Copyright 1967, by the author(s). All rights reserved.

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1 Copyright 1967, by the author(). All right reerved. Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. To copy otherwie, to republih, to pot on erver or to reditribute to lit, require prior pecific permiion.

2 STATE VARIABLES AND FEEDBACK THEORY by E. S. Kuh Memorandum No. ERL-M232 1 December 1967 ELECTRONICS RESEARCH LABORATORY College of Engineering Univerity of California, Berkeley 94720

3 STATE VARIABLES AND FEEDBACK THEORY E. S. Kuh Department of Electrical Engineering and Computer Science and the Electronic Reearch Laboratory Univerity of California Berkeley, California ABSTRACT The tate-pace characterization of a linear time-invariant ytem can be viewed in term of a general multiple loop feedback configuration. The return difference matrix and the null return difference matrix with repect to the A matrix are derived and related to the pole and zero of the tranfer function. A ueful formula for enitivity of the tranfer function with repect to an element of the A matrix i alo obtained. 1. INTRODUCTION Bode1 feedback theory for ingle-loop ytem wa firt general ized to the multiple-loop cae by Sandberg [1, 2]. Further extenion in term of the return difference matrix and enitivity wa given by Kuh [ 3, 4]. The relation between general feedback theory and the tatepace characterization of linear ytem wa firt uggeted by Kalman. He pointed out that the degree of a rational matrix which wa crucial in it realization in term of tate-pace characterization i related to the degree of a feedback ytem [ 5]. The preent paper i intended to bring together further the general multiple-loop feedback theory and the tate-pace repreentation of linear time-invariant ytem. The matrix ignal flow graph i ued to calculate and interpret the return difference matrix and the null return difference matrix. Baed on thee reult we then derive a ueful formula on enitivity for the tranfer function with repect to an element of the matrix A. ' The reearch reported herein wa upported by the National Science Foundation under Grant GK-716 and the Joint Service Electronic Program (U.S. Army, U.S. Navy and U.S. Air Force) under Grant AF-AFOSR

4 For implicity, we retrict our tudy to ingle-input, ingleoutput linear time-invariant ytem. The tate-pace repreentation i x = A x + bu * (1) \ y = c x + du where x, u and y are the tate vector, the input and the output, repectively. The matrix A, the vector b and c, and the calar d are the tate-pace parameter which characterize the ytem. The tranfer function i w() = X. - d +c^i _K>r\ (2) where i the complex frequency variable, and the imbol hat (* ) i ued to denote the Laplace tranform. The matrix ignal flow graph repreenta tion of (1) i hown in Fig. 1, where the feedback loop i clearly indicated. 2. RETURN DIFFERENCE MATRICES In feedback theory we alway focu our attention to a particular entity in the ytem which i of pecial interet. In the ignal flow graph of Fig. 1 we chooe the entity to be the branch matrix A. The return difference matrix for the branch A, denoted by F(A), can be introduced a follow: Setting the input u zero and conidering only the feedback loop, we break the loop at the input of the branch A a hown in Fig. 2. We apply a vector ignal g and calculate the returned ignal h. Clearly!} = 7- «l (3> The returned difference matrix F(A) i defined in term of the difference between g and h: From (3), we obtain < > =,- (4) F(A) =1 - A = ( 1 - A) (5) and det F(A) =-2^- (6) ~* ** n -2-

5 where Q() i the characteritic polynomial of the matrix A, and n i the order of A. Next we wih to derive the null return difference matrix. With reference to Fig. 3, we again open the loop at the input to the branch A and feed in a vector ignal g. In addition, we apply a pecial input u uch that the output y i identically zero. Expreing y in term of u and g in Fig. 3, we have y=du +ict(a / +bu) =0 (7) From (7), we obtain ta -c Ac u = d + c b The returned ignal h with the preence of both u and g, i h = - (Ag + bu) 1 be x~, t, ' **~ d + c b (8) We define the null return difference matrix F (A) a in Eq. (4) in term of the difference of the returned ignal h in Eq. (8) and the ignal g. Thu, from (8), we have where n i be F (A) =1 - - (1 - -"^-r- ) A d + c b <«V /v l- ia =i(l.a ) ** S +* S **» o A kc A = (1 - ~~ ) A (10) d + c b Now we are in a poition to introduce the following theorem which repreent a generlization of Blackmen' impedance formula o detf (A).,,. W(A) = W(» detf(a) (U) where w(0) i the tranfer function under the condition that branch A i (9) -3-

6 zero. From Eq. (2) or from the ignal flow graph of Fig. 1, we have c b d + c b w(0) = d + -fi-2- = zlzl (12) The proof of the theorem i traight forward and i omitted; it depend on the determinant identity det (1 + JG) = det (1^ + GJ) (13) Eq. (11) ha intereting interpretation. Conider the cae d = 0, A =(l- -^ 1 A (14) i a contant matrix. difference matrix can be written a Thu in Eq. (9) the determinant of the null return detf (A) = 351- (15) ~ ~ n where P() i the characteritic polynomial of the matrix A. The theorem a expreed by Eq. (11) become w(a) =&& %$- (16) ~ Q() where P(), the characteritic polynomial of A in (14), give the zero of the tranfer function and Q(), the characteritic polynomial of A, give the pole of the tranfer function t. Thee reult check with that of Brockett which he obtained baed on the concept of the invere ytem [ 6]. 'For the cae d 0, it i poible to derive an alternate formula,by conidering the branch 1^ in the ignal flow graph rather than the branch A to be the entity of interet. In thi cae w = d detl det -4-

7 3. RETURN DIFFERENCES FOR A GENERAL REFERENCE The purpoe of thi ection i to tudy the ituation when only a portion of the matrix A i of interet. Thi i the cae if we are intereted in the enitivity of the ytem with repect to, ay, an element a of the matrix A. Thu we can decompoe the matrix into two part: ij A = A' + K (17) w «v *v where A1 repreent the branch of interet and K i called the reference matrix." Typically A1 may contain a ingle nonzero element, a, then K i the matrix A under the condition a.. = 0. We write K=A (18) ~ '-'a =0 ij In Fig. 4 we redraw the ignal flow graph of Fig. 1 but we plit the branch A into ' and K. For eae in further reduction we inert two unity branche at the input and the output of the branch A1. Since we are intereted now in the branch A', we can redraw tjhe ignal flow graph of Fig. 4 by combining the branch K and the branch 1a hown in Fig. 5. The combined branch i J = ( 1 - K)"1. We may now ue the ignal flow graph of Fig. 5 to introduce the return difference matrix and the null return difference matrix by opening the feedback loop at the input to A1. The return difference matrix o obtained i clearly 1,- JA! =1, = (L- K)-lA' and i called, by difinition, the return difference matrix with repect to the branch A for the general reference K. We ue the following notation F (A) =1 - (i - K)~ A' (19) Clearly, if K i zero, then (19) i reduced to the original return difference matrix F(A)7 It i alo ueful to point out that (19) can be written a FK(A) =(l-k^l-k-a/) =(l- gfnl- A) (20) =F(K)"1 F(A) where K F(K)=i--f- t21) Similarly, uing the ignal flow graph of Fig. 4, we can introduce the null return difference matrix. Under uch condition the null return difference matrix i called the null return difference matrix with repect to the branch A for the general reference Kand i denoted by -5-

8 F (A). Similar to the derivation of Eq. (9), we find FT, (A) = 1 - K 1 - d+c^-kfh), (l,-k)-1a» (22) It i traightforward to how that det EK( A> detf (A) <v /\^ detf (K) (23) where a in Eq. (9) and (10) be d+c b K (24) 4. SENSITIVITY To obtain the enitivity of the tranfer function w with repect to an element a.. of the matrix A, we recall the formula [ 4],w 1 a.. f(a..) ro.. (25) where f(a..) i the clalr return difference and f (a..) i the calar null return difference with repect to the element a... In the previou ection we mentioned that we would chooe A* to be the matrix with the only nonzero term a., in the i-th row and the j-th column. Becaue of the imiplicity of the form of A', it i eaily recognized that and detfk(a)=f(a ) ij detf (A) =f (a..) ~ K ~ ij Subtituting (26) and (27) in (25) and uing the formula (20) and (23), and the fact K = A I., we obtain ~ ^'a..=0 ij 5W a.. detf(a). detf (A) *" **» a =0 *" /v a =0 ij y_ det F(A) detf (A) Thi formula give the enitivitie for the tranfer function with repect to all element of the matrix A. It i only neceary to calculate the determinant of the return difference matrix and the null return difference (26) (27) (28) -6-

9 matrix of the ytem under the nominal condition and under the condition a =0 to obtain the enitivity of the tranfer function with repect to ij a EXAMPLE Let the ingle-input ingle-output ytem be given by y = C1 0 ; Let u calculate the enitivity of the ytem with repect to the term a = -1. For convenience we write the matrix A in term of the element 21 ~ a 21 a A=-X * a21 * *-<! -C. The following calculation i eaily checked: i Z \-a -2 <* * <*) =4-<2 "3 +2"V F (A) =1 ^F0^ =7 (S ' T^ -7-

10 ThUS detf(a) n detf (A) _n w S a21 det F( A) det F ( A) ""2 We can alo ue the information of the determinant to write the tranfer function immediately. From (12) we have Thu from (11), w< ) w(0) = we have c b 7 2-a 1, _21 x 2 7(S ~} ( -3+2-a ) CONCLUSION In thi paper we have employed the feedback theory to the tate equation. We have found ome iginificance of the return difference matrix and the null return difference matrice. In particular, the determinant contain information of the nonzero pole and zero of the tranfer function and the enitivitie. 7. REFERENCES 1. Bode, H. W., Network Analyi and Feedback Amplifier Deign, D. Van Notrand, New York, Sandberg, I. W., "Linear Multi-Loop Feedback Sytem, " Bell Sytem Tech. J., Vol. 42, No. 2, pp , Kuh, E. S., "Some Reult in Linear Multiple-Loop Feedback Sytem, " Proc. Allerton Conf. on Circuit and Sytem Theory, Vol. 1, pp , Kuh, E. S., and Rohrer, R. A., Theory of Linear Active Network, Holden Day, San Francico, Kalman, R. E., "Irreducible Realization and The Degree of a Matrix of Rational Function, " SLAM J. Vol. 13, No. 2, pp , June Brockett, R. W., "Pole, Zero, and Feedback: State Space Interpretation, ": IEEE Tran., Vol. AC-10, No. 2, pp , April

11 Fig. 1. Matrix ignal flow graph repreentation of the tate-pace characterization of linear time-invariant ytem. F(A) = -ta oj // *N/ Fig. 2. Interpretation of the return difference matrix:

12 f (a)=i--la =i-t(ib c*. ^7-) A d+c'b fkf>j ^ Fig. 3. Interpretation of the null return difference matrix: F (A) = g - h under the condition that u i adjuted uch that y = 0.

13 A =A-K /v» *>j Fig. 4. The branch A i plitted into A' and K. A'=A-K Fig. 5. The ignal flow graph of Fig. 4 i redrawn to emphaize the effect of 4'.