Pseudo-predictor feedback stabilization of linear systems with time-varying input delays

This paper is concerned with stabilization of (time-varying) linear systems with a single time-varying input delay by using the predictor based delay compensation approach. Differently from the traditional predictor feedback which uses the open-loop system dynamics to predict the future state and will result in an infinite dimensional controller, we propose in this paper a pseudo-predictor feedback (PPF) approach which uses the (artificial) closed-loop system dynamics to predict the future state and the resulting controller is finite dimensional and is thus easy to implement. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system under the PPF are obtained in terms of the stability of a class of integral delay operators (systems). Moreover, it is shown that the PPF can compensate arbitrarily large yet bounded input delays provided the open-loop (time-varying linear) system is only polynomially unstable and the feedback gain is well designed. Comparison of the proposed PPF approach with the existing results is well explored. Numerical examples demonstrate the effectiveness of the proposed approaches.     

Observer-based output feedback control of discrete-time linear systems with input and output delays

In this paper, we study observer-based output feedback control of discrete-time linear systems with both multiple input and output delays. By generalising our recently developed truncated predictor feedback approach for state feedback stabilisation of discrete-time time-delay systems to the design of observer-based output feedback, two types of observer-based output feedback controllers, one being memory and the other memoryless, are constructed. Both full-order and reduced-order observer-based controllers are established in both the memory and memoryless schemes. It is shown that the separation principle holds for the memory observer-based output feedback controllers, but does not hold for the memoryless ones. We further show that the proposed observer-based output feedback controllers solve both the l₂ and l_∞ semi-global stabilisation problems. A numerical example is given to illustrate the effectiveness of the proposed approaches.     

H ∞ control for networked systems with random delays and packet dropouts

This paper is concerned with the control problem for a class of systems with random delays and packet dropouts which unavoidably occur during the data transmission in the network. A novel model is developed to describe the random one-step delays and multiple packet dropouts from sensors to controllers and from controllers to actuators by applying four Bernoulli distributed stochastic variables. An observer-based feedback controller is designed via a linear matrix inequality (LMI) approach which guarantees the closed-loop networked system to be exponentially stable in the sense of mean square and also achieves the prescribed H _?? disturbance-rejection-attenuation level. A simulation example shows the effectiveness of the proposed algorithm.     

Delay compensation for linear systems with both state and distinct input delays

This paper is concerned with the stabilization of linear systems with both state and distinct input delays. Nested predictor feedback controllers are designed to predict the future states such that the distinct input delays that can be arbitrarily large yet bounded are compensated completely. It is shown that the compensated closed‐loop system possesses the same characteristic equation as the closed‐loop system without distinct input delays. Both continuous‐time and discrete‐time time‐delay systems are studied in this paper. Moreover, the safe implementation problem for the continuous‐time nested predictor feedback controller is solved via adding input filters. Three numerical examples show the effectiveness of the proposed approaches.     

Stabilizing controllers using observers for uncertain systems with delays

In this paper, we study the problem of designing stabilizing controllers using observers for a class of uncertain systems with state and input delays. The uncertainties are assumed to be norm bounded. By employing a Lyapunov-Krasovskii functional approach, it is proven that the closed-loop observer-based controlled system is asymptotically stable. The observer and state feedback controllers are determined by solving two linear matrix inequalities which are independent of the delay factors. A simulation example is given to illustrate the theoretical developments.     

Design of Adaptive Block Backstepping Controllers with Perturbations Estimation for Nonlinear State‐Delayed Systems in Semi‐Strict Feedback Form

This paper is an extended study of an existing block backstepping control scheme designed for a class of perturbed multi‐input systems with multiple time‐varying delays to solve regulation problems, where the time‐varying delays must be linear with state variables. A new control scheme is proposed in this research where all the unknown multiple time‐varying delay terms in the dynamic equations can be nonlinear state functions in non‐strict feedback form, and the upper bounds of the time‐delays as well as their derivatives need not to be known in advance. Another improvement is to further alleviate the problem of “explosion of complexity,” i.e., to reduce the number of time derivatives of virtual inputs that the designers have to compute in the design of controllers. This is done by utilizing an existent derivative estimation algorithm to estimate the perturbations in the designing of proposed controllers. Adaptive mechanisms are also embedded in the controllers so that the upper bounds of perturbations and perturbation estimation errors are not required to be known beforehand. The resultant controlled systems guarantee asymptotic stability in accordance with the Lyapunov stability theorem. Finally, a numerical example and a practical application are demonstrated to verify the merits and feasibility of the proposed control scheme.     

Recursive predictor design for state and output feedback controllers for linear time delay systems

This paper presents a recursive method to design state and output feedback controllers for MIMO, block-feedforward linear systems with delays in the inputs, outputs, and interconnections between the blocks. The resulting controller is of predictor-type, which means that it contains finite integrals over past state and input values. The method is a generalization of the well-known model reduction approach for systems with input delay. A recursive procedure replaces delay terms with non-delay ones step by step, from the top of the cascade structure down. Controller gains are computed for the proxy system without delays, while the construction guarantees the same closed loop poles for the delay system and the proxy one. The observer is designed by applying the duality argument and the separation principle is also shown to apply.     

不确定时滞系统的无记忆鲁棒控制

讨论了含有时变时滞以及时变输入控制的 时滞线性系统的鲁棒可镇定性问题,就所有可结构化不确定参数以及所有非结构化不确定参 数两种情形分别得到了系统可由状态反馈控制器镇定的充分条件,并且相应给出了控制器的 设计．     

An LMI approach to design extended predictor feedback controllers for dynamical systems with input delay

The stabilisation of dynamical systems with input delay by an extended predictor feedback controller is investigated. After considering a more general case of the predictor feedback controllers, the design procedure is transformed into the stabilisation problem of the system controlled by a dynamic output feedback controller. By proposing appropriate Lyapunov-Krasovskii functionals, the predictor feedback gains are then synthesised from the stability conditions which are obtained in form of linear matrix inequalities. Such design approach is more flexible and extendable in comparison to one used by the conventional predictor feedback control. As a simulation example, the designed extended predictor feedback controller is applied on an active suspension system and is used to stabilise an unstable system with input delays. The obtained results demonstrate the effectiveness of the design method.     

Output transformations and separation results for feedback linearisable delay systems

The class of strict-feedback systems enjoys special properties that make it similar to linear systems. This paper proves that such a class is equivalent, under a change of coordinates, to the wider class of feedback linearisable systems with multiplicative input, when the multiplicative terms are functions of the measured variables only. We apply this result to the control problem of feedback linearisable nonlinear MIMO systems with input and/or output delays. In this way, we provide sufficient conditions under which a separation result holds for output feedback control and moreover a predictor-based controller exists. When these conditions are satisfied, we obtain that the existence of stabilising controllers for arbitrarily large delays in the input and/or the output can be proved for a wider class of systems than previously known.     

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.     

Necessary and Sufficient Stability Criterion and Stabilization for Positive 2‐D Continuous‐Time Systems with Multiple Delays

This paper addresses the stability and control problem of the linear positive two‐dimensional (2‐D) continuous‐time systems in Roesser model with multiple time delays. The contribution lies in two aspects. First, a simple novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for 2‐D continuous delayed systems. It turns out that the magnitude of delays has no any impact on the stability of these systems, which is completely determined by the system matrices. Second, a necessary and sufficient condition for the existence of state‐feedback controllers is proposed for general delayed 2‐D systems, which ensures the non‐negativity and the stability of the resulting closed‐loop systems. Two examples are given to validate the proposed methods.     

Optimal control for linear systems with multiple time delays in control input

This note presents the optimal linear-quadratic (LQ) regulator for a linear system with multiple time delays in the control input. Optimality of the solution is proved in two steps. First, a necessary optimality condition is derived from the maximum principle. Then, the sufficiency of this condition is established by verifying that it satisfies the Hamilton-Jacobi-Bellman equation. Using an illustrative example, the performance of the obtained optimal regulator is compared against the performance of the optimal LQ regulator for linear systems without delays and some other feasible feedback regulators that are linear in the state variables. Finally, the note establishes a duality between the solutions of the optimal filtering problem for linear systems with multiple time delays in the observations and the optimal LQ control problem for linear systems with multiple time delays in the control input.     

Stabilization of linear systems with distributed input delay and input saturation

This paper is concerned with stabilization of a linear system with distributed input delay and input saturation. Both constant and time-varying delays are considered. In the case that the input delay is constant, under the stabilizability assumption on an auxiliary system, it is shown that the system can be stabilized by state feedback for an arbitrarily large delay as long as the open-loop system is not exponentially unstable. In the case that the input delay is time-varying, but bounded, it is shown that the system can be stabilized by state feedback if the non-asymptotically stable poles of the open-loop system are all located at the origin. In both cases, stabilizing controllers are explicitly constructed by utilizing the parametric Lyapunov equation based low gain design approach we recently developed. It is also shown that in the presence of actuator saturation and under the same assumptions on the system, these controllers achieve semi-global stabilization. Some discussions on the assumptions we impose on the system are given. A numerical example illustrates the effectiveness of the proposed stabilization approach.     

H∞ Control of Discrete-Time Systems With Multiple Input Delays

This paper is concerned with the finite horizon H_infin full-information control for discrete-time systems with multiple control and exogenous input delays. We first establish a duality between the H_infin full-information control and the H₂ smoothing of a stochastic backward system in Krein space. Like the duality between the linear quadratic regulation (LQR) of linear systems without delays and the Kalman filtering, the established duality allows us to address complicated multiple input delay problems in a simple way. Indeed, by applying innovation analysis and standard projection in Krein space, in this paper we derive conditions under which the H_infin full-information control is solvable. An explicit controller is constructed in terms of two standard Riccati difference equations of the same order as the original plant (ignoring the delays). As special cases, solutions to the H_infin state feedback control problem for systems with delays only in control inputs and the H_infin control with preview are obtained. An example is given to demonstrate the effectiveness of the proposed H_infin control design     

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.     

Control discrete-time switched singular systems with state delays under asynchronous switching

This article is concerned with the problem of state feedback control for a class of discrete-time switched singular systems with time-varying state delays under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the system modes. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence a stabilising switching law is first derived to guarantee the regularity, causality and exponential stability of the closed-loop system in the presence of asynchronous switching. The stabilising switching law is characterised by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.     

Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approach

This paper addresses the problem of designing robust static output-feedback controllers for nonlinear discrete-time interval systems with time delays both in states and in control input. In the approach, we do not directly employ the Lyapunov approach, as do in most of traditional fuzzy control design approaches. Instead, sufficient conditions for guaranteeing the robust stability for the considered systems are derived in terms of the matrix spectral norm of the closed-loop fuzzy system. The stability conditions are further formulated into linear matrix inequalities so that the desired controller can be easily obtained by using the Matlab linear matrix inequality (LMI) toolbox. Finally, an example is provided to illustrate the effectiveness of the proposed approach.     

Controller design for a class of nonlinear systems with input saturation using convex optimization

In this paper, we propose a new design strategy for nonlinear systems with input saturation. The resulting nonlinear controllers are locally asymptotically stabilizing the origin. The proposed methodology is based on exact feedback linearization which is used to reformulate the nonlinear system as a linear system having state-dependent input saturation. Linear saturating state feedback controllers and soft variable-structure controllers are developed based on this system formulation. The resulting convex optimization problems can be written in terms of linear matrix inequalities and sum of squares conditions for which efficient solvers exist. Polynomial approximation based on Legendre polynomials is used to extend the methodology to a more general class of nonlinear systems. To demonstrate the benefit of this design method, a stabilizing controller for a single link manipulator with flexible joint is developed.     

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.