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.     

Stabilizing region of fractional-order proportional integral derivative controllers for interval delayed fractional-order plants

This paper concentrates on computing the stabilizing region of fractional-order proportional integral derivative (FOPID) controllers for interval delayed fractional-order plants. An interval delayed fractional-order plant is defined as a fractional-order transfer function with a time delay whose denominator and numerator coefficients are all uncertain and lie in specified intervals. In the present study, first, a theorem is proven to analyze the robust stability of the given closed-loop. Then, a method is proposed to solve the problem of robustly stabilizing interval delayed fractional-order plants by using FOPID controllers. Moreover, two auxiliary functions are presented to fulfill the additional specifications of design, ensuring better performance of the controlled system with respect to the disturbance and noise. Finally, two examples are provided to illustrate the design procedure.     

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．     

Stabilizing sets of PI/PID controllers for unstable second order delay system

In this paper, the problem of stabilizing an unstable second order delay system using classical proportional-integralderivative(PID) controller is considered. An extension of the Hermite-Biehler theorem, which is applicable to quasi-polynomials, is used to seek the set of complete stabilizing proportional-integral/proportional-integral-derivative(PI/PID) parameters. The range of admissible proportional gains is determined in closed form. For each proportional gain, the stabilizing set in the space of the integral and derivative gains is shown to be a triangle.     

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.     

Stabilization of second-order unstable delay processes by simple controllers

The stabilization of second-order unstable delay processes by simple controllers is investigated. The elementary analysis based upon the Nyquist stability criterion is carried out to establish explicit and complete stabilizability results in terms of the upper limit of time-delay size and the computational methods for determining stabilizing controller parameters. The analysis provides both theoretical understanding of such stabilization problem and practical guidelines for actual controller design. The results for other unstable delay processes studied extensively in the literature are also obtained as the special cases of the above second-order one.     

Stabilisability of nonlinear systems by means of time-dependent switching rules

Given a family of linear systems which admit an asymptotically stable convex combination, the existence of stabilising time-dependent switching rules can be proved by using the Baker–Campbell–Hausdorff formula for exponentials. The control laws obtained in this way are periodic, fast switching and independent of the initial state. We prove that under a similar assumption, the approach can be extended to provide stabilising time-dependent switching rules for the families of nonlinear vector fields, as well. However, the resulting control law, in general, is not periodic, not fast switching and it may depend on the initial state.     

IMC design for unstable processes with time delays

A modified IMC structure is proposed for unstable processes with time delays. The structure extends the standard IMC structure for stable processes to unstable processes and controllers do not have to be converted to conventional ones for implementation. An advantage of the structure is that setpoint tracking and disturbance rejection can be designed separately. A method is proposed to tune the modified IMC structure with an emphasis on the robustness of the structure. Design for some typical delayed unstable processes shows that the control structure can be tuned easily and achieve good tradeoff between time-domain performance and robustness.     

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.     

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     

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.     

Stability analysis and design of integrating unstable delay processes using the mirror-mapping technique

Robustness analysis and design for the integrating unstable delay systems are discussed in this note. The Nyquist criteria have established the exact stability margin of the novel robust control scheme, which is meaningful in the process control practice. Comparing with the existing results, the control law is designed based on the delay-approximated model (using the all-pole Padé approximation). The unstable system was mirror mapped into a minimum-phase system and then the control law was derived by the closed-loop gain shaping algorithm (CGSA). In addition, a small constant δ was introduced in the algorithm to prevent the integral cancellation limitation, which is inherent in the CGSA. The proposed scheme obtains several advantages: a concise design procedure and easy to implementation due to the simple unit feedback structure. The comparative analysis with respect to recently successful works illustrates a substantial improvement in the performance-robustness tradeoff.     

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.     

Improved quasi-uniform stability criterion of fractional-order neural networks with discrete and distributed delays

This article mainly investigates the quasi-uniform stability of fractional-order neural networks with time discrete and distributed delays (FONNDDDs). First, a novel fractional-order Gronwall inequality with discrete and distributed delays (FOGIDDDs) is established; it can be used to study the stability of a variety of fractional-order systems with discrete and distributed delays (FOSDDDs). Second, on the basis of this inequality and Leray-Schauder alternative theorem, the existence and uniqueness results for the FONNDDDs are proved. Third, an improved criterion for the quasi-uniform stability of FONNDDDs is obtained in terms of this inequality. Ultimately, one numerical example is provided to expound the effectiveness and the superiority of the proposed result.     

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.     

Wright分数阶时滞微分方程的离散化过程

引进了一种离散化方法对分数阶时滞微分方程进行离散化求解。首先考察Wright分数阶时滞微分方程;其次分析相应具有分段常数变元的Wright分数阶时滞微分方程,并应用离散化过程对模型进行数值求解;然后根据不动点理论讨论该合成动力系统不动点的稳定性;最后借助MATLAB对模型进行数值仿真,并结合Lyapunov指数、相图、时间序列图、分岔图探讨模型更多复杂的动力学现象。结果显示,提出方法成功对Wright分数阶时滞微分方程进行离散。