ACTIVE electrical networks, which require energizing sources for their operation, are most widely

Transcription

1 1 Stabiization heory For Active Muti-port Networks Mayuresh Bakshi Member, IEEE,, Virendra R Sue and Maryam Shoejai Baghini, Senior Member, IEEE, arxiv: v1 [cs.sy] 10 Jun 2016 Abstract his paper proposes a theory for designing stabe interconnection of inear active muti-port networks at the ports. Such interconnections can ead to unstabe networks even if the origina networks are stabe with respect to bounded port excitations. Hence such a theory is necessary for reaising interconnections of active mutiport networks. Stabiization theory of inear feedback systems using stabe coprime factorizations of transfer functions has been we known. his theory witnessed gorious deveopments in recent past cuminating into the H approach to design of feedback systems. However these important deveopments have sedom been utiized for network interconnections due to the difficuty of reaizing feedback signa fow graph for muti-port networks with inputs and outputs as port sources and responses. his paper resoves this probem by deveoping the stabiization theory directy in terms of port connection description without formuation in terms of signa fow graph of the impicit feedback connection. he stabe port interconnection resuts into an affine parametrized network function in which the free parameter is itsef a stabe network function and describes a stabiizing port compensations of a given network. Index erms Active networks, Coprime factorization, Feedback stabiization, Mutiport network connections. I. INRODUCION ACIVE eectrica networks, which require energizing sources for their operation, are most widey used components in engineering. However operating points of such networks are inherenty very sensitive to noise, temperature and source variations. Often there are considerabe variations in parameters of active circuits from their origina design vaues during manufacturing and cannot be used in appications without externa compensation. Due to such uncertainties and time dependent variations, which cannot be modeed accuratey, compensation of active networks or interconnections can ead to an unstabe circuit or even greater sensitivity even if the component parts are stabe. On the other hand interconnection of passive networks remains passive and stabe. For this reason stabiity is never a consideration in passive network synthesis or design. Passive network theory and design thus enjoys a rich anaytica framework devoid of the engineering compication of stabiity [1], [2]. On the other hand stabiity in anaysis of active networks is an important property [9] hence synthesis of active networks with stabiity is an important probem of circuit design. he purpose of this paper is to deveop a systematic approach to interconnection of active inear time invariant (LI) muti-port networks with the resoution of this stabiity issue in mind. Specificay, we address the foowing question. Question. Given a LI network N with a port mode G at its nomina parameter vaues, what are a possibe (LI, active) networks N c with compatibe ports, when connected to N at ports with specified (series or parae) topoogy, form a stabe network? his question is anaogous to that of feedback system theory, what are a possibe stabiizing feedback controers of a LI pant? Such a question ed to andmark new deveopments in feedback contro theory bmayuresh@ee.iitb.ac.in, Assistant Professor, Dept. of Engineering and Appied Sciences, VII Pune, India vrs@ee.iitb.ac.in, Professor, Dept. of Eectrica Engineering, II Bombay, India mshoejai@ee.iitb.ac.in, Professor, Dept. of Eectrica Engineering, II Bombay, India

2 2 in recent decades [6], [7], [8] broady known under the H approach to contro. o motivate precise mathematica formuation of the probem we need to consider stabiity property of active networks and nature of interconnections at ports. A. Stabiization probem for network interconnection In active network theory two kinds of stabiity property have been we known [4], the short circuit stabiity and the open circuit stabiity. hese can be readiy extended to muti-port networks. Primariy these stabiity properties refer to the stabiity of the source to response LI system associated at the ports where either an independent votage or a current sources are attached and the network responses are the corresponding current or votage refected at the source ports respectivey. Hence the probem of stabiization can be naturay defined as achieving stabe responses at ports by making port interconnections with another active network. Such a stabiization probem has somehow never seems to have been addressed in the iterature as far as known to authors. (Athough such a probem appears to have made its beginning in [4, chapter 11]). Modern agebraic feedback theory, whose comprehensive foundations can be referred from [6], came up with the soution of the stabiization probem in the setting of feedback systems. However the formaism of feedback signa fow graph is not very convenient for describing port interconnections of networks. Athough Bode [3] considered feedback signa fow graph to describe ampifier design the methodoogy did not easiy carry further for muti stage and muti oop ampifier design. For singe ports defining the oop in terms of port function is reativey simpe as shown in [4, chapter 11]. Hence if stabiity of muti port networks is to provide a basis for stabiization probem of such networks, the stabiization probem must be formuated directy from port interconnection rues rather than signa fow graph rues. his forms the centra motivation of the probem proposed and soved in this paper. In recent times behaviora system theory [5] considered probems of synthesis of contro systems in which feedback contro is subsumed in genera interconnection between systems by defining inear reations between variabes interconnected. Hence such interconnections are sti equivaent to signa fow graph connections. he port interconnection in networks is however of specia kind than just mathematica interconnection since connections between ports are defined between physica quantities such as votages and currents and have to foow either series or parae connection (Kirchhoff s) aws. Hence network connections at ports are instances of physicay defined contro rather than signa fow defined contro which separates physica system from the ogic of contro. Contro arising out of physica reationships between systems is aso reevant in severa other areas such as Quantum contro systems, Bioogica contro systems and Economic poicy studies. Hence this theory of circuit interconnections shoud in principe be reevant to such other types of interconnected systems as we. B. Background on systems theory, networks and coprime representation We sha foow notations and mathematica background of LI circuits from [4]. A driving point function of a LI network is the ratio of Lapace transform of source and response physica signas which wi aways be currents or votages at ports and which wi be a rationa function of a compex variabe s with rea coefficients. In singe port case a network function is aways an impedance or admittance function. In the case of muti port network the driving point function is a matrix whose (i, j)-th entry is the driving point function with source at j-th port and response at i-th port. Hence the entries can aso represent current and votage ratios. We first need to make the precise we known assumption (as justified in [4, chpater 10]) that whenever a network is specified by its driving point function it has no unstabe hidden modes. In the case of muti port networks this assumption can be made precise by assuming specia fractiona representation which depends on the notion of stabe functions as foows. A the detais of this approach are referred from [6]. he BIBO stabe LI systems have transfer functions without any poes in the RHP, the cosed right haf compex pane, (caed stabe proper transfer functions). his set of transfer functions form an agebra

3 3 which is denoted as S. Genera transfer functions of LI systems are represented as = nd 1 where n, d are themseves stabe transfer functions and moreover n, d are coprime. We refer [6], [7] for the theory of stabe coprime fractiona representation. Anaogousy we sha consider network functions aways represented in terms of fractions of stabe coprime functions. his is then an equivaent representation of network functions without hidden modes. For mutiport network functions we sha consider the douby coprime fractiona representation [6] as the hidden mode free representation. One more assumption we make for convenience is that whenever we consider network functions they wi aways be proper functions without poes at infinity. Athough this rues impedences and admittances of pure capacitors and inductors, in practice we can aways consider these devices with eakage resistance and conductance hence their modes are practicay proper. Hence with this reguarization to properness, any fractiona representation nd 1 wi have the d function in S without zeros at infinity. Finay, if H = N r Dr 1 = D 1 N are douby coprime fractions of a network function H then a functions H in a neighbourhood of H wi be defined by douby coprime fractions H 1 1 = Ñr D r = D Ñ where Ñr, D r, D, Ñ are in the neighbourhoods of N r, D r, D, N respectivey. his way we sha draw upon the rich machinery of stabe coprime fractiona theory of [6] for formuating a stabiization theory for muti port interconnection. he organization of this paper is as foows. Section 2 first reviews stabiity in singe port networks and then the stabiization probem in case of singe port networks is defined and soved. In section 3, the stabiity and stabiization probem in case of mutiport networks is defined. he stabiization probem in case of interconnected two port network is then theoreticay soved and the expression for the set of compensating networks stabiizing the interconnected network is derived. In section 4, a practica circuit exampe of two stage operationa ampifier in unity feedback configuration is considered. his exampe is soved using coprime factorization approach to obtain set of compensating networks in terms of a free parameter that stabiizes the given operationa ampifier foowed by concusion in section 5. II. SABILIY AND SABILIZAION IN SINGLE POR NEWORKS We begin with the singe port case. wo types of stabiity properties for a singe port were defined in [4] for LI active networks. Consider a singe port LI network for which the port can be excited with an independent source of a known type. he stabiity property by definition depends on the type of this source. 1) Short circuit stabiity. An independent votage source v s is connected to an active network with driving point impedance Z(s) at the port as shown in the Fig. 1. he network has a interna sources Fig. 1. One port network excited by a votage source zero (or has zero stored energy). Let V s denotes the Lapace transform of this source votage. he current i r at the port in transformed quantity denoted I r = Z(s) 1 V s where I r is the Lapace transform of the response current i r. he network is then said to be short circuit stabe if a bounded v s has bounded response i r. Mathematicay, this is equivaent to the condition, the network is short circuit stabe iff Z(s) 1 has no poes in RHP. 2) Open circuit stabiity. An independent current source i s is connected to an active network with driving point admittance Y (s) at the port as shown in the Fig.2. he network has a interna sources zero (or has zero stored energy). Let I s denotes the Lapace transform of this source votage.

4 4 Fig. 2. One port network excited by a current source he votage v r at the port in transformed quantity denoted V r = Y (s) 1 I s where V r is the Lapace transform of the response votage v r. he network is then said to be open circuit stabe if a bounded i s has bounded response v r. Mathematicay, this is equivaent to the condition, the network is short circuit stabe iff Y (s) 1 has no poes in RHP. A. Stabiization probem in singe port network Next we state stabiization probems in singe port case. For a singe port network, say N, connections at the port are series (or parae) connections of impedance (or admittance). However, if the source at the port is a votage (respectivey current) then a parae (respectivey series) connection of a compensating network has no effect on the current (respectivey votage) in N (respectivey votage across N). In such case, the connected network has no compensating or controing effect on current (respectivey votage) in (or across) N. Hence the connection of a compensating network must be appropriate. his constraint eads to two different notions of stabiization. 1) Short circuit stabiization: For stabiization of votage fed impedance, say Z, it is required to change votage across Z which is possibe ony by a series connection of compensation impedance Z c. Due to this compensation, the controed current in Z is I r = V s /(Z + Z c ). Foowing definition of short circuit stabiity, the stabiization probem envisages impedances Z c such that (Z + Z c ) 1 are stabe. However a stronger requirement is chosen to define the short circuit stabiization as foows. Probem 1 (Short Circuit Stabiization). Given a one port network with impedance function Z fed by a votage source, find a impedance functions Z c such that the impedance of the series connection Z = (Z + Z c ) satisfies i) Z 1 has no poes in RHP. ii) Z 1 is a stabe function where Z = Z + Z c for a Z in a sufficienty sma neighborhood of Z. If above conditions are satisfied by Z c then it is caed short circuit stabiizing compensator of Z. First condition ensures stabiity of the interconnected network. he second condition is important in practice and requires that the compensator ensures stabiity of the interconnection over a sufficienty sma neighbourhood of Z. 2) Open circuit stabiization: For compensation of current fed admittance, say Y, it is required to change current through Y which is possibe ony by a parae connection of compensating admittance Y c. Due to this compensation, the controed votage across the port is V r = I s /(Y + Y c ). Foowing definition of open circuit stabiity, open circuit stabiization probem envisages finding a admittances Y c such that (Y + Y c ) 1 is stabe. However a stronger requirement is chosen to define stabiization as foows. Probem 2 (Open circuit stabiization). Given a one port network with admittance function Y fed by a current source, find a admittance functions Y c such that the admittance of the parae connection Y = (Y + Y c ) satisfies i) Y 1 has no poes in RHP. ii) Ỹ 1 is a stabe function where Ỹ = Ỹ + Y c for a Ỹ in a sufficienty sma neighborhood of Y. If above conditions are satisfied by Y c then it is caed open circuit stabiizing compensator of Y.

5 5 Fig. 3. One port network with compensating admittance connected in parae B. Structure of the stabiizing compensator for singe port open circuit stabiization We first describe the coprime fractiona representation. Consider the agebra S of stabe proper network functions. A genera admittance function of a singe port network, say Y, is considered in the form Y = nd 1 where n, d beong to S, d has no zeros at infinity with the additiona property that they are coprime i.e. have greatest common divisors which are invertibe in S. his is equivaent to the fact that there exist x, y in S such that the foowing identity hods. Consider anaogousy a coprime fractiona representation Y c = n c d 1 c Y c and x c, y c in SS with the foowing identity. With this we have the foowing reationship between Y and Y c. nx + dy = 1 (1) for compensator network function n c x c + d c y c = 1 (2) Lemma 1. If a one port network with an admittance function, say Y, is fed by a current source is connected in parae across an admittance Y c then the combined network is open circuit stabe if and ony if nd c + dn c is a unit of S. ht. he admittance of the parae connection is as shown in the Fig. 3. he parae connection gives the combined admittance as Y = Y + Y c which has fractiona representation as given beow. For open circuit stabiity, Y 1 of Y. Denote then as we as Ỹ 1 Y = nd c + dn c dd c (3) must be in S for a Ỹ in a sufficienty sma neighbourhood = ñd c + dn c Ỹ 1 = dd c It foows from equation (4) that if Ỹ 1 is stabe then a roots of in RHP are canceed by RHP roots of dd c. However over a neighbourhood of n, d the pairs ñ, d are aso coprime and hence do not have a common root in RHP. Since d c and n c are aso copime, the ony roots of in RHP common with d c possibe are those common between d c and d. But as d varies over a neighbourhood of d there can be no common roots with a fixed d c over the whoe neighbourhood. Hence if Ỹ 1 is stabe over a neighbourhood of n, d then there is no possibiity of RHP root canceation between and dd c. Hence if Ỹ 1 is stabe then must not have a root in RHP or it must be a unit of S. his proves necessity. Conversey, if = nd c + dn c is a unit then in a sufficienty sma neighbourhood of n, d a are units in S. Hence Ỹ 1 are stabe functions in a neighbourhood. Hence sufficiency is proved. (4)

6 6 It is worth noting here that for proving interna stabiity of feedback systems, the crucia formuation [6] of the stabiity of the map between two externa inputs to two interna outputs is repaced by requiring stabiity over a neighbourhood of the given function Y. his aso practicay makes sense as given functions are never accuratey same as rea function, just as in contro system a pant mode is never an exact representation of the rea pant. Athough the two input-output formuation can be reconstructed for defining stabiity of the interconnection at the ports, we prefer the above approach of avoiding the feedback oop signa fow graph. In terms of given coprime representations of Y as in above emma we have a more specia coprime fractiona representations for Y c, Coroary 1. If a one port network with admittance function in fractiona representation Y = nd 1, is fed by a current source and an admittance Y c is connected in parae across the port, then Y c stabiizes Y iff there is a fractiona representation Y c = n c d 1 c in S which satisfies nd c + dn c = 1. Proof. If nd c + dn c = 1 hods for some fractiona representation Y c = n c d 1 c then we have = 1 a unit of S. Hence Y c stabiizes Y from above emma. Conversey, et Y c stabiizes Y with coprime fractiona representation Y c = n 1c d 1 1c, then = nd 1c +dn 1c is a unit of S hence Y c = n c d 1 c where n c = n 1c 1, d c = d 1c 1 are aso coprime and satisfy nd c +dn c = 1. he next theorem gives the set of a admittances Y c which form stabe interconnection with a given admittance Y when connected in parae and fed by a current source. heorem 1. If Y = nd 1 is a coprime fractiona representation of an admittance Y with the identity nx + dy = 1 hen the set of a admittance functions Y c which form open circuit stabiizing parae compensation with Y are given by the fractiona representation Y c = (y + qn)(x qd) 1 where q is an arbitrary eement of S such that x qd has no zero at infinity. Proof. Suppose Y c = (y + qn)(x qd) 1 for some q in S. hen, = n(x qd) + d(y + qn) = 1 (5) his representation of Y c is a coprime fractiona representation and by coroary (1), it foows that Y c stabiizes Y. Conversey, suppose Y c stabiizes Y then from the above coroary we have a coprime fractiona representation Y c = n c d 1 c, where n c, d c S satisfy the foowing reation. nd c + dn c = 1 (6) Hence a soutions of n c, d c in S of this identity with d c without a zero at infinity characterize coprime fractions of Y c. Such soutions are we known (see [6], [7] for proofs) and are given by the formuas as beow. n c = (y + qn), d c = (x qd) (7) his proves the formua caimed for a Y c which form an open circuit stabe combination. Note that the admittance Y and that of a stabiizing compensator Y c can not share a common poe or zero in RHP (caed a non-minimum phase (NMP) poe or zero). Even their coser proximity in RHP woud mean that the interconnected circuit has poor stabiity margin. Many such interpretations can be gathered from this agebraic characterization of stabe interconnection of one port admittances which have been impicit part of knowedge of circuit designers or are new additions to this fied.

7 7 C. Structure of the stabiizing compensator for singe port short circuit stabiization Anaogous to the singe port open circuit stabiization case, the formua for stabiizing Z c in case of short circuit stabiization probem can be stated and derived in simiar manner. We sha thus state ony the fina theorem on the structure of the stabiizing impedance Z c for this case. In the present situation of short circuit stabiization, we have an impedance Z fed by a votage source V s the current is then I r = V s /Z. he current can be controed ony when we add another impedance Z c in series which changes the current to V s /(Z + Z c ). Hence Z c is a short circuit stabiizing impedance iff (Z + Z c ) 1 is in S. he structure of such a compensator is then given by the foowing. heorem 2. If Z = nd 1 is a coprime fractiona representation of an impedance Z with the identity nx + dy = 1 hen the set of a impedance functions Z c which form short circuit stabiizing series compensation with Z are given by the fractiona representation. Z c = (y + qn)(x qd) 1 where q is an arbitrary eement of S such that x qd has no zero at infinity. Proof. Proof readiy foows from the open circuit stabiization case above by repacing Y, Y c by Z, Z c respectivey aong with their fractiona representations and noting that in the present case stabiity of the interconnected network is equivaent to the fact that (Z + Z c ) 1 is in SS. III. MULI POR SABILIZAION FOR BOUNDED SOURCE BOUNDED RESPONSE (BSBR) SABILIY In case of muti port circuits, it is required to consider both, the open and short circuit stabiity, simutaneousy due to existence of independent votage and current sources at the ports simutaneousy. We first define the stabiity in the muti-port case. Consider a inear time invariant circuit represented by the foowing equation. Y r = U s (8) where U s denotes the vector of Lapace transforms of the independent sources at the ports and Y r denotes the vector of Lapace transforms of the responses at the ports (respecting indices). represents matrix of hybrid network functions between eements of Y r and U s respectivey. We assume that is a proper rationa matrix (with each of its eements having degree of numerator poynomia ess than or at the most equa to the degree of the denominator poynomia) and has a forma inverse as a proper rationa matrix. We ca such a circuit is Bounded Source Bounded Response (BSBR) stabe if for zero initia conditions of the network s capacitors and inductors, uniformy bounded sources have uniformy bounded responses. his is the case iff the hybrid network function (matrix) is stabe i.e. has every entry beonging to S. Let M(S) represent set of matrices of respective sizes whose eements beong to S. Consider a compensation network of same number and type of independent sources as the given network of (8) to be compensated at a its port indices. In other words, we want to connect two ports of same index between the two networks ony when both ports are either votage fed or current fed. We can then connect the ports of the two networks at the index in either series or in parae as shown in the Fig. 4, as parae (respectivey series) connection of ports has no effect on the current (respectivey votage) in the individua circuits for the same votage (respectivey current) source. In other words a compensating network wi have no effect on the response of a given network if connected in parae (respectivey series) at a votage (respectivey current) source. Hence we consider the interconnection of ports in series (respectivey parae) when the common source at the port is a votage (respectivey current) source. Let a compensating network has the hybrid function matrix c. (As in case of, we assume c to be proper rationa with proper rationa forma inverse.) hen for the source vector U cs the response vector Y cr in the compensating network is given by the foowing equation. Y cr = c U cs (9) Now if the two networks are connected as shown in the Fig. 4, the independent source vector Ûs appied to the interconnection distributes in the two networks as given by the foowing equation. Û s = U s + U cs (10)

8 8 (a) Parae connection for compensation when the source is current at ith port (b) Series connection for compensation when the source is votage at ith port Fig. 4. Diagram of two possibe connections at a port whie the common response vector Ŷr of the two networks at the ports is given by the foowing equation. Ŷ r = U s = c U cs (11) Hence the source vectors refected on ports of each network are given by the foowing equations. U s = 1 Ŷ r U cs = c 1 Ŷ r (12) herefore, for the combined network we have the source response reationship given by the foowing equation. Ŷ r = ( 1 + c 1 ) 1 Û s (13) which is the hybrid representation of the interconnected network. It is thus cear that the interconnected network is BSBR stabe iff the hybrid matrix of interconnection ( 1 + c 1 ) 1 is in M(S). As in the singe port case, we formay define the stabiization probem with additiona restriction that the hybrid matrices of the interconnection arising from a in a neighborhood of are aso stabe. Probem 3 (Muti-port Hybrid Stabiization). Given a muti-port hybrid matrix function of an LI network, find a hybrid network function matrices c of the compensating network connected as in figure 4 such that 1) ˆ = ( 1 + c 1 ) 1 is in M(S). 2) ( ˆ = 1 + c 1 ) 1 is in M(S) for a in a neighbourhood of. he matrix functions c sha be caed stabiizing hybrid compensators of. A. Douby coprime fractiona representation For a comprehensive formuation of the muti-port stabiization we resort to the matrix case of coprime factorization theory over the S deveoped in [6]. his is the douby coprime fractiona representation of proper rationa functions over matrices M(S). For the proper rationa network function the right coprime representation is = N r Dr 1 where N r, D r are matrices in M(S), D r is square, has no zeros at infinity and for which there exist X, Y in M(S) satisfying the foowing identity. X N r + Y D r = I (14) Anaogousy, the eft coprime representation is = D 1 N where D, N are matrices in M(S), D is square, has no zeros at infinity and for which there exist X r, Y r in M(S) satisfying the foowing identity. N X r + D Y r = I (15)

9 9 he douby coprime representation of is then given as 1) is expressed by right and eft fractions = N r Dr 1 = D 1 N where N r, D r, N, D are matrices over M(S), D r, D are square and have no zeros at infinity, 2) here exist matrices X, Y and X r, Y r in M(S) which satisfy the foowing equation. [ ] [ ] [ ] X Y Nr Y r I 0 = (16) D N D r X r 0 I We describe the douby coprime fractiona representation of a compensating network with hybrid network function c by the respective matrices of fractions and identities by N cr, D cr, N c, D c and X cr, Y cr, X c, Y c. It is aso usefu to reca that a square matrix U in M(S) is caed unimoduar if U 1 aso beongs to M(S). his is true iff det U is a unit or an invertibe eement of S. Next, an open neighbourhood of is aso specified in terms of the douby coprime fractiona representation of. Any in a neighbourhood of is specified by a douby coprime fractiona representation D 1 with fractions 1 = Ñr D r = Ñ and matrices X, Ỹ and X r, Ỹr in M(S) satisfying the identities as given in equation (16) in which the fractions Ñr, D r, D, Ñ are in respective neighbourhoods of the fractions of. In terms of the douby coprime fractiona (DCF) representation and the notion of neighbourhoods we have the preiminary. heorem 3. Consider the hybrid port interconnection as in the Fig. 4 of a given network with a compensating network c. hen the interconnection is BSBR stabe (or c stabiizes ) iff for a given douby coprime fractions as above of there exist a douby coprime fractions of c that satisfy the foowing equation. [ ] [ ] [ ] Dc N c Nr N cr I 0 = (17) D N D r D cr 0 I Proof. he expression for the compensated network ˆ in eft (respectivey right) coprime fractions of (respectivey c ) can be obtained as shown beow. where ˆ = ( 1 + c 1 ) 1 = [(N r Dr 1 ) 1 + (D 1 c N c ) 1 ] 1 = N r ( r ) 1 N c (18) r = N c D r + D c N r (19) Simiary, we can obtain the foowing equation by using eft coprime fractions for (D 1 N ) and right coprime fractions for c (N cr D 1 cr ). ˆ = N cr ( ) 1 N (20) where = N D cr + D N cr (21) Anaogous expression hods for the compensated network function ˆ in terms of Ñ r, Dr for a in a neighbourhood of as given beow. ˆ = Ñr( r ) 1 N c (22) where r = N c Dr + D c Ñ r (23) If c stabiizes then ˆ is in M(S) for a in a neighbourhood of. If r has any RHP zeros when varies in a neighbourhood of then the poes of ˆ in RHP can appear ony from such zeros. Since such zeros aso vary continuousy with parameters of in an open neighbourhood and N c has

10 10 constant parameters, it foows that ˆ is in M(S) iff the matrix Ñ r ( r ) 1 beongs to M(S) for a in a neighbourhood of. Let z be a zero in RHP of r then there is a vector ṽ over S (aso varying with parameters) such that Ñ r ( z)ṽ( z) = ( r )( z)ṽ( z) = 0 (24) which is equivaent to both Ñr and r having a common RHP zero at z for a in a neighbourhood of. But then this impies the foowing reation for a. ( r )( z)ṽ( z) = N c ( z) D r ( z)ṽ( z) = 0 (25) Since N c has constant parameters, its zeros are stationary for variations of. Hence the equation (25) simpifies to the foowing equation. D r ( z)ṽ( z) = 0 (26) for a in a neighbourhood of. However equation (26) aong with equation (24) mean that Ñr, D r are not right coprime in any neighbourhood of. Since the coprime fractions remain coprime in an open neighbourhood of, this is a contradiction. his proves that r has no zeros in RHP or that r is unimoduar in a sufficienty sma neighbourhood of. Using identica arguments it foows that is aso unimoduar. In partucuar it foows that and r are unimoduar. Now if a stabiizing c is represented by right and eft coprime fractions N cr, D cr and D c, N c then the new fractions, right fractions, D cr 1 and eft fractions 1 r D c, 1 r N c satisfy the reations (17). his proves the necessity part of the caim. Now, et the reations (17) be satisfied between the DCFs of and c. hen the interconnection network function is ˆ = N c N r = N N cr (27) N cr 1 since = r = I. Hence ˆ is BSBR stabe. On the other hand for a sufficienty sma open neighbourhood of the perturbed fractions Ñr, D r, D, Ñ perturb r and from identity but they sti remain unimoduar. Hence the interconnection function ˆ = Ñr( r ) 1 N c = Ñc( ) 1 N r has no poes in RHP hence is BSBR stabe for a perturbations in a sufficienty sma neighbourhood. his shows that c stabiizes. his proves sufficiency. Remark 1. Entire proof above can aso be written starting from the eft coprime fractions for and the expression (20) for the interconnection function. At the same time above theorem can aso be expressed starting with DCF of 1 and estabishing the structure of c 1 which are just another hybrid port matrix functions of these networks. he structure of stabiizing compensators c now foows from the equation (17) in terms of the DCF of as foows. Coroary 2. Given a DCF (16) of the set of a stabiizing compensators c are given by any of the foowing aternative formuae. c = (X QD ) 1 (Y + QN ) c = (Y r + N r Q)(X r D r Q) 1 (28) for a Q in M(S) such that functions det(x QD ) and det(x r D r Q) have no zero at infinity. Proof. he stated formuas are a soutions of the identity (17) which shows the reationship between and a stabiizing c. he conditions on zeroes of denominator fraction matrices is to ensure that these matrices are proper when inverted.

11 11 IV. MULIPOR NEWORK SABILIZAION EXAMPLE We now show an exampe of stabiization of a practica circuit of two stage operationa ampifier in unity feedback configuration. he equivaent circuit for this two stage op amp without a compensating network is shown in the Fig.5(a). It is required to find a compensating network c such that the interconnection is stabe. he compensating network can be connected across the two ampifier stages and we can consider a port with a pair of terminas formed due to its connection. hus the equivaent circuit can be redrawn as shown in the Fig.5(b). (a) Sma signa equivaent circuit of two stage opamp (b) Sma signa equivaent circuit of two stage opamp as a two port network Fig. 5. wo stage opamp with sma signa equivaent circuit [ IM1 Let Y r = (s) ] [ VaM1 be vector of Lapace transforms of the responses and U V M3 (s) s = (s) ] be vector of I M3 (s) Lapace transforms of the independent sources.hus we have the foowing matrix equation between the excitation and response signas. [ IM1 (s) ] [ ] [ ] 11 = 12 VaM 1 (s) (29) V M3 (s) I M3 (s) his is equivaent to Y r = U s where eements of matrix can be computed using the parameter vaues associated with the equivaent circuit. he circuit can be simpified and soved using Kirchhoff s aws so that the eements of matrix are given as, [ 11 = sc x 1 + g m 1 (sc gd g m2 ) ] (30) D 1 [ sc ][ x 12 = 1 + N 1(sC gd g m2 ) ] (31) sc 2 + g m2 + g 2 D 1 where 21 = g m 1 [sc 2 + g m2 + g 2 ] D 1 (32) 22 = s(c 1 + C 2 ) + (g m2 g m1 + g 2 + g 1 ) D 1 (33) D 1 = [(C 1 + C 2 )C gd + C 1 C 2 ]s 2 + [C 2 g 1 + C 1 g 2 + (g m2 g m1 + g 1 + g 2 )C gd ]s + [g m1 g m2 + g 1 g 2 ] N 1 = s(c 1 + C 2 ) + (g m2 g m1 + g 2 + g 1 ) But this gives some of the eements of the transfer function matrix (such as 11 ) as improper with degree of numerator poynomia greater than the degree of denominator poynomia. his poses difficuty in matrix inversion.

12 12 his computationa difficuty can be resoved by adopting the reguarization procedure which incude adding the resistors either in series (or in parae) of appropriate vaues at the ports so that none of the eements of the transfer function matrix are improper. he modified equivaent circuit after reguarization is as shown in the Fig.6. Fig. 6. Modified Equivaent circuit of two stage opamp Simpifying and soving the modified equivaent circuit, the eements of matrix are as given beow. ( sc )[ x 11 = 1 + g m 1 (sc gd g m2 ) ] (34) sc x r D 1 [ 12 = N (g m2 sc gd )r 2 + N 2(sC gd g m2 ) ] (35) D 1 where 21 = g m 1 [sc 2 + g m2 + g 2 ] D 1 (36) 22 = N 2 D 1 D 1 = [(C 1 + C 2 )C gd + C 1 C 2 ]s 2 + [C 2 g 1 + C 1 g 2 + (g m2 g m1 + g 1 + g 2 )C gd ]s + [g m1 g m2 + g 1 g 2 ] N 2 = [1 + (s(c 1 + C gd ) + g 1 )r 2 ][sc 2 + g m2 + g 2 ] [sc 1 + g 1 g m1 ][ 1 + (g m2 sc gd )r 2 ] sc x N 1 = (sc x r 1 + 1)(sC 2 + g m2 + g 2 ) he vaues of various parameters are as given beow. g m1 = A/V g m2 = A/V g 1 = 1 1 R 1 = = A/V g 2 = 1 1 R 2 = = A/V C 1 = F C 2 = F C gd = F Using r = r = 0.1 Ω, matrix can be reguarized such that the D matrix in its state space mode exists which is non-singuar and thus the matrix wi have a proper eements. he matrix can now be inverted. he eements of matrix are as given beow. 10s(s )(s ) 11 = (s )(s s ) s(s ) 12 = (s )(s s ) (37)

13 13 21 = (s ) s s = 0.1(s )(s ) s s he right coprime factorization of gives matrices D r and N r respectivey. he eements of matrix D r are as given beow. he eements of matrix N r are as given beow. D r11 = (s )(s )(s ) (s )(s )(s ) D r12 = (s )(s ) (s )(s )(s ) D r21 = (s )(s ) (s )(s )(s ) D r22 = (s )(s ) (s )(s ) N r11 = 10(s )(s )(s ) (s )(s )(s ) N r12 = (s )(s ) (s )(s )(s ) N r21 = (s )(s ) (s )(s )(s ) N r22 = 0.1(s )(s ) (s )(s ) By soving the Bezout s identity X N r + Y D r = I, we get X and Y respectivey. he eements of matrix X are as given beow. he eements of matrix Y are as given beow. X 11 = (s )(s ) (s )(s )(s ) X 12 = (s ) (s )(s ) X 21 = (s )(s ) (s )(s )(s ) X 22 = (s ) (s )(s ) Y 11 = (s )(s s ) (s )(s )(s ) Y 12 = )(s ) (s )(s Y 21 = (s s ) (s )(s )(s Y 22 = (s )(s ) (s )(s )

14 14 For Q = 0 the stabiizing compensator c is given as X 1 Y. Using X and Y as computed above we get the eements of c as given beow. Now et us find ˆ which is ( c where c11 = (s s ) s c12 = (s )(s ) s c21 = (s )(s ) s c22 = (s )(s ) s ) 1. he eements of ˆ are as given beow. [ ] ˆ11 ˆ11 = ˆ 11 ˆ11 ˆN 11 = 10(s )(s )(s s )(s ) ˆD 11 = ˆD 22 = (s )(s )(s )(s )(s ) ˆN 11 ˆD 11 ˆN 11 ˆD 11 ˆN11 ˆD 11 ˆN11 ˆD 11 ˆN 12 = (s )(s )(s ) ˆD 12 = (s )(s )(s )(s ) ˆN 21 = (s )(s s ) ˆD 21 = (s )(s )(s )(s )(s )(s ) ˆN 22 = 0.1(s )(s )(s )(s )(s ) It can be seen that the eements of ˆ beong to M(S) and the compensating network c stabiizes the given network. V. CONCLUSION We have deveoped a theory for compensation of a inear active network at its ports by another inear active network such that the interconnection is stabe in the BSBR sense at these ports even when the parameters of the origina network are not exact but can be anywhere in a sufficienty sma neighbourhood. Our theory can be seen as an extension of the agebraic theory of feedback stabiization which has been we known [6] in contro theory. Whie in the feedback stabiization theory the given inear system and the controer form a feedback oop, in the case of networks connected at ports, such a oop is not readiy avaiabe. However the stabe coprime fractiona approach originay deveoped for feedback stabiization carries over to sove the probem. heory of active network synthesis cannot be deveoped without the stabiization theory and a ack of suitabe approach for synthesis of port compensation with stabiity has been possiby the main hurde. he resuting stabe interconnection is described by an affine parametrization in which the free parameter is itsef a stabe network function. his parametrization is anaogous to the we known parametrization in feedback systems theory and hence has opened doors to approach active network synthesis using anaytica methods such as H-infinity optimization.

15 15 REFERENCES [1] Gobind Daryanani, Principes of active network synthesis and design, John Wiey and Sons, [2] Anderson B. D. O., S. Vongpaniterd. Network anaysis and synthesis. A modern systems theory approach, Dover Pubications Inc., NY, [3] Bode H. W. Network anaysis and feedback ampifier design, D. Van Nostrand, Princeton, NJ, [4] Chua L. O., C.A. Desoer, E. S. Kuh. Linear and noninear circuits, McGraw-Hi Book Company, NY, [5] Poderman J.W.,Wiems, J.C. Introduction to Mathematica Systems heory A Behaviora Approach, Springer-Verag, New York, 1998 [6] Vidyasagar M. Contro system synthesis. A factorization approach, Research Studies Press, NY, [7] Doye J. C., B. A. Francis, A. R. annenbaum Feedback Contro heory, Macmian Pubishing Company, NY, [8] Zhou K. J., Doye J. C., K. Gover Robust and Optima Contro, Printice Ha, [9] S.S. Haykin, Active network theory, Addison-Wesey Pubishing Co., 1970.