Stabilization by means of periodic output feedback

Abstract

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)

Parameter Estimation of Sigmoid Superpositions: Dynamical System Approach

“…The problem of determining K(t) however is not very easy to solve, especially for nonlinear systems. Even for linear ones, a similar problem known in the literature as the Brockett problem 2 [6] has positive solutions at present for systems of second and third order [21,30]. Nevertheless, despite the obvious difficulties, we believe that the question of searching for the suitable K(t) ensuring inequality (16) for system (15) could be an achievable goal for future studies.…”

State Nullification by Memoryless Output Feedback

“…The answer to this problem is indeed complex. A sample of papers addressing the problem is Leonov [6], Moreau and Aeyels [9], [10]. These papers provide explicit examples where stabilization may not be possible with a continuous, even time-varying, feedback.…”

A time-varying matrix solution to the Brockett decentralized stabilization problem

“…The Brockett time-varying stabilization problem [1] suggests the means of using memoryless output feedback with time-varying gain matrix for system stabilization, which is one of the challenging open problems in systems and control as documented in [10]. In the general setting with no diagonal structure constraint on K(t) of the Brockett stabilization problem, there are several partial solutions reported in the literature [11]- [13]. These solutions involve either periodic scalar control gains [11], [12], [14], or some very strong conditions on the system matrix [13], which may not be satisfied in practice.…”