Linear dynamically varying LQ control of nonlinear systems over compact sets

Abstract

Abstract-Linear-quadratic controllers for tracking natural and composite trajectories of nonlinear dynamical systems evoluting over compact sets are developed. Typically, such systems exhibit "complicated dynamics," i.e., have nontrivial recurrence. The controllers, which use small perturbations of the nominal dynamics as input actuators, are based on modeling the tracking error as a linear dynamically varying (LDV) system. Necessary and sufficient conditions for the existence of such a controller are linked to the existence of a bounded solution to a functional algebraic Riccati equation (FARE). It is shown that, despite the lack of continuity of the asymptotic trajectory relative to initial conditions, the cost to stabilize about the trajectory, as given by the solution to the FARE, is continuous. An ergodic theory method for solving the FARE is presented. Furthermore, it is shown that wrapping the LDV controller around the nonlinear system secures a stable tracking dynamics. Finally, an example of controlling the Hénon map to follow an aperiodic orbit is presented.

Adaptive Synchronization of an Uncertain Complex Dynamical Network

“…Bohacek and Jonckheere [19][20] proposed the so-called linear dynamically varying method based on discrete time dynamical systems. In the following, by using Lyapunov stability theory, several novel locally and globally asymptotically stable network synchronization criteria are deduced for an uncertain complex dynamical network.…”

Relationships between Linear Dynamically Varying Systems and Jump Linear Systems

“…From this observation, using techniques similar to [6] and [7], it is possible to show that the system is locally stochastically stable; that is, if kxð0Þk is small, EðkxðkÞkÞ ! 0 as k !…”

“…If f is invertible, the uniform detectability can be weakened to detectability, which is the dual of stabilizability (see [6] for details).…”

“…One of the main results from [6] is the following: (3) is LDV stabilizable if and only if there exists a bounded function X : Y ! R nÂn with X 0 y ¼ X y b 0 that satisfies the functional discrete-time algebraic Riccati equation…”

“…It was shown in [6] that if the LDV system (3) induced by f is uniformly exponentially stabilized by the control uðkÞ ¼ F yðkÞ xðkÞ, then the nonlinear system (1) with control uðkÞ ¼ F yðkÞ xðkÞ is locally uniformly exponentially stable. By definition, locally uniformly exponentially stable means that there exist a < 1, b < y, and g > 0 such that if kxð0Þk ¼ kjð0Þ À yð0Þk < g, then kxðkÞk < ba k kxð0Þk, where a, b, and g can be taken independent of the initial condition y o , i.e., uniformly in y o and locally in x.…”