EVALUATION OF THE WEIGH‘TING FUNCTION OF A LINEAR SYSTEM BY THE METHOD OF DECONVOLUTION Thesis Ior ”no Degree OI .le. D.‘ MICHIGAN STATE UNIVERSITY Arvydas Joseph KIiore 1962 THESIS This is to certify that the thesis entitled EVALUATION OF THE WEIGHTING FUNCTION OF A LINEAR SYSTEM BY THE METHOD OF DE CONV OLUTI ON presented by ARVYDAS JOSEPH KLIORE has been accepted towards fulfillment of the requirements for PH. D degree in Electrical Engineering Major professor Date February 12, 1962 0-169 LIBRARY Michigan State University ABSTRACT EVALUATION OF THE WEIGHTING FUNCTION OF.A LINEAR SYSTEM BY THE METHOD OF DECONVOLUTION by Arvydas Joseph Kliore In an adaptive control-system, one of the basic problems is that of identifying the dynamic characteristics or mathematical model of the section of the system which varies with time in an unpredictable manner . For slowly time-varying linear systems it is proposed to con- tinuously monitor the weighting function of the system by employing the method of deconvolution to effect a step-by-step solution of the con- volution summation, the convolution summation being defined as a finite approximation of the convolution integral. Two types of error are inherent in this method of deconvolution. One is caused by imperfect knowledge of the weighting (response) function of the system, which in general must be estimated prior to the appli- cation of the deconvolution method. The second type of error is caused by the truncation of the convolution summation. The prOpogation of both types of error through iterations of the deconvolution procedure is in- vestigated, and a set of sufficient conditions for the convergence of these errors is derived in terms of certain matrices which are functions of the variations of the input function. These conditions can be applied directly if the input function is known in advance. If the input function Abstract Arvydas J. Kliore is not known, as is the case usually, the investigation of the convergence of errors in advance of the deconvolution computation is difficult. However, these conditions may be applied while the computation is in process to check the behavior of errors during the actual computation. Finally, it is shown that if the input function meets certain conditions under which the system may be considered initially quiescent, the deconvolution computation may be carried out without the effect of these errors. EVALUATION OF THE WEIGHTING FUNCTION OF.A LINEAR SYSTEM BY THE METHOD OF DECONVOLUTION By Arvydas Joseph Kliore A THESIS Submitted to the School of Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1962 ACKNOWLEDGEMENT The author wishes to express his appreciation to Professor Herman E. Koenig, his major professor, for his encouragement and guidance in the development of this thesis, and for his patience in its editing. ***********9H(—** -11- m4? SECTION I. II. III. IV. INTRODUCTION . MATHEMATICAL DESCRIPTION OF PHYSICAL SYSTEMS TABLE OF CONTENTS THE METHOD OF DECONVOLUTION . ERROR ANALYSIS CONCLUSION -iii- 0 00° 17 26 LIST OF FIGURES FIGURE Page 1. A general adaptive system . . . . . . . . . . . . .. 2 2. Representation of a system . . . . . . . . . . . . . 6 3. Decomposition of a time-function into rectangular pulses . . . . . . . . . . . . . . . . . 7 A. Input-output conventions for the convolution summation . . . . . . . . . . . . . . .. 17 5. Weighting function of a stable system with inertia effects . . . . . . . . . . . . . . . . 18 6. The form of the input function for an initially quiescent system . . . . . . . . . . . . . 63 7. Computation of a single point gi . . . . . . . . . . 70 .. iv... LIST OF APPENDICES APPENDIX Page A. Inversion of the [A] Matrix . . . . . . . . . . . . 71 B. Behavior of the Absolute Sum of the Truncated Values of g(t) . . . . . . . . . . . . .. 75 -v- I. INTRODUCTION In recent years the field of automatic control has witnessed a rapid growth of interest in the concept of adaptive control. In most general terms, an adaptive control system is one in which the input-out- put characteristics of the process to be controlleda‘re measured continu- ally and these measurements are used for continuous self-optimization of the entire system regardless of the criteria of Optimization. As is the case with ordinary feedback control systems, a concise mathematical de- finition of an adaptive system is difficult to realize, and for this reason the usual definition is conceptual rather than mathematical. Historically, the shift of interest to adaptive control was prompted by the increasing complexity of control problems, which made standard feedback control techniques inadequate, and by the simultaneous advancement of computer technology, which made possible the inclusion of complex digital or analog computers as real time elements of a control _ * system.(1 2) For example, a system whose dynamic characteristics vary with time in an unpredictable manner can only be handled from the adap- tive vieWpoint. A conceptual diagram of a general adaptive system appears in Figure l. The identification computer continuously monitors the dynamic characteristics of the process and the actuating-signal computer The superscript indicates the number of the reference in the BibliOgraphy, and the page number of a Specific reference. Thus, (1-2) refers to reference No. 1, page 2. T. Input Actuating I‘ Actuating Signal] I Output Signal Process Computer I ,1 l ‘J Identification ’] Computer - Process Characteristics Figure l. A general adaptive system generates an actuating signal, based on the input and the process charac- teristics to realize a desired output. The two basic problems in such an adaptive system are identification and actuation. The problem of identification may be considered the primary requisite of any adaptive control system, since adaptivity implies an automatic and frequent de- termination of the dynamic characteristics of the process to be con- trolled. These characteristics may be expressed as the weighting func- tion, transfer function, differential equation, or some other mathema- tical model. (1'9) Various methods of solving the identification problem have ap- (10,11) One class of identification schemes peared in the literature. requires test input signals in addition to the normal operating signals of the system. An example of such a scheme is one in which a multi- channel correlator is used to measure the process weighting function as the cross-correlation function between the process output and a binary - 3- white-noise which is added to the system input.(12’l3) A second example of an identification scheme employing a test signal is found in the Sperry adaptive autOpilot, in which a train of pulses is added to the ordinary command signals, and the reSponse of the (11+) system to this input is used to evaluate a performance criterion. Other identification procedures have been devised based entirely upon the information contained in the normal input and output signals of a process. Among these is a technique due to Kalman(15) which is considered to be the first significant contribution in the field of adap- tive control. This technique employs the solution of a difference—equa- tion representation of the input-output relationship of a process to obtain a Z—transform representation of the process transfer function. An identification scheme employing orthonormal expansion was (16) prOposed by Braun. The input of the system is expressed as an ex- pansion in Laguerre functions or eXponentials, and the coefficients in the corresponding expansion of the unit-step reSponse are computed. An orthogonal-spectrum analyzer is applied to the implementation of this procedure. In a method proposed by Mishkin and HaddadSlY) the unit step response of a process is obtained by an approximate solution of the con- volution integral when the input is assumed to be composed of a combi- nation of impulse, step, ramp or parabolic functions. This method re- quires repeated differentiation of the output function to derive the necessary Taylor-series coefficients. -h- The identification methods mentioned above are only several of the many that have been proposed. They were outlined here because they are typical of the various methods that have been published. This thesis presents a method of process identification based on a finite summation representation of the convolution integral which may be implemented through relatively simple computation. The approxi- mations inherent in this method lead to errors which are comprehensively analyzed in Section IV. II. MATHEMATICAL DESCRIPTION OF PHYSICAL SYSTEMS In the broadest sense of the term, a physical system is a col- II lection of components that are connected in some rational manner to per- form a specific function. If the exact characteristics of all components of the system are known, along with their interconnection, then the char- acteristics of the system can be completely determined analytically. However, when such information is not available, it is necessary to es- tablish the characteristics of the system from external measurements. This is the well known"black-box" approach, in which a system, no matter how complex, is assumed to have one input and one output of interest. In the work that follows the output variable of the system is assumed to be dependent only upon the one input variable, and the characteristics of the system itself. The methods of mathematically describing such unilaterally de- pendent systems constitute the major part of this section. Only those tOpics which are relevant to later development are discussed. This discussion is given here only as background for later work, and is not a rigorous development in itself. 2.1 Linear Systems Let a system be represented schematically as in Figure 1., Let r(t) be the input variable and C(t) the output, or reSponse variable of the system. In general, the output variable of the system depends on the input variable and the system characteristics. The system charac- teristics may in turn be dependent on the input variable or any of its -5- -6- r(t) System C(t) System System Input ' Output Variable Variable Figure 2. Representation of a system derivatives, and also upon one or more independent variable such as time, temperature, etc. Systems that have characteristics which are dependent on the input variable are classified as nonlinear. The fol- lowing discussion will consider only systems that can be assumed to be linear over some range of the input variable. A linear system is defined as one for which the derivatives of the input and output variables are related by a linear equation, as shown below: q P k d t ak( ) dtk k:O 3:0 dJ C(t) = bj(t) dtj r(t) (2—1) Time-Invariant Linear Systems In Eq. (2-1) the coefficients ak(t) and bJ(t) are functions of time, and this characterizes the system as being time-varying. However, the simplest mathematical analysis results when the system characteristics can be assumed to be time-invariant, and hence it is advantageous if the mathematical characterization of the system can be eXpressed in the form -7- of a differential equation with constant coefficients q p k 3 ak d k em = b, ‘1 r(t) (2-2) dt 3 dtJ =0 J=O In this discussion, a system will be assumed to be tineeinvar- iant if the coefficients in the differential equation do not change over the interval of time required for measurement. (2) 2.2 The Convolution Integral SuppoSe that an input r(t) is applied to a linear, time-invar- iant system. Furthermore, let the input be subdivided into a series of rectangular pulses of width T, as shown in Figure 3. Figure 3. Decomposition of a time-function into rectangular pulses. -8- When a unit-pulse function is defined as below, then the input function may be approximately represented by 00 r(t) ; r(tk) 13(tk)i)T k=--C’<> where I)“ T): .1. fort < t< t +T (2-3) .. k? T k — "" k = 0 otherwise. and as T—)0 00 r(t) = lim r(tk) p(tk;T)T (2'h) T-firo . k=-C<> Let the response of the system to a unit pulse, p(tk,T), at t = tk be denoted by v(t-tk,T). Then, the reSponse to a pulse of height r(tk) and width T .t t = tk is ck(t) = v(t-tk, T) r(tk) T (2-5) where v(t-tk, T) = o for tk > t Now, using the superposition property, the total response can be appro- ximated as the sum of the various pulse responses of the form of Eq. (2-5) and is given by cm :. 2km: v(.-.k,T)r(.k)T <2-6) oo k=-oo k:-oo 7-15 -9- Let g(t-tk) be defined as the limit of the unit-pulse response v(t-tk,T) when T-—9rO s(t-tk).= 11m v(t-tk, T) (2-7) T-e»0 . From the definition of an integral it follows that 00 c(t) = lim v(t-tk, T) r(tk) T T-a'O k=-c<> oo =k/. g(t — t')”r(t') dt' (2'8) -C>O and since g(t—t’)=o fort =1. [r(t)] I 3) C(s) =I, [g(t)] m ,N w» -11- The s—domain function G(s) =-§%§§— is commonly called the transfer function of the system having a weighting function g(t) 52::1 [G(s)]. 2.u PrOperties of the weighting Function 93 a Linear system(3‘l59) It can be shown, that for a general linear system, G(s) is a rational function of's G(s) = E 2 = ’4 b sp + b sp-l + ... + b s + b p p-l l *. 0 a s + a s + ... + a s + a q l o (2-lh) where for convenience aq is taken as unity. In the development that follows it is assumed that p < q. This restriction applies to a very large class of physical systems, particularly to those that are described as having "inertia" effects. Thus, when p < q, B(s)/A(s) is a prOper fraction, and if the polynomial equations A(s) = O and B(s) = 0 have no common roots, the q-th order polynomial equation (commonly designated the characteristic equation) Q Q-1 ' _ A(s) = s + aq_ls + ... + als + a0 — 0 (2-15) -12- in general has n distinct roots sl, 5 . , Sn’ with each root si ap- 2} pearing with some multiplicity mi. Thus, G(s) may be written as G(s) = 3(3) (2-16) 2 mn (s-s (s-s ... (s-s ) l 2 n The fraction 'EISI can now be eXpressed as a sum of partial fractions. For each pole s of multiplicity mk, there are mk partial fractions of k the form Mk1 , Mk2 . Mkmk : w_13 '°° 9 (s-sk)Ink (s- s k)mk s-sk and G(s) may be expressed as a sum of fractions k= (s-sk (S-Sk)mk A(s) l d'j"l . = lim . , . Mk3 s——,s J-l ° dsJ"l where B(s) The inverse Laplace transform of G(s) may now be evaluated, to yield g(t) -13- n mk -l -' s t g(t) inf: [G(s)] = ——§§1——- tInk J e k (mk- J)! ksl J=l (2'18) where Mk3 is defined as in Eq. (2-17). Stability(3-197) For the purpose of this discussion a system is defined to be absolutely stable if all the roots S1 of the characteristic equation A(s) = 0 have negative real parts. If this condition is satisfied, then from Eq. (2-18) n mk M . -J a t J t lim g(t) == lim -——¥&l—— tmk e k e “I t—><'>O taco (mk- J)! ksl i=1 and, if Gk < O for all k then lim g(t) = 0 (2-19) t-9CK) and for an absolutely stable system, g(t) vanishes with increasing t. If sJ is that root of A(s) which has a negative real part such that for all k f j '“J I < I°kI then, for t sufficiently large,the effect of all other roots is negli- gible and g(t) may be approximated by -1h- m M.. m.-J _ . iant g(t) ': —-‘l-1 t1 6 Ioflte i a- . ' (mi J)° i=1 2 K tn eXp (--a t) exp (jooit) (2-20) where m. n M. . m.-j K t - ——-ii—— t l . (mi- J). J=l and a =I°iI The Weighting Function at t = O In later work, it is necessary to specify the behavior of g(t) as t approaches zero. This information is most conveniently obtained Afrom the transfer function, G(s). If dz:[s(t)] = G(s) (3-267) then, from the initial value theorem lim g(t) = lim s G(s) (2-21) i:-+() s—acx: If b 5p + b Sp-l + ... + b s + b _ B s “ p pp-l l o G(s) — — A s q q-l s + a s + ... + a s + a Q-1 l o -15- then lim g(t) :9 lim _s__B£_sl t->O $900 A(s) Thus, if it is required that lim g(t) = o, it is necessary that t-é'O lim -§—§£§l== O. This is true only if the order of A(s) is greater than s—roo A(s) the order of s B(s), i.e., q > p+l or q 3 p+2 (2-22) This property is important in the develOpment of forthcoming sections. (its) 2.5 Time—Series Representation_9f Continuous Systems Let a continuous input function be considered as a series of pulses at t = o, T, 21:, .. , kT, , of width T and height r(kT). If T is chosen to be such that r(t) and g(t) do not change appreciably in the period T, and if the width T of the pulses is very small, then from the superposition prOperty, the response of the system may be approximated by the sum of the responses to each pulse, which may be assumed to be of the form of Eq. (2-7). Thus for kT < t < (k + l)T, the response of the system may be approximated by c(t) 2 g(t - ncv) r(nt)'1‘ (2-23) ne-oo The approximation becomes better as T becomes smaller. For T suffi-~ ciently small, it follows from Eq. (2-23) that at t = kT -16- c(kT) _. T r(nT) g[(k—n)T] (2-2u) ==-oo or, changing the order of summation, c(kT) s T g(J'T) r[(k-J)T] (2-25) J=0 In the special case where r(t) = O for t < 0, Eq. (2-2h) shows the se- quence of values of the output function c(t), at intervals of time T, as an explicit function of the corresponding values of the input function r(t) and the values of the weighting function g(t) evaluated at t = nT, and can be written as _;(03- r_g(o) o o - - e—fi ‘";(03' C(T) _ T g(T) s(0) 0 - - - r(T) C(ZT) g(ET) g(T) g(O) - - - r(2T) _; -d _—; . . __ L_ .‘__ (2-26) This time series representation of the characteristics of linear systems is useful in various numerical techniques of system analysis and syn- thesis, and forms the basis for the computational deconvolution tech- nique described in Section III. III. THE METHOD OF DECONVOLUTION For the purposes of this discussion, the term deconvolution is defined as the step-by-step solution of the convolution summation to ob— tain the weighting function g(t), or more precisely, the values of the continuous weighting function at uniformly Spaced points in time. The convolution summation is defined as the finite approximation to the con- volution integral arising from the time-series representation of the response characteristics of a linear, time-invariant system, as defined in the preceding section. 3.1 The Finite Approximation of the Convolution Summation Let a linear system having a weighting function g(t) be sub- jected to an input r(t), and let the resulting output be represented by c(t). Furthermore, let r(kT) and c(kT) be the values of r(t) and c(t) at t = tO+ kT, where tO is some arbitrary time origin, as shown in Figure A . A c(t) C(kT) ‘F///,z‘r””fl—-——-\\\\\\~‘_§;;”,Lr”//’ r(tyi s——___n /?‘rEE£lé £7 -T/‘// t -2T t -T o o to tO+T to+2T t0+kT t Figure A. Input-output conventions for the convolution summation -17- ‘ IIIrIIIFIIIIIvIEII-IIIEE -18- Let the weighting function of the system, g(t),be such that 1) lim g(t) = o t-e>o 2) 1im g(t) = o 't4>oo (3-1) - It was shown in Section II, that the conditions in Eq. (3-1) are satis- fied by an absolutely stable system with "inertia" effects. The form of such weighting function is shown in Figure 5. K g(t)4 .: ii‘T. ANT .\ “o_ T 2T 3T \\\\\\~—.;’L””,,- (NrflT) t g(kT) Figure 5. Weighting function of a stable system with inertia effects As shown in Section 2.5, if T is chosen such.that the varia- tion in r(t) and g(t) over any interval of time T is small, the response of the system may be eXpressed in the form of a convolution summation C(nT) = T g(kT) r[(n-k) T] (3-2) =O -19- where in order to simplify the notation c(nT) = c(tO + nT) r[(n-k)T ] r(tO + nT - kT) ll g(kT) g(0 + M) and T is the sampling interval. Since‘ lim g(t) = O, for any 6 >'O, there exists a positive 'be>oo integer N, such that for k >lN, g(kT) ‘< e , and if e is chosen suf- ficiently small with respect to the precision of observation, the appro- ximation can be made that g(kT) = o for k _>_ N (3-3) and the convolution summation (3—1) becomes N-l C(nT) = T g(kT) r[(n-k) T] (34) k=O Thus, under the finite assumption, the value of the output function c(t) at any sampling instant is dependent only on the values of the input function r(t) at the preceding N-l sampling instants. 3.2 The Procedure 9: Deconvolution For purposes of clarification, let it be assumed that the sy- stem is at rest prior to the time to, at which time it is desired to be- gin the deconvolution computation, i.e., r(t) = O for t < to. -20- Furthermore, let the following notation be adOpted: rk = r(tO + kT) ck = c(tO + kT) sk = g(k‘I‘) (3-5) Using this notation, the expression for the convolution summation is c = T gk rn-k (3-6) where or C (3-7) As g(t) was assumed to be such that g(O) = O, the above compu- tation is theoretically not necessary, because the result is known before- hand. However, this initial computation will be useful in the error analysis of Section IV. At t = t0 + T, the convolution summation gives -21- 0 II 1 T(rl go + I'o g1) but, since gO = O, c c = T r g or g = l l o l’ l Tr (3-8) Similarly, at t = t0 + 2T 02 = T(rl gl + rO g2) and since gl is available from the preceding computation, c r _ 2 __1. eg—Tr - 1, s1 (3-9) 0 0 Continuing the procedure, at t = t0 + 3T, the convolution summation gives = T (r2 gl + rl g2 + rO g3) c3 and since g1 and g2 have been previously computed, (3-10) Continuing in this manner, at t = t0 + iT, the output is given by the convolution summation i i-l 0i = T ri-k 3k = T (r0 gi + ri-k 3k) and k=l i-l k=l ci 1 gi = Tr ' :7 ri-k g1: (3'11) 0 O k==l where gk has been previously computed for k = l, 2, ... , i-l. _22- r Letting sk = , Eq. (3-11) may be written as I‘o i-l Ci 63'. = - Si-k gk (3'12) Tr k:l Thus, at any time t = t0 + iT, gi can be computed using the re- sults of previous computations and the apprOpriate values of the input function, together with the value of the output at t = t0 + iT. If the computation is continued through t = to + (N—1)T, all values of g(kT) from k = 1 to (N-l) will have been computed, and the entire pro- cedure may be started again at t = tO+ NT or at any arbitrary time there- after. Again, let t5 be defined as the time at which the new iter- ation is begun, and let *1 II I r(tO + kT) and O I — c(té + kT) (3-13) Furthermore, let'gk be the values of gk computed during the first iteration. If the system has been operating for a time greater than NT before the beginning of the computation, the values'g'k will be taken as estimates of the actual values. Then, at t = td the con- volution summation gives -23- N-l co = T r_k gk =0 or, solving for go, N-l Co g0 = " S-k gk (3-1I-I) Tr k=l In general,the value of gO as calculated from Eq. (3-lh) is non-zero only because of one or both of two sources of error: 1) Errors in the initial estimates of gk. 2) Truncation errors resulting from the finite summation. Since the actual value of gO is known to be zero the result of the computation in Eq. (3-lh) represents the error, which will be de- signated as E0. In order to partially compensate for the effect of these systematic errors let EO be subtracted from the calculated values of each gi. Thus, although at t = t; + T, assuming gO = O, the convolution summation states that N-l C1 = T (To 8;1 + r-k+l gk) the computation for gl will be defined as N-l gl — S--k+l gk - Eo (3'15) TrO k=2 -gh- where Ei are the previously computed or estimated values of g(kT). Simi- larly, at t = ta + 2T N-l C2 = T (r1 g1 + ro g2 + r-k+2 gk) k:3 or, using the value of gl obtained from the preceding computation k=3 (3-16) At t = ta + 3T the convolution summation gives N-l c = T (r g + r g + r g + r gk) 3 2 l l 2 o 3 -k+3 k;h and using the values of g1 and g2 computed during the two previous in- tervals N-l _ 3 _. g3 Tr ' S2gl 1g2 ' S-k+3 gk ' Eo k: (3-17) Finally, at t = ta + iT, for any i = 1, 2, ... , (N - 1), N-l -25- or i-l N-l c, A ' l _ T ' E r—k+igk + rogi + r-k+i gk k=l k=i+l All gk (k = l, 2, ... , i-l) have been computed previously in this iteration and gk (k = i+l, ... , N-l) are known either from previous iterations, or as estimates. Therefore, gi can be computed as follows (3-18) When all N-l values have been computed, the iteration is com- plete and the next iteration may be started at t = t5 + NT or at any time thereafter. The iteration cycle starts with a computation of E0, and continues with the computations of each g2.L (i = l to i = N-l) during successive sampling intervals, using the most recently computed values of gi in each computation. Thus, this computational procedure may be carried on indefinitely, completely regenerating all gi during each iteration. If the system weighting function g(t) varies slowly with time in such a manner that the variation is small over a period of time NT, the deconvolution computation provides a revised represen- tation of g(t) at intervals of time t = NT. IV. ERROR ANALXSIS The success of any computational procedure, such as that which was discussed in the preceding section, depends to a great extent on the errors that arise as a result of various inaccuracies which distinguish an actual computation from an idealization. From a strictly computational viewpoint, these sources of error may be divided into two groups; (1) the errors that are caused by arith- metical operations with finite precision and, (2) those which arise as a result of approximations, estimates, and other sources of inaccuracy present in a particular computation. The first of these is common to all computational procedures and various methods of evaluating its effect may be found in mathe- (6) matical literature, and for this reason it is not discussed. The other sources of error are characteristic of the particular computational procedure. These errors are discussed here in detail. As the deconvolution procedure is applied in an iterative manner, the behavior of errors propogated from iteration to iteration is of great importance, and the determination of conditions under which these propogated errors converge to zero is the primary objective of this error analysis. h.l Analysis 3: the Error Caused by Inaccurate Initial Estimates It is reasonable to assume that the weighting function, g(t), of a time-varying system is initially known to some degree of accuracy. -26- -27- The purpose of this section is to analyze the effect of initial error in the values of each g(kT) on the results of successive computations of the weighting function. The Propogated Error Let the deconvolution computation begin at some arbitrary time to. Furthermore, let the values of the input function r(t) be known for t = (t0 - T), (tO - 2T), ..... , [tO - (N-l)T], and let these values be designated by rmk = r(tO - kT) Also, let it be assumed that the estimated values of the weighting function g(t) at t = T, 2T, ..., kT, ..., (N-l)T are available, and are represented by + e (4-1) where gk = g(kT) and ek is the error associated with the estimate of gk. Furthermore, let gk=0 forkZN As shown in Section III, the value of the output function c(t) at t = to+ is given by the convolution summation -28- N-l c0 = T r".k gk , k=0 or N-l c -9 = r g + r g T o o -k k where c0 = c(to) , r Setting S-k = , it follows that r o N-l co go = Tro ' S-k gk =1 However, since the actual computation is performed with values of gk that are not exact, the result is: N-l go = TI: - S-k gk k=l N-l N-l C = —£L-- s s e Tro -k gk -k k -29- or S0 = go + l O ’ where N-l 1E0 = " s-k ek k=l However, as shown in Section II, for all systems under consider- ation, go as O, and thus the result of the first computation gives the error E =‘é = - s e (h-2) Now, for t = t0 + T, the convolution summation with gO = O is N-l c1 = T(ro g1 + r-k+l gk) k=2 or, solving for gl N-l g ’ cl - £3 8 l Tr ~k+l k k=2 Since IE0 is available from the previous computation let the computation of'g.l be defined as follows N-l N-l C1 = Tro ‘ S-k+l gk ' S-k+l ek ’ 1E0 k=2 k=2 Setting'gl = gl + lEl it follows that the error associated with the second computation is (h-B) Similarly, for t = t0 + ET, the convolution summation is , N-l c2 E7’= r1 g:L + ro g2 + r—k+2 5k k=3 and N-l g2 Tro 1 g1 -k+2 gk #3 Again, the actual computation for'g2 using the previously com- puted value of'gl gives N-l N-l N-l C 'g = —§L-- s p s - s E s e - E 2 Tro l°l -k+2 gk 1 1 l -k+2 k 1 o k=3 k=3 and letting g2 = g2 + 1E2’ The expression for the error associated with the evaluation of g2 is N-l E (u-u) Continuing the error analysis, at t = t0 + 3T, the convolution summation is N-l c3 = T ( r2gl + I'132 + rog3 + r-k+3 gk) k=4 OI' N-l c3 g3 = Tro ' S2g1 ’ Slg2 ' S-k+3 gk k: However, the actual computation using the previously computed values‘gl andug2 gives D32- E where S-k+3gk ’ s2 1E1 ' S1 1 2 ' l 0 (1-5) Proceeding with the computation of each successive gi, each time subtracting 1E0, it will be found that the expression relating all E1, {1 = 1, 2, .., N~l) is - _ - —‘ _' _l '__ 1 o o o o lEl lE0 sl 1 o - - - o o 1E2 -130 s2 s1 1 — — _ o o lE3 -lEO I I I ' I I | | | l I | z | I | | | I SN-3 SN-h SN-S ' ’ ' l O lEN-2 '1Eo SN-2 SN—3 SN-u ' ' ’ S1 1 L1EN—1 L_”1Eo -33- N-l where To transform Eq. (4—6) into a form convenient for analysis, pre— multiply both sides by the nonsingular matrix I—i o o - - - o 5— -1 1 o - - - o o o -1 1 - - - o o [M] = I | I I I (h-7) I I | I ' I | I I I o o o - - - 1 o o o o - - - -1 1 to obtain 1 - o o - - - o o lEl (sl-l) 1 o - - - o o lE2 (82-81) (81-1) 1 - — — o o 133 I I I I I l I I I I I I I I I I I I (SN—3‘SN—u) (SN-h-SN-S) (SN-S'SN-6) ' ' ' l O IEN-2 LEEN-Q-SN-3) (SN-3'SN-u) (SN—M‘SN—5) " ‘ ’ (51“1) l lEN-l — fl '1Eo ‘ Q2 92 - 93 = 93 I “II (II-8) I 5—ieN-2 ‘ (5-1‘3—2)eN-1 5-1 811-1 -34- Now, let the differences between successive values of the nor- malized input function be defined as s. - s. = A. (h-9) N-l lEO = - S-kek = -s_lel - s_2e2 - s_3e3 ~ ... - s-(N-l)eN-l ksl and N-l 92 = S-k+lek = s_le2 + s_2e3 + ... + s-(N-2)eN-l k=2 it follows that the first entry on the right-hand side of Eq. (h-B) is .lEO - 92 = S-lel - (s_l-s_2)e2 —(s_2-s_3)e3 - ... - [S-(N-2)-s-(N—l)]eN-l N-l = (1 — Ao)el — A_k+1 ek (u-lo) =2 Similarly, for all i = l, 2, ..., N-l, since N-l 01 = S-k+i-l ek = S-1e1 + S-2ei+l + "° + S-(N-i)eN-l =i -35- and N-l 9i+1 S-k+i ek = 5—1 ei+1 k=i+l + ... + S-(N*i-l) eN-l it follows that the i-th entry on the right-hand side of Eq. (h-B) is 0i ' 0i+1 = 5-1 e1 ' (5-1'5-2) ei+1 ‘ °°° ' [s-(N-i-l) ’ S—(N-i)]eN-l N-l (1 — Ao)ei - A_k+l ek (u-11) k=i+l substituting Eq. (u-g), Eq. (u-lo) and Eq. (1-11) into the matrix equation (h-B) a set of equations relating the errors in the first iteration to the errors in the original estimates and the normalized in- put differences Ai is obtained. ~— " " ‘7 1 o o --- o o lEl Al 1 o —-- o o lE2 A2 A1 1 ——— o o lE3 I I I I I I I I I I I I I I I l | I AIII-3 AN-u AN-S "” l O lEN-2 I. AN-2 AN-3 AN—u "' A1 1 lEN-l __ L_. .1 F— .I '— _' (1'10) ‘A-i ’A-e ”' 'A-(N-3) 'A-(N-e) e1 0 (l-AO) -A_l —-- -A_(N_u) -A_(N_3) e2 f :1 (1"‘10’ 'A-(N-S) 'A—(N-II) els I I | | I I l I I I I I I o o o --- (l-AO) -A_l eN_2 o o o --- o (l-AO) -_J eN-l (u-12) Eq. (h-l2) may be rewritten in symbolic matrix notation I A 11 I E 11 = I B 11 I e 1 (1-13) where _ _ 1 o --- o I A 11 = I1 I ___ ? I . I IiéN'z AN_3 --- 1 [B 11 _ (l'Ao) 'A-1 "’ 'A-(N-a) ? (1:10) "' ‘A-(N—s) l I I I I I __ o o --- (l-AO) __ '"E I Fe ‘1 [E] = 1 1 , and [e] = 1 l 1?2 e2 I I LlEN-J; eN-l -37- Since [A] is triangular,the inverse, [A1i1. exists(§) and Eq. (h-l3) can be solved to obtain the errors in the first iteration as explicit functions of the errors in the initial estimates and differences between the successive values of the normalized input function [EIl= III;l IIBIl [e] (1-11) = [C]l [e] where -1 IcIl [A11 [B11 During the second iteration, begun at some time t ETtO + NT, the com- putational procedure will be exactly the same, except that the error associated with each value of the weighting function gi used in the com-- putation is designated by' E ,(i,= l,2,...,IN-l), i.e., the error result- 1 i ing from the first iteration. Thus, the expression for the error propogated through the second iteration of the deconvolution procedure is of the same form aS-Eq.(h-lh), namely , -l [E]2=[A]2 I1312 [ml = 1012 [E11 (II-15) where _;El_7 [E12 = 2E2 BEN-l. -38- and the [A]2 and [B12 matrices are identical in form to [A]l and IBII, differing only in the values of the entries A1. Substituting Eq. (h-lh) into Eq. (h-lS) the errors in the second iteration are given by - (h-16) IE12= IcI2 IcIl [e] For each successive iteration the computational procedure is re- peated, using the'g-i computed in the preceding iteration. Thus, the expression for the error 'Ei associated with each gi computed during the j-th iteration is _. -l .— [E]J - [A13 IB]J [E]j_l Ic]j IE] 1 where I—E - j i [E13 = 3E2 gEN-li and [A13 and [B]J are identical in form to [A]l and [B]l. But since 1E13-1 = [C]j_l [313-2 [E] '2I— I013“2 [E]J 3 I [E] = [c]l Ie] it follows that the error associated with the j-th iteration, as an ex- plicit function of the error associated with the original estimates, is -39- IE]J.= [c]j IcIJ._l IC]j_2 IcI2 IcI1 [e] r j I I _ = < WINE I [e] (1-17) i=1 where I [01‘ == IAJE} I313 and [A]‘and [B11 are matrices identical in form.to [A]l and [B]l shown after Eq. (h-lB). Conditions for Convergence of the Propogated Error In previous discussion the nature of the initial errors ei was not defined. However, before any statement concerning the convergence of the propogated error can be made, the prOperties of the initial errors ei must be known. Since these errors arise from the imperfect knowledge of the values of the weighting function g(t) at t = T, 2T, ... (N-l)T, it is reason- able to assume that each ei is an independent random variable. Further- more, let it be assumed that each ei is normally distributed, with a mean m1 = O, and a variance CE = 0: for all i = l, 2, ..., (N-l). The error prOpogated through the j-th iteration of the deconvo- lution procedure is given by Eq. (h-l7) Letting 3 I l [01$ = ID]j (h—18) Eq. (h-l?) can be written as -h0- [E]. = [D]. [e] J or in detail E - j 1 jdll J 12 jdl,N-l e1 (1 .. .. 3E2 j 21 3 22 jd2,N-l e2 I I I I I I = I I ' I E d - - j N-l de-1,1 j N-l,2 de-l,N-l eIII-1 I_. __ IL_ _1 N-l Thus, JEi = jdik ek , for 1 = 1, 2, ... , (N-l). k=l Now , since all e can be shown that the variance of each ‘Ei 15(9-99) N-l Var (.E ) — Jdik k:l and the mean M»| A I for all 3, it follows that max - j 00 IpiI S Amax :ngI (LLB) k=N for all i = l, 2, ..., (N-l) Furthermore, it is shown in Appendix B, that for an ab- solutely stable system,given e >IO, there exists an integer P, such that for any N >>P 0°. is“ k;N or, from Eq. (h-AB) IPII < A e (II-III) max Now, let _ -1 [GI]j - [A13 IPIJ. (II-us) 2.55... then Eq. (h-h2) can be written as follows i-l IE1i = IQIi + IDIi-k [qu (II-.II6) ksl where from Appendix A r— _- l O O - - - o jal 1 O - - - O -l A = ,a a 1 .. - - I 13 J 2 J l ”___ 9)....— F‘-—-— C) jaN-2 3 N23 3 Nah Therefore, the ieth entry of [Q] in Eq. (unto) is of the form i-l qi - pi + pk ai-k (1~h7) k=l To establish a bound on the magnitude of qi’ let a.| .f a for i max 1, 2’ 000 , (N‘l) j=lg2’ooo’oo then, from Eq. (h-hh) and Eq. (umu7) q < I1 + (1-1) a ] A e (IImIIs) Ij il - max max for all j. -56- Forming the matrix product indicated in Eq. (h-hé), the neth entry of [E]i representing the error associated with gfi in the iath iteration is i-l N-l iEn = iqn + (kdnj)(kqj) (heh9) k=l j=l The bound of this error is i-l N-l IiEnI < q‘max [1 + Z :Ikdnjl] (21-50) k-l '=l where from Eq. (h-hB) qmax = [l + (Na-=2) amax] A max €32:Ijqi for 8.11;], and all i : lg 29 000 9 N‘lo Now let all [D]k be such that, for any row n N-l Ildnj 5 l " r5n i Ni I2dnJI 5 (l "' 5n) ZIldnd J=l . 5:1; Ni 2 Nu 2|de 5 <1~ s9 IkanI =1 3:1 where o < an < l (u-sl) If the conditions of Eq. (h-Sl) are satisfied, Eq. (h-SO) can be written as i-l iEnI < qmax (l - Bn k=0 )k (II-52) In the limit, as the number of iterations, i, increases without bound, OO . k llm |.E I < q (1 - 5 ) 1900 l n max n k=O OI‘ qmax lim ,E I < for n = l, 2, ... , (N-l) Thus, if the conditions of Eq. (u-sl) are satisfied, the accumulated error due to truncation approaches a limit. Furthermore, this limit may be made arbitrarily small by choice of N. Comparing the conditions of Eq. (4-23) of Section h.l with the conditions of Eq. (h-Sl) it is observed that the latter are much more restrictive, and in fact imply the former. Specifically, from Eq.(h-51) N-l Ikdnd :5 (l - s) :::> IkdnJI < l for all k, -58- and therefore 2 kdnj < Ikdnjl l N-l 2 g kdnj < Ikdnjl J=l It should be pointed out, that the conditions of Eq. (4-51) are and N- J=l sufficient conditions for convergence of the truncation error, and that in practice a less restrictive condition may bring about convergence. 4.3 Discussion of the Error Analysis In the preceding sections,the conditions for the convergence of the errors caused by inaccurate initial estimates and truncation were de- rived in terms of a coefficient matrix [D]. This matrix is a product of matrices [C]j, the entries of which are complicated functions of dif- ferences of the adjacent values of the normalized input function s(t) over the entire interval of time during which the deconvolution procedure is carried out. In order to investigate the behavior of errors for any given input function, it is necessary to derive the [A] and [B] matrices given in Eq. (h-l3) for each period of iteration, invert the [A] matrices, compute the coefficient matrices [C] = [AJ-l [B] and, finally, form.the products of the [C] matrices to Obtain the [D] matrices which contain the information required for application of the convergence criteria. Admittedly, the preceding is cumbersome to apply and requires considerable computation. However, attempts to obtain convergence -59.. conditions in terms of quantities more simply related to the input func- tion r(t) have not been successful, primarily because of the unwieldly form of the coefficients of [A]...1 (see AppendixA)° Furthermore, since the convergence criteria are based on the behavior of the input function r(t) over the entire interval of time over which the deconvolution computation is iterated, they cannot be applied in advance of the actual computation without knowledge of r(t) for the entire interval of interest. Thus, these criteria are of limited value for investigating the convergence of errors caused by truncation and in- exact initial knowledge of the weighting function in advance of the actual computation. However, the application of the error convergence criteria simultaneously with each iteration of the deconvolution com- putation would furnish information on the behavior of the error while the computation is being carried out. Thus, while in the general case, evaluation of the effects of the two types of inherent errors is difficult, in practical applications of the deconvolution procedure some quite feasible simplifying restric- tions can be imposed. Several of these are discussed next. h.h Periodic Input Functions The error analysis is considerably simplified if the input function r(t) is known to be periodic, and if the period is an integral multiple of the time NT, i.e., l, 2, .... (h-5h) r(tO + kNT) = r(to), k Under these conditions r1: r(tO +T) == r(tO +T +kNT) = krl and in general, ri = r(tO + iT) = r(tO + iT + kNT) (h-55) = kri for i = o, _t l, 3: 2,1... , : (N-l) where the pre-subscript k indicates the value of ri used in the k-th iteration of the deconvolution computation. Thus, under this condition, the values of ri occurring in every iteration of the deconvolution pro- cedure are identical to the correSponding r occurring during the first i iteration, i.e., lri = 2ri = 3ri = ..... .. = kri = . ..... (H-56) where l, 2, ... , k, ... is the number of the iteration. kri Now, since 8. = , from Eq. (h-56) it follows that, for k 1 krO i=o,il,_+_2, ...“.I'N-l lsizesi=3si= ooooooo=ksi= oooooo (II-57) Using the definition of Eq. (4-9), it follows that for all i = o, _t 1, i 2, i N-l, -61- A.1= A = A, = ...... = A. = ..... (h-58) Referring to the form of the LA] and [B] matrices shown in Eq. (h-l3), it may be noted that since the form of these matrices does not change from iteration to iteration, and since the A1 which constitute the coefficients of these matrices are identical for each iteration, the matrices [A] and [B] reSpectively will be identical for each iteration. Thus In] [A] = [9.13 = = [II] = 2 k and (h-59) [B] [B]2 = [1313 = ......= I13]k = Since [C]j is defined as _ ml [C]J — [A15 IBIJ. it follows from Eq. (4-59) that [011 = [012 = [013 = = [c]k= (II-60) and j _ _ J ID]J — I#1I[C]k - [c]l (it-61) Now, since the convergence conditions in Eq. (h-23) and Eq. (h-Sl) are stated in terms of the coefficients of the [D] matrix, the application -62- of these conditions becomes considerably simpler when the [D] matrix can be evaluated as in Eq. (h-6l). Under these conditions, the computation of the [C] matrix, which requires the inversion of the [A] matrix, must only be performed once, and the [D] matrices for successive iterations are found simply by performing the multiplication of [C] into itself the required number of times. This not only leads to a considerable computational simplification, but also allows the Convergence conditions to be applied in advance of the actual deconvolution computation. h.5 Initially Quiescent Systems From the viewpoint of the practical application of the deconvo- lution procedure, a very important simplification is obtained when the system can be considered to be quiescent prior to the commencement of the deconvolution computation. For the purposes of this discussion, a system will be con- sidered to be quiescent if the input function r(t) has a constant value or is equal to zero for all time t such that (tO - NT) :5 t < tO - T, where tO is the time at which the deconvolution computation is to be started. Let the input function r(t) be of the form shown in Figure 6 -63- r(t) r rv(t) V Figure 6 The form of the input function for an initially quiescent system Analytically, the form of such an input function is ll 56 r(t) for (tO - NT) 5 t _<_ (tO - T) (II-62) r(t) R+r(t) fort>t-T V 0 Using the convolution summationIthe output c(t) of the system at any sampling instant (tO + nT), where n = O, l, ... , N-l, is given by N-l c(tO + nT) = T g(KT) r [tO + (n-k)T] (h-63) k;0 but,since r(t) is of the form shown in Eq.(h-62) N-l n c(tO + nT) = TR g(kT)+ T g(kT) rvlt0 + (n-k)T] ' (h-6h) k=O k=0 1 or c(tO + nT) = C + cv(tO + nT) where n cv(to+ nT) = T g(kT) rvlto + (n-k)T] (h-65) kso From Eq. (h-6h) it can be seen that the output c(t) prior to t = t0 consists only of the constant C, and the varying part of the out- put cv(t) is related to the varying part of the input, rv(t),by a con- volution summation of the same form as Eq. (h-63), but having only n + 1 terms. Thus, considering only the varying parts of the output and in- put functions, the deconvolution computation may be started at t = t0 without knowledge of the initial values of g(kT), since r (t + kT) = o for k < 0. V0 Using the notation cv(to + nT) = on (h-66) and r (t + nT) = r VO n then,from the summation in Eq. (h-65), at t = t0 + T, the output cl is -65- c1 = T gk rl-k kso = T[gorl + glro] but gO = O for all systems under consideration, and hence; C s1 -- T1} (II-67) O Similarly, for t = t0 + 23 C2 = T gk r2.1: T[glrl + g2ro] and it follows that: C 1‘ s2 = 2 - --1 s1 (1-68) Tr r o In general, for t = t0 + nT, where n = l, 2, 3, ... , N-l -66- but, since all gk for k = l, 2, ... , n-l are available from previous com- putation, gn can be computed as n-l c a = n - S n Tro gk nwk (II-~69) :1 where r _ n S " 1:" n 0 Thus, for an initially quiescent system, all values of gn may be computed sequentially without any previous knowledge of the values of gk. These computed values are then available for use in subsequent item rations of the deconvolution procedure, and it is not necessary to use estimated values of’gkp'Underthese conditions the procedure is free of the effects of the error that is associated with estimation. However, the special nature of the input function r(t) which is necessary to obtain an initially quiescent system raises another problem. It may be Observed from the expressions for the computation of any gk, that in each case the computation requires multiplication by the term -—— , where rO is the value of rv(t) at t = to. It is also required that o rv(tO - T) = 0, thus rO represents the change in the input function r(t) from t = to- T to t = to, a period of time equal to the sampling interval T. If this change is small, then the reciprocal C—%— ) is subject to large inaccuracy for even a small error in the measugement of the value of r0. Thus, the change in r(t) from t = to- T to t = t0 must be large in order to make the computation of-{%— as accurate as possible. For 0 -67- example, if 1% accuracy is desired in-—%— , and the precision of measure- 0 ment of rO is 5, then 1 _ l < .01 r r + e: - r O O O 01' I“ 2 1006 This implies that in order to benefit from the simplificationtnxnmflfl:about by the consideration of an initially quiescent system, the deconvolution computation must be started immediately after introducing a large distur- bance into the input of the system which had been quiescent for a time greater than NT previous to that time. This requirement, while restric- tive, is not unreasonable in applications to some practical systems. The second type of error that has been considered, the trunca- tion error, is not eliminated by the restriction to initially quiescent systems. However, as shown in Appendix B, the contribution of the trun- cation error can be made arbitrarily small by choice of a large enough time NT. Thus, if in a practical application NT is chosen such that g(nT) for n.> N is much smaller than the precision of measurement, e, of the system, the contribution of the truncation error may be made negligible. It may be therefore concluded, that while the effects of the truncation and estimation error are difficult to evaluate in general, the process of deconvolution may be carried out essentially free of these types of error if the system is initially quiescent in the sense of the pre- ceding discussion, and if the process is truncated only after a sufficiently large time NT. V. CONCLUSION In the preceding sections, the method of deconvolution was in- vestigated as a solution to the identification prdblem inherent in any ape proach to an adaptive control system. The implementation of this method would make continuously available a representation of the weighting func- tion of a slowly timeevarying system, which then might be used in some adaptation scheme to render the overall system independent of the variation. To compute one point on the weighting function, Nol multipli- cation operations and N addition operations are required, where N is the number of sampling intervals. This computation is shown schematically in Figure 7. Thus, a total of 2N~l arithmetical operations must be per~ formed for each point. The number of samples, N, depends on the value of T chosen in the approximation of the convolution integral, since the total time NT is approximately constant for any one system under con- sideration. The storage requirements also depend on N, since 2N numbers must be in storage at all times. Thus, both the complexity of compu- tation and the storage requirements are directly proportional to the number of samples, N. However, the accuracy of the approximation, and hence the ac~ curacy of the results, becomes better as T is made smaller, and conse- quently, as the number N is increased. As a result, in any application of the deconvolution procedure, a compromise must be made between ac- curacy and the size and complexity of the required computation. No at- tempt has been made to prepare specific computation techniques for the implementation of the deconvolution method, as that is not the purpose of this work. -68- -69- The error analysis, which comprises the greater portion of this work, is important because of the iterative nature of the deconvolution method and the errors arising from the finite approximation (truncation) and the inexact knowledge of the system characteristics prior to the be- ginning of the computation. The results of the error analysis are eXpressed as a set of indirect restrictions on the allowable variation of the input signal to insure convergence of these propogated errors. Since these restrictions are expressed as conditions applying to highly complicated functions of the variation of the input signal, direct conditions applying to the input signal could not be derived analytically, and the practical application of these conditions in the general case is difficult. However, certain simplifications are introduced by imposing restrictions on the input function r(t). In particular, if the input function r(t) is periodic with a period kNT, the computation of the quantities required for the application of the error convergence criteria is greatly simplified. In addition, this restriction makes possible the application of the convergence cri- teria prior to the deconvolution computation. Finally, in the special case in which the system can be consid- ered to be initially quiescent, the deconvolution computation is free of the errors caused by inexact knowledge of the initial values of the weigh- ting function. If in addition the number of sampling intervals N is chosen to be such that g(NT) is much less than the precision of measurement, the effect of truncation is negligible, and the deconvolution computation can be carried out without the effect of these two types of inherent error. -70- flw ucfiom ofimdmm m mo Gofimpdmaou .h ondwfih _ lem ..... H+Hw aw HIHM ..... Mm NW Hm 0m XV WW w 1 1. WW I tx c $+ZIwm ..... Tm Hm ..... mmflm NIwm Two Mm .. _o I o X _ ah IHI _ , a .H - 3w I - five udmuDO Seaman-w 3: HSQCH APTENDIX A INVERSION OF THE [A] MATRIX It is desired to obtain the inverse of the matrix [A] shown below __ ‘I' l o o - - - o 0 Al 1 o - - - o 0 A2 A1 1 - - - o o [A] ~ = A3 A2 A1 - - - o o (A-l) I | I I . | I I I I I AN-3 AN-II. AN-5 " " ’ l O AN-2 AN-3 AN-II " " ' A1 1 This matrix is unit lower triangularg6) thus its inverse is also unit lower triangular and it can be obtained in a straightforward mannero By definition of an inverse, if l o o - - - o 0 al 1 o - - - o 0 a2 al 1 - - - O 0 IA] '1 = I l I I I (A-2) I I I I I I I I I l aN-3 8LN--II 811-5 " ‘ ‘ 1 O Law—2 aN-3 aN-II ' ' " 8L1 1 -71- -72- then it follows that: [111'1 [A] = [I] (A-3) or in detail 1 0 o - - - 0 1 o o - - o - - _ A - - a1 1 o o 1 1 0 0 a2 al 1 - - - 0 A2 Al 1 - - O = I I I I I I I I I I I I I I I I I l I I I I | I aN-E aN..3 awn ' ‘ ' lJ “11-2 AN_3 ANA - - 1J —1 o o - - - o—I I o 1 o - - - o o o 1 - - - o | I I I I I I I I, I I I . b) o o - - - 1_ I a + A = O, or a = - A (A-h) and a + a A + A = O, or a = - A + A (A-S) Similarly, or the first and thus -73- a3 + a2 Al + a1 A2 + A3 = O _ 3 a3 — — A3 + 2 A1A2 — Al (A-6) In the same manner, multiplication of the fifth row of [A]-1 into column of [A] yields: II 0 a1+ + a3 Al + a2 A2 + a1 A3 + AA _ 2 2 h a1+ — - Ah + 2 A1A3 - 3AlA2 + A2 + Al (A-7) (l9) Employing the same procedure, it can be shown that _ 2 2 3 5 a5 — - A5 + 2AA Al + 2A3A2 - 3A3Al - 3A2Al + IIAQAl - Al (A-8) a = A + 2A A + 2A A - 3A A2 + A2 - 6A A A 6 6 5 1 h 2 h 1 3 _ 3 2 1 3 3 2 2 h 6 + AA3A1 + A2 + 6A2Al - 5A2Al + Al (A-9) 2 a7 = - A7 + 2A6Al + 2A5A2 - 3A5Al + 2AAA3 3 2 2 2 - 6AhA2Al + hAuAl - 3A3Al - 3A3A2 + l2A3A2A1 1+3 23 57 - 5A3Al + IIAeAl - lOA2Al + 6A2Al - Al (A-lO) -7h- _ 2 A8 + 2A7Al + 2A6A2 - 3A6Al + 2A5A3 m (I) I l 3 2 6A5A2Al + IIA5Al + Ah , 6AhA3Al 2 2 h 2 - 3AhA2 + 12AuA2Al - 5AhAl - 3A3A2 6A2A2 + 12 A A2A - 20A A A3 + 3 1 3 2 1 3 2 1 5 u 3 2 2 h 2 6 8 + 6A3Al - A2 - 10A2Al + 15A2Al - 7A2Al - Al (A-ll) The form of the expression for terms up to a may be found 10 in Reference 18 and for terms greater than a in Reference 19. 10 APPENDIX B BEHAVIOR OF THE.ABSOLUTE SUM OF THE TRUNCATED VALUES OF g(t) It was shown in Section II, that for any absolutely stable linear system, g(t) will, in the general case, have the following form when t is sufficiently large g(t) = Ktn exp (~at) exp Gmit) I (B-l) where n is a positive integer or zero; a,a§. are positive constants, Under the assumption that g(t) is of the above form for t _>_ NT, it follows from Eq. (B-l) that, since |exp (j (nit) I = l, |g(t)l : K tn eXp (-at) (B-2) and n ngI = K(kT) eXP (-akT) (B-3) Consequently the summation of the absolute values of the truncated portion of g(t) is 00 00 ngl = K (kT)n exp (-akT) (B-h) =N k=N In order to evaluate Eq. (B-h) it is necessary to apply (7) Maclaurin's Integral Theorem -76- To apply this theorem, the following must hold: Ing < ngl " 1 (B-5) Substituting Eq. (B-3) into Eq. (B-S) [(k+1)T]n exp [-a(k+1)T] < l (kT)n exp (-akT) ‘ and it follows that (35% n 5 as (a) or 1n (53:3) 5 33 (13-6) If Eq. (B-6) is satisfied for all k '2: N, the hypothesis of Maclaurin's Integral Theorem is satisfied and C" (>0 (1:13)“ exp (~akT) _<_ T xn exp (-aTx) dx (13-7) k:N ex - n-l , anEl(TaNT) [(aNT)n+ n(aNT) + ... + n.] -77- For any finite n, the bracketed eXpression in Eq. (B-7) is of less than eXponential order in (aNT), and thus the term eXp(-aNT) domi- nates. Thus,for any E >'O, there exists an integer P'such that for any N > P ”13 ('aNT) [(aNI‘)n + n(aNT)n-l + + n!] < E n+1 K a T or 00 ngI < 6 (B-8) k:N Thus, Eq. (B-8) states that for any absolutely stable linear system, 00 ngl is bounded and can be made arbitrarily small by choice of N. =N REFERENCES Books 1. E. Mishkin and L. Braun, Jr., “Adaptive Control Systems," McGraw- Hill Book Company, New York, 1961. 2. W. W. Seifert and C. W. Steeg, "Control Systems Engineering," McGraw- Hill Book Company, New York, 1960. 3. M. F. Gardner and J. L. Barnes, "Transients in Linear Systems," J. Wiley and Sons, New York, 1942. h. J. G. Truxal, "Control System Synthesis," McGraw-Hill Book Company, New York, 1955. 5. S. Seshu and N. Balabanian, "Linear Network Analysis," J. Wiley and Sons, New York, 1953. 6. A. S. Householder, "Principles of Numerical Analysis," McGraw- Hill Book Company, New York, 1959. 7. F. H. Miller, "Calculus," J. Wiley and Sons, New York, l9h8 8. R.S. Burington, "Handbook of Mathematical Tables and Formulas," Handbook Publishers, Inc., Sandusky, Ohio, 1953. 9. D.A.S. Fraser, "Statistics - An Introduction," J. Wiley and Sons, New York, 1958. Bibliographies 10. J. A. Aseltine, et al., "A.Survey of Adaptive Control Systems," I. R. E. Transactions on Automatic Control, PGAC-6 December, 1958, p. 102. ' 11. R. Stromer, "Adaptive or Self-Optimizing Control Systems - A Bibliography", I. R. E. Transaction on Automatic Contrdl, PGAC-h, May 1959: Po 65. -78- -79.. Periodicals and.Technical Reports 12. 13. 1h. 15. 16. 17. l8. 19. G. W. Anderson, et al., "A.SelfeAdjusting System for Optimum Dynamic Performance," IRE National Convention Record, part A, pp. 182-l90,l958. D. C. Kowalski, "Statistical Stability'Monitor,? Report No. ll307R, Bendix Corporation - Research Laboratories Division, May,l960. S. S. Osder, "Adaptive Flight Control System," Proceeding of the Self.Adaptive Flight Control Systems Symposium, WADC T. R. 59-h9, pp. 81-122, January,l960. R. E. Kalman, "Design of a Self-Optimizing Control System," Trans. A.S.M;E., Vol. 80, p. #68, February,l958. L. Braun, Jr., “0n.Adaptive Control Systems," IRE Transactions on Automatic Control, PGAC-h, No. 2, p. 30, November,l959. E. Mishkin and R. A. Haddad, "On the IdentificatiOn and Command Problems in Computer—Controlled.Adaptive Systems," Report No. R-767-59, Microwave Research Institute, Polytechnic Institute of Brooklyn, September, 1959. E. I. Jury and F. W. Semelka, "Time-Domain Synthesis of Sampled Data Control Systems,“ Trans. A.S.M.E., Vol. 80, No. 8, November,l958. E. I. Jury and F. W. Semelka, "A.Method for Determining the Coef- ficients of the Modified Z-Transform.EXpansion," Internal Memo No. l, Sampled-Data Systems Group, Electronics Research Laboratory, University of California, Berkeley, California, August, 1956. ROOM USE ONLY "0”” ”SE 0le ll‘: ‘1‘]!!! III III .I