Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

This paper focuses on the problem of delay-dependent robust stochastic stability analysis and controller synthesis for Markovian jump systems with state and input delays. It is assumed that the delays are constant and unknown, but their upper bounds are known. By constructing a new Lyapunov-Krasovskii functional and introducing some appropriate slack matrices, new delay-dependent stochastic stability and stabilization conditions are proposed by means of linear matrix inequalities (LMIs). An important feature of the results proposed here is that all the robust stability and stabilization conditions are dependent on the upper bound of the delays. Memoryless state feedback controllers are designed such that the closed-loop system is robustly stochastically stable. Some numerical examples are provided to illustrate the effectiveness of the proposed method.     

A full sensing window Random-Access algorithm for messages with strict delay constraints

We consider the Channel Multiple-Access problem for messages with strict delay constraints. The constraints are represented by an upper bound on the transmission delays. For this problem, and for binary collision-noncollision feedback per slot, we present a simple full sensing window Random-Access algorithm. We analyze the algorithm and we compute the fraction of maintained traffic and the expected delay for the successfully transmitted packet, for various input Poisson intensities and various values of the bound on the transmission delays.     

一类多时滞非线性系统的自适应鲁棒镇定

本文针对一类具有多时滞状态扰动的非线性系统,讨论了其自适应鲁棒镇定问题。在本文中,多时滞状态扰动的上界未知,通过设计自适应律估计上界的值。基于Lyapumov-Krasovskii函数设计了鲁棒控制器,使闭环系统的鲁棒镇定问题可解。一个数值例子的仿真验证了结论的正确性。     

Adaptive robust control of uncertain nonlinear systems with nonlinear delayed state perturbations

The problem of robust stabilization of uncertain nonlinear dynamical systems with multiple time delays is considered. In the paper, the upper bound of the nonlinearity and uncertainty, including delayed states, is assumed to be a linear function of some parameters which are still assumed to be unknown. Here, we do not require that the nonlinear terms including delayed states are linear norm-bounded in the states. An improved adaptation law with σ-modification is employed to estimate the unknown parameters, and a class of memoryless adaptive robust state feedback controllers is proposed. It is also shown that the proposed adaptive robust controllers can guarantee the uniform asymptotic stability of uncertain nonlinear time-delay systems. Finally, as a numerical example, an uncertain time-delay ecosystem with two competing species is given to demonstrate the validity of the results.     

A full sensing window Random-Access algorithm for messages with strict delay constraints

We consider the Channel Multiple-Access problem for messages with strict delay constraints. The constraints are represented by an upper bound on the transmission delays. For this problem, and for binary collision-noncollision feedback per slot, we present a simple full sensing window Random-Access algorithm. We analyze the algorithm and we compute the fraction of maintained traffic and the expected delay for the successfully transmitted packet, for various input Poisson intensities and various values of the bound on the transmission delays.This work was supported by the U.S. Office of Naval Research, under Contract ONR-N14-86-K-0742.     

Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time‐delay systems

In this paper, an improved linear matrix inequality (LMI)‐based robust delay‐dependent stability test is introduced to ensure a larger upper bound for time‐varying delays affecting the state vector of an uncertain continuous‐time system with norm‐bounded‐type uncertainties. A quasi‐full‐size Lyapunov–Krasovskii functional is chosen and free‐weighting matrix approach is employed. Less restrictive sufficient conditions are derived for robust stability of time‐varying delay systems with norm‐bounded‐type uncertainties. Moreover, the investigation of the stabilization problem with memoryless state‐feedback control is presented such that the stabilizability criteria are obtained in terms of matrix inequalities, which can be solved via utilizing a cone complementarity minimization algorithm. Finally, the problem of output feedback stabilization for square systems is also taken into consideration. The output feedback stabilizability criteria are derived in the form of linear matrix inequalities, which are convex and can be easily solved using interior point algorithms. A plenty of numerical examples are presented indicating that the proposed stability and stabilization methods effectively improve the existing results. Copyright © 2006 John Wiley & Sons, Ltd.     

Conditions for the delay-independent stabilization and the decay rate assignability of linear systems

In a previous paper (Amemiya 1983) two sufficient conditions were introduced for the delay-independent stabilization of linear systems by means of state-variable feedback including no delays.

In this paper the concept of delay-independently upper or lower bound of decaying rate assignability (DIUDA or DILDA) are introduced and it is proved that the two conditions previously obtained are necessary and sufficient condition for DIUDA and necessary condition for DILDA of linear systems by means of state-variable feedback including no delays. The conditions obtained are also applicable to systems including time-varying delays.     

On the Achievable Delay Margin Using LTI Control for Unstable Plants

Handling delays in control systems is difficult and is of long-standing interest. It is well known that, given a finite-dimensional linear time-invariant (FDLTI) plant and controller forming a strictly proper stable feedback connection, closed-loop stability will be maintained under a small delay in the feedback loop, although most closed loop systems become unstable for large delays. One previously unsolved fundamental problem in this context is whether, for a given FDLTI plant, an arbitrarily large delay margin can be achieved using LTI control. Here, we adopt a frequency domain approach and demonstrate that, for a strictly proper real rational plant, there is a uniform upper bound on the delay that can be tolerated when using an LTI controller, if and only if the plant has at least one closed right half plane pole not at the origin. We also give several explicit upper bounds on the achievable delay margin, and, in some special cases, demonstrate that these bounds are tight.     

Containment control for second-order multi-agent systems with time-varying delays

This paper considers the containment control problem for second-order multi-agent systems with time-varying delays. Both the containment control problem with multiple stationary leaders and the problem with multiple dynamic leaders are investigated. Sufficient conditions on the communication digraph, the feedback gains, and the allowed upper bound of the delays to ensure containment control are given. In the case that the leaders are stationary, the Lyapunov–Razumikhin function method is used. In the case that the leaders are dynamic, the Lyapunov–Krasovskii functional method and the linear matrix inequality (LMI) method are jointly used. A novel discretized Lyapunov functional method is introduced to utilize the upper bound of the derivative of the delays no matter how large it is, which leads to a better result on the allowed upper bound of the delays to ensure containment control. Finally, numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.     

具有有界时滞的网络控制系统的镇定

针对具有有界时滞且时滞上界大于一个采样周期的网络控制系统,研究了系统建模和状态反馈镇定问题.在分析有界时滞的所有可能性的基础上,提出一种能够用于处理时变控制律问题的网络控制系统数学模型,进而将该系统的镇定问题转化为镇定一系列模型的鲁棒控制问题.根据 Lyapunov 方法,给出了保证闭环系统稳定的状态反馈控制器.仿真算例验证了所提出方法的有效性.     

带传输滞后的线性离散系统的状态反馈镇定

针对存在传输滞后的线性离散系统的状态反馈镇定问题,给出了系统可镇定的一个内部限制条件.为克服这一限制条件,提出了两种方法:一种是充分利用滞后状态的信息,另一种是设计带有递推动态的状态反馈控制器.研究结果表明,若系统在没有传输滞后时能通过状态反馈被镇定,则存在传输滞后时一定也能通过设计新的控制器使系统被镇定.     

不确定多时滞离散切换系统的输出反馈镇定

针对一类具有多重状态时滞和凸多面体不确定性的离散切换系统,研究该系统在任意切换信号下基于状态观测器的输出反馈鲁棒镇定问题.首先通过状态变量的替换将多时滞项转变成不含时滞项,再利用分段Lyapunov函数和线性矩阵不等式方法,导出凸多面体不确定多时滞离散切换系统基于观测器的输出反馈鲁棒镇定的充分条件,并将此条件转化成为一组线性矩阵不等式的可行性问题,同时给出基于状态观测器的输出反馈控制器.仿真结果表明设计方法是可行有效的.     

具有短时延的网络控制系统的一种鲁棒控制方法

针对具有短时延的网络控制系统, 本文提出了一种基于鲁棒控制的方法来解决该类系统的稳定化控制问题. 考虑状态反馈控制律, 将闭环网络控制系统描述为一个离散时间线性不确定系统模型, 其中的不确定部分反映了时延的时变特性对系统动态的影响. 得到了该闭环网络控制系统的渐近稳定性条件, 且该条件建立了闭环网络控制系统稳定性与两个时延参数, 即允许时延上界和允许时延变换范围, 之间的定量关系. 进一步的, 还给出了稳定化反馈控制器的设计步骤. 最后通过一个示例验证了本文所提出方法的有效性.     

Event-triggered control for stochastic systems with multiple delays

This paper investigates the stabilization problem for continuous-time stochastic systems with multiple delays under continuous event-triggered mechanisms, of which both static case and dynamic case are considered individually. In order to avoid zeno phenomenon in every sample path, a suspension time after each successful execution is forced for our event-triggered mechanisms, resulting in intermittent detection of system states. Under such control strategy, we deduce mean square exponential stability of stochastic systems with multiple delays by means of Hanalay-type inequality and obtain a delay-dependent-based and less-conservative stabilization criterion without involving the upper bound of time delays. Besides, a co-design procedure is proposed for linear controller and event-triggered mechanisms. In the end, an illustrative example is presented to show effectiveness of the proposed co-design procedure and contrasts the system performance under static and dynamic event-triggered mechanisms.     

Stabilization of linear system with input saturation and unknown constant delays

In this paper, we study the stabilization of linear critically unstable systems subject to input saturation and multiple unknown input delays. We find tight upper bounds for delays which are inversely proportional to the maximal magnitude of open-loop eigenvalues on the imaginary axis. For delays satisfying these upper bounds, linear low-gain state and finite dimensional dynamic measurement feedbacks are constructed to solve the semi-global stabilization problem. The effectiveness of the proposed design is illustrated by an example.     

Absolute stability and stabilization for Lurie networked control systems

This paper investigates the problem of absolute stability and stabilization for networked control systems (NCSs) with the controlled plant being Lurie systems (Lurie NCSs), in which the network‐induced delays are assumed to be time‐varying and bounded. By considering the relationship between the network‐induced delay and its upper bound, an improved stability criterion for networked control system is proposed. Furthermore, the resulting condition is extended to design a state feedback controller by employing an improved cone complementary linearization (ICCL) algorithm. A numerical example is worked out to illustrate the effectiveness and the benefits of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Decentralized robust control of uncertain Markov jump parameter systems via output feedback

This paper addresses the problem of decentralized robust stabilization and control for a class of uncertain Markov jump parameter systems. Control is via output feedback and knowledge of the discrete Markov state. It is shown that the existence of a solution to a collection of mode-dependent coupled algebraic Riccati equations and inequalities, which depend on certain additional parameters, is both necessary and sufficient for the existence of a robust decentralized switching controller. A guaranteed upper bound on robust performance is also given. To obtain a controller which satisfies this bound, an optimization problem involving rank constrained linear matrix inequalities is introduced, and a numerical approach for solving this problem is presented. To demonstrate the efficacy of the proposed approach, an example stabilization problem for a power system comprising three generators and one on-load tap changing transformer is considered.     

Design of delay-range-dependent robust controller for uncertain genetic regulatory networks with interval time-varying delays

This paper investigates the problem of robust stabilization for genetic regulatory networks with interval time-varying delays, which are subject to norm-bounded time-varying parameter uncertainties. The time delays including lower and upper bounds of delay are assumed to appear in both the mRNA and protein. The regulatory functions are assumed to be globally Lipschitz continuous. The resulting delay-range-dependent robust controller with interval range is designed in terms of improved bounding technique. A sufficient condition for the solvability of the problem is obtained via a linear matrix inequality (LMI). When this LMI is feasible, an explicit expression of a desired state feedback controller is also given. The theory developed in this paper is demonstrated by two numerical examples.     

Stability and Stabilization of Networked Control Systems Under Limited Communication Capacity and Variable Sampling

This paper investigates the problem of quantized feedback control for networked control systems (NCSs) with time‐varying delays and time‐varying sampling intervals, wherein the physical plant is a continuous‐time, and the control input is a discrete‐time signal. By using an input delay approach and a sector bound method, the network induced delays, the signal quantization and sampling intervals are presented in one framework in the case of the state and the control input by quantization in a logarithmic form. We exploit a novel Lyapunov functional with discontinuity, taking full advantage of the NCS characteristic information including the bounds of delays, the bounds of sampling intervals and quantization parameters. In addition, it has been shown that the Lyapunov functional is decreased at the jump instants. Furthermore, we use the Leibniz‐Newton formula and free‐weighting matrix method to obtain the stability analysis and stabilization conditions which are dependent on the NCS characteristic information. The proposed stability analysis and stabilizing controller design conditions can be presented in term of linear matrix inequalities, which have less conservativeness and less computational complexity. Four examples demonstrate the effectiveness of the proposed methods.     

Stabilization of discrete-time linear systems with a time delay in the feedback loop

A discrete-time, linear, time-invariant control system with a fixed time delay in the feedback loop is considered. We investigate the problem of feedback stabilization and present some general criteria. Simple necessary and sufficient conditions for stabilization, which take the form of algebraic inequalities involving the system parameters, are developed for time delays whose values do not exceed 2, as well as for one-dimensional systems with an arbitrary time delay.