Abstract. We study the dynamical equations of nonlinear inductor-capacitor circuits. We present a novel Lagrangian description of the dynamics and provide a variational interpretation, which is based on the maximum principle of optimal control theory. This gives rise to an alternative method for deriving the dynamic equations. We show how this generalized Lagrangian description is related to generalized Hamiltonian models discussed in the literature by means of a Legendre transformation. Some distinctive features of the present approach are that it is applicable to circuits with arbitrary topology and that the variational principle and the resulting equations do not involve nonphysical inductor charges or capacitor fluxes.
SYSTeMS, Universiteit Gent, Technologiepark 9, 9052 Zwijnaarde, Belgium { Luc. Moreau,Dirk. Aeyels}@rug.ac. be 1 I n t r o d u c t i o n We consider linear time-invariant continuous-time systems k ( t ) = A z ( t ) + bu(t), y(t) = cz(t)(1) with 2-dimensional state z E R2, scalar input U E R, and scalar output y E R. The matrices A,b and c are constant and of appropriate dimension. We discuss the problem of making system (1) exponentially stable by means of a static time-varying output feedback u(t) = k ( t ) y ( t ) . Easily verifiable necessary and sufficient conditions for this problem to be solvable are presented. Moreover the proof of the sufficiency part is constructive; that is, it supplies the required feedback gain k ( t ) . This paper thus solves an open problem posed by Brockett [l] for the particular case of scalarinput scalar-output second-order systems. We assume throughout the paper that b # (0 O)T and c # (0 0). C o n s t r u c t i o n of static periodic o u t p u t feedbackwith @(t) the solution of the matrix differential equa-Compare with [2]. Expressed in these new coordinates, the closed-loop system becomesIn the present section, we restrict attention to controllable systems. We may then assume without loss of c2 c2generality that (1) is in the controller form al ,) 4 t ) + ( : ) 4 t h + (ul -u2--(-) ) exp(-czkl sinwt), aZz(t) = --exp(c2kl sinwt) + -+ u2 + kocz. (11)Notice that in the new coordinates < the system matrix depends on w and t only through the product w t . Averaging theory may then be applied: system (7) and hence the original closed-loop system (4) is exponentially stable for w sufficiently large, if the following averaged system is exponentially stable; see, e.g., [3, Theorem(2)We assume that c2 # 0.We study the influence of the output feedback 21.(12)(15) 011 P12with parameters k0,kl and w E R. Substituting (3) in(2), we obtain the closed-loop systemWe introduce new state-coordinates C according to CI c1 c2 c2'This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility rests with its authors. of the University of Ghent.with 'Corresponding author. Supported by BOF grant 011D0696 0-7803-5250-5/99/$10.00 0 1999 IEEE 108 E* = & JdPn exp(*c2kl sin t)dt. (16)
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