Linear Dynamically Varying Versus Jump Linear Systems

The connection between linear dynamically varying (LDV) systems and jump linear systems is explored. LDV systems have been shown to be useful in controlling systems with "complicated dynamics". Some systems with complicated dynamics, for example Axiom A systems, admit Markov partitions and can be described, up to finite resolution, by a Markov chain. In this case, the control system for these systems can be approximated as Markovian jump linear systems. It is shown that (i) jump linear controllers for arbitrarily fine partitions exist if and only if the LDV controller exists; (ii) jump linear controllers stabilize the dynamical system; (iii) jump linear controllers are approximations of the LDV controller. it is usually difficult to determine whether a partition is Markovian. This paper proceeds as follows: In the next section the nonlinear tracking problem for systems with complicated dynamics is presented along with LDV controllers that solve this tracking problem. In section 3 jump linear systems are introduced along with some standard results. Section 4 shows how under certain conditions the nonlinear tracking problem described in section 2 may be described as a jump linear control problem. However, it is shown that this approach has difficulties in that stability of the closed loop nonlinear system cannot be easily proved. Section 6 consists of the main results. 2 'Ikacking Systems with Complicated Dynamics via LDV control