Attaining Mean Square Boundedness of a Marginally Stable Stochastic Linear System With a Bounded Control Input

Abstract

ABSTRACT. In this article we construct control policies that ensure bounded variance of a noisy marginally stable linear system in closed-loop. It is assumed that the noise sequence is a mutually independent sequence of random vectors, enters the dynamics affinely, and has bounded fourth moment. The magnitude of the control is required to be of the order of the first moment of the noise, and the policies we obtain are simple and computable. § 1. INTRODUCTION Stabilization of stochastic linear systems with bounded control inputs has attracted considerable attention over the years. This is due to the fact that incorporating bounds on the control is of paramount importance in practical applications; suboptimal control strategies such as receding-horizon control Hokayem et al., 2009], and rollout algorithms [Bertsekas, 2000], among others, were designed to incorporate such constraints with relative ease, and have become widespread in applications. However, the following question remains open: when is a linear system with possibly unbounded additive stochastic noise globally stabilizable with bounded inputs? In this article we shall provide sufficient conditions that give a positive answer to this question with minimal hypotheses.Bounded input control has a rich and important history in the control literature [Lin et al., 1996;Stoorvogel et al., 2007;Sussmann et al., 1994;Yang et al., 1997Yang et al., , 1992. The deterministic version of the bounded input stabilization problem was solved completely in a series of articles [Sussmann et al., 1994;Yang et al., 1992] In the presence of affine stochastic noise the linear system ( ) becomes x t+1 = Ax t + Bu t + w t , where (w t ) t∈ 0 is a collection of independent (but not necessarily identically distributed) random vectors in d with possibly inter-dependent components at each time t. With an arbitrary noise it is clearly not possible to ensure mean-square boundedness; for instance, if the noise has a spherically symmetric Cauchy distribution on d , then given any initial condition x 0 ∈ d , the second moment of x 1 does not even exist. Similarly, if the second moment of the noise becomes unbounded with time, it is not possible to control the second moment of the process (x t ) t∈ 0 . It is necessary to assume, at least, that the noise has bounded variance.

Stochastic Stabilization of Partially Observed and Multi-Sensor Systems Driven by Unbounded Noise Under Fixed-Rate Information Constraints

“…The two algorithms run in parallel in a synergic fashion. Note that a two-stage algorithm for simultaneous estimation of the process order and parameters, conceptually similar to that proposed in this technical note, has already been investigated in [22] and [23]. In comparison to the result presented there, formal conditions ensuring the convergence of the estimation procedure are provided in the present work.…”

“…Variable-rate coding schemes for time-varying channels were considered in [17]. Authors in [22] studied conditions for stabilization when the control actions are uniformly bounded, the controlled multi-dimensional system is marginally stable and is driven by noise with unbounded support. The multi-sensor setting with unbounded noise has not been studied to our knowledge.…”

“…Various identification techniques targeting fractional-order models, have recently emerged both in the time [1], [2], [9]- [11], [13] and in the frequency [5], [21] domain. Instrumental variable method has been utilized in [22]- [24], Kalman filtering in [18] and the use of modulating functions was considered in [7]. On-line methods for estimation of the commensurate order are given in [16].…”

On 1-norm stochastic optimal control with bounded control inputs

“…In the case of ρ(A) = 1 with the deterministic part of the system (2) Lyapunov stable, the sole assumption of bounded control inputs is insufficient, and another constraint must be embedded into (18) in order to ensure the mean-square boundedness of the state; see [12] for details.…”

“…These papers, however, assume either compactly supported disturbances or only probabilistic input and state constraints, whereas [7] and [12] deal exclusively with stability in the presence of hard input constraints. In this paper we prove in a much simpler way a slight generalization of one of their stability results.…”