Discrete-time Homogeneous Systems Research Articles

Discrete-time homogeneity: Robustness and approximation

In this paper we study robustness properties of discrete-time homogeneous systems. We also analyse stability and robustness properties of a more general class of nonlinear discrete-time systems that can be approximated by homogeneous ones. We apply the results to the investigation of stability properties of discretized continuous-time systems.

Distributed LQR control for discrete-time homogeneous systems

ABSTRACTThis paper investigates the distributed linear quadratic regulation (LQR) controller design method for discrete-time homogeneous scalar systems. Based on the optimal centralised control theory, the existence condition for distributed optimal controller is firstly proposed. It shows that the globally optimal distributed controller is dependent on the structure of the penalty matrix. Such results can be used in consensus problems and used to find under which communication topology (may not be an all-to-all form) the optimal distributed controller exists. When the proposed condition cannot hold, a suboptimal design method with the aid of the decomposition of discrete algebraic Riccati equations and robustness of local controllers is proposed. The computation complexity and communication load for each subsystem are only dependent on the number of its neighbours.

Value Iteration for (Switched) Homogeneous Systems

In this note, we prove that dynamic programming value iteration converges uniformly for discrete-time homogeneous systems and continuous-time switched homogeneous systems. For discrete-time homogeneous systems, rather than discounting the cost function (which exponentially decreases the weights of the cost of future actions), we show that such systems satisfy approximate dynamic programming conditions recently developed by Rantzer, which provides a uniform bound on the convergence rate of value iteration over a compact set. For continuous-time switched homogeneous system, we present a transformation that generates an equivalent discrete-time homogeneous system with an additional ldquosamplingrdquo input for which discrete-time value iteration is compatible, and we further show that the inclusion of homogeneous switching costs results in a continuous value function.

Growth rate of switched homogeneous systems

We consider discrete-time homogeneous systems under arbitrary switching and study their growth rate, the analogue of joint spectral radius for switched linear systems. We show that a system is asymptotically stable if and only if its growth rate is less than unity. We also provide an approximation algorithm to compute growth rate with arbitrary accuracy.

GROWTH RATE OF SWITCHED HOMOGENEOUS SYSTEM

We consider discrete-time homogeneous systems under arbitrary switching and study their growth rate, the analogue of joint spectral radius for switched linear systems. We show that a system is asymptotically stable if and only if its growth rate is less than unity. We also provide an approximation algorithm to compute growth rate to an arbitrary accuracy.

Global stabilization of discrete-time homogeneous systems

The feedback stabilization of continuous time Δ r -homogeneous systems has been studied by several authors, see for instance (Dayawansa and Martin, Proceedings of the 28th C.D.C., Tampa, Florida, December, 1989, Hermes, J. Differential Equations 92 (1991) 76–89, Hermes, Systems Control Lett. 24 (1995) 7–11, Kawski, Proceedings of the 27th C.D.C., Texas, 1988, Texas, Kawski, Control Theory and Advanced Technology, vol. 6, MITA Press, Tokyo, 1996, pp. 497–516, Sepulchre and Aeyels, Math. Control Signal Systems 9 (1996) 34–58). In (Kawski, Analysis of Controlled Dynamical Systems, Birkhauser, Basel, 1991, Kawski, Proceedings of IFAC NOLCOS, 1995), Kawski gave an intrinsic concept of homogeneity with respect to general smooth dilations. In this note, we define the homogeneity of transition maps w.r.t. continuous dilations and we show how one can construct a global stabilizing feedback control for discrete-time homogeneous systems which are locally asymptotically stabilizable.