Stabilisation of unstable FOPDT processes with a single zero by fractional-order controllers

In this article, stabilisation of unstable first-order-plus-dead-time (FOPDT) processes with a single zero by fractional-order (FO) controllers is investigated. A Nyquist stability criterion-based approach is adopted to derive the conditions for stability. Sufficient stabilisability conditions by FO [proportional integral] controllers and FO-lead–lag controllers are established. In addition, robust stability of the system with these FO controllers is investigated. To illustrate the results, some examples are provided.     

Enhanced modified Smith predictor for second-order non-minimum phase unstable processes

In this paper, a modified Smith predictor design is proposed for enhanced control of non-minimum phase unstable second-order time-delay processes with/without zero. The proposed method involves the design of two controllers, i.e. set-point tracking controller and disturbance rejection controller. Set-point tracking controller is designed as a proportional-integral-derivative (PID) in series with a lag filter using direct synthesis method. The disturbance rejection controller is designed as a PID in series with a lead/lag filter based on direct synthesis method. Set-point weighting is considered for minimising the overshoots. The proposed method is applied by simulation on several second-order unstable processes. Robustness studies have been carried out using the small-gain theorem. The method gives good nominal and robust control performances. Significant improvement in the disturbance rejection is obtained with the proposed method when compared to the recently reported methods in the literature.     

Frequency-domain design of pid controllers for stable and unstable systems with time delay

An approach for tuning PID-type controllers is developed for single input single-output, linear time-invariant systems, based on an extension to the method of D-partition. This method permits design for simultaneous minimum gain and phase margin requirements. It also allows design for specified maximum gain and phase cross-over frequencies of the controlled system. The technique can be applied to systems with stable or unstable plants as well as to irrational systems with significant time delay. Another advantage of the method is that it can be used for various controller configurations including derivative in the feedback path. Three examples illustrate the tuning method.     

Finite-time stability of a class of nonlinear fractional-order system with the discrete time delay

This paper investigates the finite-time stability problem of a class of nonlinear fractional-order system with the discrete time delay. Employing the Laplace transform, the Mittag-Leffler function and the generalised Gronwall inequality, the new criterions are derived to guarantee the finite-time stability of the system with the fractional-order 0 < α < 1. Further, we propose the sufficient conditions for ensuring the finite-time stability of the system with the fractional-order 1 < α < 2. Finally, based on the modified Adams–Bashforth–Moulton algorithm for solving fractional-order differential equations with the time delay, we carry out the numerical simulations to demonstrate the effectiveness of the proposed results, and calculate the estimated time of the finite-time stability.     

Improved PID controller design for unstable time delay processes based on direct synthesis method and maximum sensitivity

In this paper, a proportional-integral-derivative controller in series with a lead-lag filter is designed for control of the open-loop unstable processes with time delay based on direct synthesis method. Study of the performance of the designed controllers has been carried out on various unstable processes. Set-point weighting is considered to reduce the undesirable overshoot. The proposed scheme consists of only one tuning parameter, and systematic guidelines are provided for selection of the tuning parameter based on the peak value of the sensitivity function (M_s). Robustness analysis has been carried out based on sensitivity and complementary sensitivity functions. Nominal and robust control performances are achieved with the proposed method and improved closed-loop performances are obtained when compared to the recently reported methods in the literature.     

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.     

Stabilization of all-pole unstable delay processes by simple controllers

The stabilization of a class of all-pole unstable delay processes of arbitrary order with single unstable pole by means of simple controllers is investigated in details. Complete stabilizability conditions are established and the computational methods for determining stabilizing controller parameters presented. They provide theoretical understanding of such a stabilization problem and can also serve as practical guidelines for actual controller design.     

Variable structure controllers for unstable processes

A variable structure control (VSC) method for unstable industrial processes is proposed. The proposed control method is able to provide a highly satisfactory system performance and to tackle with robustness issues of the processes in the presence of uncertainties. An ITAE-based numerical tuning algorithm for acquiring optimal control parameters, and a direct auto-tuning mechanism for the proposed controller are also provided. The performance of the proposed VSC method is illustrated on some unstable process models including a continuous stirred tank reactor (CSTR), in order to show its effectiveness, validity and feasibility.     

Design of PID controllers for interval plants with time delay

First of all, the box theorem is extended to the interval plants with the fixed delay. An approach is presented to design the PID controller for interval plants with the fixed delay, which can obtain all of the stabilizing PID controllers. Then, using Hermite–Biehler theorem, extreme point results are provided by the virtual quasi-polynomials. When two virtual and two vertex quasi-polynomials corresponding to a Kharitonov-like segment plant are stable under a particular PID controller, it is sufficient that the same PID controller can stabilize this Kharitonov-like segment plant. The virtual quasi-polynomials are obtained in a simple way, and they are expressed in terms of the controller and the Kharitonov polynomials of the interval plants. A PID controller stabilizes interval plants with the fixed delay if it simultaneously stabilizes thirty-two quasi-polynomials. The example is given to illustrate the proposed method.     

Analytical design of two-degree-of-freedom control scheme for open-loop unstable processes with time delay

This paper proposes an analytical two-degree-of-freedom control scheme for open-loop unstable processes with time delay, which leads to the remarkable improvement of regulatory capacity for both of reference input tracking and load disturbance rejection. Firstly a conventional proportional or plus derivative controller is deliberately employed for stabilizing the setpoint response. Then the setpoint tracking controller is analytically derived in terms of the integral-squared-error (ISE) performance specification. By proposing the desired closed-loop complementary sensitivity function for rejecting load disturbances, the corresponding controller, i.e. disturbance estimator, is inversely figured out. Hence the nominal setpoint response is decoupled from the load disturbance response by virtue of the open-loop control manner for the setpoint tracking. In consequence, both of them can be optimized simultaneously and separately. At the same time, robust stability analysis for the proposed control structure is provided in the presence of the process multiplicative uncertainty. Accordingly the on-line tuning rule for the single adjustable parameter of each controller is suggested to cope with the process unmodeled dynamics in practice. Finally, several illustrative examples are included to demonstrate the superiority of the proposed method.     

Distributed formation control of fractional-order multi-agent systems with absolute damping and communication delay

The distributed formation control of fractional-order multi-agent systems is mainly studied under directed interaction topology in this paper. First, the control algorithm with absolute damping and communication delay is proposed to achieve the formation control. Then, some sufficient conditions are derived by using the matrix theory, graph theory and the frequency-domain analysis method. Finally, based on the numerical method of predictor–corrector, several simulations are presented to illustrate the effectiveness of the obtained results.     

An admissible-behaviour-based analysis of the deadlock in Petri-net controllers

This paper addresses the problem of verifying the discrete control logic that is typically implemented by programmable controllers. Not only are the logical properties of the controller studied during verification, the behaviour of the overall controlled system is also examined. An approach that combines the calculation of the safety-oriented interlock controllers in terms of supervisory control theory (SCT), the corresponding calculation of the admissible behaviour of the system, and the specification of the desired system operation by Petri nets is proposed. A potential deadlock in the controlled system is then verified by taking the admissible-behaviour model as a process model. The analysis of the simultaneously operated supervisory-control-based interlock controller and the Petri-net-based sequential controller is performed with a C-reachability graph. The paper focuses on the calculation of the graph, and the approach is illustrated with an example of a simple manufacturing cell.     

几种不稳定滞后对象的预测PID 控制

针对几种不稳定滞后过程,给出一种预测PID控制器的结构形式．该控制器具有内环和外环两种控制器：内环控制器主要用于稳定系统;外环控制器具有预测PID控制的结构形式,主要用于消除输入干扰的影响和改善控制系统的动态性能．这种控制器结构简单,可调参数少,且参数的调节方便、直观．仿真结果表明,在干扰和模型失配的情况下,此类预测PID控制器仍具有良好的控制性能和鲁棒稳定性能．     

Stabilization of pure unstable delay systems by the mirror mapping technique

In this note, a novel robust control design for a class of pure unstable systems with time delay was addressed. In the algorithm, the time delay was not neglected as usual and was approximated as a stable component using improved Padé technique. The unstable system was mirror mapped into a stable system and subsequently the closed-loop gain shaping algorithm is used to design the robust control law. The proposed controller was easy to tune which had only one adjustable parameter with engineering sense. By virtue of the Nyquist stability criterion, the exact stabilization analysis has been obtained for the closed-loop system. Simulation results show that the control effects can tackle the input load disturbances and the delay perturbation more effectively. The method has the advantages of simple and efficient design.     

Design of observer-based controllers for a class of discrete systems

Design of observers and observer-based controllers for linear, discrete control systems with fast and slow modes are considered in this paper. The time separation is expressed in terms of an inequality relating norms of system submatrices. It is shown that reconstruction of fast and slow states can be accomplished by a full-order observer, the gains of which are computed using a two-stage procedure. Then, provided that the fast subsystem is asymptotically stable, it is shown that a low-order observer-based controller can be designed with independent gain matrices to stabilize the original discrete system. The theoretical analysis is illustrated by a ninth-order boiler system model.     

不同阶分数阶混沌系统的同步与参数辨识

针对不确定分数阶混沌系统的同步和参数辨识问题,提出一种新的方法,即用不同阶分数阶系统来同步和参数辨识.利用主动控制和预控制量方法,基于分数阶混沌系统稳定性理论和自适应控制理论,设计控制器,实现不同阶分数阶混沌系统之间的同步和参数辨识.理论和仿真结果实现了不同阶Chen 系统间的同步和辨识,表明了该方法的有效性.     

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay

The stabilization of exponentially unstable linear systems with time-varying input delay is considered in this paper. We extend the truncated predictor feedback (TPF) design method, which was recently developed for systems with all poles on the closed left-half plane, to be applicable to exponentially unstable linear systems. Assuming that the time-varying delay is known and bounded, the design approach of a time-varying state feedback controller is developed based on the solution of a parametric Lyapunov equation. An explicit condition is derived for which the stability of the closed-loop system with the proposed controller is guaranteed. It is shown that, for the stability of the closed-loop system, the maximum allowable time-delay in the input is inversely proportional to the sum of the unstable poles in the plant. The effectiveness of the proposed method is demonstrated through numerical examples.     

一阶时滞不稳定过程的复合PID控制

针对一阶时滞不稳定过程讨论了一类复合ＰＩＤ控制及其参数整定公式．该方法基于分步设计的思想，在比例控制器镇定基础上对闭环所构成的广义稳定对象，设计二级ＰＩＤ控制器．在二级控制器设计中，通过引入时滞二阶稳定模型优化ＰＩＤ参数．本文同时讨论了等价的二自由度ＰＩＤ控制器．     

The simplest fuzzy PID controllers: mathematical models and stability analysis

This paper reveals mathematical models for the simplest fuzzy PID controllers which employ two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. Mathematical models are derived via left and right trapezoidal membership functions for each input, singleton or triangular membership functions for output, algebraic product triangular norm, different combinations of triangular co-norms and inference methods, and center of sums (COS) defuzzification method. Properties of these structures are studied to examine their suitability for control application. For the structure which is suitable for control, bounded-input bounded-output (BIBO) stability proof is presented. An approach to design fuzzy PID controllers is given. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controllers.