An exploration of the Razumikhin stability theorem with applications in stabilization of delay systems

Abstract

In this paper, we establish a new evolution property of Lyapunov functions that satisfy the Razumikhin Stability Theorem conditions. The derived property is shown to be a valid alternative criterion to prove the asymptotic stability of a delay system without calculating the derivative of the Lyapunov function, and it is especially useful in solving stabilization problems for time delay systems when the delay is unknown. As an example application of this derived property, we investigate the stabilization of continuous-time linear systems with an unknown time-varying delay, by proposing a switched low gain feedback control technique that utilizes the derived property. Existing literature on stabilization of continuous time delay systems all require some knowledge, such as an upper bound, of the delay. In this work, by proposing a switched low gain feedback control law utilizing the derived property, we are able to achieve asymptotic stabilization of the system without any knowledge of the delay. Furthermore, the proposed switched low gain feedback law also significantly improves the closed-loop system performance in terms of overshoot and convergence speed, in comparison with the traditional fixed gain low gain feedback design. Simulation study verifies the theoretical results.

Introduction

In this work, we derive a new evolution property of Lyapunov functions that satisfy the Razumikhin Stability Theorem conditions, which is especially useful in solving stabilization problems of time delay systems when the delay is unknown. As an example application of the derived property, we consider the stabilization of continuous-time linear systems with an arbitrarily large bounded time-varying delay in the input, which, to the best of our knowledge, is an open challenging research problem that has not been solved in the literature.

Time delay is ubiquitous in engineering systems. Examples include the computation time required by the controller and the transmission time in a remotely controlled system. It is well known that time delay can not only degrade the system performance, but sometimes also destroy the system stability if not dealt with appropriately. In the past few decades, there has been much research effort spent on designing delay-dependent controllers for systems with delays in the input and/or the state, such as Artstein (1982), Bellman and Cooke (1963), Gu, Kharitonov, and Chen (2003), Krstic (2009), Krstic (2010a), Krstic (2010b), Mazenc, Mondie, and Niculescu (2003) and Mazenc, Mondie, and Niculescu (2004). Among the various stabilization approaches for linear systems with delayed input, the model reduction technique (Artstein, 1982, Mayne, 1968) induces a predictor feedback law that achieves finite spectrum assignment of the closed-loop system by using a prediction of the future state. Further development of the technique to a broader class of systems with state delays was discussed in Jankovic (2009). To deal with the uncertainties in the system parameters and in the delay, adaptive versions of the predictor feedback law were developed in Bresch-Pietri, Chauvin, and Petit (2012) and Bresch-Pietri and Krstic (2010) to achieve local regulation of general linear systems when the information on the range of the delay is available. It is noted that the predictor state feedback control law includes an infinite dimensional distributed term, which makes the feedback law more difficult to implement than a finite dimensional one.

To overcome the difficulty in implementation, simplified predictor-based feedback was developed in the form of truncated predictor feedback and delay independent truncated predictor feedback, both of which are finite dimensional (see Lin and Fang (2007), Lin (2007), and Zhou, Lin, and Duan (2012)). Low gain feedback design techniques such as eigenstructural assignment based approach (Lin, 1998, Lin and Fang, 2007) and parameteric Lyapunov equation based approach (Zhou et al., 2012) were developed to construct the feedback gain matrix with a low gain parameter. For some classes of linear systems, simpler form of the predictor-based feedback law without explicitly using the delay can be constructed to achieve stabilization, and thus the requirement of the knowledge of the delay can be relaxed. For example, for the delay independent truncated predictor feedback technique, which was originally proposed (Lin and Fang, 2007, Zhou et al., 2009) for systems with all open loop poles at the origin, only an upper bound of the delay is required to determine the value of the low gain parameter even when the input delay is time-varying. However, to the best of our knowledge, the stabilization of continuous time delay systems in the absence of any knowledge of the delay remains a challenging open problem even for the seemingly simple systems such as a chain of integrators, for which only global regulation has been achieved recently by adaptively tuning the low gain parameter in the truncated predictor feedback law and carrying out partial differential equation based analysis of the resulting closed-loop system (Wei & Lin, 2019).

In the traditional low gain feedback, the low gain parameter is set as a constant. Recently, in Su, Wei, and Lin (2019), in the discrete-time setting, we proposed a new switched low gain feedback law and fully eliminated the requirement of the knowledge of an upper bound of the delay. It is natural to think about whether we could extend the methodology to continuous-time delay systems. However, different from the discrete-time setting, where, without the requirement of knowing the delay, the calculation of $\Delta V$ (difference of Lyapunov function values at two contiguous steps) could still be obtained accurately by postponing the calculation of $\Delta V$ by one time step, in continuous-time setting, the calculation of $\dot{V}$ requires the exact information of delay. Therefore, it is technically much more challenging to design the appropriate switched low gain feedback law for stabilization of continuous-time delay systems with unknown delay. To overcome this difficulty, in this paper, we explore the Razumikhin Stability Theorem and derive a new evolution property of Lyapunov functions that satisfy the conditions of Razumikhin Stability Theorem. This derived property is essential for us to design appropriate switching conditions for switched low gain feedback control of continuous-time linear systems with unknown input delay. Then, with the carefully designed switched low gain feedback control law for continuous-time delay systems, we are able to fully eliminate the requirement of the knowledge of the time-varying delay.

Our results on stabilization of the continuous-time delay systems with unknown time delay is also largely inspired by Theorem 2 in Zhou et al. (2009), where an upper bound ${\gamma }^{\ast }$ is established such that the closed-loop system stability will be achieved by a fixed gain low gain feedback control law with $\gamma <{\gamma }^{\ast }$. Different from Zhou et al. (2009), delay upper bound $D$ is unknown in this study, and thus the upper bound of the feasible low gain parameter ${\gamma }^{\ast }$ cannot be established. Therefore, if delay upper bound $D$ is unknown, the fixed low gain feedback design in Zhou et al. (2009) cannot guarantee the asymptotic stability of the closed-loop system. To overcome this difficulty, we use a control law with a structure similar to the fixed gain low gain feedback control law in Zhou et al. (2009), and we start with initial guesses of the feasible low gain parameter $\gamma$ and an initial estimate of the upper bound on the time-varying delay $\hat{D}$. We then carefully design a switching criterion for $\gamma$ and $\hat{D}$, by using the derived evolution property of Lyapunov functions that satisfy the conditions of the Razumikhin Stability Theorem. If the guessed values of $\gamma$ and $\hat{D}$ are not appropriate for the control law to stabilize the system, the switching condition will be triggered, and $\gamma$ will be decreased and $\hat{D}$ will be increased. We will prove that, there are at most finitely many switches in total as the system evolves, and the final values of $\gamma$ and $\hat{D}$ will be appropriate for stabilizing the closed-loop system. It might be useful to point out that, the derived upper bound for the feasible low gain parameter ${\gamma }^{\ast }$ in Zhou et al. (2009) is conservative, and under our proposed switched low gain feedback control law, the value of low gain parameter $\gamma$ at steady state is usually several orders of magnitude larger than ${\gamma }^{\ast }$ with much better closed-loop performance in terms of overshoot and convergence rate, as demonstrated in the simulation study. Therefore, it is clear that not only the requirement of the knowledge of the time-varying delay is completely eliminated under our new switching design, but the typical conservativeness of the traditional low gain feedback design is also reduced to improve the closed-loop performance significantly.

We would like to point out that, stabilization of the continuous-time delay systems with unknown delay described in this paper is an example application of the derived property, and it is believed that the derived property is helpful for solving control problems for other time delay systems. It is also noted that, Razumikhin Stability Theorem is one of the main tools for stability analysis of time delay systems. Our work explores how the Razumikhin Stability Theorem can be effectively utilized in the control design process to solve open challenging control problems for time delay systems, to the best of our knowledge.

The main contributions of this work are two folds. First, we establish a new evolution property of Lyapunov functions that satisfy the Razumikhin Stability Theorem conditions. The derived property is shown to be a valid alternative criterion to proving the asymptotic stability of a delay system without calculating the derivative of the Lyapunov function, and is especially useful in solving stabilization problems for time delay systems when the delay is unknown. Second, as an example application of this derived property, we solve the stabilization problem for continuous-time linear systems with an unknown time-varying delay by proposing a new switched low gain feedback control design. In comparison with the relevant previous literature, our method eliminates the requirement of any knowledge of the delay and significantly improves the closed-loop system performance in terms of overshoot and convergence speed in comparison with the traditional fixed gain low gain feedback design.

The remainder of the paper is organized as follows. In Section 2, we explore the Razumukhin Stability Theorem and establish one useful property that could be used later for the switched low gain feedback control design. In Section 3, as an example application of the derived property, we formulate a control problem and present the main results on stabilization of continuous-time delay systems in the absence of prior knowledge of the delay. Section 4 summarizes the simulation results. Section 5 concludes the paper.