The paper examines the problem of systems redesign within the context of passive electrical networks by considering the problem of multi-parameter and topology changes, and their representation. This representation may be used to investigate the impact of such changes on properties such as characteristic frequencies. The general problem area is the modelling of systems, whose structure is not fixed but evolves during the system life-cycle. The specific problem we are addressing is the study of effect of changing the topology of an electrical network that is changing individual elements of the network into elements of different type and value, augmenting / or eliminating parts of the network and developing a framework that allows the study of the effect of such transformations on the natural frequencies. This problem is a special case of the more general network redesign problem. We use the Impedance-Admittance models and we establish a representation of the different types of transformations on such models. The representation of the structural transformations is given in terms of the companion pencil that preserves the natural topologies of the RLC network

Karcanias, N.

Leventides, J.

Livada, M.

City Research Online

MATRIX PENCIL REPRESENTATION OF STRUCTURAL TRANSFORMATIONS OFPASSIVE ELECTRICAL NETWORKSNicos Karcanias, John Leventides and Maria LivadaSystems and Control Research Centre, School of Engineering and Mathematical Sciences,City University London, Northampton Square, London EC1V 0HB,N.Karcanias@city.ac.ukABSTRACTThe paper examines the problem of systems redesign withinthe context of passive electrical networks by considering theproblem of multi-parameter and topology changes, and theirrepresentation. This representation may be used to investigatethe impact of such changes on properties such as characteris-tic frequencies. The general problem area is the modellingof systems, whose structure is not fixed but evolves duringthe system life-cycle. The specific problem we are addressingis the study of effect of changing the topology of an electri-cal network that is changing individual elements of the net-work into elements of different type and value, augmenting /or eliminating parts of the network and developing a frame-work that allows the study of the effect of such transforma-tions on the natural frequencies. This problem is a specialcase of the more general network redesign problem. We usethe Impedance-Admittance models and we establish a rep-resentation of the different types of transformations on suchmodels. The representation of the structural transformationsis given in terms of the companion pencil that preserves thenatural topologies of the RLC network.Index Terms— Systems Theory, Networks Theory1. INTRODUCTIONThe problem of redesigning autonomous (no inputs or out-puts) passive electric networks [1], [2] aims to change thenetwork (natural frequencies) by modification of the typesof elements, possibly their values, interconnection topologyand possibly addition, or elimination of parts of the network.As such, this is a problem that differs considerably from astandard control problem, since it involves changing the sys-tem itself without control and aims to achieve the desirablesystem properties, as these may be expressed by the naturalfrequencies by system re-engineering. In fact, this probleminvolves the selection of alternative values for dynamic ele-ments (inductances, capacitances) and non-dynamic elements(resistances) within a fixed interconnection topology and/oralteration of the network interconnection topology and possi-ble evolution of the (increase of elements, branches). The aimof the paper is to define an appropriate representation frame-work that allows the deployment of control theoretic toolsfor the re-engineering of properties of a given network whenthere are multi-parameter variations within a fixed, or variablecardinality network topology. We use impedance and admit-tance modelling [2], [3] for passive electrical networks andconsider here systems with no sources (autonomous descrip-tions), since our current interest is on the shaping of naturalfrequencies. The emphasis here is on the study of the differentrepresentations of the passive network that enable the inves-tigation of the transformations on such models as structuraltransformations. The problem considered here is:• Define the representation of changes of a many dy-namic, or non-dynamic elements with preservation, oralteration of existing topologies without changes inthe overall nodal or loop cardinality of the networkand define a framework for studying natural frequencyassignment.The overall aim is to explore the structure and representationsof the Impedance-Admittance modelW (s) and introduce ap-propriate representation of the above transformations whichenable the study of the shaping of natural frequencies. Matrixrepresentations of the above transformations are introducedas additions of structural transformations on theW (s)model.For RL (resistor-inductor) or RC (resistor-capacitor) networksthe corresponding impedance or admittance models becomematrix pencils and it has been shown that the single param-eter variation problem is equivalent to Root Locus problems[4]. The general case of RLC networks is considered andwe introduce the notion of companion pencil, sF + G, thathas the same non-zero structure withW (s) and preserves thenatural topological properties of the network. We establishthe representation of cardinality preserving transformationsas additive transformations on sF +G [5] and show that nat-ural frequency assignment may be studied within the exterioralgebra framework of the Determinantal Assignment Problem[6], [7], [8].2. PASSIVE NETWORKMODELS ANDTOPOLOGIES2.1. Impedance and Admittance ModelsIn the network loop analysis method, the variables are se-lected such that the vertex law is automatically satisfied.Here, we consider only planar graphs with b branches andn vertices. We then consider the variables associated witheach of the meshes and we define them as loop variables. Theoverall system is reduced to a number of meshes, which are[q = (b  n + 1) [3],[9], referred to as loop cardinality ofthe network. If we denote by (f1, f2, ..., fq) the set of theLaplace transforms of the loop currents and by (us1, ..., usq)the set of Laplace transforms of equivalent voltage sources,then the loop or impedance model is defined by [10]:Z(s)f(s) = us(s) (1)where Z(s) has elements zii(s) expressing the sum ofimpedances in loop i and zij(s) is the sum of impedancescommon between loops i and j. This is known as the loop orimpedance model and it is an integral-differential symmetricmatrix and Z(s) is the network impedance matrix.Alternative modeling is the method using the across vari-ables from each vertex to some reference vertex. The numberof vertex equations is in general p = (n   1) and will be re-ferred to as nodal cardinality of the network. If we denote by(u1, u2, ..., un) the Laplace transforms of the node voltagesand by (is1, ..., isn) the set of Laplace transforms of equiv-alent current sources, then the node or admittance model isdefined by [2]:Y (s)u(s) = is(s) (2)where: yii(s) is the sum of admittances in node i; yij(s) isthe sum of admittances common between nodes i and j. Thisis referred to as the node or admittance model and it is anintegral-differential symmetric matrix and Y (s) is referred toas the network admittance matrix.2.2. The Autonomous Natural Impedance-AdmittanceModel and TopologiesWhen we consider networks with no inputs (no current, orvoltage sources) the resulting admittance, or impedance net-work models may be described in a unifying way as:{pB + p 1C +D}x(t) = 0 (3)where p, p   1 are respectively the differential, integral op-erators respectively and x(t) is the vector of nodal voltages,or loop currents. Such a description may be referred to asthe natural autonomous network description and the operatorW (s) = sB + s 1C +D will be called the natural networkoperator. Note that for the case of admittance we have thatB is a matrix of A-type elements (i.e. mass, inertance, ca-pacitance), C is the matrix of T-type elements (i.e. spring,inductance) and D is a matrix of D type elements (i.e. re-sistance). For the case of impedance the reverse holds true.Hence, B is the matrix of T-type elements, C is the matrixof A-type elements and D is the matrix of D-type elements.The symmetric operator W (s) is thus a common descriptionof Y (s) and Z(s) matrices. The operatorW (s) describes thedynamics of the network and of special interest are the prop-erties of its zeroes. Network modeling uses the system graph,which is the basic topological structure that generates the sys-tem equations. We may introduce some additional topologies,which are linked to the specifics of the Node and Loop analy-sis. The detailed topological structures that emerge depend onthe nature of the elements in the network. The mass, inertiaand capacitance store energy by virtue of their across- vari-ables (velocity, voltage) and they are referred to as A-typeenergy storage units [2]. Springs and inductances store en-ergy by virtue of their through- variables and are called T-typeenergy- storage devices. The dampers and resistances dissi-pate energy and will be called D-type elements.2.2.1. The Vertex TopologyEvery network may be represented in terms of a set of ver-tices, or nodes and all branches between two vertices may berepresented by an admittance function. The nature of the el-ements in the branches of the natural vertex graph defines anelement dependent topology, which is characterized by adja-cency type matrices. If we set the external sources to zero, thereduced graph will be referred to as the kernel vertex graph.For a given kernel vertex graph we define A-vertex sub-graphby eliminating from the kernel vertex graph all T- and D-typeedges. Similarly, we define the T-vertex sub-graph by elim-inating all A- and D-type edges and the D-vertex sub-graphby eliminating all A- and T-type edges. The sub-graph ofthe natural vertex graph obtained by eliminating all T-, D-,A- type elements represents the location of the through vari-able sources and will be called the source-vertex sub-graph,or simply S-vertex sub-graph.2.2.2. The loop topologyThe loop topology is a notion dual to that of the vertex topol-ogy and it is based on the following principle: Every networkof n vertices and b edges (branches) may be represented byq = (b  n+ 1) loops leading to independent equations. Allbranches common between two loops may be represented byan impedance function. Specification of the values of throughvariables for the loops defines the values of all across vari-ables in the network. In a similar way to the case of nodalanalysis, we may define the loop topology based on the ker-nel loop graph and its sub-graphs the A-loop sub-graph, theT-loop sub-graph, the D-loop sub-graph and the source-loopsub-graph [2].3. THE LINEARISATION OF THE AUTONOMOUSNATURAL IMPEDANCE-ADMITTANCE MODELStarting from the integral-differential model of (3), describedby the operator W (s) the natural question that arises is howwe can transform it to an equivalent first-order, matrix pencildescription, which preserves the topology of the network. Weintroduce a new set of variables, bx = [x, x˜]t, p 1x = exwhichreduces (3) to a first order description given by equation (5)which has an associated matrix pencil sF +G defined by (6)and referred to as the network matrix pencil which is defined:B 00 I  pxpx˜ =  D  CI 0  xx˜ (4)sF +G =sB+D C I sI (5)Note that the above autonomous differential description pre-serves the topological properties of the network as these arerepresented by the B, C, D matrices, but its dimensionalityis not necessarily minimal (dimensionality of sF + G). Thepencil derived is structured, but not symmetric in the generalcase and it will be referred to as the companion pencil of thenetwork. The zeroes of W (s) define the natural frequenciesof the network. Note that:     sB+D C I sI      = sk   sB + s 1C +D  or|sF +G| = sk. |W (s)| ,W (s) 2 k⇥k(s) (6)Remark (1): The non-zero natural frequencies of the networkare given by the zeroes of the pencil and thus this pencil maybe used for the study of assignment of natural frequenciesunder different types of transformations. For the special caseswhere the network is characterized only by A- and D- typeelements or T- and D- type elementsfW (s) = sB +D,cW (s) = bsC +D,bs = s 1 (7)which are symmetric matrix pencils [5]. These pencils arederived from passive networks and thus inherit the passivityproperties [4], [2].4. NETWORK TRANSFORMATIONSThe general modeling for passive electrical networks providesa description of networks in terms of symmetric, integral, dif-ferential operator, W (s) = sB + s 1C + D. It is clearthat the network may be represented by the triple of matri-ces structural transformations {C,B,D}. The study of thestructural changes on the network may be expressed as trans-formations on the matrices {C,B,D}. The general classesof structural transformations which may preserve, or alter thecardinality of the network, and may also change its differenttypes of topology are defined below.4.1. Classification of Structural TransformationsType 1: Changing the values of the components of the sys-tem without changing the topology as this is described by{C,B,D} tipple.Type 2: Altering the nature of components by transformationson tipple without changing the element cardinality of the net-work.Type 3: Modifying the networks topology and changing thecardinality of elements by removing components / subsys-tems.Type 4: Augmenting the networks topology and changing thecardinality of elements of the system by adding subsystemsto the existing topology of the network. In the following wefocus on Cases 1, 2 preserving the loop, or nodal cardinalityand thus the dimensionality of {C,B,D}. These transforma-tions are then expressed as:Definition 1: Given the triple of matrices {C,B,D} we con-sider transformations on the network matrices of the typeC 0 = C ± c(x, b), B0 = B ± l(x, b), D0 = D ± r(x, b) (8)which preserve the physical elements cardinality (loop, ornodal cardinality) and depend on the real parameter x 2 <and the position vector b 2 <k. In fact, consider the changesc(x, b), l(x, b), r(x, b) which have the general form f(x, b)[4] where:f(x, b) = xbbT for b = ei , or b = ei   ej , i 6= j (9)4.2. ExamplesConsider the electrical network of Figure (1). The networkvariables are the loop currents I1, I2, I3. The impedancemodel expresses the impedances in the three loops and thushas the form of (9). We now assume that in this network wechange the corresponding topology by adding the elementsL4, R5, C3 as shown in Figure (2).Figure 1: Initial RLC networkSpecifically the transformations are:• Add a resistor to loop 1• Add an inductance common to loops 1 and 2• Add a capacitor to loop 2Z(s) =0@ C1 1  C1 1 0 C1 1 C1 1 + C2 1  C2 10  C2 1 C2 11A s 1+0@ R1  R1 0 R1 R1 +R2 +R3  R30  R3 R3 +R41A+0@ 0 0 00 L2 00 0 L31A s == s 1C +D + sB(10)Using the formulation (11) the above transformations can beexpressed formally with modification to the correspondingmatrices as shown below:Figure 2: Augmented RLC network(i) For the A-type elements:C 0 = C +1C3b3bt3 ,where b2 = e2 =⇥0 1 0⇤t (11)The above expresses the addition of capacitor C3 to loop 2.Hence, we have:C 0 =24 C1 1  C1 1 0 C1 1 C1 1 + C2 1 00 0 C3 135++C3 124 0 0 00 1 00 0 035 ==24 C1 1  C1 1 0 C1 1 C1 1 + C2 1 + C3 1 00 0 C3 135 (12)(ii) For the D-type elements:D0 = D +R5b1bt1 ,where b1 = e1 =⇥1 0 0⇤t (13)The above expresses the addition of resistor R3 to loop 1.Hence, we have:D0 =24 R1  R1 0 R1 R1 +R2 +R3  R30  R3 R3 +R435++24 R5 0 00 0 00 0 035 ==24 R1 +R5  R1 0 R1 R1 +R2 +R3  R30  R3 R3 +R435 (14)(iii) For the T-type elements:B0 = B + L4b12bt12 ,where b12 = e1   e2 (15)The above expresses the addition of inductance to thebranch common to loops 1 and 2. Hence, we have:B0 = B + L424 1  1 0 1 1 00 0 035 =24 L1 0 00 L2 00 0 L335++24 L4  L4 0 L4 L4 00 0 035 =24 L1 + L4  L4 0 L4 L2 + L4 00 0 L335(16)Summarizing, the transformed network is described by thecorresponding matrices B0, C 0 , D0, which lead to the newimpedance matrixW (s) describes the above transformations,is given by: fW (s) = s 1C 0 + sB0 +D0.Remark (2): The presence of an element of A , T , D type is expressed by an entry in the corresponding matrixC,B, T respectively. In specific:• If an element is present in the ith loop (node), then itsvalue is added in the ith position of the respective ma-trix.• If an element is common to the ith and jth loop then itsvalue is added to the ith and jth loop diagonal entries,as well as subtracted from the (i, j) and (j, i) position ofthe corresponding matrix.Theorem (1): Consider a network described by the triple C,B, D with natural operatorW (s) = sB+s 1C+D and cor-responding companion pencil sF +G. Any network preserv-ing cardinality transformation (combination of Type 1 and 2)may be represented by a triple {C⇤, B⇤, D⇤} and it resultsin a companion pencil sF 0 +G0 defined by:sF 0 +G0 = sF +G+sB⇤+D⇤ C⇤0 0 = (sF +G) + (sH +K) (17)Thus, structural transformations that preserve network cardi-nality are expressed as structured additive perturbations on thecompanion matrix pencil. This allows for the development ofa framework for determinantal assignment of natural frequen-cies discussed next. A version of this problem was recentlyconsidered in [5].5. CONCLUSIONSThe paper has examined the problem of redesign of passiveelectric networks as a problem describing the structure evo-lution of systems linked to changes in the nature of topology,and values of the physical elements. Four different types ofstructural transformations have been defined and for the twowhich preserve the network cardinality it has been shown thatthese transformations are expressed as additive structuredtransformations on the companion pencil. The assignment ofnatural frequencies of the network may then be formulatedas a spectrum assignment of matrix pencils under additivetransformations and may be studied within the framework ofDeterminantal Assignment introduced in [6], [7]. Amongstthe problems under investigation is the study of spectrumassignment under special families of (H,K) transformations,the characterization of the fixed frequencies (if any) andthe derivation of conditions for arbitrary assignment of suchfrequencies.6. REFERENCES[1] N. Karcanias, “Structure evolving systems and controlin integrated design,” Annual reviews in Control, 2008.[2] N. Karcanias, “Passive network redesign: System struc-ture evolution,” Tech. Rep., City University London,2011.[3] S. Seshu and M. Reed, Linear Graphs and Electricalnetworks, Addison- Wesley, 1961.[4] N. Karcanias T. Berger, G. Halikias, “Effects of dy-namic and non-dynamic element changes in rlc net-works,” International Journal of Circuit Theory andApplications, 2014.[5] Leventides J. and Karcanias N., “Zero assignment ofmatrix pencils by additive structured transformations,”Linear Algebra and its Applications, vol. 431, no. 8, pp.1380–1396, 2009.[6] N. Karcanias and C. Giannakopoulos, “Grassmann in-variants, almost zeros and the determinantal zero andpole assignment problem,” Int. J. Control, 1984.[7] J.Leventides and N.Karcanias, “Global asymptotic lin-earisation of the pole placement map: A closed formsolution for the constant output feedback problem,” Au-tomatica, vol. 31, no. 9, pp. 1303–1309, 1995.[8] B. Laios N. Karcanias and C. Giannakopoulos, “Thedecentralised determinantal assignment problem: Fixedand almost modes and zeros,” Int. J. Control, vol. 48,pp. 129–147, 1988.[9] F R. Gantmacher, The Theory of Matrices, vol. I and II,Chelsea, New York, 1959.[10] M. Marcus and H. Minc, A Survey of Matrix Theory andMatrix Inequalities, Allyn and Bacon, 1964.

Matrix pencil representation of structural transformations of passive electrical networks

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