Transitivity of Commutativity for Linear Time-Varying Analog Systems. Mehmet Emir KOKSAL

Transcription

1 Transiiviy of Commuaiviy for Linear Time-Varying Analog Sysems Mehme Emir KOKSAL Deparmen of Mahemaics, Ondokuz Mayis Universiy, 5539 Aakum, Samsun, Turkey Absrac: In his conribuion, he ransiiviy propery of commuaive firs-order linear imevarying sysems is invesigaed wih and wihou iniial condiions. I is proven ha ransiiviy propery of firs-order sysems holds wih and wihou iniial condiions. On he base of impulse response funcion, ransiiviy of commuaion propery is formulaed for any riple of commuaive linear ime-varying relaxed sysems. Transiiviy proves are given for some special combinaions of firs and second-order linear ime-varying sysems which are iniially relaxed. Keywords: Commuaiviy, Differenial equaions, Iniial condiions, Time-varying sysems, linear sysems, Impulse response. AMS Subjec Classificaion: 93C05, 93C5, 93A30. Inroducion As he main branches of applied mahemaics, differenial (and inegral) equaions arise in many areas of sciences and engineering including acousics, elecromagneic, elecrodynamics, fluid dynamics ec. There is a grea deal of papers on he heory, echnique and applicaions of differenial equaions. Especially, hey are used as a major ool in order o achieve many developmens in real engineering problems by modelling, analyzing and solving naurel problems. For example, an inerdisciplinary branch of applied mahemaics and elecricelecronics engineering, hey play a pioneering role in sysem and conrol heory, ha deal wih he behavior of dynamical sysems wih inpus, and how heir behavior is modified by differen combinaions such as cascade and feedback connecions which is. When he cascade connecion in sysem design is considered, he commuaiviy concep places a prominen role o improve differen sysem performances.

2 Consider a sysem A described by a linear ime-varying differenial equaion of he form n A A: a i () di d i y A() = x A (); i=0 () (x y) represens inpu-oupu pair of he sysem a any ime R, a i () are ime-varying coefficiens wih a n () 0, n A 0 is he oder of sysem; and y (i) A ( ) R, i = 0,,, n A are he iniial condiions a he iniial ime R. When wo sysems of his ype are inerconneced sequenially so ha he oupu of he former feeds he inpu of he laer, i is said ha hey are conneced in cascade []. If he order of connecion does no affec he inpu-oupu relaion of he combined sysem AB or BA, i is said ha sysems A and B are commuaive. Figure : Cascade connecion of he differenial sysem A and B If he combined sysem has an overall inpu-oupu relaion invarian wih he sequence of connecion, i is said ha hese sysems are commuaive [2]. In [2], J. E. Marshall has proven ha for commuaiviy, eiher boh sysems are ime-invarian or boh sysems are imevarying. In addiion, he has revealed he commuaiviy condiions of firs-order sysems. Laer Marshall s work, a grea deal of researches has been done on commuaiviy. In [3-5], he necessary and sufficien condiions for commuaiviy of second-order sysems are presened. Koksal has presened he general commuaiviy condiions for ime-varying sysems of any order and reformulaed he previous resuls obained for seccond-order sysems in he forma of general condiions [6]. In his work, he general condiions are used o show ha any sysem wih consan forward and feedback gains is commuaive wih he sysem iself, which is an imporan fac for he feedback conrol heory. In 985, Koksal prepared a echnical repor which is a survey on commuaiviy [7]. In his 2

3 repor, an ieraive formula is derived and an explici formula is given for he enries of he coefficien marix expressing he firs se of commuaiviy condiions and he second se of commuaiviy condiions, respecively. Hence, he heorem saing hese condiions is formally proved. Morover, explici commuaiviy resuls for fourh-order sysems are obained. Finally, commuaiviy of Euler s sysem is proved. The conen of he published bu undisribued work [7] can be found in he exhausive journal paper of M. Koksal inroduced he basic fundamenals of he subjec [8]. His paper covers almos all he previous resuls excep he ones relaed wih iniial condiions and sensiiviy. This paper is he firs uorial paper ha has appeared in he lieraure. More han one decade no publicaion had been appeared in he lieraure unil he work in 20 [9]. This reference is he second basic journal publicaion afer he firs appeared in 988 [8]. In [9], anoher generic paper by he same auhor has presened explici commuaiviy condiions of fifh-order sysems in addiion o reviews of commuaiviy of sysems wih nonzero iniial condiions, commuaiviy and sysem disurbance, commuaiviy of Euler sysems. Research on commuaiviy has no confined o analog sysems only; here has been some lieraure on he subjec for discree-ime sysems as well [0, ]. Hence, he research is coninuing on boh digial and analog area [2]. In [2], explici resuls for finding all he second order commuaive pairs of a firs-order linear ime-varying sysem have been given and he derived heoreical resuls have been verified by an example. Abou he commuaiviy of coninuous ime linear ime-varying sysems, [3] has been he las paper appearing in he lieraure; i deals wih driving necessary and sufficienly condiions for he decomposiion of a second order linear ime-varying sysem ino wo cascade conneced commuaive firs-order linear ime-varying subsysems. Afer a shor inroducion of he lieraure in his secion, he ransiiviy propery of commuaiviy is inroduced in Secion 2. Transiiviy propery of commuaive firs-order sysems wih and wihou iniial condiions is sudied in Secion 3. In Secion 4, ransiiviy propery of commuaiviy for relaxed sysems of any order is formulaed on he base of impulse response funcion. Secion 5 covers he verificaion of he general resuls formulaed in Secion 4 on he base of impulse response funcion for firs-order sysems. Secion 6 illusraes he resuls presened in Secions 4 and 5. Secion 7 includes ransiiviy proves for some combinaions of firs and second-order relaxed linear ime-varying sysems. Finally, he paper finishes wih conclusions and fuure work which compose he las secion, Secion 8. 3

4 2. Transiiviy Propery of Commuaiviy Logically or mahemaically, ransiiviy is a propery of a binary relaion such ha whenever one elemen is relaed o a second elemen, and he second elemen is relaed o a hird elemen, hen he firs elemen is also relaed o he hird elemen. Le A, B, C be dynamical sysems described by linear ime-varying differenial equaions of he form () wih heir special inpu-oupu variables, orders of complexiy, and coefficiens. If A and B are commuaive among hemselves, furher, B and C are also commuaive amoung hemselves, wha can i be said abou he commuaiviy of sysems A and C. If A and C are also commuaive among hemselves, his propery is called ransiiviy propery of commuaiviy, in oher words commuaiviy is a ransiive relaion. For commuaive sysems A, B, C saisfiying ransiiviy, all he riples ABC, ACB, BAC, BCA, CAB, CBA yield he same funcionally equivalen cascade conneced sysem and hence he same inpu-oupu propery. The preference of an individual sysem connecion depends on is relaive performance characerisics such as sensiiviy, disurbance, robusness, and ec., wih respec o he ohers. I is well-known (or rivial o show) ha linear scalar sysems, ha is sysems of order 0, are always commuaive among hemselves wheher ime-invarian or ime-varying; and ransiiviy always hold for such sysems. Furher, iniially relaxed ime-invarian sysems are always commuaive. When iniial condiions exis, which is valid for non-scalar (s or higherorder) sysems, he commuaiviy of such sysems (even including a scalar sysem) is no auomaic and requires some condiions. Hence, he following discussions are devoed mainly o commuaiviy of sysems a leas one of hem is of order one or higher. 3. Transiiviy for Firs-order Sysems Le A, B, C be firs-order linear ime-varying sysems defined by a differenial equaion of he form (). Le a, a 0 ; b, b 0 ; and c, c 0 are ime-varying (in general) coefficiens of hese sysems. Assume also ha (A, B) and (B, C) are commuaive pairs. The firs commuaiviy condiions [7-9] for (A, B) and (B, C) yields [ b b 0 ] = [ a 0 a 0 ] [k k 0 ], (2a) 4

5 [ c c 0 ] = [ b 0 b 0 ] [l l 0 ], (2b) respecively; where k, k 0, l, l 0 are some consans. Insering he values of b, b 0 as expressed in (2a) ino Eq. (2b) and rearranging, we obain [ c c 0 ] = [ a 0 a 0 ] [ k l k 0 l + l 0 ]. (2c) This equaion already saisfies he firs commuaiviy condiion for Sysems A and C; hence, A and C are commuaive pairs under zero iniial condiions. For he case of exisence of non-zero iniial condiions, he second commuaiviy condiion [9] for (A, B) and (B, C) yields y A ( ) = y B ( ) 0, (3a) a 0 ( ) a ( ) = b 0( ) ; (3b) b ( ) y B ( ) = y C ( ) 0, (4a) b 0 ( ) b ( ) respecively. Hence, Eqs. (3) and (4) lead o wrie = c 0( ), (4b) c ( ) y A ( ) = y C ( ) 0, (5a) a 0 ( ) a ( ) = c 0( ). (5b) c ( ) The resul in Eq. (5) clearly shows he saisfacion of he second commuaiviy condiion for Subsysems A and C. Hence, A and C are also commuaive when iniial condiions exis, and ransiiviy propery is valid for non-relaxed firs-order linear ime-varying sysems as well. In fac, Eq. (3b) is no saisfied for all sysems A and B saisfying commuaiviy condiions (2a) under relaxed condiions. Using (2a) we see ha Eq. (3b) requires a 0 ( ) a ( ) = b 0( ) b ( ) = k a 0 ( ) k 0 k a ( ) k 0 a k 0 ( ) = ; (6a) a ( ) 5

6 Or equivalenly k 0 = k. (6b) Similarly, using (2b), we see ha Eq. (4b) requires b 0 ( ) b ( ) = c 0( ) c ( ) = l b 0 ( ) l 0 l b ( ) l 0 b l 0 ( ) =, (7a) b ( ) l 0 = l. (7b) Using (2c) wih (6b) and (7b), we can easily show ha (5b) is saisfied; Tha is: c 0 ( ) c ( ) = = [k l a 0 ( ) + k 0 l + l 0 ] k l a ( ) ( k )l ( l ) k l a 0 ( ) a ( ) 6 = k 0 l l 0 k l a 0 ( ) a ( ) = a 0( ), (8) a ( ) which is exacly Eq. (5b); hence, A and C are commuaive under non-zero condiions as well if he consans k, l, k 0,, l 0 saisfy (6b) and (7b). These are explici commuaiviy condiions under non-zero iniial saes equivalenly replacing by (3b) and (4b) in addiion o (2a) and (2b) which are necessary and sufficien for relaxed case. Wih hese condiions 5b is also saisfied, so ha ransiiviy is valid. We sae he resuls obained so far by wo heorems: Theorem : The necessary and sufficien condiions ha wo firs-order linear ime-varying sysems A and B described by differenial equaions of he form () are commuaive under zero iniial condiions are ha i. The coefficiens of Sysem B mus be expressible in erms of he coefficiens of hose of ii. A as in Eq. (2a) where k 0 and k 0 are arbirary consans. Furher, Sysems A and B are commuaive under arbirary non-zero iniial condiions as well if and only if Condiion (i) is saisfied wih k 0 = k (ha is k 0 and k 0 canno be chosen arbirarily), and heir non-zero iniial condiions mus be equal. Noe ha his heorem is he firs heorem expressing he commuaiviy condiions for unrelaxed sysems by using he arbirary consans relaing he coefficiens of wo sysems for heir commuaiviy in relaxed condiions. Noe also ha Eqs. (6b) and (7b) are independen of

7 he iniial ime. So, commuaiviy does no depend on he iniial ime for firs-order nonrelaxed sysems. Theorem 2: The ransiiviy propery of commuaiviy for firs-order linear ime-varying sysems is always valid for boh relaxed and un-relaxed commuaive sysems ha is for sysems wih zero and non-zero iniial condiions. 4. Transiiviy for High-order Sysems In his secion, higher-order linear ime-varying sysems wih zero iniial condiions are considered; scalar (0-order) and firs-order sysems are also included in he scope. To prove he ransiiviy propery of commuaiviy, he impulse response funcion is used. The resuls are general in he sense he subsysems of each commuaive pair may have a differen order han is parner. Le A be a liear ime-varying iniially relaxed sysem having an impulse response h A (, τ); by definiion his is he response o a uni impulse occurring a ime τ and observed a ime τ [4]. And he response of A for an arbirary inpu u A () is expressed by superposiion inegral y A () = h A (, γ)u A (γ)dγ. (9a) Consider now he cascade connecion of A and B as shown in Fig. 2 and denoe his connecion by AB. Smilar o Eq. (9a), for he oupu of B we wrie y B () = h B (, γ)u B (γ)dγ. (9b) I is obvious form Fig., and using Eq. (9a) τ u B (τ ) = h B (τ, γ)u(γ)dγ. (9c) Figure 2: Cascade connecion AB of Sysems A and B. 7

8 Finally, insering Eq. (9c) in Eq. (9b), and observing y B () in Fig. 2, we obain τ y( ) = h B (, τ) [ h A (τ, γ)u(γ)dγ] dτ. (9d) This is he response of AB for any inpu u() applied for.to find he impulse response h A (, ) of he connecion AB, we subsiue u() = δ( ) in Eq. (9d) and arrive τ h AB (, ) = h B (, τ) [ h A (τ, γ)δ(γ )dγ] dτ = h B (, τ)h A (τ, )dτ, (0a) 0 where he second equaliy resuls from he propery of impulse funcion. In a similar resul, he impulse response h BA (, τ) of he connecion AB can be wrien as h BA (, ) = h A (, τ)h B (τ, )dτ. (0b) Since equal impulse responses yield equal inpu-oupu pairs, for commuaiviy of A and B i mus be rue ha h AB (, ) = h BA (, );, R. (a) Using Eqs. (0a, b); (a) can be wrien as h B (, τ)h A (τ, )dτ = h A (, τ)h B (τ, )dτ ;, R, (b) or equivalenly [h B (, τ)h A (τ, ) h A (, τ)h B (τ, )] dτ = 0;, R. (c) In fac, eiher of Eqs. (b) or Eq. (c) express necessary and sufficien condiions in erms of impulse responses for relaxed sysems A and B o be commuaive. On he oher hand, for commuaiviy of B and C under zero ininial condiions, he neccessary and suffifcien condiion can be obained from () by changing A B and B C, he resul is 8

9 h BC (, ) = h CB (, );, R, (2a) h C (, τ)h B (τ, )dτ = h B (, τ)h C (τ, )dτ ;, R, (2b) or equivalenly, [h C (, τ)h B (τ, ) h B (, τ)h C (τ, )] dτ = 0;, R. (2c) For he proof of he commuaiviy of C and A, ha is he ransiiviy propery of commuaiviy, i is required o show h CA (, ) = h AC (, );, R, ha is, (3a) h A (, τ)h C (τ, )dτ = h C (, τ)h A (τ, )dτ ;, R, (3b) or equivalenly, [h A (, τ)h C (τ, ) h C (, τ)h A (τ, )] dτ = 0;, R. (3c) In he ligh of he presence of equaliies in (b) and (2b), he work lef for he proof of ransiiviy is o show he validiy of (3b); if so hen (3a) holds. This process will end wih he proof of ransiiviy of linear ime-varying relaxed sysems of higher orders. I worh s o remark ha Eqs. (0a, b) are firs used by J. E. Marshall o prove his asserion menioned in Inroducion (Eqs. (2, 3) in [2]). Laer, hey are used by he auhor for sudying commuaiviy of Euler sysems (Eqs. (34a, b) in [7]). When we consider a scalar sysem A: a 0 ()y A () = x A () (4a) for which is response h A (, ) o a uni impulse δ( ) is obviously h A (, ) = y A () = a 0 () x A() = a 0 ( ) δ( ). (4b) 9

10 Then, le B be a ime-varying sysem of s order or higher wih impulse response h B (, ). Eqs. (0a) and (0b) lead o h AB (, ) = h B (, τ) δ(τ ) a 0 ( ) dτ = δ( τ) h BA (, ) = a 0 (τ) h B(τ, )dτ = h B (, ) a 0 ( ), (5a) h B (, ) a 0 (). (5b) From hese equaions, i is obvious ha a scalar sysem ha can be commuaive wih a s or higher order sysem mus be a ime-invarian sysem; ha is a 0 () = a 0 ( ) = consan for all. In oher words, a scalar ime-varying sysem canno commuaive wih any order of higher order (s, 2nd ) ime-varying sysem. The only commuaive pairs of i are he scalar sysems. Though commuaiviy of a consan gain sysem (scalar ime-invarian sysem) is always valid wih ime-varying sysem of any order, if he iniial condiions exis he commuaiviy is possible only if he consan gain is ; ha is Sysem A is an ideniy. The validaion of ransiiviy using he equivalence of impulse responses h AC (, ) and h CA (, ) for firs-order sysems are considered in Secion 5. Bu he exension of his resul for sysems a leas one is 2nd order or higher is no sraigh forward and appears o be an unsolved problem. Insead of jumping his problem fully, we consider he special case ha ransiiviy holds condiionally. Hence, he following discussion is no resriced by order limiaion. Le A, B, C be linear ime-varying sysems wih impulse responses h A, h B, h C saisfying h B (, τ)h A (τ, ) h A (, τ)h B (τ, ) = 0, h C (, τ)h B (τ, ) h B (, τ)h C (τ, ) = 0. (6a) (6b) Eq. (6a) implies ha Eqs. (a, b, c) are saisfied, hence (A, B) is a comuaive pair. In a similar manner (6b) implies ha (B, C) is also a commuaive pair. Muliplying (6b) by h A (τ, ) and insering he equivalence of h B (, τ)h A (τ, ) as obained from (6a), we find h C (, τ)h B (τ, )h A (τ, ) h B (, τ)h C (τ, )h A (τ, ) = 0, h C (, τ)h B (τ, )h A (τ, ) h C (τ, )h A (, τ)h B (τ, ) = 0, (7a) (7b) 0

11 Since h B (τ, ) 0, i follows ha [h C (, τ)h A (τ, ) h A (, τ)h C (τ, )]h B (τ, ) = 0. (7c) h C (, τ)h A (τ, ) h A (, τ)h C (τ, ) = 0. (8a) This is sufficien o wrie h C (, τ)h A (τ, )dτ h A (, τ)h C (τ, )dτ = 0. (8b) Comparing wih Eqs. (0a) and (0b), he firs inegral in (8b) is he impulse response of AC and he second one is ha of CA; hence, (8b) yields h AC (, ) = h AC (, ). (9) So, ha A and C are commuaive. This proves he ransiiviy proper of higher-order sysems under he se assumpions. 5. Verificaions for firs-order sysems This secion verifies he ransiiviy propery of firs-order linear ime-varying sysems in heir relaxed modes as presened in Secion 3 by using he general resuls obained for sysems of any order in Secion 4. Consider a firs-order linear ime-varying sysem A defined by a () d d y A() + a 0 ()y A () = x A (), (20) wih zero iniial condiions. The impulse response h A (, ) of his sysem o an impulse δ( ) occurring a = is obviously expressed as [4, 5] a 0 (γ) h A (, ) = a ( ) e 0 a (γ) dγ = a ( ) ef A () f A ( ), (2a) where f A (γ) = a 0(γ) a (γ) dγ, d dγ f A(γ) = a 0(γ) a (γ). (2b) (2c)

12 Hence, for Sysems B and C of he same ype of A, i can be wrien ha b 0 (γ) h B (, ) = b ( ) e 0 b (γ) dγ = b ( ) ef B () f B ( ), h C (, ) = f B (γ) = b 0(γ) b (γ) dγ, d f dγ B(γ) = b 0 (γ), b (γ) c ( ) e c0 (γ) 0 c (γ)dγ = c ( ) ef C () f C ( ), (22a) (22b) (22c) (23a) f C (γ) = c 0(γ) c (γ) dγ, (23b) d dγ f C(γ) = C 0(γ) C (γ). (23c) For commuaiviy of A and B, consider he verificaion of Eq. (a) (or equivalenly Eqs. (b, c)). Using he relaions in Eq. (2a) beween he coefficiens of A and B in Eq. (22b) and hen recognizing (2b), we obain f B (γ) = b 0(γ) b (γ) dγ = k a 0 (γ) + k 0 dγ = a 0(γ) k a (γ) a (γ) dγ k 0 k a (γ) dγ = f A (γ) k 0 k a (γ) dγ. (24a) Using his resul in Eq. (22a), we obain where h B (, ) = k a ( ) e f A () k 0 k a () d f A ( )+k 0 k a ( ) d0, = k a ( ) ef A () f A ( ) e g( ) g() = k e g( ) g() h A (, ), (24b) g() = k 0 k a () d. (24c) By using (5b) in (7a), we obain 2

13 h AB (, ) = e g(τ) g() h k A (, τ)h A (τ, )dτ. (25) Before proceeding furher, we show he ransiiviy propery of h A (, ); in fac using (2a) h A (, τ)h A (τ, ) = a (τ) ef A () f A (τ) a ( ) ef A (τ) f A ( ) = a (τ) a ( ) ef A () f A ( ) = a (τ) h A(, ). (26) Same propery holds for he remaining firs-order sysems B and C as well. Anoher relaion we need in he fuure proves is arrived as follows: eg(τ) dτ = a (τ) k 0 ek a (τ) dτ dτ = a (τ) k k 0 e k 0 k a (τ) dτ = k k 0 e g(τ) = k k 0 [e g() e g( ) ]. (27a) Similarly, we obain e g(τ) k dτ = [e a (τ) g( ) e g() ]. k 0 (27b) Using (26) and (27a) in (25), we have e g() = k h AB (, ) = e g(τ) g() k a (τ) h A(, )dτ h A (, ) eg(τ) dτ a (τ) = e g() k h A (, ) k k 0 [e g() e g( ) ] = k 0 h A (, ) [ e g( ) g() ]. (28) Using (24b) in (0b) and using he formula (26) and (27b), we obain sequenially h BA (, ) = h A (, τ) k e g( ) g(τ) h A (τ, )dτ = e g( ) g(τ) h k a (τ) A (, )dτ 3

14 = eg( ) k e g( ) = h A (, ) e g(τ) a (τ) dτ k h A (, ) k k 0 [e g( ) e g() ] = k 0 h A (, ) [ e g( ) g() ]. (29) Obviously, AB and BA have he same impulse response as seen in Eqs. (28) and (29), respecively; so A and B are commuaive. Remark : Alhough Eq. (c), a propery of impulse response of he commuaive sysems A and B, seems o mislead o he conclusion of he inegran being zero; ha is, (, τ, ) h B (, τ)h A (τ, ) h A (, τ)h B (τ, ) = 0. (30) This is sricly wrong. In fac, using (24b) for h A (, ), and (26) laer, we have (, τ, ) = k e g(τ) g() h A (, τ)h A (τ, ) h A (, τ) k e g( ) g(τ) h A (τ, ) = 0 = k h A (, τ)h A (τ, ) [e g(τ) g() e g( ) g(τ) ] = h k A (, ) [eg(τ) g() e g(0) g(τ) ] = 0. (3a) a (τ) Since, h A (, ) 0 and a (τ) 0, his implies e g(τ) g() = e g() g(τ), (3b) or equivalenly e 2g(τ) = e g()+g( ) ; g(τ) = 2 [g() + g( )]. (3c) Wih (24c), he las equaion is equivalen o a (τ) dτ = 2 [ a () d + a ( ) d ], (3d) which is impossible for an arbirary funcion a ( ) And for general values of τ. 4

15 Similar resul obained in Eqs. (25-29) can be obained for commuaive pair (B, C); eiher by he parallel derivaions made for (A, B), or by replacing A B, B C, so a b and k, k 0 by l, l 0, respecively. For he commuaiviy of A and C, o show ha h AC (, ) = h CA (, ), he replacemens B C, (k, k 0 ) (k l, k 0 l + l 0 ) in Eqs. (25-29) complee he verificaion of ransiiviy for iniially relaxed firs-order linear ime- varying sysems by using impulse responses. 6. Example This example is given for illusraion of he resuls presened in Secions 4 and 5. Consider sysem A describe by ( + )y A + ( + 2)y A = x A, <, (32) where a () = +, a 0 () = + 2; f() for A is compued by using (2b) as Then, by (2a), is impulse response is compued as f A () = + 2 d = ln( + ). (33) + h A (, ) = e ln(+)++ln( +) + Le subsysem B be described by + = e ++ln + = + + e +. (34) 2( + )y B + (2 + 5)y B ; <, (35) where b () = 2( + ), b 0 () = which can be obained from a and a 0 by using Eq. (2a) wih k = 2, k 0 = ; hence, (A, B) is a commuaive pair. Using (22b), f() for B can be obained as f B () = d =.5 ln( + ). (36) 2( + ) Then, by (22a), he impulse response of B is compued as h B (, ) = e.5 ln(+)++.5 ln( +) 2( + ) = e ++.5 ln + + 2( + ) 5

16 = ( 2( + ) e + + ).5 0 ( + ).5 = ( + ) 0.5 2( + ).5 e +. (37) For compuaions of h AB (, ) and h BA (, ), firs consider he inegrands in Eqs. (0a), (0b) and (b). h B (, τ)h A (τ, ) = (τ + )0.5 2( + ).5 e +τ τ + e τ+ = 2( + ).5 (τ + ) 0.5 e +, (38a) h A (, τ)h B (τ, ) = + e +τ (+) 0.5 2(τ+).5 e τ+ = (+) 0.5 2(+)(τ+).5 e +. (38b) Obviously, hese inegrands are no idenical and Eq. (30) in connecion o Remark is no saisfied for he general values of τ R. The impulse response of AB and BA are compued as in (0a) and (0b), ha is he inegral of he inegrands in (38a) and (38b), respecively. The resuls are as follows: h AB (, ) = h B (, τ)h A (τ, )dτ = e + (τ + ) 0.5 2( + ) e + = 2( + ).5 (τ + ) 0.5 dτ = e + ( + ).5 [( + )0.5 ( + ) 0.5 ] = e +0 + [ ( + + ) 0.5 ], (39a) h BA (, ) = ( + ) 0.5 e + 2( + ) (τ + ).5 dτ = ( + ) 0.5 e +0 2( + ) (τ + ) = (+) 0.5 e + (+) [ e +0 ( +) 0.5 (+) 0.5] = + [ (+ + )0.5 ]. (39b) Obviously, h AB (, ) h BA (, ). When Sysem C is obained from B by (2b) wih l = 0.5, l 0 = 3.5, is coefficiens are found as c = ( + ), c 0 = (2 + 5)( 0.5) = +. Then, f() for C is compued from (23b) as f c () = + d = + 2ln( + ). (40a) Using (23a), he impulse response of C is now obained as 6

17 h C (, ) = e +2 ln(+)+ 2 ln( +) ( + ) = + e + ( + + ) 2 ( + )2 = ( + ) 3 e +. (40b) The impulse response of he connecion AC is compued by Eq. (0a) modified by replacing B by C, and using Eqs. (34) and (40b) for A and C; we obain ( + ) 2 h AC (, ) = h C (, τ)h A (τ, )dτ = (τ + ) 3 e +τ = ( + ) 2 e + 0 (τ + ) 4 dτ = ( + )2 e + 3 = [ ( + ) 3 ( + ) 3] ( + ) 2 e + 3 τ + e τ+dτ (τ + ) 3 = e [ ( 3( + ) + ) ]. (4a) Noe ha he impulse response of he connecion CA can be obained similarly by using Eq. (0a) by replacing B by C and using Eqs. (40b) and (34) for C and A; he resul is h CA (, ) = h A (, τ)h C (τ, )dτ = = e + 0 ( + )( + ) 3 (τ + )2 dτ (τ + )2 e +τ + ( + ) 3 e τ+dτ e + = 3( + )( + ) 3 (τ + )3 = e + 3( + )( + ) 3 [( + ) 3 ( + ) 3 ] = e [ ( 3( + ) + ) ]. (4b) This is he same impulse response expressed in (4a). Hence, i is verified ha (A, C) is a commuaive pair. 7. Transiiviy for Some s and 2 nd Order Relaxed Sysems 7

18 Le A, B, C be linear ime-varying sysems of orders 2,, ; respecively. Le hey be represened by he general form of () wih coefficiens a 2, a, a 0 ; b, b 0 ; c, c 0 ; inpus by x A, x B, x C ; oupu by y A, y B, y C. I is rue ha A has 2 nd or lower order commuaive pairs B if and only if [8-0] a d d [a 0 4a 2 + 3a 2 2 8a a 2 + 8a a 2 4a 2 a 2 6a 2 ] k = 0, (42a) a b k a [ b ] = a b a 0.5 [ k ], 2 (2a a 0 2) [ a 0 k 0 4 ] (42b) where k 2, k, k 0 are arbirary consans. We assume ha B is a finie order commuaive pair of A; hence k 2 = 0 and 0.5 [ b a 2 0 ] = [ b a (2a a 0 2) ] [ k ]. (43) k 0 4 On he oher hand, i is rue ha B has s or lower order commuaive pairs C, if [8] [ c c 0 ] = [ b 0 b 0 ] [ l l 0 ], (44) where l and l 0 are arbirary consans. Insering he values of b and b 0 from Eq. (43),) ino Eq. (44), and rearranging, we obain [ c 0.5 a 2 0 c ] = [ a (2a a 2) ] [ k l ]. (45) k 0 l + l 0 4 I is obviously rue ha his equaion is in he form of (43). So C is among he firs-order commuaive pairs of A. Hence, i has been proven ha if (A, B) and (B, C) are commuaive pairs so is (A, C). Hence, ransiiviy is valid. Le s ge one sep up and consider ransiiviy propery beween wo second order and one firsorder sysems. Le A, B, C be 2 nd, s, 2 nd order sysems, respecively, wih coefficien a 2, a, a 0 ; b, b 0, c 2, c, c 0. Assume (A, B) and (B, C) are commuaive pairs and show (A, C) is also commuaive. Eq. (43) is valid since (A, B) is a commuaive pair. Similar equaion 8

19 0.5 [ b c 2 0 ] = [ b c (2c c 2) 0 ] [l ] (46) l 0 4 is also valid since (B, C) is a commuaive pair. Furher, (42a) is valid and he similar equaion for C is wrien as c d d [c 0 4c 2 + 3c 2 2 8c c 2 + 8c c 2 4c 2 c 2 6c 2 ] l = 0. (47) Since B is of order, k in (43) and l in (46) are no zero; hence (42a) and (47) yield a 0 = [A 0 + 4a 2 + 3a 2 2 8a a 2 + 8a a 2 4a 2 a 2 6a 2 ], (48a) c 0 = [C 0 + 4c 2 + 3c 2 2 8c c 2 + 8c c 2 4c 2 c 2 6c 2 ], (48b) respecively, where A 0 and C 0 are arbirary consans. From Eq. (43), he firs line, we obain a 2 = b 2 2 k a 2 = 2b b 2, a 2 = 2(b 2 + b b ) 2. (49a) k k Using hese resuls in Eq. (43), he second line, we obain a = b (2b 0 2k 0 + b ) 2 a = b k (2b 0 2k 0 + b ) + b (2b 0 + b ) 2. (49b) k Insering he values of a 2, a 2, a 2, a, a in Eq. (49) ino Eq. (48a) and simplifying, we obain a 0 = A 0 + [ b 2 0 k 0 ] + b b 0 k 2 k. (49c) Similar resuls can be obained saring from (45) and using (48b): c 2 = b 2 l 2, (50a) c = b (2b 0 2l 0 +b ) l 2, (50b) 9

20 c 0 = C 0 + [ b 2 0 l 0 ] + b b 0 l 2 l. (50c) To prove he commuaiviy of A and C, i is sufficien o show ha a c m a [ c ] = a c a 0.5 [ m ], (5) 0 2 (2a a 2) [ a 0 m 0 4 ] which is similar equaion o (42b), where m 2, m, m 0 are some consans. I is rue ha, he coefficiens of C can be expressed as in (5) by proper choice of consans m 2, m, m 0 ; in fac, insering in coefficiens a i in Eq. (49) and coefficien c i in Eq. (50) ino Eq. (5), we find ha m 2 = k 2 l 2, (52a) m = 2k (k 0 l 0 ) l 2, (52b) m 0 = C 0 A 0 k 2 l 2 + [ k 0 l 0 l ] 2, (52c) which are all consans. Hence, Eq. (5) ogeher wih Eq. (42) imply ha C is a second-order commuaive pair of A, ha is A and C are also commuaive. This proves he ransiiviy propery of (A, B) and (B, C) o (A, C). We express he resul by a heorem: Theorem 3: The commuaiviy propery beween hree subsysems of which a mos 2 of hem second order and he oher(s) are firs-order always saisfy he ransiiviy propery under zero iniial condiions. 8. Conclusions Transiiviy propery of commuaiviy is defined for commuaive linear ime-varying sysems of any order. The ransiiviy resuls are presened for firs-order sysems wih and wihou iniial condiions. The sudy is carried on wo approaches; one is based on use of impulse response funcion and he oher depends on use of general condiions se by he auhor and ohers for commuaiviy of linear ime-varying sysems. 20

21 One illusraive example is included o show he validiy of he resuls. However, general ransiiviy propery of commuaive high order sysems have no been proved or disproved and his remains as an unsolved problem ye. REFERENCES [] R. Boylesad and L. Nashelsky, Elecronic Devices and Circui Theory, Prenice Hall, New Jersey, 203. [2] E. Marshal, Commuaiviy of ime varying sysems, Elecronics Leers, 3 (977) [3] M. Koksal, Commuaiviy of second order ime-varying sysems, Inernaional Journal of Conrol, 36 (982) [4] S. V. Salehi, Commens on Commuaiviy of second-order ime-varying sysems, Inernaional Journal of Conrol, 37 (983) [5] M. Koksal, Correcions on Commuaiviy of second-order ime-varying sysems, Inernaional Journal of Conrol, 38 (983) [6] M. Koksal, General condiions for he commuaiviy of ime-varying sysems, IASTED Inernaional Conference on Telecommunicaion and Conrol (TELCONi84), 984, pp [7] M. Koksal, A Survey on he Commuaiviy of ime-varying sysems, METU, Gazianep Engineering Faculy, Technical. Repor no: GEEE CAS-85/, 985. [8] M. Koksal, An exhausive sudy on he commuaiviy of ime-varying sysems, Inernaional Journal of Conrol, 47 (988) [9] M. Koksal and M. E. Koksal, Commuaiviy of linear ime-varying differenial sysems wih non-zero iniial condiions: A review and some new exensions, Mahemaical Problems in Engineering, 20 (20) -25. [0] M. E. Koksal and M. Koksal, Commuaiviy of cascade conneced discree ime linear ime-varying sysems, 203 Auomaic Conrol Naional Meeing TOK 203, (203) p [] M. Koksal and M. E. Koksal, Commuaiviy of cascade conneced discree-ime linear ime-varying sysems, Transacions of he Insiue of Measuremen and Conrol, 37 (205) [2] M. E. Koksal, The Second order commuaive pairs of a firs-order linear ime-varying sysem, Applied Mahemaics and Informaion Sciences, 9 (205) -6. [3] M. E. Koksal, Decomposiion of a second-order linear ime-varying differenial sysem 2

22 as he series connecion of wo firs-order commuaive pairs, Open Mahemaics, 4 (206) [4] C. A. Desoer, Noes For A Second Course On Linear Sysems, Van Nosrand Rheinhold, New York, 970. [5] Chi-Tsong Chen, Linear Sysem Theory and Design, New York Oxford, Oxford Universiy Press,