485 CORRESPONDENCE 1965 Forced Response of Linear TimeVarying Systems c HAVING FREQUENCY RESPONSE Gliwl Fig. 1 . I The utility of error coefficients in studying thebehavior of constantparameter linear systems has been well established [l]. This concept can be extended easily to timevaryinglinearsystemsindeterminingthe H(5, t ) , response toanyinput,knowing where H(s, t j isZadeh’ssystemfunction [21-[11. of a T h e generalinput-outputrelation time-varying linear system can be written as Control system with one zero memory nonlinearity. U p , t)ez(t) K(p, t)edt) = (1) where p = d / d t , e l ( t )is theinput ande2(t),the output. In other words, L(p, t)W(t,T ) = k(P, (2) f)8(T) where W ( t , T ) is the impulse response and S ( T ) thedeltafunction.Themethodsuggested here consists in considering the output of the system as a superposition integral, given by (for signals applied to t=O) et(t) = Fig. 2. Approximate and exact region of stable initial conditions for example problem. A T)el(t - 7)dT (3) and in expanding e l ( t - T ) in terms of Taylor’s series so as to get (assuming that the first n-derivatives of el(f) exist) ez(tj = IiTith theassumptionthat X ( e ) is reThere is one crossing of 1/G+) and --\‘(e) (which occurs onthenegative real axis) placed by a linear element as in the precedwith2 ing paragraph, the response e ( t ) can be calculated from knowledge of the initial conditions of the linear part;e ( t ) nil1 contain a n l/GCjw) = - ;V(e) = 0.1 oscillatorycomponentcaused bythetwo e = 12.7 poles on the j axis. The steady amplitude of oscillation can be found from the evaluw = 0.1. (3) ation of the residues of thetwoneutrally stable poles. Now set The peak of the steady-state response of a = O.le = - 0.lc (4) e is now set equal to the limit cycle ampli-,V(e) locus a tt h e tudereadalongthe andsubstitutein ( 1 ) takingtheLaplace -iV(e), 1 / G ( j w ) crosing. equation retransform sults which contains all of the initial conditions of thelinear-partdifferentialequaO.O1s2c(O) (O.ls2 O.Ols)t(O) tions, and which expresses a surface in the e(s) = -(5) initial condition space. Every point on the (s2 0.01) (S 0.1) surface of initial conditions produces steadystate oscillation in thesystem(subjectto where C(0) hasbeenassumedzerointhe the approximation inherent in the describing interest of simplicity. Byevaluatingthe function method and the degree with which residue at s = O . l j , the peak steady state e ( t ) the variable e is represented by the linearis found to be J-a~(t, +J ~ ( t7 ), [ e l ( t ) - Te:(t) -r Since \ye are interested only in the forced response of the system, letting t--t CO, += et(t) = J-, ~ ( t , [ e l ( t ) - Te:(t) 7) 3 Let us now define h + + + + c,, = (-l)= ( +m F ( t , T)T%!T. (6a) J4 Substituting from (6a), (5) may be written as e2(t) = coel(t) + cle<(b) +- - . (6b) which gives the forced response to any input e l ( t ) . Also, by Zadeh’s definition, the system function H(s, t ) is given by [2]-[4] ized system). A new surface of initial conditions is obtained for eachcrossing of the l / G ( j w ) and - X ( e ) loci. T h e following exampleillustratesthe method. Let the nonlinear element be saturation (with saturation occurringat one unit of e ) , and let the linear part beexpressed by e ... =c + 0.16 + 0.01~ (1) c=-a. Then -e s2 G(s) = - (s) = a + 0.1s + 0.01 . s3 (2) By setting (6) equal to 12.7 [from ( 3 ) ] ,the approximationtotheboundarycurve of stable initial conditions is obtained. 4 plot of the 6(0) vs. c(0) stability boundary approxi2. Thestability mationisshowninFig. boundary as determined from analog simulation of (1) is shown for comparison. H. N. SCOFIELD -4strionics Lab. Marshall Space Night Center or in general Huntsville, Ala. = c1 Manuscript received June 1 , 1965. TR.4KSACTIONS IEEE 486 +- an Lt - H ( S , H asn (-1),J t) r”i$’(t,r)dr ON AUTOMLIATIC CONTROL 1 2t -m - c,. (10) Thus, given H(s, t ) we can find the response t o a n y forcing function e l ( t ) using (6b) and(10). The coefficients CO, cl, . . . are all functions of time,andbycomparison with their counterparts in the case of constant parameter linear systems, these may be called time-varyingerror coefficients. However, the use of these coefficients may not be very pronounced in the synthesis of linear time-varying systems. are all independent of Since CO, t l , . * el(t), the forced response of any input may be quickly determined using (6b). 3 e&) = - - - e& 412 = coj(t) 1) Let us consider a system,described by the differential equation, + 2et P 12.5‘ , The authors gratefully acknowledge the kind help of Dr. hl. D. Srinath and 11.A . L. Thathachar. I t isclear that the function c ( t ) is viewed as a sum of ramp functions. The approximation of a typical nonlinear character1. I n Fig. 1, c ( t ) is isticisshowninFig. approximated by c [ t ) = K j t ) - (t - 0.2j.L.jt - 0.2). (2) R. BALASUBRAMANIAS~ B. L. DEEKSHATULU One can convert such functions to stan- Dept. of Elec. Engrg. Indian Institute of Science Bangalore, India REFEREKCES [I] B. C. Kuo. Anlotnalic Control Syslems. Englervood Cliffs, S. J.: Prentice-Hall. pp. 151-153, 1962. 121 L.A. Cadeh. ‘Frequency analysis of variable networks, Proc. I R E , vol. 35. pp. 291-299. March low 131 *Circuit,analysis of linear varying parameter networks. J . A p p l . Phys.. voL 21. pp. 117111 77. Kovember 1950. [41 W. Kaplan. O p e r a i w m l .tierhods for Linear S33ferns. Reading. Mass.: Addison-\%’esley, 1962, pp. 515-525. dard polynomial form by having availablea table of coefficientsfor approximating the function (t-tr;)G(t-ttl;). Such a table is used to find the contribution of each ramp function to the Laguerre coefficients. If A o , A I , A 2 , . . . , A,, represent the Laguerre spectra of c ( t ) , one may write + 12e2‘+ 2 Hence u = Lt H + +2 4 27 + 32e‘ t) = 36(1 + 2 4 H(s, I) = st0 a -His, as 3 6(1 6 6 ~ 3 +4et e&) = ___ 6(1 2e9 + Ak 0.4 1.00000 0,00000 it can be easily checked t h a t Hence co = L1 H(s, t) st0 = =f ( t ) 21 0.2 0.81873 -0.98246 0.01637 -0.01528 0.01439 -0.17iO1 0,67032 -0.93845 0.05363 -0.04648 0.04004 2.77646 GENERAL Insystemsengineeringitissometimes advantageous to represent functional a relationship in terms of polynomials. However, one may encounter a functional relationship c ( t ) defined for 0 < t < a which cannot be conveniently represented by a polynomial. In such cases, it is possible to approximatethefunctionsbythe piecewise linear representation 1‘1 1 - The approximation of a nonlinear characteristic curve. 0.6 0.8 0.54881 -0.87810 0.09879 -0.07910 0.09393 0.30335 0,44933 -0,80879 0.14379 0.01549 0.14666 -0.09790 1 .o I. -1.00000 0.00000 0.00000 0.00000 The results may be checked by direct evaluation. The response to any input f(t) may be determined in a similar manner. 2) For the system given by Fig. 1. TABLE I + + + (2P + 6t + 2)]e?(t) & ( t ) is thekth normalized Laguerre pol>-nomial and can be expressed as [3] This correspondence describes a method of evaluating the Laguerre coefficients of a continuous (or piecewise continuous) function which can be approximated by a piecewise linear function. The function is viewed as a sum of ramp functions. Table I can be used to find the contribution of each ramp function to the Laguerre coefficients. 27 32e‘ 3 4et e&) = ___ 6(1 2et) 36(1 2 4 + + (3) Tables of Laguerre Coefficients for Representation of Piecewise Linear Functions and e 2 ( t )n-henf(t)=t would be [t2p2 (3t2 4t)p c AtLdt). L O tI Thus e2(t)when f(t) is a unit step would be + + C(I) G 1 R. Balasubramanianis now withtheDept. of Elec. Engrg.. University of New Brunswick. Fredericton. N.B.. Canada. First we shall find the coefficients CO, cl, . . . of this system and then find the forced response when f(t) is a unit step input, or a ramp input. It can be readily checked that H(s, the t ) of this system is 61 (t-tk). * -.._-_. Examples Go = * .\CKSOWLEDGYEST . 1 where L’(t-tk) is the unit step function and Kr; is a coefficient of therampfunction and for any otherf(t) + clj’(t) + e?f”(t) + - OCTOBER 0.36388 -0.53576 0.18394 0.12263 0.07664 4.37469 The characterizing coefficients Laguerre spectrum of c ( t ) are [ 3 ] A k = Joic(t)L,G(t)e-tdt ilk of the (5) where e--L is the weighting factor. c ( t ) is a continuous (or piece-wise continuous) function such as (3): An alternative expression for (4)is [2] “ c(t) E Kr;(t- Ir;) rr(t - t k ) (1) L O Thus whenf(t) = t hlanuscriiot received Aueust 20. 196-4.: revised March 22. 1965 and July 22. 1965. This work is a part of the research work under the Grant-in-Aid :>rugram supported by the Department of Electrical Engineering, Ohio State Unlx-erslty. Columbus, Ohio. The first fen- Laguerre polynomials are given as follows