Symbolic Determination of Jacobian and Hessian Matrices and Sensitivities of Active Linear Networks by Using Chan-Mai Signal-Flow Graphs

Every network synthesis procedure normally includes a first-order or (more rarely) secondorder network sensitivity analysis. The main problem here is the evaluation of the corresponding firstor second-order derivatives of network functions with respect to the circuit element values. These derivatives form the network Jacobian (J) and Hessian (H) matrices, respectively. A variety of methods exist for such an evaluation but most of them are intended for the sensitivity of one network transfer function only. Besides this in many cases it is desirable to find the symbolic expressions of the sensitivities because such a presentation facilitates the element value influence determination. An other useful and important application of the matrices J and H is in the tasks for optimization of synthesized networks with respect to their sensitivities or other parameters (Korn & Korn, 1968; Wilde,1978).


Introduction
Every network synthesis procedure normally includes a first-order or (more rarely) secondorder network sensitivity analysis.The main problem here is the evaluation of the corresponding first-or second-order derivatives of network functions with respect to the circuit element values.These derivatives form the network Jacobian (J) and Hessian (H) matrices, respectively.A variety of methods exist for such an evaluation but most of them are intended for the sensitivity of one network transfer function only.Besides this in many cases it is desirable to find the symbolic expressions of the sensitivities because such a presentation facilitates the element value influence determination.An other useful and important application of the matrices J and H is in the tasks for optimization of synthesized networks with respect to their sensitivities or other parameters (Korn & Korn, 1968;Wilde,1978).
As it is well known all linear active networks can be modeled by using passive elements and nullator-norator pairs (nullors).The presented paper deals with the application of Chan-Mai signal-flow graphs (CMG) to the determination of the matrices J and H elements, having in mind the peculiarities of nullors and their influence on the passive element network admittance matrix and on the corresponding CMG.The method developed here is an improved and enlarged version of the approach in (Nenov, 2004).One demonstrates that the method reduces to the obtaining of two (for the elements of J) or four (for the elements of H) isomorphic Chan-Mai signal-flow graphs.

Chan-Mai signal flow graph
It was introduced in graph theory in 1967 (Chan & Mai, 1967).Compared with other kinds of oriented graphs (especially Mason and Coates graphs) the Chan-Mai graph (CMG) holds out a simplest way to the representation the relationships between the dependent and independent quantities in an algebraic equation set.In order to make easier the understanding of the following sections of the paper further we give the procedure for drawing of CMG and the basic formulae related.
is a square matrix with real or complex entries and are the vectors of the dependent and of the independent variables, respectively.The CMG consists of n vertices with sink signals y 1 , y 2 , …, y n , n vertices with source signals x 1 , x 2 , …, x n and maximum n 2 edges with transmission coefficients a ji directed from the vertex x i toward the vertex y j ; i,j = 1, 2, …, n -Fig. 1.The calculations on the base of a CMG are connected with the following definitions (Chan & Mai, 1967, Donevsky & Nenov, 1979): i.By removing all outgoing from the vertex x i edges and by adding the edges with transmission coefficients y j from the vertex x i directed toward the vertices y j , j=1, 2, …, n one obtains the graph CMG,i; ii.A separation (S) contains all vertices of CMG and a part of edges so that every vertex is incident to only one incoming and one only outgoing edge.The product of the transmission coefficients of all edges in a separations represents the corresponding separation product (SP); iii.Two edges with transmission coefficients a ij and a ji form a symmetrical pair.iv.An edge which does not belong to a symmetrical pair is an asymmetrical edge.
f o r 0 ( 1) for 0 or In ( 4) and ( 5) r is the number of the separations in CMG, m is the number of the separations in CMG, i , N a,k is the number of all asymmetrical edges in k-th separation of CMG, N s,k is the number of all symmetrical pairs in k-th separation of CMG, N aq is the number of the asymmetrical edges in q-th separation of CMG,i, N s,q is the number of the symmetrical pairs in q-th separation of CMG,i, whereas SP q (CMG,i) and SP k (CMG) are the separation products of q-th separation of CMG,i and the separation products of k-th separation of CMG, respectively.

Nullor network Chan-Mai signal-flow graph
Suppose that an equivalent nullor network N with m+1 nodes, r passive branches and g nullors is given and the nodal equation of its passive part N p (the part of N which is obtained by removing all nullors) is pp p is the nodal matrix of N p and ,1 ,2 , ,1 ,2 , ... ; ...
are the nodal voltage and the nodal current vectors of N p , respectively.Additionally we assume that between the nodes of all node pairs in N only one element or more than one but parallel connected elements exist.
The equation ( 1) can be represented graphically by using a CMG G p (Chan & Mai, 1967).Further, taking into account the peculiarities of the nullators and the norators (Davies, 1966) the graph G p can be transformed into the graph G of the actual network N according to the following Rule 1: i.When a nullator is connected between the node k in N and the ground node m+1 one removes all vertices going out from the vertex V k of G p ; ii.When a norator is connected between the node k in N and the ground node m+1 one removes all vertices coming into the vertex I k of G p ; iii.When a nullator is connected between the nodes k and l in N one unites the vertices V k and V l in G p ; iv.When a norator is connected between the nodes k and l in N one unites the vertices I k and I l in G p .
The so obtained graph CMG G corresponds to the matrix equation where Y is an (nn) nodal admittance matrix of N, V is the nodal voltage vector of N and I is the nodal current vector of for n=m-g.

Jacobian matrix determination
The matrices in ( 9) have the form: In the common case every element Y ji in ( 7) is an algebraic admittance sum ; , {1,2,..., }; {1,2,..., }, where y s is the admittance of s-th branch of the network N p .
The vectors V and I correspond to the unknown (dependent) variables and to independent variables of N, respectively and consequently Let us suppose that the admittance y s changes its value to Usually the admittance y s takes part in several (but no more then four) elements of ( 7) and then all these elements change their values (Nenov, 2004 Note that the values of the derivatives in ( 16) are 1 or - By substituting Y ' and V ' in ( 9) instead Y and V, respectively, it follows Having in mind that the equation ( 20) yields Then we obtain .sss s dy dy y and the Jacobian matrix (Korn & Korn, 1968).for the change of the admittance y s is   .
and according to (20) ÷ ( 22) The expressions (22) show that in order to find the vector J s it is necessary to follow the Rule 2: i. Find the vector V by using the CMG G; ii.Evaluate the vector V s ; iii.Draw a new CMG G s where the source vertices are the elements of the vector J s and the sink vertices are the elements of the vector V s ; iv.Find the source vertex variables in G s .The network N in Fig. 2 is given, where m=6; r=9; g=2.Here obviously V 2 =V 3 =V 23 ; V 6 =0 and we wish to find the vector www.intechopen.comAccording to (4) and ( 5) for the separations of the graph G (Fig. 6) we obtain and for the unique separation of the graph G 23 (Fig. 7): 23,1 1 1 4 5 ,23,1 ,23,1 ;2 ; Then the formulae ( 4) and ( 5) yield 14 23 23 4 () Having in mind ( 26) ÷ ( 29) we have Than by substituting ( 35) in (36) and by taking into consideration (33) from ( 4) and ( 5) we obtain the vector J 3 :

Hessian matrix determination
In many practical cases it is necessary and useful to find not only the first-order derivatives of a network function or variable (for example voltage V w ) among n variables with respect to some parameter (for example y s ) but their second-order derivatives with respect to the same or to an other parameter (for example y t ), too.
The matrix formed from all possible second-order derivatives of V w with respect to the simultaneous changes of two parameters is the Hessian matrix or briefly Hessian (Korn & Korn, 1968, Wilde, 1978).Obviously for a network one exists a variety of Hessian matrices -every one matrix corresponds to a definite network function or variable.
The results obtained in section 3. can be applied to the derivation of a Hessian matrix as it will be explained below.By differentiating the vector J s i n ( 2 Because the elements in Y depend linearly on the network element admittances and their derivatives with respect to the parameter y s equal 1, -1 or 0 it holds The last result compared with the formulae ( 27) and ( 28) shows that we can find the vector  2 V/ y s  y t in principle by using the same approach as for  V/ y s in section 3.
However here we must pay attention to the obtaining of the matrix K st : In the common case the matrices K s and K t contain more than one nonzero element (1 or -1).Hence we can expressed each of them as a sum of no more then four addends The products in ( 44) are square matrices with only one nonzero element which is a definite element of Y -1 .Let, for example, the nonzero element for the left-side matrix in ( 44) is on i-th row and on j-th column and the similar element for the right-side matrix is on k-th row and on l-th column.Then it is easy to see that the corresponding product in (44) contains the element z u,v ; u, v Є{1, 2, …, n} on i-th row and on l-th column, where z u,v is an element of Y -1 .The upper (lower) sign of this element holds for equal (non equal) signs of nonzero elements of K s,a and K t,b in (44), respectively.
The matrix Y -1 can be evaluated by using an auxiliary CM graph G 0 too.For this purpose let us consider the equation This means that if the CM graph G 0 corresponds to (45) the multipliers of e 1 , e 2 , …, e n for every element of X are elements of Y -1 .Note that in real cases a limited number of the elements of Y -1 are necessary only.Hence for determination of an element of the Hessian matrix H st we can form the following: Rule 3: i. Draw the CM graph G p of the nullor network under consideration; ii.Transform the graph G p into the graph G, according to the Rule 1 in section 2. and compose the vectors V and I; iii.Determine the vector V from G; iv.Write the matrices K s and K t ; v. Determine the matrix Y -1 by using the auxiliary CM graph G 0 ; vi.Determine the matrix K st ; vii.Determine the matrix V st ; viii.Draw a CM graph G st in accordance with V st ; ix.Determine the elements of the vector  2 V/ y s  y t from G st .Note that by following the above sequence we obtain 2n elements of n Hessian matrices simultaneously, because ∂ 2 V/∂y s ∂y t =∂ 2 V/∂y t ∂y s -Fig.14.

Example B
Suppose that we want to determine the vector  2 V/ (sC 3 ) (sC 4 ) for the network N in Fig. 2.
Because the items i, ii and iii of the Rule 3 were fulfilled in the Example A we have to continue further: Here the matrices K 3 and K 4 are 31 32 33 13 24 0000 0 0 00 0000 0 0 00 ; ; 0100 0 0 00 0000 0 100 000 0 000 0 ; 000 0 000 1 We can find the nonzero elements of K 34 by using the auxiliary CM graph G 0 drawn in Fig. 15.By comparing ( 47) with ( 49) one settles we need only these addends of elements x 2 and x 4 in (47) that content the quantities e 4 and e 3, e 4 , respectively.According to the Chan-Mai procedure we draw the graphs G 0,2 and G 0,4 -Fig.16 and Fig. 17    From Fig. 18 ÷ Fig. 19 we obtain the products Note that with the exception of the sink and source quantities the graph G 0 is isomorphic to the graph G in Fig. 4.That is why the expressions (33) remain valid for the denominator in (4) also.Than for the elements of the vector (47) from ( 51) and (33) it follows: Now we return to ( 42) and ( 49) and obtain

First and second-order quadratic sensitivity sums
The sensitivity is an important parameter for the evaluation of practical suitability of electrical networks.For this purpose usually one uses the first-order sensitivity and the second-order sensitivity, defined by the well known formulae (Cederbaum, 1984;Chua & Lin, 1975) .respectively and where F is a network function or variable and x, y are changeable network element parameters.
Obviously, the derivatives in these expressions can be determined according to the above described method based on Chan-Mai signal-flow graphs.Besides very often we are interested in a global index as a quadratic sum of sensitivities (first-or second-order): where i є {1, 2, …, n} .
Without loss of generality further we assume that the functions F i are the elements of the voltage vector V. Then the sum (60) can be derived with the help of the expressions of the corresponding Jacobian matrix subvectors J i and of the voltage vector V: If from the elements of the Hessian matrices H i one forms the vector

Conclusions
A topological method for obtaining the Jacobian and Hessian matrices and their use for quadratic first-or second-order sensitivity sums calculation of active networks is presented.
It is based on the replacement of the investigated network N by using a nullor equivalent circuit and on the representation of the circuit passive part N p by a Chan-Mai signal-flow graph G p .The Jacobian and the Hessian matrix elements of the nullor network can be obtained by means of the some dependent variables of some Chan-Mai graphs derived from G. The substantial advantage of the method consists in the use mainly of isomorphic graphs.Two examples illustrate the proposed method.

Fig. 1 .
Fig. 1.Chan-Mai Signal-Flow Graph Jacobian and Hessian Matrices and Sensitivities of Active Linear Networks by Using Chan-Mai Signal-Flow Graphs 233 www.intechopen.comSymbolic Determination of Jacobian and Hessian Matrices and Sensitivities of Active Linear Networks by Using Chan-Mai Signal-Flow Graphs 235 Example A Fig. 2. Nullor Network NIn Fig.3the CMG G p of the passive part of N is drawn(Nenov, 2004).Further following the Rule 2 we reach to the graph G in Fig.4for

Fig. 6 .
Fig. 6.Separations S 1 , S 2 and S 3 of CM Graph G 11 33 3()     J YV Y V (36)and following Rule 2 one draws the CM graph G J3 (Fig.8).Obviously, the graphs G and G J3 have one and the same structure and consequently the expressions (33) hold for the source vertex quantities in (29) also.But for the nominator polynomials in (4) we have to draw according the Rule 2 four new CM graphs -G J3,1 , G J3, 23 , G J3, 4 and G J3, 5 -Fig.9.