Investigating stability for driver advisory train cruise control systems with aperiodically sampled measurements

Stability of neural-network based train cruise advisory control with aperiodical measurements

This paper is concerned with the delay-dependent stability analysis of linear systems with a time-varying delay. Two types of improved Lyapunov-Krasovskii functionals (LKFs) are developed to derive less conservative stability criteria. First, a new delay-product-type LKF, including single integral terms with time-varying delays as coefficients is developed, and two stability criteria with less conservatism due to more delay information included are established for different allowable delay sets. Second, the delay-product-type LKF is further improved by introducing several negative definite quadratic terms based on the idea of matrix-refined-function-based LKF, and two stability criteria with more cross-term information and less conservatism for different allowable delay sets are also obtained. Finally, a numerical example is utilized to verify the effectiveness of the proposed methods.

Design of optimal PID controller for multivariable time-varying delay discrete-time systems using non-monotonic Lyapunov-Krasovskii approach

Relaxed mixed convex combination lemmas: Application to stability analysis of time-varying delay systems

Many stability and stabilization problems of nonlinear time-varying systems lead to asymptotic behavior of (–K + L)-type systems, which consist of a K-function and an L-function. The stability of these systems is of fundamental importance for a series of stabilization problems of time-varying nonlinear control systems. Even though the asymptotical stability of such systems has been used widely and (in most cases) implicitly, we do not find a rigorous proof, in the literature, and the existing proof for a particular case is questionable. Under quite general conditions, we prove that the solution of these systems tends to 0 as t . Some generalizations are also obtained. As an immediate consequence, a general theorem is obtained for the stabilization of time-varying systems. Using the new framework, we examine several stability and stabilization problems. First of all, for cascade systems, two sets of sufficient conditions are obtained for uniformly asymptotical stability and globally asymptotical stability, respectively. Then we consider the stability of ISS and IISS systems. A new concept, namely, strong IISS, is proposed. Several stability properties for autonomous systems are extended to time-varying systems. Finally, we consider stabilization via detection. A rigorous proof is given for a smooth state feedback time-varying system with weak detectability to be stabilizable by means of an observer.

One of challenging issues on stability analysis of time-delay systems is how to obtain a stability criterion from a matrix-valued polynomial on a time-varying delay. The first contribution of this paper is to establish a necessary and sufficient condition on a matrix-valued polynomial inequality over a certain closed interval. The degree of such a matrix-valued polynomial can be an arbitrary finite positive integer. The second contribution of this paper is to introduce a novel Lyapunov-Krasovskii functional, which includes a cubic polynomial on a time-varying delay, in stability analysis of time-delay systems. Based on the novel Lyapunov-Krasovskii functional and the necessary and sufficient condition on matrix-valued polynomial inequalities, two stability criteria are derived for two cases of the time-varying delay. A well-studied numerical example is given to show that the proposed stability criteria are of less conservativeness than some existing ones.

The Lyapunov–Krasovskii functional (LKF) method is an effective tool to obtain delay-dependent stability criteria of time-varying delay systems. However, in the method, the requirement that there must exist an appropriate LKF for a system with a delay varying in a delay interval is too strict and may lead to considerable conservativeness. In order to get less conservative stability criteria, a novel stability analysis method called a delay-mode-based LKF (DMBLKF) method is first presented for nonlinear delay systems. Compared with the LKF method, the main advantage of the DMBLKF method lies in relaxing the aforementioned requirement and leading to a less conservative criterion in stability analysis. Then, with the help of the novel method, new stability criteria of linear delay systems are derived. Finally, the effectiveness of the proposed method and the obtained criteria is illustrated through two numerical examples.

Complete LKF approach to stabilization for linear systems with time-varying input delay

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems.

Recent work by Mazenc and Malisoff provided a trajectory-based approach to proving stability of time-varying systems with time-varying delay. It uses a contraction lemma, instead of Lyapunov-Krasovskii or Razumikhin functions. Here, we use their lemma, and a Lyapunov function for an undelayed system, to provide a new method to prove stability of linear continuous-time, time-varying systems with time-varying bounded delays. No constraint on the upper bound of delay is imposed, nor do we need any differentiability of the delay. Instead, we use an upper bound on an integral average of the delay. We prove input-to-state stability under disturbances.

This paper is concerned with the problem of delay-derivative-dependent stability analysis for Markovian jumping neutral-type interval time-varying delay systems with mixed delays. The first, based on the Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality approach, a delay-range-dependent condition for neutral-type Markovian jumping systems (NMJSs) with time-varying delays is obtained, which can guarantee global asymptotical stability of these NMJSs. Unlike the previous methods, the upper bound of the discrete delay and neutral delay derivative is taken into consideration even if this upper bound is larger than or equal to $$1(\dot{h}(t)<1,\dot{\tau}(t)<1)$$ . It is proved that the obtained results are less conservative than the existing ones. In our result, the time-varying delays are only assumed to be bounded. This has undoubtedly extended its application range. To better handle the problem on stability for neutral control systems, in which time-varying delay was involved, a stability criterion with less conservatism was put forward. Moreover, because our results in this paper are all based on the linear matrix inequality (LMI) approach, we can utilize Matlab’s LMI Control Toolbox to verify the global stability of correlation systems conveniently. As far as we are aware, this paper seems to be the first to discuss stability problems for neutral-type Markovian jumping systems with delay-derivative-dependence and delay-range-dependence.

This note analyzes the stochastic stability for time-varying neutral stochastic hybrid delay system, which includes the stability in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> th( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p\geq 2$</tex-math></inline-formula> )-moment, the asymptotical stability in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> th( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p\geq 2$</tex-math></inline-formula> )-moment, the exponential stability in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> th( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p\geq 2$</tex-math></inline-formula> )-moment, and the almost surely exponential stability. One modified version of generalized delay integral inequality, the Lyapunov–Krasovskii function, and the stochastic analysis are used. The proposed methodology can surmount the analytical difficulty, which stems from the coexistence of neutral term, stochastic disturbance, bounded time-varying delay, and a sign-changed time-varying coefficient in the diffusion condition. An example is given to show the effectiveness of the theoretical results obtained.

The design of a hypersonic vehicle controller has been an active research field in the last decade, especially when the vehicle is studied as a time-varying system. A time-varying compound control method is proposed for a hypersonic vehicle controlled by the direct lateral force and the aerodynamic force. The compound control method consists of a time-varying linear quadratic regulator (LQR) control law for the aerodynamic rudder and a sliding mode control law for the lateral thrusters. When the air rudder cannot continuously produce control force and torque, the direct lateral force is added to the system. To solve the problem that LQR cannot directly obtain the analytical solution of the time-varying system, a novel approach to approximate analytical solutions using Jacobi polynomials is proposed in this paper. Finally, the stability of the time-varying compound control system is proven by the Lyapunov–Krasovskii functional (LKF). The simulation results show that the proposed compound control method is effective and can improve the fast response ability of the system.

&lt;abstract&gt;&lt;p&gt;This work develops some novel approaches to investigate the stability analysis issue of linear systems with time-varying delays. Compared with the existing results, we give three innovation points which can lead to less conservative stability results. Firstly, two novel integral inequalities are developed to deal with the single integral terms with delay-dependent matrix. Secondly, a novel Lyapunov-Krasovskii functional with time-varying delay dependent matrix, rather than constant matrix is constructed. Thirdly, two improved stability criteria are established by applying the newly developed Lyapunov-Krasovskii functional and integral inequalities. Finally, three numerical examples are presented to validate the superiority of the proposed method.&lt;/p&gt;&lt;/abstract&gt;

In this paper, the stability analysis problem of time-varying delay systems is studied. Based on recent research on Lyapunov-Krasovskii functionals (LKFs), relaxed conditions have been found to weaken some matrices in terms of LKFs, which means that some integral terms cannot be strictly positive definite. Therefore, motivate by this consideration, a new relaxed condition is constructed to weaken some matrices in constructed LKF. Then, for the sake of obtaining more information of time delays, dynamic delay interval (DDI) method was introduced to address the double integral terms with time-varying delay. Furthermore, some recent technologies are employed to process derivatives of LKF. As a consequence, a less conservative result is obtained. The examples given by previous papers are used to demonstrate the superiority of our work.