Stability Boundary Locus Research Articles

An interactive design strategy for fractional order PI controllers in LabVIEW

This paper presents an interactive design for fractional order PI (FOPI) controller based on inverse Fourier transform method (IFTM) in accordance with stability region of a closed-loop control system in LabVIEW, which is a powerful graphical program. Stability boundary locus (SBL) method is used to obtain the stability region including all stabilising FOPI controller parameters in (Kp, Ki) plane. The time response of the closed-loop control system with FOPI controller is then obtained by IFTM using the stabilising controller parameters selected from stability region. Changing the selected fractional order controller parameters in stability region, users can observe the step response of the system interactively.

An interactive design strategy for fractional order PI controllers in LabVIEW

This paper presents an interactive design for fractional order PI (FOPI) controller based on inverse Fourier transform method (IFTM) in accordance with stability region of a closed-loop control system in LabVIEW, which is a powerful graphical program. Stability boundary locus (SBL) method is used to obtain the stability region including all stabilising FOPI controller parameters in (Kp, Ki) plane. The time response of the closed-loop control system with FOPI controller is then obtained by IFTM using the stabilising controller parameters selected from stability region. Changing the selected fractional order controller parameters in stability region, users can observe the step response of the system interactively.

Robust PID Load Frequency Controller Design with Specific Gain and Phase Margin for Multi-area Power Systems

In interconnected power systems, the load frequency control (LFC) is considered a hugely beneficial ancillary service. The goal of the LFC in an interconnected power system is to limit the frequency of each area within certain bounds and to maintain the tie-line power flows within some pre-specified latitudes by balancing the power outputs of the generators so as to satisfy ever changing load demands. In the classical control theory, PID controller is said to be robust if it provides some specific gain and phase margin. In this paper, a novel methodology is proposed for the robust PID controller design having specific gain and phase margins for LFC in a multi-area power system. The proposed technique is based on stability boundary locus and PID controllers are designed for four-area power system having different types of turbines. The simulations are carried out using MATLAB and effectiveness of the proposed methodology is verified by the comparison with a recently published approach.

PID Tuning Method for Integrating Processes Having Time Delay and Inverse Response

In this paper, a PID tuning method for integrating processes having time delay and inverse response is presented. The method is based on the stability boundary locus method and geometrical center (WGC) approach. The systematic procedure of the method is first to obtain the stability region in the PI controller parameters (proportional gain: kp and integral gain: ki) plane according to derivative gain (kd) using the stability boundary locus method and then to find the weighted geometrical center point of this region. The WGC controllers are obtained by using different values of kd. Simulation examples have demonstrated that PID controller designed by using the proposed method gives good results.

Design of PI Controller using Optimization Method in Fractional Order Control Systems

In control systems, controller preference and design is an important issue for meeting the desired design criteria. In this paper, PI controller design was performed by using optimization method for fractional order systems. First, all the PI controller parameters that make the control system stable are calculated by using the stability boundary locus method. However, each controller parameter selected in the stability region may not be able to optimally control the system. Optimal controller parameters that provide the best control from the PI controller parameters that make the system stable by using the optimization method are obtained. In the optimization process, the optimal PI controller parameters are calculated by using the integral performance criterion based on the error. Simulation studies have been done for closed loop control system including a fractional order transfer function with time delay. It has been shown that the presented method can be successfully applied to fractional order control systems.

DESIGN, ANALYSIS AND EXPERIMENTAL VERIFICATION OF A NOVEL NONLINEAR PI CONTROLLER

In this study, a novel variable gain PI controller structure is introduced. The proposed controller structure consists of a sector-bounded nonlinear function of the relative error value in cascade with a linear fixed-gain PI controller. The stability analysis of the closed loop system is examined through Popov stability criterion, Routh-Hurwitz stability method and stability boundary locus method for both second-order and higher-order systems. In addition, the performance of the controller against parameter variations and disturbances is investigated through some simulations for second order systems. An experimental study, an active suspension system, is conducted to examine the performance of the controller for higher order systems. In the literature, there are similar controllers, but the proposed one is superb in terms of effectiveness and stability. The new controller prevents the saturation of the controller signal. Simulation results and experimental studies reveal that proposed controller structure is quite effective for both lower and higher order systems.

Fractional order PID controller for power control in perturbed pressurized heavy water reactor

This paper presents an interval fractional-order proportional integral derivative (INFOPID) controller design for the power control of a highly nonlinear Pressurized Heavy Water Reactor (PHWR) under step-back condition. A single robust INFOPID controller is designed utilizing stability boundary locus method for eight nuclear reactor models considering eight interval conditions for each reactor model of the PHWR. The interval models have been obtained using Edge theorem. Simulation results show that the proposed INFOPID controller applied for active step-back in the reactor gives an efficient set point tracking performance, defined for variation in initial power level or control rod drop. The performance of the proposed INFOPID controller is evaluated using different integral error criterion by comparing its performance with existing methods in the literature. The proposed controller is found to give better results than the existing techniques.

PI Kontrolörlü Otomatik Gerilim Regülasyon Sisteminin Kararlılık Analizi

The system voltage is one of the most important parameters, which determines the power quality. The stability of the system voltage is critical for the power system. This paper investigates the stability analysis of the automatic voltage regulation (AVR) system controlled by a PI controller related to the controller gains. For this purpose, a graphical-based technique called as the stability boundary locus method is proposed in the paper to obtain the stable parameters space of PI controller gains. Stability of closed region computed on parameters space determines from roots of characteristic equation of AVR system. The time-domain simulations are performed in Matlab/Simulink environment to validate the theoretical results.

Practical Tuning Algorithm of PDµ Controller for Processes with Time Delay

In this paper, a practical tuning algorithm of fractional order PD controller for processes with time delay using the weighted geometrical center (WGC) method is presented. This method is based on calculating of the stabilizing PDµ controller parameters region which is plotted using the stability boundary locus in the (kd,kp) plane and computing the weighted geometrical center of stability region.The important advantages of the proposed method are both calculating of controller parameters without using complex graphical methods and ensuring the stability of closed loop system. From the examples, it can be easily seen that this simple tuning method can perform quite reliable results in that unit step response.

Design of PIλDμ controller for global power control of Pressurized Heavy Water Reactor

In this paper, a robust stabilizing controller design method is presented for global power control of a Pressurized Heavy Water Reactor (PHWR) under step-back condition scheme using a Fractional Order Proportional Integral Derivative (PIλDμ) controller resulting into robust performance. The method is applicable to design a controller for One Non Integer Order Plus Time Delay (NIOPTD-I) plant which satisfies design specifications such as phase margin and gain crossover frequency. Stability boundary locus method is used in (Kp, Ki, Kd) parameter space for NIOPTD-I plants to obtain stability region. The robust performance is obtained by satisfying flat phase condition at gain crossover frequency where phase is almost constant for large span of frequencies. The simulation result of the proposed PIλDμ controller shows active step-back control to the insertion of the rod with no undershoot and with the robust performance, hence safe to the plant for gain variations from 500% lower side to 1000% upper side. The PIλDμ controller with a plant shows that 30% and 50% global power drop from initial 100% is achieved in a reasonable time without undershoot.

Design of FOPID controller for fractional-order plants with experimental verification

In this paper, a design method is proposed for a robust stabilizing fractional order proportional integral derivative (FOPID) controller for One Non-Integer Order Plus Time Delay plant (NIOPTD-I). The NIOPTD-I model is obtained after reducing the higher order continuous time process model. The FOPID controller is designed by using stability boundary locus method, which satisfies user defined frequency domain specifications phase margin and the gain crossover frequency. A robust control system is designed against gain variations by achieving flat phase condition which is due to zero slope at the gain crossover frequency of the phase plot. Thus robust stabilizing FOPID controller for NIOPTD-I fractional order plant is designed. The applicability of the proposed method is illustrated with simulation example and experimental validation with the liquid level control system.

Gain and phase margins based delay-dependent stability analysis of single-area load frequency control system with constant communication time delay

This paper presents a comprehensive stability analysis of a single-area load frequency control (LFC) system with constant communication delays. First, an exact method that takes into account both gain and phase margins is proposed to determine stability delay margins in terms of system and controller parameters. The method implements an elimination procedure to transform transcendental characteristic equation into a standard polynomial of the crossing frequency. The real roots of this new standard polynomial exactly match with the purely imaginary roots (crossing frequencies) of the original characteristic equation with transcendental terms. Secondly, an effective and simple graphical method is implemented to compute all stabilizing Proportional Integral (PI) controller gains for a given time delay. The approach is based on extracting stability region and the stability boundary locus in the PI controller parameter space having user defined gain and phase margins, and relative stability. The time-domain simulation studies indicate that the proposed scheme improves dynamic performance gain and phase margins are included in delay-dependent stability analysis of single-area LFC with communication delays.

Comprehensive gain and phase margins based stability analysis of micro‐grid frequency control system with constant communication time delays

This study presents a comprehensive delay-dependent stability analysis of a micro-grid system with constant communication delays. First, an exact method that takes into account both gain and phase margins (GPMs) is proposed to determine stability delay margins in terms of system and controller parameters. The method implements an elimination procedure to transform transcendental characteristic equation into a standard polynomial of the crossing frequency. The real roots of this new standard polynomial exactly match with the purely imaginary roots (crossing frequencies) of the original characteristic equation with transcendental terms. Second, an effective and simple graphical method is implemented to compute all stabilising proportional integral (PI) controller gains for a given time delay. The approach is based on extracting stability region and the stability boundary locus in the PI controller parameter space having user defined GPMs, and relative stability. The time-domain simulation studies indicate that the proposed schemes give better desired dynamic performance as compared with the recently developed schemes for micro-grid with communication delays.

Robust control of static Var compensator-based power oscillation dampers using polynomial control in power systems

A static Var compensator (SVC) installed in a power transmission network can be effectively exploited to enhance the damping of low frequency electromechanical oscillations. The application of robust control theory offers more reliable and robust damping controller to achieve desired damping level considering variations in the operating conditions of power system. This paper presents a new approach to design a robust proportional-integral (PI) controller for stabilizing power system oscillations. The variability in operating conditions is captured using an interval polynomial and then, Kharitonov’s theorem is used to design the desired damping controller. The proposed method is based on plotting the stability boundary locus in the ( kp-ki) plane and then computing the stabilizing values of the parameters of a PI controller. Besides stabilization, computation of stabilizing PI controllers that achieve user specified gain margin (Gm), phase margin (Pm) and bandwidth is studied simultaneously. This novel method enables designers to make the convenient trade-off between stability and performance by choosing the proper margins and bandwidth specifications. In addition, the most appropriate stabilizing input signal is selected using Hankel singular value (HSV) and right half plane-zeros (RHP-zeros) for the SVC-based supplementary damping controller. The effectiveness and robustness of the proposed controller are demonstrated using eigenvalue analysis and time-domain simulation for a 16 machine 68-bus test system. The simulations and analysis are implemented in matrix laboratory environment and power system analysis toolbox.

Computation of robustly stabilizing PID controllers for interval systems.

The paper is focused on the computation of all possible robustly stabilizing Proportional-Integral-Derivative (PID) controllers for plants with interval uncertainty. The main idea of the proposed method is based on Tan’s (et al.) technique for calculation of (nominally) stabilizing PI and PID controllers or robustly stabilizing PI controllers by means of plotting the stability boundary locus in either P-I plane or P-I-D space. Refinement of the existing method by consideration of 16 segment plants instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example in which the uncertain mathematical model of an experimental oblique wing aircraft is robustly stabilized.

The effects on stability region of the fractional-order PI controller for one-area time-delayed load–frequency control systems

One of the controllers used in load–frequency control systems is the PI controller, taking account of time delay originating from measurement and communication. In control systems, along with the use of the fractional-order controller, computing parameter space exhibited stable behaviour on the controller parameters and analysing its efficiency have become a significant issue. This study focuses on computing the effects of the fractional integral order ( α) on the stable parameter space for the control of a one-area delayed load–frequency control system in the case of a fractional-order PI controller. The effect of time delay on the stable parameter space is also investigated at different fractional integral orders ( α) in the time-delayed system with fractional-order PI controller. For this purpose, a characteristic equation of the delayed system with the fractional-order PI controller is obtained, and the stable parameter spaces of the controller are computed according to the fractional integral order ( α) and time delay ( τ) values using the stability boundary locus method, which is graphics based. Moreover, the generalized modified Mikhailov criterion is used for testing the stability region on the Kp − Ki plane. The obtained results verified that the stability region on the Kp − Ki plane change depending on the α and τ.

Decentralized PID load frequency control for perturbed multi-area power systems

In this paper, a new approach is proposed for the robust PID controller design for load frequency control (LFC) problem in a multi-area power system having different turbines. The proposed approach is based on Kharitonov’s theorem and stability boundary locus. In this paper, first of all, four-area power system model for LFC problem is presented in the form of a linear uncertain system with fixed lower and upper bounds in the coefficients. Then, stability analysis is carried out using Kharitonov’s theorem and lastly a robust controller is designed for each control area of multi-area power system. The beauty of the proposed approach is that the fixed PID controller is taking care of uncertainty in the each control area. The efficiency and efficacy of the proposed approach is tested through two examples and results are compared with few recently proposed approaches.

An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators

This paper introduces an integer order approximation method for numerical implementation of fractional order derivative/integrator operators in control systems. The proposed method is based on fitting the stability boundary locus (SBL) of fractional order derivative/integrator operators and SBL of integer order transfer functions. SBL defines a boundary in the parametric design plane of controller, which separates stable and unstable regions of a feedback control system and SBL analysis is mainly employed to graphically indicate the choice of controller parameters which result in stable operation of the feedback systems. This study reveals that the SBL curves of fractional order operators can be matched with integer order models in a limited frequency range. SBL fitting method provides straightforward solutions to obtain an integer order model approximation of fractional order operators and systems according to matching points from SBL of fractional order systems in desired frequency ranges. Thus, the proposed method can effectively deal with stability preservation problems of approximate models. Illustrative examples are given to show performance of the proposed method and results are compared with the well-known approximation methods developed for fractional order systems. The integer-order approximate modeling of fractional order PID controllers is also illustrated for control applications.

Stability Region in the Parameter Space of PI Controller for a Single-Area Load Frequency Control System With Time Delay

This letter proposes a graphical method to compute the stabilizing values of proportional-integral (PI) controller parameters for a single-area load frequency control (LFC) system with time delay. The method is based on the stability boundary locus. The stability region is displayed in controller parameter space and time-domain simulation studies are carried to verify the accuracy and effectiveness of the proposed method.

A Model Identification Method for Tuning of PID Controller in a Smith Predictor Structure

In this paper, a model identification method is presented for tuning of PID parameters in Smith predictor structure based on internal model control. The purposed method fits stability boundary locus (SBL) of a plant model with SBL of a FOPDT (First Order Plus Dead Time) or a SOPDT (Second Order Plus Dead Time) model to determine the controller parameters. Two examples are employed to illustrate the applicability and simplicity of the purposed method in both tuning of controller parameters and identification of model parameters for an integer order plant and a fractional order plant.