Simultaneous compensation of input and state delays for nonlinear systems

The problem of compensation of arbitrary large input delay for nonlinear systems was solved recently with the introduction of the nonlinear predictor feedback. In this paper we solve the problem of compensation of input delay for nonlinear systems with simultaneous input and state delays of arbitrary length. The key challenge, in contrast to the case of only input delay, is that the input delay-free system (on which the design and stability proof of the closed-loop system under predictor feedback are based) is infinite-dimensional. We resolve this challenge and we design the predictor feedback law that compensates the input delay. We prove global asymptotic stability of the closed-loop system using two different techniques—one based on the construction of a Lyapunov functional, and one using estimates on solutions. We present two examples, one of a nonlinear delay system in the feedforward form with input delay, and one of a scalar, linear system with simultaneous input and state delays.     

Nonlinear control of feedforward systems with bounded signals

The stabilization problem for a class of nonlinear feedforward systems is solved using bounded control. It is shown that when the lower subsystem of the cascade is input-to-state stable and the upper subsystem not exponentially unstable, global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the "upper" part of the state only. This is made possible by invoking the bounded real lemma and a generalization of the small gain theorem. Thus, stabilization is achieved with typical saturation functions, saturations of constant sign, or quantized control. Moreover, the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback. Finally, a new class of stabilizing control laws for a chain of integrators with input saturation is proposed. Some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems.     

Consensus of high-order multi-agent systems with large input and communication delays

We study in this paper the consensus problem for multi-agent systems with agents characterized by high-order linear systems with time delays in both the communication network and inputs. Provided that the open-loop dynamics of the agents is not exponentially unstable, but may be polynomially unstable, and the communication topology contains a directed spanning tree, a truncated predictor feedback approach is established to solve the consensus problem. It is shown that, if the delays are constant and exactly known, the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays. If it is further assumed that the open-loop dynamics of the agents only contains zero eigenvalues, the delays are allowed to be time-varying and unknown. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.     

Design of stabilizing controllers of upper triangular nonlinear time-delay systems

In this paper, it is considered the state feedback controller design for a class of upper triangular nonlinear systems with simultaneous input and state delays. By using the state transformation of nonlinear systems, the problem of designing controller can be converted into that of designing a dynamic parameter, which is dynamically regulated by a dynamic equation. Then, by appraising the nonlinear terms of the given systems, a dynamic equation can be delicately constructed. At last, with the help of Lyapunov stability theorem, it is provided the stability analysis for the closed-loop system consisting of the designed controller and the given systems. Both discrete delays and continuous delays with integral form are considered here. Different from many existing control designs for upper triangular nonlinear systems, neither forwarding recursive nor saturation computation is utilized here, and thus our design procedure is simpler. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

Exponential stability of nonlinear time-delay systems with delayed impulse effects

The problem of exponential stability for nonlinear time-delay systems with delayed impulses is addressed in this paper. Lyapunov-based sufficient conditions for exponential stability are derived, respectively, for two kinds of delayed impulses (i.e., destabilizing delayed impulses and stabilizing delayed impulses). It is shown that if a nonlinear impulsive time-delay system without impulse input delays is exponentially stable, then under some conditions, its stability is robust with respect to small impulse input delays. Moreover, it is also shown that for a stable nonlinear impulsive time-delay system, if the magnitude of the delayed impulses is sufficiently small, then under some conditions, the delayed impulses do not destroy the stability irrespective of the sizes of the impulse input delays. The efficiency of the proposed results is illustrated by three numerical examples.     

Adaptive robust stabilisation for a class of uncertain nonlinear time-delay dynamical systems

The problem of adaptive robust stabilisation is considered for a class of uncertain nonlinear dynamical systems with multiple time-varying delays. It is assumed that the upper bounds of the nonlinear delayed state perturbations are unknown and that the time-varying delays are any non-negative continuous and bounded functions which do not require that their derivatives have to be less than one. In particular, it is only required that the nonlinear uncertainties, which can also include time-varying delays, are bounded in any non-negative nonlinear functions which are not required to be known for the system designer. For such a class of uncertain nonlinear time-delay systems, a new method is presented whereby a class of continuous memoryless adaptive robust state feedback controllers with a rather simpler structure is proposed. It is also shown that the solutions of uncertain nonlinear time-delay systems can be guaranteed to be uniformly exponentially convergent towards a ball which can be as small as desired. Finally, as an application, an uncertain nonlinear time-delay ecosystem with two competing species is given to demonstrate the validity of the results.     

Bounded control of feedforward nonlinear systems subject to input time‐delay

This article is concerned with the global stabilization problem of a family of feedforward nonlinear time‐delay systems whose linearized system consists of multiple distinct oscillators. To fully utilize the delayed information and maintain the state decoupling property in the controller design, the considered nonlinear feedforward system is first transformed into a new system which contains time delays in both its input and states based on a novel model transformation containing time delays, and then the stabilizing saturated controller for the transformed system is designed based on the recursive design method. Meanwhile, explicit stability conditions are also provided. When the linearized system is a cascade of multiple oscillators and multiple integrators, a modified saturated feedback control utilizing not only the current state but also the delayed state is also established for the corresponding global stabilization problem. Two examples, including a practical one, are given to show the effectiveness and superiority of the proposed approaches.     

Control of large scale interconnected systems with input and state delays using decentralized adaptive state observers

This paper investigates the problem of decentralized model reference adaptive control (MRAC) for a class of large scale interconnected systems with both state and input delays. The upper bounds of the interconnection terms are considered to be unknown. Time varying delays in the nonlinear interconnection terms are bounded and nonnegative continuous functions and their derivatives are not necessarily less than one. For exact prediction, a decentralized adaptive state observer is designed and a nested predictor based approach is established to predict the future states of the input delay compensation. It is shown that the solutions of uncertain large‐scale time‐delay interconnected systems converge uniformly exponentially to a desired small ball. The effectiveness of the proposed approaches is illustrated by two examples.     

A fast identification algorithm for systems with delayed inputs

A fast identification algorithm is proposed for systems with delayed inputs. It is based on a non-asymptotic distributional estimation technique initiated in the framework of systems without delay. Such a technique leads to simple realisation schemes, involving integrators, multipliers and piecewise polynomial or exponential time functions. Thus, it allows for a real-time implementation. In order to introduce a generalisation to systems with input delay, three simple examples are presented here. The first illustration is a first-order model with delayed input and noise. Then, a second-order system driven through a transmission line is considered. A third example shows a possible link between simultaneous identification and generalised eigenvalue problems.     

Minimum-energy control for time-varying systems with multiple state and input delays

This paper considers the minimum-energy control problem for a class of time-varying systems with multiple state and input delays. First, a state transform matrix is presented. By using the transform matrix, a system with multiple state and input delays is converted to a formal equivalent one without delay. Then, the optimal problem of the novel system is solved by using the maximum principle. Analytical expressions of the optimal control law and optimal performance are given by two formulas with respect to the state transform matrix. At last, an algorithm is given to solve the analytical expression of the state transform matrix. Simulation results show that the design algorithm is efficient and easy to implement.     

Finite-time stability and stabilization of time-delay systems

Finite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artstein’s model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input.     

On Asymptotic Stabilizability of Linear Systems With Delayed Input

This paper examines the asymptotic stabilizability of linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open-loop system is not exponentially unstable (i.e., all the open-loop poles are on the closed left-half plane). A simple example shows that such results would not be true if the open-loop system is exponentially unstable. It is further shown that such systems, when subject to actuator saturation, are semiglobally asymptotically stabilizable by linear state or output feedback.     

Relationships between asymptotic stability and exponential stability of positive delay systems

This paper explores the relations between asymptotic stability and exponential stability of continuous-time and discrete-time positive systems with delay. A system is said to be positive if its state and output are non-negative whenever the initial condition and input are non-negative. Two results are obtained. First, if a positive system is asymptotically stable for all bounded (further continuous, for continuous-time systems) time-varying delays, then it is exponentially stable for all such delays. In particular, if a positive system is asymptotically stable for a given constant delay, then it is exponentially stable for all constant delays. Second, if the involved delays are unbounded, then the positive system may be not exponentially stable even if it is asymptotically stable.     

Robust stabilizers for an implicit dynamical delay system of the neutral type

A feedback stabilization problem is investigated for a class of imperfectly known implicit systems with discrete and distributed delays. The imperfections acting on the systems, which may be time‐, state‐, delayed state‐, and/or input‐dependent, are modelled as additive nonlinear perturbations influencing a known set of nonlinear functional differential equations of the neutral type. Sufficient conditions, which include a delay‐dependent matrix inequality and a delay‐dependent stability criterion involving some bounding parameters for the uncertainty in the system, are presented and a class of robust feedback stabilizers, including both memoryless and those with memory, are designed to guarantee a prescribed stability property for the class of implicit systems. Copyright © 2005 John Wiley & Sons, Ltd.     

随机观测时滞系统的最优估计

研究了在观测中存在Markov跳跃时滞的离散时间系统的线性最小方差状态估计问题. 首先, 通过引入跳跃时滞的示性函数, 将带有跳跃时滞的观测方程转化为带有乘性噪声的定常时滞系统. 进一步采用状态扩维的方法, 将定常时滞系统转化为无时滞的Markov跳跃系统. 最后, 基于得到的无时滞系统, 采用Hilbert空间已有的几何论知识, 设计线性最优状态估计器, 得出基于Riccati方程的滤波器的表达式, 并证明了所得滤波器的渐渐收敛性.     

Neural observer-based adaptive compensation control for nonlinear time-varying delays systems with input constraints

This paper describes a neural network state observer-based adaptive saturation compensation control for a class of time-varying delayed nonlinear systems with input constraints. An advantage of the presented study lies in that the state estimation problem for a class of uncertain systems with time-varying state delays and input saturation nonlinearities is handled by using the NNs learning process strategy, novel type Lyapunov-Krasovskii functional and the adaptive memoryless neural network observer. Furthermore, by utilizing the property of the function tan h²(?/?)/?, NNs compensation technique and backstepping method, an adaptive output feedback controller is constructed which not only efficiently avoids the problem of controller singularity and input saturation, but also can achieve the output tracking. And the proposed approach is obtained free of any restrictive assumptions on the delayed states and Lispchitz condition for the unknown nonlinear functions. The semiglobal uniform ultimate boundedness of all signals of the closed-loop systems and the convergence of tracking error to a small neighborhood are all rigorously proven based on the NN-basis function property, Lyapunov method and sliding model theory. Finally, two examples are simulated to confirm the effectiveness and applicability of the proposed approach.     

Delay‐range‐dependent control of nonlinear time‐delay systems under input saturation

This paper describes a delay‐range‐dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time‐delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay‐range‐dependent and delay‐dependent stabilization problems for linear and nonlinear time‐delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay‐rate‐independent condition for control of nonlinear systems in the presence of input saturation with unknown delay‐derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L₂ gain reduction from disturbance to output is devised to address the problem of the stabilization of input‐constrained nonlinear time‐delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time‐delay network and a large‐scale chemical reactor under input saturation and varying interval time‐delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive control for a class of nonlinear systems with time-varying delays in state and input

This paper is concerned with the adaptive stabilization problem of uncertain input delayed systems.A solution to this problem is given for a class of uncertain nonlinear systems with time-varying delays in both state and input.An adaptive asymptotically stabilizing controller,which can guarantee the stability of the closed-loop system and the convergence of the original system state,is designed by means of the Lyapunov-Krasovskii functional stability theory combined with linear matrix inequalities (LMIs) an...     

Global tracking of nonlinear systems with simultaneous unknown time‐varying state and input delays

This paper presents the design of a robust control law for a class of nonlinear dynamical systems subjected to parametric uncertainty and simultaneous unknown, variable state and input delays. A novel controller is developed, which consists of a filtered tracking error and the integral of previous values of control input where the limits of integration are dependent on the known bound of the input delay. Lyapunov‐Krasovskii functionals–based stability analysis guarantees a global uniformly ultimately bounded tracking result where sufficient conditions on controller gains and maximum allowable delay are derived. The performance and robustness of the controller are evaluated by simulation on a two‐link robot manipulator for different combinations of time‐varying state and input delays.     

不确定状态滞后系统时滞相关鲁棒H∞控制

研究了不确定状态滞后系统的鲁棒H∞控制问题,假定参数不确定性时变未知但有 界.基于LMI方法,提出了一种新的鲁棒无记忆状态反馈H∞,控制器的设计方法,得出的结 论与时滞大小有关,相对于时滞无关的结论具有较少的保守性.