Fixed-time output regulation of linear delay systems by smooth time-varying control

This article investigates the fixed-time output regulation problem (FxTORP) for linear systems in the presence of input delay. A linear controller consisting of the linear periodic delayed feedback (PDF) gain and the feedforward gain obtained by solving regulator equations is designed, such that FxTORP is addressed. If only the measurable output can be used for feedback, a linear observer with periodic coefficient and artificial delay is designed so that its state converges to the state of the augmented system at a prescribed finite time. Based on the estimated state, the output regulation problem can also be solved by using observer-based output feedback. The most significant advantages of this article are that the PDF gain can be taken as smooth and the output regulation problem is achieved within a prespecified regulation time. Finally, a simulation example is given to substantiate the validity of the proposed approaches.     

Comment on ‘ Model reduction by minimizing the weighted equation error ’

A method for the control of time delay systems is presented. The systems are represented by moving models established by the use of block pulse functions. This approach permits the linear feedback control law to be applied to non-linear systems. The moving model representation and the optimal feedback control of time delay systems are presented to demonstrate that the method exhibits satisfactory results.     

有限时间二次型数值算法研究及其应用

为了实际需要和学术发展的要求,研究了以倒立摆为控制对象,通过闭环网络形成的反馈控制系统的随机传输时延的最优控制问题。在求解有限时间最优控制律过程中,通过矩阵Raccati方程的离散变换,利用Matlab中计算无限时间二次型最优控制器的LQR函数,从而求出有限时间LQR问题的数值解。通过仿真结果证明,研究的方法能够使倒立摆系统最终稳定,从而说明提出的算法对于求解有限时间LQR问题是有效的。     

OPTIMAL REJECTION WITH ZERO STEADY‐STATE ERROR OF SINUSOIDAL DISTURBANCES FOR TIME‐DELAY SYSTEMS

This paper studies the problem of optimal rejection with zero steady‐state error of sinusoidal disturbances for linear systems with time‐delay. Based on the internal model principle, a disturbance compensator is constructed to counterbalance the external sinusoidal disturbances, so that the original system can be transformed into an augmented system without disturbances. Then, with the introduction of a sensitivity parameter and expanding power series around it, the optimal disturbance rejection problem can be simplified to the problem of solving an infinite sum of a linear optimal control series without time‐delay or disturbance. The optimal control law for disturbance rejection with zero steady‐state error consists of accurate linear state feedback terms and a time‐delay compensating term, which is an infinite sum of an adjoint vector series. In the presented approach, iteration is required only for the time‐delay compensation series. By intercepting a finite sum of the compensation series, we obtain an approximate physically realizable optimal control law that avoids complex calculation. A numerical simulation shows that the algorithm is effective and easy to implement.     

一类奇异时滞系统的奇异二次指标最优控制问题

利用基本的代数等价变换，将一类奇异滞后系统的奇异二次指标最优控制问题转化为正常状态滞后系统的非奇异二次指标最优控制问题，并讨论了二的等价性，在一些常规条件下，给出了问题的解，并把最优控制综合为最优状态反馈。     

Optimal control of linear time-delay systems

A method is presented whereby an optimal control may be obtained for a linear time-varying system with time delay. The performance criterion is quadratic with a fixed, finite upper limit, and results in a set of differential equations with boundary conditions whose solution yields an optimal feedback control. A numerical technique is developed for the solution of the differential equations, and two examples are worked.     

Stabilization of linear systems with time‐varying input delay by event‐triggered delay independent truncated predictor feedback

This article deals with the problem of stabilization of linear systems with time‐varying input delay by an event‐triggered delay independent truncated predictor feedback law, either of the state feedback type or the output feedback type. Only the information of a delay bound rather than the delay itself is required in the design of both control laws and event‐triggering strategies. For both the state feedback case and the output feedback case, an admissible delay bound that guarantees the stabilizability of a general linear system is established, and the Zeno behavior is shown to be excluded. For linear systems with all open‐loop poles at the origin or in the open left‐half plane, stabilization can be achieved for a delay under an arbitrarily large bound.     

Optimal filtering for linear systems with state and observation delays

In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman–Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman–Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman–Bucy filter available for linear systems without delays and two versions of the extended Kalman–Bucy filter for time delay systems. Copyright © 2005 John Wiley & Sons, Ltd.     

基于Riccati-It方程设计线性时滞系统鲁棒控制器的方法

利用Riccati-Ito型矩阵代数方程研究线性时不变系统的状态反馈镇定问题，对时滞系统给出了一种状态反馈镇定方法，这种镇定方法适应于许多时滞系统，其设计方案具有滞后无关性及对于许多系统滞后项系数任意性的完全适应性．本文第二部分研究时不变线性不确定系统的鲁棒镇定，利用Riccati-Ito型矩阵代数方程给出了这类系统鲁棒镇定控制器的设计方法．     

无线通讯网络功率和流量的预联合控制

To mitigate the loop delay in distributed wireless networks,a predictive power and rate control scheme is pro- posed for the system model that also accounts for the conges- tion levels and input delay instead of state-delayed in a network. A measurement feedback control problem with input delay is formulated by minimizing the energy of the difference between the actual and the desired signal-to-interference-plus-noise ratio (SNR)levels,as well as the energy of the control sequence.To solve this problem,we present two Riccati equations for the con- trol and the estimation for the time delay systems.A complete analytical optimal controller is obtained by using the separation principle and solving two Riccati equations,where one is back- ward equation for stochastic linear quadratic regulation and the other is the standard filtering Riccati equation.Simulation re- sults illustrate the performance of the proposed power and the rate control scheme.     

An optimal control approach to robust tracking of linear systems

In our early work, we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem. If the system is linear, then the optimal control problem becomes a linear quadratic regulator (LQR) problem, which can be solved by solving an algebraic Riccati equation. In this article, we extend the optimal control approach to robust tracking of linear systems. We assume that the control objective is not simply to drive the state to zero but rather to track a non-zero reference signal. We assume that the reference signal to be tracked is a polynomial function of time. We first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic Riccati equation. Because the state feedback is not always available in practice, we also investigated the output feedback. We show that if we place the poles of the observer sufficiently left of the imaginary axis, the robust tracking problem can be solved. As in the case of the state feedback, the observer and feedback can be obtained by solving two algebraic Riccati equations.     

无线通讯网络功率和流量的预联合控制

To mitigate the loop delay in distributed wireless networks, a predictive power and rate control scheme is proposed for the system model that also accounts for the congestion levels and input delay instead of state-delayed in a network. A measurement feedback control problem with input delay is formulated by minimizing the energy of the difference between the actual and the desired signal-to-interference-plus-noise ratio (SNR) levels, as well as the energy of the control sequence. To solve this problem, we present two Riccati equations for the control and the estimation for the time delay systems. A complete analytical optimal controller is obtained by using the separation principle and solving two Riccati equations, where one is backward equation for stochastic linear quadratic regulation and the other is the standard filtering Riccati equation. Simulation results illustrate the performance of the proposed power and the rate control scheme.     

The Liapunov's second method for continuous time difference equations

Among many other cases such as economic and lossless propagation models, continuous time difference equations are encountered as the internal dynamics in a class of non‐linear time delay systems, when controlled by a suitable state feedback which drives the output exponentially to zero. The Liapunov's second method for these infinite dimensional systems has not been extensively investigated in the literature. This paper has the aim of filling this gap. Liapunov's second method theorems for checking the stability and the asymptotic stability of this class of infinite dimensional systems are built up, in both a finite and an infinite dimensional setting. In the finite dimensional setting, the Liapunov function is defined on finite dimensional sets. The conditions for stability are given as inequalities on continuous time. No derivatives are involved, as in the dynamics of the studied systems. In the infinite dimensional setting, the continuous time difference equation is transformed into a discrete time system evolving on an infinite dimensional space, and then the classical Liapunov theorem for the system in the new form is written. In this paper the very general case is considered, that is non‐linear continuous time difference equations with multiple non commensurate delays are considered, and moreover the functions involved in the dynamics are allowed to be discontinuous, as well as the initial state. In order to study the stability of the internal dynamics in non‐linear time delay feedback systems, an exogenous disturbance is added, which goes to zero exponentially as the time goes to infinity. An example is considered, from non‐linear time delay feedback theory. While the results available in the literature are inconclusive as far as the stability of that example is concerned, such stability is proved to hold by the theorems developed in this paper, and is validated by simulation results. Copyright © 2003 John Wiley & Sons, Ltd.     

Optimal filtering for linear state delay systems

In this note, the optimal filtering problem for linear systems with state delay over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the optimal estimate equation similar to the traditional Kalman-Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman-Bucy filter. The resulting system of equations for determining the error variance consists of a set of equations, whose number is specified by the ratio between the current filtering horizon and the delay value in the state equation and increases as the filtering horizon tends to infinity. In the example, performance of the designed optimal filter for linear systems with state delay is verified against the best Kalman-Bucy filter available for linear systems without delays and two versions of the extended Kalman-Bucy filter for time-delay systems.     

Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation

An application of a finite spectrum assignment method for time-delay systems to a feedback control of Mach number in a wind tunnel is presented. The linearized model of Mach number control is a system of three state equations with a delay in one of the state variables. The proposed feedback is a linear combination of state variables and weighted integrals of some of the state variables over a period equal to time delay. The spectrum of the closed-loop system is finite and consists of three eigenvalues that can be placed arbitrarily. Four possible variants of the feedback control law are presented. The calculation of feedback coefficients is very simple, which makes it possible to update the controller's parameters on-line with the changes of the operating point. An extensive numerical simulation of the system dynamics and the feedback controllers was done and led to several conclusions. In particular, a very good settling time under problem constraints has been obtained, and several nice features of the proposed feedback controller have been demonstrated.     

Robust optimal control of time‐delay systems based on Razumikhin theorem

This paper presents an optimal control approach for the general robust control design problem of linear time delay systems, which considers parameter uncertainties as well as state delay. It is shown that the robust control problem can be transformed into an optimal control problem with the amouof plant uncertainties involved in the performance index. A stability criterion has been developed under which the uncertain dynamical system can not only achieve stability, but also acquire the guaranteed level of performance for regulation. A suitable linear state feedback control law is also characterized via Lyapunov stability theory to ensure performance robustness of the closed‐loop system. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

时变时滞不确定系统的鲁棒输出反馈控制

研究了时变时滞不确定系统基于状态观测器的动态输出反馈实现鲁棒镇定的分 析和综合问题.所研究的系统不仅同时包含时变状态时滞和时变控制时滞,而且包含时变未 知且有界不确定参数.提出了确保该系统可通过输出反馈鲁棒镇定的充分条件,并将该充分 条件转化为线性矩阵不等式(LMI)问题,最终通过求解两个LMI来构造输出反馈控制律.     

Optimal linear estimation for continuous stochastic systems with random observation delays

This paper is concerned with the linear minimum mean square error estimation for Itô‐type differential equation systems with random delays, where the delay process is modeled as a finite‐state Markov chain. By first introducing a set of equivalent delay‐free observations and then defining two reorganized Markov chains, the estimation problem of random delayed systems is reduced to the one of delay‐free Markov jump linear systems. The estimator is derived by using the innovation analysis method based on the Itô differential formula. And the analytical solution to this estimator is given in terms of two Riccati differential equations that are of finite dimensions. Conditions for existence, uniqueness, and stability of the steady‐state optimal estimator are studied for time‐invariant cases. In this case, the obtained estimator is very easy to implement, and all calculation can be performed off line, leading to a linear time‐invariant estimator. Copyright © 2011 John Wiley & Sons, Ltd.     

基于微分对策的最优状态观测器和最优状态反馈控制器的设计

研究了线性系统基于二次型指标的最优状态观测器和最优状态反馈控制器的设计问题．将观测状态的状态反馈和状态误差的输出反馈分别作为两个对局方,应用微分对策理论研究了系统的最优控制问题．给出了最优状态观测器和基于状态观测器的最优状态反馈控制的存在性条件．将系统的最优状态观测器和最优控制器的设计问题转化为一对Riccati方程的求解问题．研究表明最优状态观测器在一般情况下不存在．并进一步研究了基于状态观测器的次优控制问题,给出了基于LMI的优化算法．     

Computation of optimal control for linear systems with delay

A numerical algorithm is presented to calculate an optimal control recursively for linear multivariable systems with delay. The algorithm is based on the method of steepest descent in Hilbert space. The optimal control of a multivariable system with delay for a quadratic criterion function is given by the Riccati partial differential equations. These are simultaneous partial differential equations which are difficult to solve numerically. In most computational algorithms, errors are inevitable, a vast memory is required, and a lot of computational time is needed. This makes it impractical to use available computers for these algorithms. The algorithm presented here gives optimal control effectively without a large memory requirement. It also provides a practical computational method for obtaining the optimal control of multivariable systems with delay. Several numerical computations are performed to show how the effectiveness of the algorithm compares with other methods.