans- fer function (DOTF) is defined. This new DOTF has been used to analyse linear discrete SISO control sys- tems. Several open loop and closed loop control sys- tems are investigated with different forms of transfer functions using step input. The results are found to match exactly with the solutions obtained by the con- ventional z-transform analysis. Also, the accuracy of sample values derived by this method is entirely inde- pendent of m, or the value of h, for a fixed time period T. So, if we have unequal sampling intervals for any input signal, the output sequence is also spaced in a similar fashion, provided the DOTFs are deduced con- sidering the same pattern of unequal sampling inter- vals. The present technique can analyse any plant having a specific DOTF for any number of inputs in one mathematical operation using eqn. 19. Since delta functions are defined in time domain, no inverse trans- formations are necessary to obtain the solution.

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