Domains of PID controller coefficients which guarantee stability and performance for LTI time-delay systems

In this work, the computation of the regions in the space of coefficients of a PID controller, inside which the controller stabilizes an LTI time-delay plant and guarantees a performance specification for the closed-loop system, is addressed. To ascertain the sensor noise attenuation, as a sample performance requirement, the H_∞-norm of the weighted complementary sensitivity function is kept below a desired bound. The problem of minimization of the H_∞-norm is transformed into the simultaneous stabilization of a family of quasipolynomials. Then, the stable domains of the PID coefficients are computed for these quasipolynomials using the parameter space approach. This work extends the previous results on the computation of the stability domains of a PID controller to ensure the desired performance specification for the closed-loop system.     

无模型SISO时滞系统的PID参数稳定域研究

针对无精确模型的单输入单输出(SISO)时滞系统, 利用频率响应数据, 给出确定PID参数稳定域的解析方法. 首先通过继电反馈频域辨识法得到系统的频率响应数据, 然后基于该数据确定控制参数的奇异边界线和非奇异边界线, 并判断边界线的哪一侧具有更少的不稳定极点, 从而给出能使闭环系统稳定的控制参数区域. 该方法避免了模型辨识的复杂计算过程, 为无模型SISO时滞系统的PID控制器设计和调节提供了一条简单有效的途径. 仿真结果验证了方法的有效性.     

时延网络控制系统的稳定性

针对网络控制系统中存在的不确定时延 ,首先讨论了系统的建模问题 .系统中传感器采用时间驱动 ,执行器与控制器采用事件驱动 ,传感器的数据采用单包传输 .假设传输时延小于采样周期 ,网络控制系统可以建模为一类具有不确定性的线性离散时延系统 .利用Lyapunov方法 ,给出了闭环系统渐近稳定的充分条件 ,基于相应的线性矩阵不等式可行解 ,可以求解状态反馈控制律 .最后 ,用仿真例子验证了该方法的有效性     

Synthesis of PID controllers for integral processes with time delay

This article deals with the problem of determination of the stabilizing parameter sets of Proportional‐Integral‐Derivative (PID) controllers for first‐order and second‐order integral processes with time‐delay. First, the admissible stabilizing range of proportional‐gain is determined analytically in terms of a version of the Hermite–Biehler Theorem applicable to quasi‐polynomials. Then, based on a graphical stability condition developed in parameter space, the complete stabilizing regions in an integral‐derivative plane are drawn and identified graphically, not calculated mathematically, by sweeping over the admissible range of proportional‐gain. An actual algorithm for finding the stabilizing parameter sets of PID controllers is also proposed. Simulations show that the stabilizing regions in integral‐derivative space are either triangles or quadrilaterals. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust stability of uncertain singular time-delay systems via LFT approach

The robust stability problem of continuous-time singular systems with multiple state delays and bounded parametric uncertainties is considered. Both commensurate and non-commensurate delays are investigated. On the basis of the linear fractional transformations (LFTs) framework and μ-analysis, a systematic approach is derived to convert the robustness problem to a robust nonsingularity problem. Some explicit conditions are proposed to guarantee the uncertain singular time-delay system being regular, impulse-free and stable independent of delay. Two illustrative examples are given to show the feasibility of the proposed technique.     

不确定奇异时滞系统的保性能控制

利用线性矩阵不等式方法,讨论不确定性奇异时滞系统的保性能控制问题,给出了问题可解的一个充分条件和保性能鲁棒控制器的设计,以及相应的可保性能指标.最后举例说明了所提出方法的正确性.     

A fuzzy PID controller for nonlinear and uncertain systems

 In order to control systems that contain nonlinearities or uncertainties, control strategies must deal with the effects of these. Since most control methods based on mathematical models have been mainly focused on stability robustness against nonlinearities or uncertainties, they are limited in their ability to improve the transient responses. In this paper, a nonlinear fuzzy PID control method is suggested, which can stably improve the transient responses of systems disturbed by nonlinearities or unknown mathematical characteristics. Although the derivation of the control law is based on the design procedure for general fuzzy logic controllers, the resultant control algorithm has analytical form with time varying PID gains rather than linguistic form. This means the implementation of the proposed method can be easily and effectively applied to real-time control situations. Control simulations are carried out to evaluate the transient performance of the suggested method through example systems, by comparing its responses with those of the nonlinear fuzzy PI control method developed in [9].     

含时变滞后的不确定系统的时滞相关型鲁棒控制设计

研究含时变滞后的不确定系统的控制综合问题. 基于Lyapunov方法提出了一种含滞后补偿的鲁棒控制设计方法. 闭环稳定性条件由一组线性矩阵不等式表示. 在这些条件中给出了稳定性和滞后以及其导数之间的关系. 实例显示, 利用提供的方法所给出的结果比以往文献给出的结果保守性小.     

All stabilizing PID controllers for time delay systems

This paper presents a method to compute the entire set of stabilizing PID controller parameters for an arbitrary (including unstable) linear time delay system. The main contribution is to handle the infinite number of stability boundaries in the (k_d,k_i)-plane for a fixed proportional gain k_p. For retarded open loops, it is shown that the stable region in the (k_d,k_i)-plane consists of convex polygons. Concerning neutral loops, a new phenomenon is introduced. For certain systems and certain k_p, the exact stable region in the (k_d,k_i)-plane can be described by the limit of a sequence of polygons with an infinite number of vertices. This sequence may be well approximated by convex polygons. Moreover, the paper describes a necessary condition for k_p-intervals potentially having a stable region in the (k_d,k_i)-plane. Thus, the set of stabilizing controller parameters can be calculated after gridding k_p in these intervals. A Matlab tool implementing the presented method is available.     

Stability analysis of hybrid switched nonlinear singular time-delay systems with stable and unstable subsystems

The issue of exponential stability of a class of continuous-time switched nonlinear singular systems consisting of a family of stable and unstable subsystems with time-varying delay is considered in this paper. Based on the free-weighting matrix approach, the average dwell-time approach and by constructing a Lyapunov-like Krasovskii functional, delay-dependent sufficient conditions are derived and formulated to check the exponential stability of such systems in terms of linear matrix inequalities (LMIs). By checking the corresponding LMI conditions, the average dwell-time and switching signal conditions are obtained. This paper also highlights the relationship between the average dwell-time of the switched nonlinear singular time-delay system, its stability and the exponential convergence rate of differential and algebraic states. A numerical example shows the effectiveness of the proposed method.     

A PI/PID controller for time delay systems with desired closed loop time response and guaranteed gain and phase margins

In this article we present a graphical tuning method of PI/PID controller for first order and second order plus time delay systems using dominant pole placement approach with guaranteed gain margin (GM) and phase margin (PM). The stability equation method and gain phase margin tester have been used to portray constant GM and PM boundaries. The PID controller parameters have been obtained for different dominant poles and plotted graphically in the parameters plane of controller within the specified GM and PM regions. To demonstrate the effectiveness and confirm the validity of the proposed methodology, three examples with numerical simulations are presented.     

Robust non-fragile fractional order PID controller for linear time invariant fractional delay systems

A fractional order PID controller is designed to stabilize fractional delay systems with commensurate orders and multiple commensurate delays, where the time delays in the system may belong to several distinct intervals. Moreover, the controller parameters should belong to given intervals. In order to stabilize the system, the D-subdivision method is employed to choose the stabilizing set of the controller parameters from their available values. Furthermore, the nearest values of the obtained stabilizing set to their mean values are selected as the controller parameters so that a non-fragile controller is concluded. Two numerical examples evaluate the proposed control design method.     

Stability conditions for integral delay systems

In this paper we consider a special class of integral delay systems arising in several stability problems of time‐delay systems. For these integral systems we derive stability and robust stability conditions in terms of Lyapunov–Krasovskii functionals. More explicitly, after providing the stability conditions we compute quadratic functionals and apply them to derive exponential estimates for solutions, and robust stability conditions for perturbed integral delay systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Graphical computation of gain and phase margin specifications-oriented robust PID controllers for uncertain systems with time-varying delay

This paper proposes a novel graphical method to compute all feasible gain and phase margin specifications-oriented robust PID controllers to stabilize uncertain control systems with time-varying delay. A virtual gain-phase margin tester compensator is incorporated to guarantee the concerned system with certain robust safety margins. The complex Kharitonov theorem is used to characterize the parametric uncertainties of the considered system and is exploited as a stability criterion for the Hurwitz property of a family of polynomials with complex coefficients varying within given intervals. The coefficients of the characteristic equation are overbounded and eight vertex Kharitonov polynomials are derived to perform stability analysis. The stability equation method and the parameter plane method are exploited to portray constant gain margin and phase margin boundaries. The feasible controllers stabilizing every one of the eight vertex polynomials are identified in the parameter plane by taking the overlapped region of the plotted boundaries. The overlapped region of the useful region of each vertex polynomial is the Kharitonov region, which represents all the feasible specifications-oriented robust PID controller gain sets. Variations of the Kharitonov region with respect to variations of the derivative gain are extensively studied. The way to select representative points from the Kharitonov region for designing robust controllers is suggested. Finally, three illustrative examples with computer simulations are provided to demonstrate the effectiveness and confirm the validity of the proposed methodology. Based on the pre-specified gain and phase margin specifications, a non-conservative Kharitonov region can be graphically identified directly in the parameter plane for designing robust PID controllers.     

Switched nonlinear singular systems with time‐delay: Stability analysis

This paper presents an approach to the stability analysis of a class of nonlinear interconnected continuous‐time singular systems with arbitrary switching signals. This class of interconnected subsystems consists of unknown but bounded state delay and nonlinear terms, and each subsystem can be globally stable, unstable, or locally stable. By constructing a new Lyapunov‐like Krasovskii functional, sufficient conditions are derived and formulated to check the asymptotic (exponential) stability of such systems with arbitrary switching signals. Then, some new general criteria for asymptotic (exponential) stability with average dwell‐time switching signals are also established. The theoretical developments are demonstrated by two numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

典型大时变时滞系统的鲁棒fuzzy-PID控制及应用

针对典型大时变时滞系统, 设计新的鲁棒fuzzy_PID控制器, 实现系统参数大范围失配情况下的闭环鲁棒稳定, 并且使闭环系统达到设定值无静差跟踪及满意的动态性能. 现场应用效果良好.     

Computing stability domains in the space of time delay and controller coefficients for FOPDT and SOPDT systems

In this paper, a method is proposed to compute the maximum allowable time delay for first-order plus dead time and second-order plus dead time systems, in order to maintain stability. Designing first order controllers for such systems to preserve stability for a longer time delay is the main aim of this paper. The procedure uses the properties of the phase diagram of the open-loop transfer functions of such systems. The effects of the variation of the controller coefficients on the maximum allowable time delay of the systems are investigated and the stability domains in the space of the uncertain delay and the controller coefficients are computed. The results can be used to design first-order robust controllers for first and second order processes containing uncertain delays.     

带有非线性扰动的时变时滞系统的稳定性准则

研究了带有非线性扰动的时变时滞系统的稳定性问题.基于时滞分割方法,提出了保守性更小的系统稳定性分析准则.利用一个自由参数将时滞区间分割为2个子区间,进而构造了含有时滞分割点状态信息和3重积分项的Lyapunov-Krasovskii泛函,并采用自由矩阵积分不等式界定泛函导数中的交叉项.基于Lyapunov稳定性定理,得到了以线性矩阵不等式描述的时滞相关型稳定性准则.数值算例表明该稳定性准则能够得到更大的时滞上界,与已有结果相比具有更小的保守性.     

Exponential stability of stochastic singular delay systems with general Markovian switchings

In recent years, Markovian jump systems have received much attention. However, there are very few results on the stability of stochastic singular systems with Markovian switching. In this paper, the discussed system is the stochastic singular delay system with general transition rate matrix in terms of uncertain and partially unknown transition rate matrix. The aim is to answer the question whether there are conditions guaranteeing the underlying system having a unique solution and being exponentially admissible simultaneously. The proposed results show that all the features of the underlying system such as time delay, diffusion, and general Markovian switchings play important roles in the system analysis of exponential admissibility. A numerical example is used to demonstrate the effectiveness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

切换线性时滞系统的稳定性判据

考虑了一类切换线性时滞系统的稳定性问题.基于Lyapunov函数方法和矩阵测度的概念,分别给出了切换系统时滞独立以及时滞依赖的渐近稳定性和指数稳定性判据,设计了相应的镇定切换律.最后,通过数值算例验证了所提算法的正确有效性.