THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING AN APPROXIMATE METHOD FOR ANALYTICALLY EVALUATING THE RESPONSE OF TIME-VARIABLE LINEAR SYSTEMS Elmer G. Gilbert July, 1960 IP-446

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TABLE OF CONTENTS Page LIST OF FIGURES............................................ iii ABSTRACT................................ iv INTRODUCTION.............................................. RESPONSE DETERMINATION FOR T.V.L.S................ THE APPROXIMATE REPRESENTATION......................... METHODS FOR EVALUATION OF THE APPROXIMATE REPRESENTATION.... 5 ORTHONORMAL APPROXIMATING FUNCTIONS....................... 10 EVALUATION OF THE ai(t)............1............ 13 THE APPROXIMATE REPRESENTATION IN TERMS OF EXPONENTIAL FUNCTIONS.......................................... 14 RESPONSE TO STOCHASTIC INPUTS............................... 15 Examples...................................... 19 CONCLUSION.......*............................. 24 ACKNOWLEDGEMENT..............-, 24 REFERENCES................................................ 35 ii

LIST OF FIGURES Figure Page 1 Physical Interpretation of Approximation to T.V.L.S......................................... 25 2 Feedback Control System with Finite Width Sampling.. 26 3 Periodic Functions of Normal Solution............... 27 4 Step Responses for Exact and Approximate Systems.... 28 5 Form of h(t,T) for t >> 1 29 6 h(tT) and h*(t,T) for t = -2, -1, 0, 1, 2......... 30-34 iii

ABSTRACT For all but the most elementary time-variable linear systems solutions of the response problem must be particular solutions obtained by machine computation. Thus the insight into system response characteristics which is gained from analytic procedures is lost. The paper proposes an approximate representation of the time-variable weighting function h(t,T) (system response at time t to a unit impulse at time t-T) which permits a straightforward analytic evaluation of response for both deterministic and stochastic inputs. Although the evaluation of the approximate representation may require machine computation, once the approximate representation is obtained, no further machine calculations are necessary. N The approximation has the form h*(t,T) = ai(t) =i(T) where the Oi(T) are arbitrary approximating functions. Methods are given for evaluating the ai(t). Several families of orthonormal approximating functions are presented which simplify both the evaluation and utilization of the approximation. For example, when Oi(T) have rational Laplace transforms the ai(t) are computed directly from the solutions of certain time-variable linear differential equations. For stationary stochastic inputs simple series expressions for the output correlation function are derived and it is shown that propitious choice of the Oi(T) greatly reduce the number of integrations which must be made in evaluating the series. Two example problems illustrate the feasibility of the proposed procedures. iv

INTRODUCTION Methods for response determination of time-variable linear systems (T.V.L.S. for short) are well developed and have been described by many authors. (12) However, except in rather special situations, application of these methods has been restricted, mainly because of the very great computational difficulties. Consequently, for most practical systems, solutions of the response problem must be particular solutions obtained by machine computation, and the insight into system response characteristics which is gained from analytic procedures is lost. This paper investigates the nature of the computational difficulties and proposes an approximate representation for T.V.L.S. which admits analytic determination of response for both deterministic and stochastic inputs. Altbough the evaluation of the approximate representation may require machine computation, once the representation is obtained, no further computer calculations are required for a large class of inputs. To begin, some general matters concerning the response of T.V.L.S. are reviewed and the approximate representation is presented. Then, methods for evaluation and utilization of the representation are considered. Finally two illustrative examples are described. RESPONSE DETERMINATION FOR T.V.L.S. The T.V.L.S. considered are represented by -L

-2where x(t) is the response, f(t) is the input, La(p,t) = Cn(t)pn +... + aoo(t), LP(p,t) = m(t)pm +... + Po(t), and p = d/dt. Straightforward generalizations, such as making x and f vectors, are possible but are avoided for conciseness. If it is assumed that the T.V.L.S. is initially at rest so that the response is due entirely to f(t), x(t) can be given by the integral t l -m h~t,)f(t —;rr (2) where h(t,T) is the time-variable weighting function or system response at time t to a unit impulse at time t-T. Initial conditions, if present, can be introduced by a suitable equivalent input. (3) Familiar restrictions on system characteristics have a one-to-one correspondence to simple conditions on h(t,T), for example: non-prediction and h=O, T < 0; time invariance and h(t,T) = h(t-T); periodicity with period T and h(t,T) = h(t + T,T); stability and j Ih(t,T) IdT < M < om, -o < t < 0 ) -00 The weighting function can be determined from the Green's function(5) and for m < n takes the form ht,7J- 0 o, To t~l xl(t),...,xn(t) are a set of linearly independent solutions of the homogeneous equation Lax = 0 and C1,...,Cn are obtained from xl,...,xn in a complicated way. Even,for the relatively few equations of low order where the xi(t) are known and tabulated, substitution of (3) in (2) generally yields impossibly difficult integrations.

-3Various transform methods have been proposed for the solution of (1),(67) the most familiar being that of Zadeh(8) which evaluates response as a superposition of exponential responses. Zadeh's approach is to define the time-variable transfer function and show that 5iZst)Ffi)r'i (5) where F(s) is the Laplace transform of f(t)* and C is a contour properly chosen in the s-plane. The method has the advantage that it maintains the same basic philosophy which is employed in the Laplace transform solution of time-invariant linear systems, including the use of residues in the evaluation of (5); it has the disadvantage that it is rarely possible to express H(st) in terms of known functions. Thus any computational advantage implied by (5) is of limited practical value. THE APPROXIMATE REPRESENTATION The preceding paragraphs substantiate the fact that the basic difficulty in the analysis of T.V.L.S. is not theoretical but computational. Since lengthy computation does appear to be unavoidable, it would seem better to direct the computational effort toward results of * Upper-case-letter funtions denote the Laplace transform of corresponding lower-case-letter functions.

-4a general, rather than a specific nature. To be more explicit, it would be better to obtain a representation for h(t,T) which would permit simple integration of (2) for a large class of inputs, than to obtain particular solutions of (1) for several different forcing functions. While it has only been possible to find such representations as approximations to h(t,T), accuracy of the representations generally can be made good enough for most engineering applications. The approximation of h(t,T) which has been found most useful is * N J1 (t,2^) = EII aE4?ct,(V ^(6) Lti It has the advantage that the approximate response is given simply as x*t ~- f ((t-7)1) _ - The approximating functions Xi(T) are purposely chosen to make the N integrations straightforward, the complexity of the problem being embodied in the ai(t), which if necessary can be presented in tables or graphs. An alternative approach to the evaluation of x*(t) is to observe that H (sot) ^-X 4a~ m, (8) so that ^ C ~

-5Often, the complex integral can be evaluated by means of residues or tables of inverse Laplace transforms. Another advantage of (6) is that it has a simple physical interpretation. Noting that each integral in (7) corresponds to the response of a time-invariant linear filter gives the realization, shown in Figure 1, a parallel set of time-invariant linear filters followed by a set of time-variable gains. This realization has led Cruz and Van Valkenburg(9) to employ (6) and similar representations in a procedure for the synthesis of T.V.L.S. For purpose of analysis the physical interpretation helps in attaining a good qualitative understanding of general response characteristics. At any given instant of time the response x*(t) consists of a simple linear combination of responses of N time-invariant linear filters. Furthermore, a translation in time of an input function f(t) produces only a corresponding translation in time of the outputs of the N time-invariant linear filters. Thus the effect of applying the same input at different times is easily appraised. The details of applying (7) and (9) when f(t) is deterministic are obvious and will not be developed further. When stochastic inputs are considered the choice of the 0i(T) is crucial; therefore, dissussion of such inputs will follow a description of several useful families of iY. First, methods for evaluating the approximation will be presented. METHODS FOR EVALUATION OF THE APPROXIMTE REPRESENTATION - "'Usually, the functiqnp(T) are chosen quite arbitrarily before the evaluation of the approximate representation begins. However,

-6when L and L in (1) are periodic with period T, there is a natural choice for the %i(T) which results from the Fourier series expansion of certain periodic functions having their origin in the Floquet theory. () The Floquet theory for periodic linear differential equations shows that the linearly independent solutions of the homogeneous equation LAx = 0 may be written as ci) -- Lt) he, - I, (10) where the qi(t) are periodic with period T. This result presumes that the characteristic exponents Pi are distinct, an assumption which causes no great loss of generality. If the solutions (10) are substituted in (3), h(t,T) takes the form where the ri(t) are periodic with period T. Since under very reasonable conditions on La and L, ri(t) is continuous and has a sectionally continuous derivative, it is possible to express ri(t) as the complex Fourier series - o itkt ()=- I tk7Cf T (12) Substitution of (12) in (11) then gives Lh t,)k (13) = O,'<.Q.

-7which, except for the change in summing notation, is the same as (6).. 2ikt The time-variable gains are the periodic functions rikqi(t)e; 2ntk the time-invariant filters have impulse responses e(Pi T Note that the time-invariant filters have rational Laplace transforms 2<Tk -1 (s - Pi + ) and are stable if the T.V.L.S. is stable (the Pi <0~,i=l,...,n). In terminating the infinite series (13) the relative size of the various terms must be considered. Certainly, when k gets sufficiently large the rik become small. Also, many of the qi(t)ePiTri(t-T) may be negligibly small so that response is essentially determined by a few "dominant modes." When the ai(t) and ~i(t) consist of a large constant part plus a small variable part, the Pi Xi the characteristic roots of the invariant system formed by setting the small variable parts of the ai and Pi to zero. In addition, the fluctuations of the ri(t-T) become so small that all but the k=O terms in (13) may be neglected. While the above procedure leads to a reasonable choice for the ii(T) and provides a direct method for computing the ai(t), it is restricted to periodic T.V.L.S. and the reduction of h to normal form (11) may be difficult. Moreover, even for periodic systems, it may be convenient for computational purposes to specify the Oi(T) in different form. If the Oi(T) are specified arbitrarily, there are a number of possible procedures available for choosing the ai(t). For the purposes considered here it seems most convenient to minimize the integral-square error,oo

-8Although it is not necessary or always desirable to assume that the (i(T) are orthogonal, it will be done to simplify the presentation. Later, a number of useful orthogonal approximating functions will be described. Since orthogonal function approximations are well known and have been used successfully for the approximation of both timeinvariant linear systems(l0,ll) and T.V.L.S(9)the presentation which follows serves mainly to introduce notation. For convenience it is assumed that the i (T) are orthon:ormal, i.e., If )qvrX -s I Ck -Jo - O i k (15) Minimization of E(t) then gives 00 ark &hat r)t&21~ (16) and N i EMn'-rk,(t J)Vd2- 4 a6t) (17) 00 It must be assumed that f h2(t,T)dT exists, a requirement which is -00 satisfied if the approximated system is stable ( f lh(t,T) ldT < MS < oo) and has a uniformly bounded weighting function (Ih(t,T)l < Mh < o). These conditions are satisfied in most practical problems, at least for the range of t which must be considered. Evaluation of ai(t) is usually difficult and will be examined in a subsequent section. One important fact is obvious - if h is piliodic with period T then the ai are periodic with period T.

-9The usual extensions of the above results are possible. For example, the weighted-integral-square error co E^) - IA/6)[h24 (2J4rY (18)'" OID can be minimized. The problem of finding suitable functions Oi(T) orthonormal with respect to W(T) then exists. Actually, the approximation obtained by terminating (13) is a minimum weighted-integral-squareerror approximation where the weight e -2it is applied to the approximation of the i-th mode. This results from the minimum integral-squareerror property of the Fourier series. Another extension is that of the constrained approximation. A constrained approximation permits certain properties of (1) to be represented exactly in h*(t,T). An example of a useful constraint would be I hat r)^^\^rdr = f t)l (19) This constraint would assure x(t) = x*(t) when f(t) = constant. It is also possible to represent the functions ai(t) in a series of functions 5k(t) so that (20) L -, I but, such a representation seems to have more value in synthesis than in analysis.(9) Little can be said in a general way about the error in response Ix - x*l and how it is influenced by error jh - h*1 in the representation. However, if the input is bounded by Mf and both h and h* are stable, then Ix- x*I is bounded. This can be seen from

-1000 60 00 OACO 00 co < iTh( t M ~< IFI hh yt, 4) Ilr f |. M MS t /,() ^o (21) If either h or h* is unstable then it can be shown that jx - x*j can become unbounded if the bounded input is chosen properly. Thus stability is a crucial factor in representability. (12) ORTHONORMAL APPROXIMATING FUNCTIONS Three primary considerations should be observed in choosing the orthonormal approximating functions oi(T): 1) the complexity of the computations for the ai(t), 2) the complexity of the integrations in (7), 3) the number N of terms in the approximation for a given accuracy. Usually, it is not possible to make all computations simple and N small, and still achieve good approximation accuracy. Where the compromises are made depends on the system being considered and on the intended applications of the approximate representation. A few of the mote useful orthonormal ses are given below. A great variety of other sets which have been used in signal theory and synthesis are also applicable. (15,14)

-11The simplest orthogonal functions are those which are nonoverlapping functions of T, e.g., the set = (7 ) -/(22) where UO(T) is the unit impulse at T = 0 and T1 < T2 <... < TN. This is the set proposed by Cruz.(9 15) He lets the coefficients ai(t) h(t,Ti) and realizes the Xi(T) by a tapped delay line. The representation is not a minimum-integral-square-error approximation because the Xi(T) cannot be normalized. An orthonormal set which is nonoverlapping is (7)- O,'- (-) r a 2ri - r, (i-)Tz r2 ilr (23) Although computation of the a (t) is now more difficult, a stepwise rather than an impulse representation is obtained. The set d (jo) 0 0 l-0)2 TAF, W2 TL7i f = S )2 (H ri-/)2nrafz?;) t3f (24) is orthonormal because the functions are non-overlapping in the frequency domain. Since the i (T) corresponding to (24) are complex and non-zero for negative T, the functions are non-realizable, a limitation which does not limit their usefulness in analysis. Usually, the non-overlapping functions given by (23) and (24) lead to a fairly large N (good accuracy requires small AT or Af) and

-12difficult computations for the ai(t). However, because the functions are non-overlapping, the influence of the ai(t) cn response is easily intepreted. From the standpoint of general simplicity, functions ji(s) which are rational in s are most advantageous. If it is assumed that i (s) has poles at sls2,...,si orthogonalization in the frequency domain yields.11) (s t,).(S-S. (S+,.) -25 (25) ~sg (5S,)(5[ ) -... (5,s_,) -Si 26) -- &-(5+) +5a) - +Si ) S"-(24S)s^5, (5+s) (5+s, ) -- Ss_.) Z -) 5 X where 8si < 0. Formula (25) is used when si is real; formula (26) and (27) are used when si and si+l are complex conjugates. In several special cases the functions ~i(T) corresponding to (25) may be expressed in terms of familiar functions. For example, when all si = -1 ),() =Ji L., (22) r >o (28) = 0 ) <'O where Li(Y) are the Laguerre polynomials of order i. If si = i - ((2^a e Ji (a ), x(o ~== C) it ft < ~ (29) i~~,~.~~

-13where Ji(2,21y) are Jacobi polynomials of order i.(16) In any case, applying the inverse Laplace transform to Oi(s) gives (providing none of the si are equal) ~k- (30) e constants dik have been tabulated by Katz ) for a number of dif The constants dik have been tabulated by Kautz(10) for a number of different choices of si. When the i (s) are rational, both the evaluation of the ai(t) and the integrals in (7) and (9) become relatively easy. Also, freedom of choice of the Si often provides good approximation accuracy with small N. EVALUATION OF THE ai (t) The evaluation of the ai(t) given by (16) is difficult for many of the same reasons that cause difficulties in the response evaluation of T.V.L.S. Therefore machine computation is usually required. Evaluation of ai(t) by a digital computer could be carried out in two steps: 1) numerical solution of the adjoint differential equation(5) for h(t,7) as a function of T. 2) numerical integration of (16). Differential analyzer evaluation of ai(t) using techniques similar to those proposed by Gilbert(ll) for the approximation of time-invariant systems appears promising and is presently being investigated. When the ji(s) are rational functions of s a simplification occurs., Substituting (30) in (16) gives <. n.

-14-. = Sk),(t)~= hdik 5h(tr)e d7 dik H(-k,t)(31) But as Zadeh shows H(s,t) satisfies the differential equation La Ip+)- ts,2)) ) If the T.V.L.S. described by h(t,T) is stable and Resi < 0, i=l,...,N, H(-si,t) exists and is given as the solution of (32) with pkH(-si,t) = 0 for k=0,l,...,n-l at t = t1. The time t1 must be taken in the distant past from the values of t for which H(-si,t) is desired. To compute all the ai(t), N equations like (32) must be solved. If many values of t are to be considered this is considerably lesswork than solving the adjoint equation for h(t,T) for every value of t. The main limitation on effective use of (31) is an accuracy requirement. For large i, ai(t) tends to be much smaller in magnitude than the individual terms of the sum. Thus small errors in the H(-si,t) may cause large errors in the ai(t) THE APPROXIMATE REPRESENTATION IN TERMS OF EXPONENTIAL FUNCTIONS If the n((s) are rational in s (30) can be substituted into (6) yielding

-15 - it -0 C as a sum of exponentials rather than a sum of orthonormal functions. Thus (7) and (9) become andA/ ( c F(sl t The resulting simplification in response evaluation is often worthwhile. From (33) and (31) it is clear that b, it) = g asc~t), vk(36) A/ bi(t) could also be derived by substituting the non-orthogonal series (33) in (14) and choosing the bi(t) to minimize E(t). RESPONSE TO STOCHASTIC INPUTS It is clear how the approximate representation (6) is utilized in evaluating res nse to deterministic inputs. The determination of response characteristics, e.g., the correlation function

-16- of x*(t), when the input is a random process requires elaboration. As in the case of deterministic inputs, it will be seen that the functional form of (6) is advantageous. Let ensemble averages be represented by a bar. Then by using (7) the correlation function of x* can be written as f (t,,t,)= oesstition (37) ad (38)simpif e to =.,,ta t) b4 k 4 1t7 a'I - 7;l (38) i'1 k l -o-fl where \\rff is the& correlation function of the input: VtOt^tS = - *,}+(t (39) Evaluation of the integrals in (38) is usually a difficult matter. Freedom in choosing the Oi(T) is sometimes helpful, particularly when rff(t1-Tl, t2-T2) can be written in the form rj(tl,t2)2(TiT2) If the input process is stationary \ff(tlt2) = rff(t2-tl) and (38) simplifies to -*(tt) =DII1 tL A Avo3olAoAt)&*(r~ ), L

-17By letting T3= T2 T1 and by defining rk@n) (41) (40) can be written as YA ^tltL) - E ctt,~ak~t~J Ys tL -t) 4 (42) i:,k l' Equations (41) and (42) have frequency domain equivalents 17ks) ki) ~~(43) and cz^tJ/) -- ^q*-O (44) where Tff(jo) is the spectral density of the input process. Certainly, if'Yff(jw) and. i(jio) are rational in jo there should be no trouble in evaluating the integrals in (44) by means of residues. Because rik(jco) Tff(jo) = rki(-jiw) ~ff(-jw), only the sign of (t2-tl) is changed in the integral when k and i are interchanged. Thus the total number of integrations required in (44) is.(1) and not N. 2 The number of integrations can be reduced still further by propitious choice of the ji (jo). The Laguarre functions -(S +) ~5 (s-i)' (45)

-18illustrate the point since for them 235+ (X C'y'^k4+I ^~(46) Hence, there are only 2N-1 functions rik(s) as i and k take on all values between 1 and N. Because of the result noted in the previous paragraph, this leads to N separate integrations in the evaluation of (44). The advantage is offset somewhat by the fact that the Laguarre functions have multiple poles, making computation of the residues of Tff(s) rik(s) more difficult. The non-overlapping functions given in (23) provide an even greater simplification. In these functions (^^ ^,BT~ -jW ^T-i)A k / )- AT < j e Substitution of (47) in the integrals of (44) gives Xx (48) ttYe, to)/ q(t, ( )O(t,) (- t, + LT-T) (48) where 2(49) Thus only one integration is needed.

-19Expressions for other correlations can be derived in a manner similar to that outlined above. This has been done for the cross-correlation of x*(t) and g(t) (a member of a random process) and the cross correlation of x*(t) and y*(t) where y*(t) is the outN put a filter having a weighting function i bi(t) Oi(T) and an input g(t). Since these results do not contribute any new knowledge of a general nature and are lengthy, they are not given here. Examples The first example is the second-order control system shown in Figure 2. This example has been treated previously by Farmanfarma(17) and Gilbert.(18) The variable gain constant is given by - o0.6 a t-k L L K(t= ^ ^,o (50) k integer and represents finite width sampling. Writing the differential equation between the input fi and the output xo gives 3fi +- ^* +K&K -KPf.t) (51) Since the coefficients in (51) are periodic (T=l) the approximation (13) is applicable. However, the multiplication of fi(t) by K(t) yields functions rl(t) and r2(t) which are zero for.6 ~ t-k < i,, and have jump discontinuities at t-k =.6, 1. This means that a fairly large number of tertI must be included in (13) for good accuracy.

-20This difficulty can be avoided, at least for inputs which do not vary too rapidly, by writing (51) in terms of the error, x = fi-Xo: i +s 5 K+) ) =+Sf;=f) (52) In what follows the error x and the equivalent input f(t) = fi(t) + 5fi(t) are the x(t) and f(t) in (1). Reducing the weighting function for (52) to normal form(2) yields lzt,>)=/,2ssw )e'} ~(t-r)-5rt)e'~t-o e where r1(t) and r2(t) are functions with period T=l given by -I. oyt -2. osYt ^ -e -^c, o2S t ^,k) - C Y-JY, ost - 5 - t9/,+.or Yt.t 1-.o t (54) In this problem ql(t) = r2(t) and q2(t) = rl(t). As Figure 3 indicates rl(t) and r2(t) are functions whose variations are fairly small compared to their average value. Thus a reasonable approximation can be obtained by neglecting all but the k=0 terms in (13): h *tt7) =.'i3?e =-,? I/~ T C (-6)

-21Since the variations in rl(t) and r2(t) are nearly sinusoidal with period T=l, addition of the k= +1, -1 terms should give an excellent approximation. Actually the k = +1, -1 terms are not too important if the input f(t) has most of its energy well below c = 2g, If f (t) = u l(t) the unit step function at t=O, f(t) ul(t) + 5uo(t). Although f(t) contains the doublet ul(t), the approximate response x*(t) is still quite accurate as can be seen from Figure 4. Even better results are obtained for smoother inputs fi(t) such as the ramp. For the second example consider the differential equation Xi + (2-t) — t = ) h(57) It can be verified easily that for this equation (5(59 with f^aYl^ijd^i (60) o That h and H are expressible in terms of elementary functions indicates that Equation (57) was chosen with care.

-22Before discussing the approximation of h(t, T) several properties of h(t,T) should be noted. First, - t r ~ - x.1, 00 t finite. Because (61) holds only for t finite, stability of (57) is not guaranteed. In fact, solutions of (57) which diverge as t - + o can be exhibited. Still, for finite t Equation (21) holds. coo Thus as long as t remains finite no fundamental difficulties in approximating h should arise. The fu Second for i hon large hFiure has simplf ed asymptotic, representations. As t center near t -, /-t, 0 where The function n(d) is shown in Figure 5. Thus for large t, h(t,T) is a pulse like function centered near T = t. For t < -1, T > l/-t,

-23t 2 t- x2 the asymptotic expansions of f e 2 dX and f e 2 dX give o o h(t') -' __ = O? 0v (64) Thus for large negative t the T.V.L.S. acts like a first order timevariant system followedby a gain (-t-1). Because h(t,T) gives a delay-like behavior for large t, it is impossible to select a small number of pproximating functions which will give good approximation accuracy for all t. However, if t<-2, h(t, T) is ell behaved and most any reascnble set-cf apprcimating functions, will give good accuracy. For convenience the functions(i(s) in (25) are taken with si = ic. By taking c. = 1/2 the "length" of the approximating functions is made approximately equal to the "length" of h(t,T). Application of (31) gives (for dik and tables of 0i(T) see Lanning and Battin(l9)). a,(t) g S{/SYt) a -IF t(/5~ t) C 3f(2 tS a3 - LEIf(l 3 - J3f ) + Y,(2.- ) (65) at) - f /1-.t) of/(Z -t)..,,f(r-) ys f(-t)

-24For positive arguments the function (X) has been tabulated (2 which aids in the computation of the ai(t). Figure 6 shows h and h* for t = -2, -1, 0, 1, 2 and N = 4. Excellent approximation accuracy is apparent. As would be expected approximation accuracy steadily de1 teriorates as t increases beyond 2. However, since (X) --- as X ~ oo, a, - - 3 t ) a2 - X o, i > as t -- oo. Thus, by means of (30) it can be shown that as t - oo. CONCLUSION Methods, which are adaptable to machine computation, have been developed for approximating the weighting function h(t,T) of a T.V.L.S. Since the form of the approximation is simpler than that given for h(t,T) by classical theory, analytic solution of the response problem is possible for both deterministic and stochastic inputs. Freedom in the choice of the approximating functions 0i(T) often leads to simplified evaluation or utilization of the approximation. While the amount of effort required to evaluate the response of a T.V.L.S. is still considerable, the introduction of workable analytic procedures should prove valuable in a number of important problems. CKNOWLEDGEMENT The author gratefully acknowledges the support of the Institute of Science and Technology, The University of Michigan.

TIME INVARIANT TIME VARIABLE FILTERS GAINS Figur 1.Pyial) nn a p(t) S ~- (P) ~- a,(t) f(t) X't) aN~~~~~~~~~~~~~~~~~~~~~r.ir PhiaItpetooApomtntTV S Figure 1. Physical In-berpreta-bion of Approxima-bion -to T.V.L.S.

-26f() +/ x(t) Kt xoM(t) Ffe a C rK yt) p(P+5) Figure 2. Feedback Control System with Finite Width Sampling,

0.8 0.7 0.6 0. 0.4 ~~~ 0.3 0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure. Periodic Functions of Normal Soluion. Figure 3. Periodic Functions of Normal Solution.

1.0 0.8 0.6\ OA 0.2 0= 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t ac- Figure 4. Step Responses for Exact and Approximate Systems.

1.4 1.0 0.8 0.6 0.4 0.2 -4. - -2. -. 0 2 3 4 Figure. Form of h(t,) r. Figure 5. Form of h(tT) for t >> 1.

0.2 1~ Figure 6a. h() and t = -2 0 2 3 4 Figure 6a. h(tT) and h*(t,T) for t w-2.

0.2 0 I 2 3 4 Figure 6bo h(t,T) and h*(t,T) for t -1.

0.4 h~~-h4.. 0 1 2 3 4 5 Figure 6c. h(t,T) and h*(t,T) for t O.

0.6 1~~ h~ t= I 0.4 2 0 0 F 2 3 4 Figure 6d. h(t,T) ana h*(tT) for t = 1.

1.41~~~~~~~~~ 12 1.0 t 2 0. 0.6 OA 0.2 0, 06 ~~_~~~~_~____________________ 0 1 23 45 Figure 6e. h(t,T) and h*(t,T) for t = 2.

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