Fragulis, George F.

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Evaluation of the impulsive solution space of linear multivariable homogeneous implicit systems

Institute of Mathematics AS CR, v. v. i.

KybernetikaGeorge F. FragulisEvaluation of the impulsive solution space of linear multivariable homogeneousimplicit systemsKybernetika, Vol. 30 (1994), No. 6, 629--638Persistent URL: http://dml.cz/dmlcz/125488Terms of use:© Institute of Information Theory and Automation AS CR, 1994Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czKYBERNETIKA — VOLUME 30 (1994), NUMBER 6, PAGES 629-638 EVALUATION OF THE IMPULSIVE SOLUTION SPACE OF LINEAR MULTIVARIABLE HOMOGENEOUS IMPLICIT SYSTEMS GEORGE F. FRAGULIS A closed formula is given, which allows the determination of the impulsive solutions of Linear Homogeneous Matrix Differential Equations (L.H.M.D.E.) directly in terms of finite and infinite spectral data of the associated polynomial matrix. Specifically the notions of finite and infinite Jordan pairs for a general polynomial matrix are defined and it is pointed out the strong relationship among them and the impulsive solutions of L.H.M.D.E. 1. INTRODUCTION In the recent years there has been a growing interest in the system-theoretic problems of generalized state space or singular systems due to the extensive applications of this kind of systems in large-scale, singular perturbation theory, circuits, robotics, economics, demography, control theory and other areas. Generalized state space systems are systems described by: Epx(t) = Ax(t) + Bu(t), (1) where p = |L is the differential operator, E G 3? rx r ranksRE1 < r, A G 5Rrxr, B G 3trXm, whereas (regular) state space systems are the systems described by: px(t) = Ax(t) + Bu(t) (2) which represent a particular case of (1) with E = Ir, the identity matrix. Regular state space systems were studied explicitly in the past decades (see e.g. [10]). On the other hand many researchers in the recent years explored special properties of generalized state space systems and found many connections between them and regular ones. Generalized state space systems were studied in the frequency domain (see [11,14-16]) as well as in time-domain (see [2-6 ,8-9]) . Generalized state space systems as (1) represent a particular case of Polynomial Matrix Descriptions (PMDs) i. e. physical systems whose dynamics can be described by a linear matrix differential equation having the form: . , . „ , , „ , , , , A(p)p(t) = B(p)u(t), (3) 630 G.F. FRAGULIS where A(p) = £ Aipi e » r x r [p], A,e 3?"*', i = 0 ,1 ,2 , . . .,qx > 1 with r a n k g j ^ < i=Q r, B(p) = JZ B,p{ € Wxm[p], B, G 3T x m , j = 0,1, 2 , . . . , a > 0, /?(<) the pseudo-t'=0 state of the system and u(t) the control input to the system. In a recent paper [13] various known results regarding the smooth and impulsive solutions of homogeneous generalized state space systems have been translated to the more general case of PMDs. A new treatment of PMDs using Algebraic methods is given in [12]. Among other topics the author gives a formula for the determination of the impulsive so-lutions of L.H.M.D.E. in terms of the notions of fast and slow states at t = 0~ which are introduced. Campell in [2-3] has found the solution of systems of the form (1) using the notion of Drazin inverse. Also in [4-6] Cobb used a different approach which utilizes the Weierstrass canonical form. Comparing those methods to ours we remark the following: Each method has its advantages and disadvantages for different problems. The main advantage of our method is that it treats systems of the form (3) which seem to be generalizations of systems of the form (1). The possible disadvantage is that the proposed solution is associated with the evalua-tion of several matrices which are derived by means of some previously established algorithms. On the other hand the approach based on the canonical decomposition which separates the system behavior in the finite and infinite frequencies provides deep insight into the structure of singular systems in systems analysis. However, it does not always provide a useful framework for actual computation. The canonical form also uses a change of the internal variable which is unconvenient in practical situations since the original variables are chosen to have their own significance. The main drawback of the method proposed by Cambell is the computational complexi-ty of the proposed algorithm. Our approach differs from that given in [12] because we give a formula such that the set of the Dirac impulses and its derivatives which are used in the determination of the impulsive solutions of L.H.M.D.E., as well as, the set of finite and infinite spectral data of the associated polynomial matrix are presented in a - easy to use - closed form. 2. MAIN RESULTS Consider the linear, homogeneous matrix differential equation: A(p)P(t) = 0, t>0 (4) and let sj(ss(s) be the Smith-McMillan form at s = oo of A(s) = Ao + Ais + • • • + ^ S ? 1 : . r i n * ' L s?1-*-1 s H If A(s) has at least one zero at s = oo then the Laurent expansion of A_ 1(s) can be written according to [12]: A~1(s) = K~ s«-+#~_ s""--2+.. + i / i s+ / fp+i l - i s _ 1 +t f - 2 s"2 +. . . = HPoi(s)+H,Pr(s), (6) Evaluation of the Impulsive Solution Space of Linear Multivariate Homogeneous Implicit ... 631 where r7P„i(s) G 9£rxr[s] is the polynomial part of ^4_1(s) and H*rr(s) G 3?r><r(s) is the strictly proper part of A~1(s). Let (C,J) and (Coo,</oo) be a finite and an infinite Jordan pair respectively of the polynomial matrix A(s). Then the matrices (C, J ) , (Coo, Joo) satisfy the conditions [17]: AqiCJqi + Л ^ - i C J ' 1 - 1 + • • • + AoC = 0 Є 3 ř r " x " , rankQn =n C CJ CJn~ AoC-J^+AiC-J*}-1 +AqiC- = 0^ Coo C„oJo C o o / Г 1 , rankgi(5„ = H, (7) (8) (9) (10) where n = degdet A(s), ji = £ (§) + 1) = £ % + (r - k), where %, j = j=k+l j=k+l h +1,..., r ate the orders of the zeros at s = oo of A(s). Then we can write (see [7]) A~1(s) = C~[sJoo -I^'BOZ +C[sL - J f ' - B , where B, B°° are constant matrices defined by: [ / o o]-t /« ,^-1][^-f ,[o,o, . . . )o, /^, (11) W h e l e : r C CooJ^" C J CooJ^r 5.-2 = K = C J Coo CT-1 A . C J ' - 1 , -v^^.Cooj: (12) (13) (14) Considering now the Laplace transformed equation (4) we obtain: /3(s )=A- 1 ( s )a(s )G5R r x l , where a(s) G ^ x ^ s ] is the initial condition vector associated with the initial values of j3(t) and its (qi - l)-derivatives at t = 0" i.e. 0(O~), /?(1)(0 ) , • • - , /?(?1 1}(0 ) given by [1]: a(S) = [S^lr , S"l-2Ir, ...,Slr,Ir]*P (15) F-Aqi 0 Aqi — i л ? 1 w: •• л . J L ŕ""1^ J (iб) 632 G. F. FRAGULIS After some matrix manipulations between A x(s) and d(s) we finally arrive at the following special form for /3(s) which appeared originally in [12]: ß(a) = [8*r+n-lIт, sЧ-+^'2Ir,...,sIr,Ir | s _ 1 7 r , s 2 / r , . . . ] 0 Я~ Я Î r - ( î l - l ) Я Í r - ( î l -2 ) Я_ ( î l_i) я_ ( 5 l_ 2 ) я_ f ł II-(?1+1) II-(íl-l) я_„. Я 0 Я_j я_2 (17) + þ " 7r,s~ Jr , . . . ] я~ ?r Я î r - 1 0 Я î r ... o ... o - 2 7r , . . . , s7 r ,Ir] Я Î r - ( î l - l ) Я ľ r - ( ? i - 2 ) ... #~ ?r Я- ( g i _2) . Я _ ( î l _ i ) Я - ( ? 1 - з ) II-(îi-2) • •• Я i ••• Я 0 " Il-?1 Я _ ( î l _ i ) ••• я . -1 Я - ( î l + i ) я. •u ••• я - -2 F = /3Po i(s) + ß., are respectively the polynomial and the strictly proper part of /?(s). We examine the form of /3Poi(s). Consider the identity: A (s)A(s) = L, which can be written also: A~ (s)A(s) = 1 + . . . + Я . s + [AqisЧ + J 4 S l _ 1 s " -1 + . . . + Л0] =Ir [H~rsÍT + H~risi'-1 .. His  Ho + H-is + .. (18) Evaluation of the Impulsive Solution Space of Linear Multivariate Homogeneous Implicit ... 633 from which after multiplying the matrices involved, we obtain the relations: H7Atl = 0 ^A.-1 + ̂ . A . =° IIí/i + Hîr_гA2 + HгЛ-___ + . . . + **>__(,.__,>-._ = 0 Я.Лo + Я o Л . + . . . + Я _ ( , _ _ 1 ) Л , _ = 0 ЯoЛo + Я__Л. + . . . + Я _ f ł A f l = / ' Я-iЛo + Я _ 2 Л . + • • • + я _ ( w + г ) л f l = 0 (19) The first <_! equations of (19) can be written in matrix form as: Я-Я -0 я~ ^ ? r - ( ? . - l ) J / ? r - ( ? . - 2 ) 0 ••• 0 0 ••• 0 Now in light of (20),(16) we have: /_„ 0 -4ţ__i Л__ Ai A 2 ДPo.(_) = [_£•+«•-1 _., . . . , _£+!, _îгjr, | ««'--/,, . . . , SІr, _.] 0 0 0 0 0 0 0 0 0 Я Î r - î . Я ?r-(„,+l) я-îr Я-(..-2) - Я_(____) lI-(ïi-З) lI-(îi-2) Яj Я Q Г я___. я~ - Í - . . L = _«- - 1 7r , . . . , s J r , / r Il-(ï.-2) Я_(___3 ) - Я - ( î i - l ) Il-(5i-2) Я l Я 0 (20) 634 G. F. FRAGULIS = [ I I í ; - , , ^ - 1 + • • • + II-(«i-2)« + II-d-i) I • • •] [... | H~ _2s^~l + ... + H0s + H-i | H~ _ 1 s^ ' -1 + ... + HlS + //o] E (21) = [Ht_v..., Hx,Ho | //-i, / / - . , . . . , // .,] 0 s^- 1 / , s î i - l 7 r S 3 i - 2 7 r . .. s J r s ? i - 2 / r s -ři-3/ r . .. 7 , s7r Ir or equivalently: ßP„i(s) = L A(s)F, where: L = [#£__, • • • ,IIi,IIo | H-i,H-,,...,ff_( ,] 6 sj_rxK?,+íi-i)] Д ( _ ) = 0 0 ••• 0 S^~lIT 0 0 ••• S^~lIr S^-2Ir s ? i - l / r s í i - 2 / r . . . s / r s ? , - 2 / r s ? i - 3 / r . . . / r SІr Ir Ir 0 (22) (23) _3_KÍr+íi-i)]xr„jsj (24) Remark 1 . A(s) has the form (24) in the case <_r > <_i. In the case gi > gV A(s) has the following form: Д(_) = 0 0 0 0 0 • • sìr-Чr _«--> s ? r - : ^ o sîr-Чr $îr-г IГ • SІr Iт /r 0 SІГ lт 0 0 Ir 0 0 0 (25) Evaluation of the Impulsive Solution Space of Linear Multivariate Homogeneous Implicit ... 635 (26) The matrix L as defined in (23) can be written also: L = [-Ce.j£'-15- )... )-C-J=.J3o=1 -CooBoo | CB,CJB,...,CJ^~2B] = [-C~J^-1_?oo,...1-CooJooBoo,-C»5oo |0,0,...,0] +[0,..., 0,0 | CB, CJB,..., CJii~2B] - -C°°[JlZ~\JlZ~'->---,J°°J» I 0,. . . ,0]diag[5«,5»,. . . ,£~] +C[0,..., 0 | U, J, J2,..., J*'-2]diag[B, B,..., B] - r r i ^ i f I £ r _ 1 4 r 2 ••• Ioo /„ I 0 0 ••• 0 - L - C ~ | G J | 0 0 . . . 0 0 I j n j . . . j ,x-xblock diag[5=o, B~,..., S» | B, B,...,B] = L. If we take the inverse Laplace transform L_ of A(s) as defined in (24) and taking in mind that: L_ [sJ] = 6^3\t),j = 0,1,.. . , we obtain: 0 0 0 0 0 • • Й ( Ғ - - І ) ( Í ) / Г б(---'Ҳt)ir б&-~Ҳt)ir б^-^^u б(<ч~2Ҳt)ir • б(^-2Ҳt)ir б^-~Ҳt)ir • • • 6<'Ҳt)Ir •• 6{t)Iг 6(t)Ir 0 б('Ҳt)Ir 6(t)Ir 6(t)Ir 0 0 0 0 0 д(0 = Ľ_ [д(s)] = Thus we have found: #...(<) = L~- [̂ ••(*)] = L A(<)F = po6(t) + Pi6M(t) + ... + P?r_xffiT (27) •i) (t), (28) where /?,• t = 0 , 1 , . . . , qT~i are r x 1 vectors obtained after some manipulations in the terms of (28). From equation (28) and the definition of F in (16) it follows that if the initial conditions /?(0") , . . . , /? ( ? r - 1 )(0 _ ) are appropriate then p(t) has an "impulsive behavior" at t — 0" which consists of a Dirac impulse 6(t) and its (qr — 1) distributional derivatives. In other words when the initial conditions are imposed on f3(t) and its (qT_i) derivatives at t = 0~, P(t) may exhibit an impulsive behavior at t = 0" which is a consequence of the fact that (4) forces j3{t) and /3(,)(t), i = 1,2,..., qT_i to satisfy certain constraints at t — 0~. The exact derivation of these constraints and their relation to the structure at s — oo of A(s) are examined in [13] explicitly. From equation (28) it is also clear that the impulsive solutions of L.H.M.D.E. are closely related — because of the form of L in (16)- to the finite and infinite Jordan pairs (C, J) and (C°°, J°o) respectively, of the polynomial matrix A(s). 636 G. F. FRAGULIS 3. ILLUSTRATIVE EXAMPLE Consider the following differential equation: I " J * 1 6(0 1 _ [ o €.(*) J " [ o and the corresponding polynomial matrix ( in the s-domain): A(s) = 8+1 s2 0 1 with A~ (s) = .+1 .+1 and S~(j)(«) = s2 0 0 1 i.e. г = 2, n = 1, џ = 92 + 1 = 1 + 1 = 2, ci = 2, $2 = 1. We can find the finite and infinite Jordan pairs ofi4 _ 1 («) (see [12,13]): C = -1 1 0 1 0 1 0 0 J = [-l], c~ = The forms of B, £ » are respectively: 5 = [1,-1] , 5 ~ = Mj ? • From (26) we have: J L=[ -C 0 0 ,C ] 1 - 1 1 0 - 1 0 0 Ii ^oo 0 0 B 0 0 0 0 0 1 0 0 0 0 1 - 1 0 - 1 1 - 1 0 - 1 0 0 From (27) we have Д(í) = 0 0 6(t) 0 0 6(t) 0 0 0 6(t) 0 0 0 <5(ť) 0 From (16) we obtain: F = A2 0 M A2 ß(0~) ßM(0~) 0 1 0 0" Г б(o-) -0 0 0 0 б(o-) 1 0 0 1 6(1)(o-) 0 0 0 0 [ tftø-). 6(0-) o 6(0") + &\o~] o Finally from (28) we obtain that the so-called impulsive solutions of the differen-Evaluation of the Impulsive Solution Space of Linear Multivariahle Homogeneous Implicit ... 637 t ial e q u a t i o n are given by: (3^(t) = LA(t)F 0 - 1 1 - 1 0 - 1 0 0 0 0 6(t) 0 0 0 0 6(t) 6(t) 0 0 0 0 6(t) 0 0 б(o-) 0 б(o-)+41}(o-; 0 = И<)Ы0-)1 M i l It is clear from the above t h a t the s y s t e m yields an impulsive solution in t h e case 6(o-) ^o. 4. C O N C L U S I O N S In this p a p e r t h e d e t e r m i n a t i o n of t h e impulsive solut ions of Linear Homogeneous M a t r i x Differential E q u a t i o n s has been invest igated. By a d o p t i n g the definitions of finite a n d infinite J o r d a n pairs of the associated polynomial m a t r i x , a closed formula for t h e d e t e r m i n a t i o n of t h e impulsive solut ions has been presented. A C K N O W L E D G E M E N T The present work was supported by the Greek General Secretariat of Industry, Research and Technology under the contract PENED-89ED37 code 1392. (Received November 30, 1993.) REFERENCES [1] F . M . Callier and C.A. Desoer: Multivariable Feedback Systems. Springer-Verlag, New York 1982. [2] S. L. Cambell: Singular Systems of Differential Equations. Pitman, London 1980. [3] S. L. Cambell, C. D. Meyer and N. Rose: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31 (1976), 3, 411-425. [4] D. Cobb: Feedback and Pole placement in descriptor-variable systems. Internat. J. Control 33 (1981), 6, 1135-1146. [5] D. Cobb: A further interpretation of inconsistent initial conditions in descriptor vari­able systems. IEEE Trans. Automat. Control AG-28 (1983), 9. [6] D. Cobb: On the solution of linear differential equations with singular coefficients. J. Differential Equations 46 (1982), 310-323. [7] N. Cohen: Spectral analysis of regular matrix polynomials. Integral Equations Oper­ator Theory 6 (1983), 161-183. [8] F. Lewis: A survey of linear singular systems. Circuits Systems Signal Process. 5 (1986), 1. [9] D. G. Luenberger: Dynamic systems in descriptor form. IEEE Trans. Automat. Control AC-22 (1977), 312-321. 638 G. F. FRAGULIS [10] H. II. Rosenbrock: State-Space and Mult ivar ia te Theory. Nelson, London 1970. [11] H. H. Rosenbrock: Structural Properties of Linear dynamical systems. Internat. J. Control 20(1974), 2, 191-202. [12] A . I .G . Vardulakis: Linear Multivariable Control: Algebraic Analysis and Synthesis Methods. Wiley, New York 1991. [13] A . I . G . Vardulakis and G. F. Fragulis: Infinite elementary divisors of polynomial ma-trices and impulsive solutions of linear homogeneous matrix differential equations. Circuits Systems Signal Process. 8 (1989), 3, 357-373. [14] G. Verghese: Infinite-frequency Behaviour in Dynamical Systems. Ph.D. Dissertation, Dept. Electrical Engineering, Stanford Univ. 1978. [15] G. Verghese, B. C. Levy and T. Kailath: A generalized state-space for singular systems. IEEE Trans. Automat. Control AC-26 (1981), 4, 811-830. [16] G. Verghese and T. Kailath: Impulsive behaviour in dynamical systems: Structure and significance. In: Proc. 4th Internat. Symp. Mathematical Theory of Networks and Systems, Delft 1979, The Netherlands, pp. 162-168. [17] L. Gohberg, P. Lancaster and L. Rodman: Matrix Polynomials. Academic Press, New York 1982. Dr. George F. Fragulis, Department of Mathematics, Faculty of Science, Aristotle Uni-versity of Thessaloniki 54006. Greece. 

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Evaluation of the impulsive solution space of linear multivariable homogeneous implicit systems

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Institute of Mathematics AS CR, v. v. i.