Control of time delay processes with uncertain delays: Time delay stability margins

In this paper, the stability of time delay processes that have uncertain delays is considered, and the maximum allowable perturbation which may occur in the time delays so as to maintain stability are determined. In particular, the characteristic equations of time delay systems are quasipolynomials, whose roots determine the stability of such systems, and the root-locus of these equations in specified desired regions is investigated. A numerical algorithm is presented for the calculation of the time delay stability margins in the space of time delays for such systems, and the size of the stability hyperspheres in this space is computed. To illustrate the procedure, the algorithm is applied to process control systems with uncertain delays and the allowable perturbations in the time delays of these systems are then computed.     

Stabilization of control loops consisting of FOPDT process and parameter-dependent PID controller

In this paper, problem of stability analysis of the control loops consisting of first-order plus dead time (FOPDT) processes and proportional-integrative-derivative (PID) controllers is studied, where the controller coefficients are functions of one or more independent parameters. An effective procedure is presented to determine a stability region in the independent parameters space. This method does not require complex numerical calculations such as solving nonlinear equations. It is based on usage of a two-valued indicator function and by using that, a stability region is easily determined. In order to clarify that, why the stability region needs to be specified in the “independent parameters space” an optimal method is given to design the PID controller for the FOPDT processes, as an instance. In this optimal method the controller coefficients are obtained as the functions of a free parameter, where this parameter needs to be chosen by the designer such that it should be near to the maximum operating frequency of the system, besides on the other hand the closed-loop system to be stable. In the end, two illustrative examples are given in order to show the usefulness and effectiveness of the proposed method, and to compare the obtained stability regions with the whole stability regions.     

基于动态反馈控制器的网络控制系统的稳定性分析

考虑实际模型的不确定性,针对一类具有不确定线性参数的网络控制系统,给出具有网络诱导时延及不确定线性参数的被控对象模型,设计其动态反馈控制器,建立基于动态反馈控制器的网络控制系统闭环模型.采用Lyapunov方法,给出保证该系统稳定的最大允许时延上界.仿真结果表明系统模型及控制器设计的合理性,所得最大允许时延上界保守性更弱.     

Proportional-derivative controllers for stabilisation of first-order processes with time delay

This paper considers the problems of determining the complete stabilising set of proportional-derivative controllers for a first-order process with time delay. First, by employing a version of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete set of all stabilising proportional-derivative parameters for first-order processes with constant time delay are obtained. Next, we provide an approach to design a robust PD controller to stabilise a first-order process with uncertain time delay, which lies inside a known interval.     

Stabilisability analysis of high-order unstable processes by fractional-order controllers

In this article, the sufficient stabilisability criteria for unstable time-delay processes by fractional-order controllers are investigated. The process is a high-order system with a single unstable pole, which also has a stable zero. The adopted approach to derive the sufficient conditions for stability is based on the well-known Nyquist stability criterion. Stabilisability is studied by applying fractional-order proportional integral and proportional derivative controllers and the results are given in terms of the maximum allowable value of time delay. Additionally, limitations on the parameters of the controllers which must be taken into account in the controller design are proposed.     

线性离散时滞系统鲁棒严格耗散控制

提出线性离散时滞系统的耗散控制问题,研究无记忆状态反馈控制律存在的条件及 相应的控制器设计方法,以使相应的闭环系统渐近稳定,同时具有严格(Q,S,R)-耗散性.进 一步考虑耗散的不确定性,研究不确定系统鲁棒严格耗散控制的分析与综合问题.结果表明, 鲁棒耗散控制器存在的条件和综合问题等价于线性矩阵不等式(LMI)的可解性问题.     

Sliding mode control design for uncertain time‐delay systems subjected to a class of nonlinear inputs

A sliding mode controller is developed for uncertain time‐delay systems with a class of nonlinear inputs. Two main results are derived in this paper. The first result is the presentation of a new delay‐dependent stability condition of uncertain time‐delay systems. In a comparison example, this stability condition is shown to be less conservative than the ones reported recently. The second result is to present a new sliding mode control for uncertain time‐delay systems subjected to a class of nonlinear inputs. The stability of time‐delay systems with unmatching condition in the sliding mode is also discussed. Two illustrative examples are included to demonstrate the superiority of the obtained results. Copyright © 2003 John Wiley & Sons, Ltd.     

Time delay margin for islanded microgrid with parameter uncertainty using fractional order proportional-integral-derivative controller

In this article, an islanded microgrid (MG) consisting of the diesel generator (DEG), the photovoltaic panel (PV), the wind turbine generator (WTG), the battery energy storage system (BESS), and the control unit is considered. In the islanded MGs, the control signals are exchanged on the open communication network, which results in the time delays in the input and the output of the islanded MG central controller (MGCC). The time delay has a destructive effect on the islanded MG stability. Thus, finding the maximum allowable time delay bound (MADB) is a significant issue. Since it is shown that the fractional order systems have larger stability region and more robustness rather than the corresponding integer order systems, in this research, we propose the fractional order proportional-integral-derivative (FOPID) controller as the MGCC to achieve a larger MADB value. As another innovation, in this article, a method is presented in which the MADB of the islanded MG system is determined considering the parametric uncertainties related to the damping coefficient (D) and the inertia constant (H). It is shown that the percentage improvement in the MADB of the uncertain MG system with the designed FOPID controller over the integer order proportional-integral-derivative (IOPID) controller is 9.64%. The accuracy of the proposed method is verified by simulation results in Matlab.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

Robust stabilization of uncertain systems with unknown input delay

This paper is concerned with the robust controller design for uncertain input-delayed systems. The time delay is assumed to be an unknown constant. A controller with delay feedback for the robust stabilization of the system is proposed. The stability criterion of the closed-loop system is derived in terms of linear matrix inequalities (LMIs). Examples show that in many cases our method can give less conservative results than those by the existing methods. Moreover, for the same cases, our controllers have lower feedback gains than the existing ones.     

State feedback controller design of networked control systems with interval time‐varying delay and nonlinearity

This paper proposes a method for robust state feedback controller design of networked control systems with interval time‐varying delay and nonlinearity. The key steps in the method are to construct an improved interval‐delay‐dependent Lyapunov functional and to introduce an extended Jessen's inequality. Neither free weighting nor model transformation is employed in the derivation of system stability criteria. It is shown that the maximum allowable bound on the nonlinearity could be computed through solving a constrained convex optimization problem; and the maximum allowable delay bound and the feedback gain of a memoryless controller could be derived by solving a set of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

Force Reflecting Bilateral Control of Master–Slave Systems in Teleoperation

In this paper, a simple structure design with arbitrary motion/force scaling to control teleoperation systems, with model mismatches is presented. The goal of this paper is to achieve transparency in presence of uncertainties. The master–slave systems are approximated by linear dynamic models with perturbed parameters, which is called the model mismatch. Moreover, the time delay in communication channel with uncertainties is considered. The stability analysis will be considered for two cases: (1) stability under time delay uncertainties and (2) stability under model mismatches. For the first case, two local controllers are designed. The first controller is responsible for tracking the master commands, while the second controller is in charge of force tracking as well as guaranteeing stability of the overall closed-loop system. In the second case, an additional term will be added to the control law to provide robustness to the closed-loop system. Moreover, in this case, the local slave controller guarantees the position tracking and the local master controller guarantees stability of the inner closed-loop system. The advantages of the proposed method are two folds: (1) robust stability of the system against model mismatches is guaranteed and (2) structured system uncertainties are well compensated by applying independent controllers to the master and the slave sites. Simulation results show good performance of the proposed method in motion tracking as well force tracking in presence of model mismatches and time delay uncertainties.     

Integral partitioning approach to robust stabilization for uncertain distributed time‐delay systems

In this paper, the problems of robust delay‐dependent stability analysis and stabilization are investigated for distributed delay systems with linear fractional uncertainties. By introducing an integral partitioning technique, a new form of Lyapunov functional is constructed and improved distributed delay‐dependent stability conditions are established in terms of linear matrix inequalities. Based on the criterion, a design algorithm for a state‐feedback controller is proposed. Following similar lines, we extend these results to uncertain distributed delay systems. The results developed in this paper can tolerate larger allowable delay than existing ones in the literature, which is illustrated by several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Stabilization Parametric Region of Distributed PID Controllers for General First-Order Multi-Agent Systems With Time Delay

The stabilization problem of distributed proportional-integral-derivative (PID) controllers for general first-order multi-agent systems with time delay is investigated in the paper. The closed-loop multi-input multi-output (MIMO) framework in frequency domain is firstly introduced for the multi-agent system. Based on the matrix theory, the whole system is decoupled into several subsystems with respect to the eigenvalues of the Laplacian matrix. Considering that the eigenvalues may be complex numbers, the consensus problem of the multi-agent system is transformed into the stabilizing problem of all the subsystems with complex coefficients. For each subsystem with complex coefficients, the range of admissible proportional gains is analytically determined. Then, the stabilizing region in the space of integral gain and derivative gain for a given proportional gain value is also obtained in an analytical form. The entire stabilizing set can be determined by sweeping proportional gain in the allowable range. The proposed method is conducted for general first-order multi-agent systems under arbitrary topology including undirected and directed graph topology. Besides, the results in the paper provide the basis for the design of distributed PID controllers satisfying different performance criteria. The simulation examples are presented to check the validity of the proposed control strategy.      

Predictor-based control for an uncertain Euler–Lagrange system with input delay

Controlling a nonlinear system with actuator delay is a challenging problem because of the need to develop some form of prediction of the nonlinear dynamics. Developing a predictor-based controller for an uncertain system is especially challenging. In this paper, tracking controllers are developed for an Euler–Lagrange system with time-delayed actuation, parametric uncertainty, and additive bounded disturbances. The developed controllers represent the first input delayed controllers developed for uncertain nonlinear systems that use a predictor to compensate for the delay. The results are obtained through the development of a novel predictor-like method to address the time delay in the control input. Lyapunov–Krasovskii functionals are used within a Lyapunov-based stability analysis to prove semi-globally uniformly ultimately bounded tracking. Experimental results illustrate the performance and robustness of the developed control methods.     

Reliable memory feedback design for a class of non‐linear time‐delay systems

This paper is concerned with the robust controller design of uncertain time‐delay systems with unknown nonlinearity and actuators failures. New methods for designing stabilizing controllers and reliable controllers are proposed. The stability criteria of the closed‐loop system, which are dependent on the magnitudes of the delay and its derivative, are derived in the form of linear matrix inequalities. Numerical and simulation results are provided to demonstrate the effectiveness of the proposed results, as well as the reduction of conservativeness when compared with existing ones. Copyright © 2004 John Wiley & Sons, Ltd.     

一类不确定时延网络控制系统的稳定性分析

针对网络控制系统中时延不确定因素,将时延的不确定性转换为系统状态方程系数矩阵的不确定性,基于LMI和Lyapunov理论,研究了一类具有时延的网络控制系统的稳定性问题,并运用Lyapunov方法和线性矩阵不等式对不确定时延系统进行了稳定性分析。并以Riccati不等式和线性矩阵不等式的形式给出了系统稳定的充分条件,使得在不确定时延情况下,系统达到理想稳定性。最后,通过Matlab工具箱对数值结果进行仿真,在时延相关情况下,求得系统达到闭环稳定时最大允许的时延,表明了设计方法的有效性。     

一类不确定线性时滞系统的输出反馈鲁棒镇定

研究一类不确定线性时滞系统的输出反馈鲁棒镇定问题,其中不确定性不必满足匹配条件。以二次Lyapunov泛函保证系统的渐近稳定性,利用线性矩阵不等式给出了系统可以利用动态输出反馈鲁棒镇定的充分条件。当此条件成立时,基于线性矩阵不等式的解构造了全阶动态输出反馈镇定控制器。     

Design and Implementation of Robust-Fixed Structure Controller for Telerobotic Systems

The controller design for bilateral teleoperation systems involves a delicate trade-off between performance and stability. To achieve high performance, high order robust controllers may not be feasible for real-time implementation because of hardware and computational limitations. The main purpose of this paper is to achieve stability and transparency in the presence of time delay in communication channel as well as model uncertainty. To address these problems, a novel robust fixed-structure controller is proposed for uncertain bilateral teleoperation systems. Here, the traditional conventional Proportional-Integral-Derivative (PID) controller is employed to achieve the requirements. The simplicity and straightforward design are the significant advantageous of the proposed method. Robust stability analysis of the proposed technique is also provided. Results demonstrate that the structure is effective in providing stability and transparency in teleoperation systems.     

Use of predicted information to improve the control performance of systems with uncertainties

This paper develops a two-loop robust control strategy for systems associated with uncertainty. The inner-loop is designed offline for robust stability. The outer-loop is used online to improve the control performance based on predicted information, taking uncertainties and constraints into account. The structure of the two-loop robust controller is illustrated. A practical control algorithm is developed for the systems which can be described using an uncertain first-order plus time delay model. The benefits of the two-loop robust control strategy are illustrated by presenting the validation results and comparing with an anti-windup PI controller.