Some simplifications of the graphical Nyquist criterion

The Nyquist criterion for feedback stability is reformulated in a very simple fashion. Using the motion of positive and negative crossings, a number of encirclements of a point by the Nyquist plot can be quickly computed. With this reformulation, it is a simple matter to determine the value of the feedback gain for which the system is stable     

Liquid level control: Simplicity and complexity

From a superficial perspective, the control of liquid level in a vessel is a “non-problem.” Well-known process control heuristics advise using a proportional-only controller with a gain K_c = 2. However many plant operators are uncomfortable with the resulting “offset” (“steady-state error” or “droop”) between the process variable and the controller setpoint that is inherent in a proportional controller because of the lack of integral action to drive the error to zero. Therefore level controllers with proportional-integral (PI) action are often found in process units.The purpose of this paper is to illustrate that PI control does not provide effective attenuation of flow disturbances and that it actually amplifies them. We also show that PI level controllers can give confusing tuning results. In most closedloop systems, increasing controller gain makes the system more oscillatory (smaller closedloop damping coefficient). When PI controllers are used to control liquid level, the exact opposite occurs (increasing controller gain makes the system less oscillatory).     

Relay feedback tuning of robust PID controllers with iso-damping property.

A new tuning method for proportional-integral-derivative (PID) controller design is proposed for a class of unknown, stable, and minimum phase plants. We are able to design a PID controller to ensure that the phase Bode plot is flat, i.e., the phase derivative w.r.t. the frequency is zero, at a given frequency called the "tangent frequency" so that the closed-loop system is robust to gain variations and the step responses exhibit an iso-damping property. At the "tangent frequency," the Nyquist curve tangentially touches the sensitivity circle. Several relay feedback tests are used to identify the plant gain and phase at the tangent frequency in an iterative way. The identified plant gain and phase at the desired tangent frequency are used to estimate the derivatives of amplitude and phase of the plant with respect to frequency at the same frequency point by Bode's integral relationship. Then, these derivatives are used to design a PID controller for slope adjustment of the Nyquist plot to achieve the robustness of the system to gain variations. No plant model is assumed during the PID controller design. Only several relay tests are needed. Simulation examples illustrate the effectiveness and the simplicity of the proposed method for robust PID controller design with an iso-damping property.     

基于稳定裕量的二阶时滞系统PID控制器参数稳定域

基于广义Hermite-Biehler定理,由二阶时滞对象的逆Nyquist曲线,可确定PID控制器比例增益的稳定范围;在积分和微分增益平面上,运用一组不等式可确定该二维平面上参数的稳定区域,从而给出了确定二阶时滞系统PID控制器参数稳定域的方法.在此基础上,针对稳定裕量指标,也给出相应的PID控制器稳定域的处理方法....     

Computation of stabilizing PID gain regions based on the inverse Nyquist plot

Based on the inverse Nyquist plot, a method is presented to determine the stabilizing PID gain regions for linear systems. It does not depend on the integral and the derivative parameters. The two stable conditions are derived from a generalization of the Hermite-Biehler theorem. Two kinds of key points can be obtained from the inverse Nyquist plot. The PID gain would be divided into several regions by abscissas of these key points. According to the number of intersection of the inverse Nyquist plot and the vertical line, the stabilizing PID gain regions can be determined. Examples are given to show the benefits of the proposed method.     

Robustness and digital stabilization of the glucoregulatory system

A pre-programmed controller that works over a family of single-input single-output systems representing the diabetic subject was achieved, based on the recent practical fourth-order digital model of the experimentally validated glucoregulatory system. The controller was chosen such that it contains an integral action to ensure that the steady-state error vanishes. Its zeros are so selected as to match the undesired complex-conjugate poles of the unstable metabolism. The controller utilizes a nonlinear reference: ramp function succeeded by step input. The closed-loop gain and the coefficient of the ramp function are tuned successively such that desired dynamics are obtained using the root-locus portrait. Three sets of well-behaved responses and their corresponding internal states were obtained with variable peak undershoot. The robustness tracking property of the controller was tested using the bounding polynomials suggested by Kharitonov's stability criterion.     

Guaranteed dominant pole placement with PID controllers

Pole placement is a well-established design method for linear control systems. Note however that with an output feedback controller of low-order such as the PID controller one cannot achieve arbitrary pole placement for a high-order or delay system, and then partially or hopefully, dominant pole placement becomes the only choice. To the best of the authors’ knowledge, no method is available in the literature to guarantee dominance of the assigned poles in the above case. This paper proposes two simple and easy methods which can guarantee the dominance of the two assigned poles for PID control systems. They are based on root locus and Nyquist plot respectively. If a solution exists, the parametrization of all the solutions is explicitly given. Examples are provided for illustration.     

Simple Tuning Rules for Integrating Processes with Large Time Delay

In the present article, controllers of the modified Smith predictor are designed for pure integrating, integrating plus first order and double integrating processes with large time delays. The direct synthesis approach is used to tune the set point tracking controller. Suitable values of the desired closed loop time constant that satisfies both robust stability and robust performance conditions are provided based on extensive simulation studies. The PD controller, which is used for load disturbance rejection, is designed so as to achieve a specified slope (ψ) of the Nyquist curve at the gain crossover frequency. The gain crossover frequency that corresponds to a phase margin of 45 degrees is suggested for all three considered integrating processes. Furthermore, ψ is recommended as 45 degrees for double integrating processes and 90 degrees for pure integrating and integrating plus first order process models. Two simulation examples are considered to illustrate the usefulness of the proposed tuning rules.     

指定圆形极点配置区域输出无偏的输出反馈协方差控制器设计

系统输出无偏性及稳态状态协方差是随机系统的两个重要稳态性能指标，系统的动态品质主要决定于系统的闭环极点位置。本文给出的输出反馈协方差控制器能确保系统：１）闭环极点位于指定圆形配置区域；２）稳态协方差满足既定的上限约束；３）稳态输出无偏。     

对角优势的常数阵实现

本文考虑Nyquist阵列法中的对角优势问题,讨论了由常数阵实现对角优势的条件与算法。 符号约定: G(s):系统的过程传递函数阵G(s)∈C~(mxm) Q(s):包括补偿器在内的开环传递函数阵Q(s)∈C~(mxm) K:前置补偿器阵,在本文中 K∈R~(mxm) q_(?),k_(?)分别为Q(s),K阵的第j列向量 ‖·‖_s向量的欧氏范数 〔·〕~T:矩阵或向量的转置 λ(·):矩阵(·)的特征值     

基于输入整形的时滞系统最小拍控制

基于Ｎｙｑｕｉｓｔ准则,按照给定的相位稳定裕度设计ＰＩ控制器使一阶时滞对象稳定,然后引入输入整形控制器对输入进行整形,实现了一阶时滞对象的最小拍控制。在确定采用周期后,整个控制系统的设计尺需按相位稳定裕度的需求解析算出ＰＩ控制器的比例增益即可,设计简单,性能良好。仿真结果表明,该方法相当有效,实现的最小拍控制无纹波。     

An admittance shaping controller for exoskeleton assistance of the lower extremities

We present a method for lower-limb exoskeleton control that defines assistance as a desired dynamic response for the human leg. Wearing the exoskeleton can be seen as replacing the leg’s natural admittance with the equivalent admittance of the coupled system. The control goal is to make the leg obey an admittance model defined by target values of natural frequency, peak magnitude and zero-frequency response. No estimation of muscle torques or motion intent is necessary. Instead, the controller scales up the coupled system’s sensitivity transfer function by means of a compensator employing positive feedback. This approach increases the leg’s mobility and makes the exoskeleton an active device capable of performing net positive work on the limb. Although positive feedback is usually considered destabilizing, here performance and robust stability are successfully achieved through a constrained optimization that maximizes the system’s gain margins while ensuring the desired location of its dominant poles.     

On frequency-domain criterion of finite-time convergence of second-order sliding mode control algorithms

Transient processes in the systems controlled by the twisting second-order sliding mode (SOSM) control algorithm and certain generic SOSM given by the describing function are analyzed in the frequency domain. The analysis is based on the approximate describing function method. The relationship between the frequency response (Nyquist plot) of the plant, the shape of the negative reciprocal describing function of the controller, and the transient process convergence rate is investigated. A simple criterion of the existence of finite-time convergence is proposed. It is shown that the convergence rate in a system controlled by a SOSM controller depends on the angle between the high-frequency asymptote of the Nyquist plot of the plant and the low-amplitude asymptote of the negative reciprocal of the describing function of the controller, which is named the phase deficit.     

In-line mixer pH system for the identification of titration curve

An in-line mixer pH system is proposed to identify the titration curve before changes in the input stream affect the main pH neutralization reactor. It is operated under integral control and shows oscillatory responses for a wide range of controller integral gain. The bifurcation diagram for the change of the controller integral gain is very similar to that of the quadratic map which is often used in illustrating the chaos phenomena of nonlinear processes. Various bifurcation diagrams are obtained as operating conditions of the in-line mixer system such as the set point of pH and components of the input stream change. The integral control keeps oscillations be around the set point, providing plentiful information about the pH process. The oscillations can be used to identify the titration curve of the input acid stream. Simulation and experimental results show that the chaotic operations of a small in-line mixer can be positively used for the identification of the titration curve.     

Gain scheduled state feedback velocity control of hydrostatic drive transmissions

In this paper, a velocity tracking controller for hydrostatic drive transmissions is developed. The solution is based on a state-dependent model that incorporates nonlinear characteristics of the system. A full state feedback controller is devised and the gains are scheduled on measured speed and pressures, together with approximated volumetric flow. The effects of uncertainties, especially those related to equilibrium values of pressures, are eliminated by utilizing so-called D-implementation. This technique eliminates the need for equilibrium values, which are model based and thus uncertain.To demonstrate the efficacy of the controller, the solution is implemented on a 4.5-ton wheel loader. For comparison purposes, a constant gain state feedback controller with integral action is devised, and also a linear PID controller is tuned. The results show that the benefits of the devised controller are significant when it is compared to these two controllers. Moreover, the controllability of the machine is maintained in every situation.     

MIMO controller synthesis with integral-action integrity

A controller synthesis method is presented for closed-loop stability and asymptotic tracking of step input references with zero steady-state error. Integral-action is achieved in two design steps starting with any stabilizing controller and adding a PID-controller in a configuration that guarantees robust stability and tracking. The proposed design has integral-action integrity, where closed-loop stability is maintained even when any of the proportional, integral, or derivative terms are removed or the entire PID-controller is limited by a constant gain matrix. The integral constant can be switched off when integral-action is not wanted.     

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.     

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.     

PID控制器增益的稳定范围研究

基于逆Nyquist曲线,提出了一种线性系统在PID控制下确定增益稳定范围的方法,为PID控制器增益的稳定提供了一条快速而有效的途径。由逆Nyquist曲线上的实部为极值的点,将PID增益分割成若干区间。再运用广义的Her—mite-Biehler定理得出一个推理和二个条件,通过纵向直线与逆Nyquist曲线的交点数,可获得系统在PID控制下增益稳定的区间。仿真实例验证了该方法的有效性。该方法应用简便,能有效解决PID控制下增益的稳定范围。