Computer Applications in Electrical Engineering Vol. 4 Analysis o the regularity pointwise completeness pointwise generacy o descriptor linear electrical circuits Tadeusz Kaczorek Białystok University o Technology 5-5 Białystok Wiejska 45A e-mail: Kaczorek@isep.pw.edu.pl A method or analysis o the regularity pointwise completeness pointwise generacy o descriptor linear electrical circuits composed o resistances capacitances inductances voltage (current) sources is proposed. It is shown that every descriptor electrical circuit is a linear system with regular pencil. Conditions or the pointwise completeness pointwise generacy o the descriptor linear electrical circuits are established. The considerations are illustrated by examples o descriptor electrical circuits. KEYWORDS: regularity pointwise completeness pointwise generacy descriptor electrical circuits. Introduction Descriptor (singular) linear systems have been considered in many papers books [ 5-8 4-8 6-8 5]. The eigenvalues invariants assignment by state output eedbacks have been investigated in [6 7] the minimum energy control o descriptor linear systems in []. In positive systems inputs state variables outputs take only non-negative values [9 5]. Examples o positive systems are industrial processes involving chemical reactors heat exchangers distillation columns storage systems compartmental systems water atmospheric pollution models. A variety o models having positive linear behavior can be ound in engineering management science economics social sciences biology medicine etc. The positive ractional linear systems some o selected problems in theory o ractional systems have been addressed in monograph [7]. Descriptor stard positive linear systems by the use o Drazin inverse has been addressed in [ 6 4 6 8]. The shule algorithm has been applied to checking the positivity o descriptor linear systems in []. The stability o positive descriptor systems has been investigated in [5]. Reduction decomposition o descriptor ractional discrete-time linear systems have been considered in [6]. A new class o descriptor ractional linear discrete-time system has been introduced in [8]. The minimum energy control o positive ractional continuous-time discrete-time linear systems has been addressed in [8 9 ]. Stabilization o positive electrical circuits by state-eedbacks has
T. Kaczorek / Regularity pointwise completeness pointwise generacy been considered in [9] decoupling zeros o positive electrical circuits in []. The zeroing problem o state variables in ractional descriptor electrical circuits by state-eedbacks has been analyzed in []. The ractional descriptor observers or ractional descriptor continuous-time linear systems have been proposed in [5]. The pointwise completeness pointwise degeneracy or stard ractional linear systems have been investigated in [-5-4 -4]. The Drazin inverse o matrices has been applied to ind the solutions o the state equations o the ractional descriptor continuous-time linear systems with regular pencils in [4]. In this paper a method o analysis o the regularity pointwise completeness pointwise generacy o the descriptor electrical circuits will be proposed. The paper is organized as ollows. In section three methods o analysis o descriptor linear continuous-time systems are recalled. The regularity o the descriptor linear electrical circuits is addressed in section. It is shown that every descriptor electrical circuit is a linear system with regular pencil. The pointwise completeness o descriptor electrical circuits is addressed in section 4 the pointwise generacy in section 5. Concluding remarks are given in section 6.. Preliminaries Consider the descriptor continuous-time linear system Ex Ax Bu (.) dx n m nn where x x u are the state input vectors E A dt nm B. It is assumed that the matrix E is singular the pencil o (E A) is regular i.e. det E det[ Es A] or some s C (.) where C is the ield o complex numbers. Method. (Weierstrass-Kronecker decomposition) It is well-known [7] that i the condition (.) is satisied then there exist nn nonsingular matrices P Q such that I n A PEQ PAQ N I (.) n n n where N is a nilpotent matrix with nilpotency index ( N N n n ) A n n n. The matrices P Q can be ound by the use o the ollowing elementary row column operations [7]:
T. Kaczorek / Regularity pointwise completeness pointwise generacy. multiplication o i-th row (column) by a nonzero scalar c. This operation will be denoted by Li c ( Ri c). addition to the i-th row (column) o the j-th row (column) multiplied by a L i j p( s) polynomial p(s). This operation will be denoted by ( Ri j p( s) ). interchange o the i-th j-th rows (columns). This operation L will be denoted by i j R ( i j ). Lemma.. The characteristic polynomial o the descriptor system (.) the characteristic polynomial o the matrix A are related by det[ Es A] k det[ I s A ] (.4) n where n n k det P det Q det( PQ). Proo. From (.) (.) we have In s A det[ Es A] det P Q Ns I n det P detq det In s A det Ns In k det I n s A Ns I n n. since det (.5) Method. (Shule algorithm) Perorming elementary row operations on the equation (.) or equivalently on the array E A B (.6) we obtain E A B (.7) A B E x A x B u (.8a) A x B u (.8b) rn where E has ull row rank r ranke. Dierentiation with respect to time o (.8b) yields A x Bu. (.9) The equations (.8a) (.9) can be written in the orm E A B x x u u A B. (.)
T. Kaczorek / Regularity pointwise completeness pointwise generacy E I det then rom (.) we have A x Aˆ ˆ ˆ x B u Bu (.a) where Aˆ E A A Bˆ E B A Bˆ E. (.b) A B E I det then perorming elementary row operations on the array A E A B A B (.) E we eliminate the linearly dependent rows in the matrix we obtain A E A B B (.) A B B E x A x B u B u (.4a) A x Bu Bu (.4b) Dierentiation with respect to time o (.4b) yields A x Bu Bu. (.5) The equations (.4a) (.5) can be written in the orm E A B B x x u u u A B B. (.6) E I det then rom (.6) we have A x Aˆ ˆ ˆ ˆ x Bu Bu Bu (.7a) where Aˆ Bˆ E A A E A B Bˆ E B A Bˆ A B E B (.7b) 4
T. Kaczorek / Regularity pointwise completeness pointwise generacy E I det then we repeat the procedure. It is well-known [ 7 6] that A i the condition (.) is met then ater inite number o steps we obtain the stard system equivalent to the descriptor system (.)... Examples. Regularity o descriptor linear electrical circuits We start with simple examples o descriptor linear circuits next the considerations will be extended to general case. Example.. Consider the descriptor electrical circuit shown in Fig. with given resistance R capacitances C C C source voltages e e. Fig... Electrical circuit o Example. Using Kirchho s laws or the electrical circuit we obtain the equations du e RC u u (.a) dt du du du C C C (.b) dt dt dt e u u (.c) The equations (.) can be written in the orm (.) where u RC x u e u e E C C C A B. u (.) The assumption (.) or the electrical circuit is satisied since RC det E C C C (.a) 5
det T. Kaczorek / Regularity pointwise completeness pointwise generacy RC s Es A C s C s C s s R C C C s C C C Thereore the electrical circuit is a descriptor linear system with regular pencil. Method. Perorming on the matrix RC s Es A Cs Cs Cs (.4) the ollowing elementary column operations R C C R C C R R we obtain the matrix RC C s RC s (.5) C C RC C C Perorming on the matrix (.5) the ollowing elementary row operations C L L C C the elementary column operations R R we obtain the desired matrix R C C C C In s A Es A PEs AQ (.6a) Ns In where C C C A RC C C C C N n (.6b) 6
T. Kaczorek / Regularity pointwise completeness pointwise generacy C C C RC P C Q (.6c) C R C C C C C C C R C C C C The matrices (.6c) can be ound by perorming the elementary row column operations on the identity matrix I. Perorming the elementary row operations C L L C C on the matrix I we obtain P the elementary column operations R C C R R C C RC R R R the matrix Q. C R C C C C Method. E In this case the matrix E has already the desired orm where RC E C C C (.7) it has ull row rank i.e. ranke. Taking into account that A (.8a) RC E det C C C RC C C A (.8b) rom (.) we obtain RC C C C x x u u (.9) x Aˆ x Bˆ u Bˆ u (.a) 7
T. Kaczorek / Regularity pointwise completeness pointwise generacy where RC RC RC Aˆ C C C R C C R C C R C C R C C (.b) RC RC ˆ B C C C R C C R C C RC ˆ C B C C C. C C C C C Method. (Elimination method) From (.c) we have u e u. (.) Substituting (.) into (.a) (.b) we obtain RC d u u e d e C C C dt u u e C dt e (.) d u u e d e A B B dt u u e dt e (.a) where 8
A T. Kaczorek / Regularity pointwise completeness pointwise generacy RC RC RC C C C R C C R C C RC RC RC B C C C (.b) R C C R C C RC B C C C C C. C C The characteristic polynomial o the matrix A has the orm s RC RC C C C det Is A s s (.4) RC C C s R C C R C C o the matrix A (given by (.6b)) C C C s C C C detis A RC C C C C s s RC C C s o the matrix  (given by (.b)) s R C R C ˆ det I s A s R C C R C C s R C C R C C s R C R C C C C s s s R C C C s R C C R C C (.5) (.6) 9
T. Kaczorek / Regularity pointwise completeness pointwise generacy Note that the additional eigenvalue s has been introduced in Method by the dierentiation with respect to time o the equation (.8b). From (.b) (.4) (.5) (.6) it ollows that the spectrum o the electrical circuit is the same or the three dierent methods it is equal to C C C s s. RC ( C C) Example.. Consider the descriptor electrical circuit shown in Fig.. with given resistances R R R ; inductances L L L source voltages e e. Using Kirchho's laws we can write the equations di di e Ri L Ri L dt dt (.7a) di di e Ri L Ri L dt dt (.7b) i i i (.7c) Fig... Electrical circuit o Example. The equations (.7) can be written in the orm (.) where i L L R R x i e u e E L L A R R i The assumption (.) or the electrical circuit is satisied since L L det E L L B. (.8)
T. Kaczorek / Regularity pointwise completeness pointwise generacy L s R L s R det Es A L s R L s R L L L L L s R L L R L L R L L s R R R R R Thereore the electrical circuit is a descriptor system with regular pencil. Method. Perorming on the matrix L s R Ls R Es A Ls R Ls R L the elementary column operations R L R R we obtain L L L L R (.9) (.) R R L RL s L L R RL RL L L L L L s. (.) L L L L L L Next perorming on the matrix (.) the elementary operations R L L R L R L R L L we obtain L L R L RL RL RL RL s L L L RL RL RL RL R L s L L L L L (.)
T. Kaczorek / Regularity pointwise completeness pointwise generacy inally R R R L L the desired orm In s A Es A PEs AQ (.a) Ns In where R L RL RL R L RL L L L A N R L RL RL R L R L n L L L (.b) R L RL L L L L L L L L L R L RL L L L L P Q (.c) L L L L L L L Perorming the elementary row operations on I we obtain the matrix P the elementary column operations the matrix Q. Method. E The matrix E given by (.8) has already the desired orm where L L E L L (.4) ranke. Taking into account that A (.5) L L E det L L L L L LL A (.6) rom (.) we obtain
where T. Kaczorek / Regularity pointwise completeness pointwise generacy L L R R L L x R R x u (.7) x A ˆ x B ˆ u (.8a) L L R R Aˆ L L R R R L L RL RL RL R L L RL RL RL R L L L L L L L ˆ B L L L L L L L L L L L L. (.8b) Method. From (.7c) we have i i i. (.9) Substituting (.9) into (.7a) (.7b) we obtain L L L d i R R R i e L L L dt i R R R i e d i i e A B dt i i e (.a) where L L L R R R A L L L R R R R L L R L R L R L RL RL R L L RL L L L L L L B L L L L L L (.b) (.c)
4 T. Kaczorek / Regularity pointwise completeness pointwise generacy L L L L L. The characteristic polynomial o the matrix (.b) has the orm s R L L RL RL R L detis A R L R L s R L L R L s R L L R L L R L L s R R R RR o the matrix A (given by (.b)) det I s A R L R L R L R L R L s L L L R L R L R L R L R L s L L L (.) s R L L R L L R L L s R R R RR o the matrix  (given by (.b)) det ˆ R L L R L R L s R L R L L R L Is A s R L R L L RL s (.) s s R L L R L L R L L s R R R RR (.) Note that in (.) the additional eigenvalue s has been introduced in Method by the dierentiation with respect to time o the equation (.8b). From (.9) (.) (.) (.) it ollows that the spectrum o the electrical circuit is the same or the three dierent methods it is determined by the zeros o the polynomial (.)... General case Note that the electrical circuit shown in Fig.. contains one mesh consisting o branches with only ideal capacitors voltage sources the one shown in
T. Kaczorek / Regularity pointwise completeness pointwise generacy Fig.. contains one node with branches with coils. The equations (.) (.7) are two dierential equations one algebraic equation. In general case we have the ollowing theorem. Theorem.. Every electrical circuit is a descriptor system i it contains at least one mesh consisting with only ideal capacitors voltage sources or at least one node with branches with coils. Proo. I the electrical circuit contains at least one mesh consisting o branches with ideal capacitors voltage sources then the rows o the matrix E corresponding to the meshes are zero rows the matrix E is singular. I the electrical circuit contains at least one node with branches with coils then the equations written on Kirchho's current law or these nodes are algebraic ones the corresponding rows o E are zero rows it is singular. Theorem.. Every descriptor electrical circuit is a linear system with regular pencil. Proo. It is well-known [7 5 7] that or a descriptor electrical circuit with n branches q nodes using current Kirchho's law we can write q algebraic equations the voltage Kirchho's law n q dierential equations. The equalities are linearly independent can be written in the orm (.). From linear independence o the equations it ollows that the condition (.) is satisied the pencil o the electrical circuit is regular. Remark.. The spectrum o descriptor electrical circuits is independent o the method used o their analysis. 4. Pointwise completeness o descriptor electrical circuits Consider the descriptor electrical circuit described by the equation (.) or u( t) t. Deining the new state vector x n n x Q x x x x n n n (4.) using (.) or u( t) t (.) we obtain x ( t) A x ( t) (4.) A t x ( t) e x t (4.) where x Q x (4.4a) 5
T. Kaczorek / Regularity pointwise completeness pointwise generacy Note that [7] x Q x n n Q x x Q Q q k n n Q. (4.4b) ( k ) k x ( t) N x () t (4.5) (k ) where is the k-th derivative o the Dirac impulse. From (4.) or x ( ) t we have where Q Q Q x x( t ) Q x ( t ) (4.6) nn nn Q Q. Deinition 4.. The descriptor electrical circuit (.) is called pointwise n complete or t t i or every inal state x there exist initial conditions n x satisying (4.4a) such that x x( t ) ImQ. Theorem 4.. The descriptor electrical circuit is pointwise complete or any n t t every x satisying the condition Proo. Taking into account that or any x rom (4.) (4.5) or t t we obtain A t ( ) ImQ (4.7) A t A t det e e e A t A x e x t x ( t ). (4.8) n Thereore rom (4.6) it ollows that there exist initial conditions x such that x x( t ) i (4.7) holds. Example 4.. (Continuation o Example.) In this case rom (.6c) we have RC Q Q Q Q R C C C C R C C C C Q C C (4.9) 6
T. Kaczorek / Regularity pointwise completeness pointwise generacy a RC x ImQ a b (4.) R C C C C a b R C C C C or arbitrary a b. The eigenvalues o the matrix A (given by (.6b)) are s C C C s using the Sylvester ormula [5] we obtain R C C C since e R C st st A t st C C C e Z Z e e A I s C C C s s s Z I A R C A I s A C C C s s s Z (4.a) Thereore the descriptor electrical circuit shown in Fig.. is pointwise complete or any t t every x satisying (4.). R C 5. Pointwise generacy o descriptor electrical circuit (4.b) Consider the descriptor electrical circuit described by equation (.) or u( t) t. Deinition 5.. The descriptor electrical circuit (.) is called pointwise n degenerated in the direction v or t t i there exists nonzero vector v such that or all initial conditions x Im Q the solution o (.) satisies the condition T v x (T denotes transpose) (5.) Theorem 5.. The descriptor electrical circuit (.) is pointwise degenerated in the direction v deined by 7
T. Kaczorek / Regularity pointwise completeness pointwise generacy vt Q (5.) or any t all initial conditions x ImQ where Q Q are determined by (4.6) (4.4) respectively. Proo. Substitution o (4.) into (4.6) yields A t x Qe x (5.) T T A t v x v Q e x (5.4) since (5.) holds or all x Q x ImQ. Example 5.. (Continuation o Example. 4.) From (5.) (4.9) we have T v [ ] (5.5) since RC. (5.6) v T Q [ ] [ ] R ( C C) C C R ( C C) C C Thereore the descriptor electrical circuit shown in Fig.. in pointwise generated in the direction deined by (5.5) or any t any values o the resistance R capacitances C C C. 6. Concluding remarks The regularity pointwise completeness pointwise generacy o descriptor linear electrical circuits composed o resistances capacitances inductances voltage (current) sources have been investigated. Three basic methods o the analysis o the descriptor linear systems with regular pencils have been presented. It has been shown that every descriptor electrical circuit is a linear descriptor system with regular pencil. Conditions or the pointwise completeness pointwise generacy o the descriptor electrical circuits have been established illustrated by simple electrical circuits. The considerations can be extended to the ractional descriptor electrical circuits []. This work was supported under work S/WE//. 8
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