Regional Stabilization of Input-Delayed Uncertain Nonlinear Polynomial Systems

Abstract

This paper addresses the problem of local stabilization of nonlinear polynomial control systems subject to time-varying input delay and polytopic parameter uncertainty. A linear matrix inequality approach based on the Lyapunov-Krasovskii theory is proposed for designing a nonlinear polynomial state feedback controller ensuring the robust local uniform asymptotic stability of the system origin along with an estimate of its region of attraction. Two convex optimization procedures are presented to compute a stabilizing controller ensuring either a maximized set of admissible initial states for given upper bounds on the delay and its variation rate or a maximized lower bound on the maximum admissible input delay considering a given set of admissible initial states. Numerical examples demonstrate the potentials of the proposed stabilization approach.

Local Stabilization for Multiple Input-Delay Systems Subject to Saturating Actuators: The Continuous-Time Case

“…Remark 1: As in literature [3], [14], we assume that ui(t) = 0 (i ∈ I[1, r]) for t < 0. It is worth mentioning that this control setting is often encountered in many control systems subject to actuator delays and transmission delays.…”

“…Note that the solutions of the system (1) are independent on the values of x(t) for t < 0. Therefore, as with [3], [14], we define the initial condition as x(t) = x0 within the interval [−h, 0] for analyzing the stability by means of the L-K approach.…”

“…Remark 2: In Assumption 1, the existence and uniqueness of the scalars t * i (i ∈ I[1, r]) can be ensured if τi(t) (i ∈ I[1, r]) are slowlyvarying with τi(t) < 1. The existence and uniqueness of t * i are also guaranteed in networked control systems, where the modeled delays are piecewise-continuous and do not grow at the sampling instants [3], [14]. In addition, it is worth pointing out that, a somewhat strong requirement ( 5) is imposed for convenience of subsequent analysis.…”

“…In the past several decades, time-delay systems have gained considerable research attention from both communities of dynamical system and control engineering. In general, the motivation for the ever-increasing research interest is twofold: 1) many control systems intrinsically contain time delays in the state, the input or the output [3], [5], [7], [17], [18], [23], [25]; and 2) some strategies and phenomena can be modeled as time delays with examples including sampling control and packet dropouts [5], [26], [28]. For the stability analysis of time-delay systems, it has been acknowledged that the Lyapunov-Krasovskii (L-K) approach plays a vitally important role that enables the use of the linear matrix inequalities (LMIs), thereby facilitating the control design.…”

“…Inspired by the above discussions, this paper takes one substantial step further by looking into the local stabilization problem for multiple input-delay systems with actuator saturations where the open-loop systems are allowed to be exponentially unstable. As in [3], [14], we redefine the initial conditions and estimate the solution bounds within initial time-intervals via the delay-dependent L-K approach. However, the main results of this paper are by no means a simple generalization of the existing ones (involving a single input delay) such as [14].…”