Linear quadratic regulation for linear time-varying systems with multiple input delays

This paper studies the classic linear quadratic regulation (LQR) problem for both continuous-time and discrete-time systems with multiple input delays. For discrete-time systems, the LQR problem for systems with single input delay has been studied in existing literature, whereas a solution to the multiple input delay case is not known to our knowledge. For continuous-time systems with multiple input delays, the LQR problem has been tackled via an infinite dimensional system theory approach and a frequency/time domain approach. The objective of the present paper is to give an explicit solution to the LQR problem via a simple and intuitive approach. The main contributions of the paper include a fundamental result of duality between the LQR problem for systems with multiple input delays and a smoothing problem for an associated backward stochastic system. The duality allows us to obtain a solution to the LQR problem via standard projection in linear space. The LQR controller is simply constructed by the solution of one backward Riccati difference (for the discrete-time case) or differential (for the continuous-time case) equation of the same order as the plant (ignoring the delays).     

具有时变时滞的离散系统的H∞控制

主要研究了控制输入和外部输入中带有时变时滞的离散系统的有限时间H无穷控制问题. 首先通过定义一个取值为0或1的二元变量将具有时变时滞的单通道离散系统转化为多通道定时滞离散系统, 进一步利用该系统H无穷控制与相应倒向随机系统平滑估计之间的对偶关系解决该问题,通过与原系统维数相同的Riccati差分方程给出了该问题的解,并且给出了因果与严格因果H控制器存在的充分必要条件.     

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.     

时变采样周期网络控制系统的鲁棒容错控制器设计

研究具有时变采样周期网络控制系统的执行器失效的完整性问题．假设系统任意两个连续采样间隔具有上界,利用输入时延法,将时变采样周期网络控制系统等价转化为连续时变时延网络控制系统．在此基础上,基于时延条件,应用Ｌｙａｐｕｎｏｖ稳定性理论和线性矩阵不等式（ＬＭＩｓ）方法证明了鲁棒容错控制律的存在条件,设计了鲁棒容错控制器,并给出了系统完整性条件下的最大允许时延的估计方法．仿真结果验证了所提方法的可行性和有效性．     

Linear quadratic tracking problem for discrete‐time systems with multiple delays in single input channel

The linear quadratic tracking problem for discrete‐time systems with multiple delays in single input channel is considered. In this paper, we provide an approach without resorting to system state augmentation. The optimal tracking control is given in terms of the current state, the previous inputs, and the output of an auxiliary backward deterministic delay system which is formulated for the first time in this paper. The solution relies on a Riccati difference equation of the same dimension as the plant (ignoring the delays). The key to our development is the establishment of a duality between the optimal tracking control and the optimal smoothing estimation of an associated stochastic backward system as well as the introduction of the auxiliary backward deterministic delay system. An analysis of the computational complexity of the proposed approach and its comparison with that of the augmentation method, which is to incorporate the delayed inputs into the augmented state, are provided. An example is given to demonstrate the effectiveness of the results. Copyright © 2009 John Wiley & Sons, Ltd.     

Exponential stability of singular systems with multiple time-varying delays

This paper deals with the class of continuous-time singular linear systems with multiple time-varying delays in a range. The global exponential stability problem of this class of systems is addressed. Delay-range-dependent sufficient conditions such that the system is regular, impulse-free and α-stable are developed in the linear matrix inequality (LMI) setting. Moreover, an estimate of the convergence rate of such stable systems is presented. A numerical example is employed to show the usefulness of the proposed results.     

Stabilization of linear systems with simultaneous state,actuation, and measurement delays

This paper considers the problem of stabilizing continuous-time linear systems with time delays. Specifically, a fixedorder (i.e. full- and reduced-order) dynamic compensation problem is addressed for systems with simultaneous state, input, and output delays. The principal result involves sufficient conditions for characterizing fixed-order dynamic controllers for delay systems via a system of modified coupled Riccati equations. The controllers obtained are delay independent and hence apply to systems with arbitrary unknown delay.     

Robust stability bounds for multi-delay networked control systems

In this paper, the robust stability of a perturbed linear continuous-time system is examined when controlled using a sampled-data networked control system (NCS) framework. Three new robust stability bounds on the time-invariant perturbations to the original continuous-time plant matrix are presented guaranteeing stability for the corresponding discrete closed-loop augmented delay-free system (ADFS) with multiple time-varying sensor and actuator delays. The bounds are differentiated from previous work by accounting for the sampled-data nature of the NCS and for separate communication delays for each sensor and actuator, not a single delay. Therefore, this paper expands the knowledge base in multiple inputs multiple outputs (MIMO) sampled-data time delay systems. Bounds are presented for unstructured, semi-structured, and structured perturbations.     

Observer-based distributed coordinated tracking control for multiple simple-pendulum network systems with time-varying delays

For networked control systems consisting of multiple simple-pendulums driven by corresponding DC motors with time-varying communication and input delays, a distributed coordinated controller via observer-based output feedback is designed to solve the tracking problem under both fixed and jointly-connected switching topologies. Firstly, the linearized dynamic model for multiple nonlinear simple-pendulum network systems with DC motors is presented, and the distributed coordinated tracking problem considering time-varying input delays is mathematically described. Then the distributed observer-based tracking control protocol with time-varying communication and input delays is proposed, and simultaneously, both the observer gain and feedback gain are designed. Two examples are simulated to demonstrate consensus tracking for three types of states, i.e., swing angles, angular velocities of multiple simple-pendulums and armature currents of DC motors can be completed utilizing the developed coordinated tracking control with fixed and jointly connected topologies.     

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 sampled-data H ∞ output tracking control for a class of nonlinear networked systems with stochastic sampling

In this article, the problem of robust sampled-data H _∞ output tracking control is investigated for a class of nonlinear networked systems with stochastic sampling and time-varying norm-bounded uncertainties. For the sake of technical simplicity, only two different sampling periods are considered, their occurrence probabilities are given constants and satisfy Bernoulli distribution, and can be extended to the case with multiple stochastic sampling periods. By the way of an input delay, the probabilistic system is transformed into a stochastic continuous time-delay system. A new linear matrix inequality-based procedure is proposed for designing state-feedback controllers, which would guarantee that the closed-loop networked system with stochastic sampling tracks the output of a given reference model well in the sense of H _∞. Conservatism is reduced by taking the probability into account. Both network-induced delays and packet dropouts have been considered. Finally, an illustrative example is given to show the usefulness and effectiveness of the proposed H _∞ output tracking design.     

基于广义扩张状态观测器的遥操作系统同步控制

针对机械臂遥操作系统中存在的时变时延问题,提出了基于广义扩张状态观测器的控制方法,实现了遥操作系统稳定并且主从机械臂关节角位置同步的控制目标。首先通过反馈线性化,将遥操作系统的主从机械臂动力学模型转化为一个关于位置跟踪误差和时延的状态空间模型。针对该多输入多输出的干扰不匹配模型,设计了广义扩张状态观测器和相应的控制律,从而消除了时变时延以及其它扰动引起的不确定性对系统的影响,并对系统进行稳定性和抗扰性分析。最后,通过仿真验证了所设计的控制方法的有效性。     

多步观测时滞的控制

Based on an innovation analysis method in the Krein space, a sufficient and necessary condition is given for the existence of the solution of H1 control problem for a linear continuous-time system with multiple delays. By introducing a re-organized innovation sequence, the H1 control problem with delayed measurements is converted into a linear quadratic (LQ) problem and a delay-free H2 estimation problem in the Krein space. The controller is given in terms of two forward Riccati equations and a backward Riccati equation.     

Input-to-state stability for networked control systems via an improved impulsive system approach

This paper presents a novel impulsive system approach to input-to-state stability (ISS) analysis of networked control systems (NCSs) with time-varying sampling intervals and delays. This approach is based upon the new idea that an NCS can be viewed as an interconnected hybrid system composed of an impulsive subsystem and an input delay subsystem. A new type of time-varying discontinuous Lyapunov-Krasovskii functional, which makes full use of the information on the piecewise-constant input and the bounds of the network delays, is introduced to analyze the ISS property of NCSs. Linear matrix inequality based sufficient conditions are derived for ISS of NCSs with respect to external disturbances. When applied to the approximate tracking problem for NCSs, the derived ISS result provides bounds on the steady-state tracking error. Numerical examples are provided to show the efficiency of the proposed approach.     

时变时滞非线性系统自适应神经网络控制

对于一类具有未知时变时滞和虚拟控制系数的不确定严格反馈非线性系统,基于后推设计提出一种自适应神经网络控制方案.选取适当的Lyapunov-Krasovskii泛函补偿未知时变时滞不确定项.通过构造连续的待逼近函数来解决利用神经网络对未知非线性函数进行逼近时出现的奇异问题.通过引入一个新的中间变量,保证了虚拟控制求导的正确性.仿真算例表明,所设计的控制器能保证闭环系统所有信号是半全局一致终结有界的,且跟踪误差收敛到零的一个邻域内.     

Linear quadratic regulation for systems with time‐varying delay

In this paper we study the linear quadratic regulation (LQR) problem for discrete‐time systems with time‐varying delay in the control input channel. We assume that the time‐varying delay is of a known upper bound, then the LQR problem is transformed into the optimal control problem for systems with multiple input channels, each of which has single constant delay. The optimal controller is derived by establishing a duality between the LQR and a smoothing estimation for an associated system with a multiple delayed measurement. Copyright © 2009 John Wiley & Sons, Ltd.     

On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays

This paper is devoted to the analysis of the ?_∞ and L_∞ gains for positive linear systems with interval time-varying delays. Through exploiting the monotonicity of the state trajectory, we first prove that for positive systems with constant delays, the ?_∞ and L_∞ gains are fully governed by the system matrices but independent of the delay size. Moreover, for positive systems with bounded time-varying delays, by comparing with the nominal systems with constant delays, it turns out that the ?_∞ and L_∞ gains are exactly the same as that of the constant delay systems. The results in this paper reveal that the ?_∞ and L_∞ gains of positive linear systems are not sensitive to the magnitude of time delays and hence the computation of ?_∞ and L_∞ gains of positive systems with bounded time-varying delays can be reduced to computing the ?_∞ and L_∞ gains of the corresponding delay-free systems. Both continuous-time and discrete-time cases are considered in this paper.     

时变时滞连续切换奇异系统的一致有限时间稳定分析

讨论一类含有时变时滞的连续时间切换奇异系统的一致有限时间稳定、有限时间有界和状态反馈镇定问题. 在给定任意的切换规则下, 运用多Lyapunov 函数和平均驻留时间方法设计使得闭环系统一致有限时间稳定和有限时间有界的状态反馈控制器, 同时给出基于线性矩阵不等式表示的控制器存在的充分条件. 最后通过数值算例表明了所提出方法的合理性和有效性.

     

Infinite horizon LQR for systems with multiple delays in a single input channel

This paper is concerned with the linear quadratic regulation (LQR) problem for both linear discrete-time systems and linear continuous-time systems with multiple delays in a single input channel. Our solution is given in terms of the solution to a two-dimensional Riccati difference equation for the discrete-time case and a Riccati partial differential equation for the continuous-time case. The conditions for convergence and stability are provided.     

Reduction model approach for linear time-varying systems with input delays based on extensions of Floquet theory

We solve stabilization problems for linear time-varying systems under input delays. We show how changes of coordinates lead to systems with time invariant drifts, which are covered by the reduction model method and which lead to the problem of stabilizing a time-varying system without delay. For continuous time periodic systems, we can use Floquet theory to find the changes of coordinates. We also prove an analogue for discrete time systems, through a discrete time extension of Floquet theory.