Edge Theorem Research Articles

Robust PID control of a tuned mass damper system and its stability analysis validation by Kharitonov and Edge theorems

ABSTRACT A robust control and stability analysis of a tuned mass damper attached to benchmark the model of mass-spring-damper system via different Kharitonov-related approaches, including the post-Kharitonov Edge Theorem, is presented in this paper. The control of uncertain parameters of the given system is demonstrated via robust Proportional-Integral-Derivative (PID) system tuning techniques in MATLAB. Usually, the Kharitonov theorem is sufficient to check the stability analysis of a controlled interval system. A detailed stability analysis of the given system by forming the Kharitonov rectangular parameter boxes and testing the positivity of Hurwitz determinants of the enclosed segments connecting opposite corners of the box is explained here. Stability checking is validated by the analysis based on the Hurwitz matrices of two Kharitonov polynomials. The multi-linear affine system model in the overall system plant is controlled and analysed via Edge theorem.

GWO based Robust Stabilization of DC Motor Fractional Order Speed Control System with Interval Coefficients

Robust stability analysis (RSA) is of significant concern for the robust behaviour of real-world control system applications. A stabilization strategy that assures stability and exhibits robust performance for a specified limit of system perturbations is necessary. This article presents an optimal robust stabilization method for a closed loop fractional order proportional integral derivative (FOPI^λD^µ) system involving DC motor with interval parametric uncertainty. To determine the optimum value of parameters for a FOPI^λD^µ controller to control the speed of a DC motor, Grey Wolf Optimizer (GWO), Genetic Algorithm (GA), Nelder-Mead (NM), Jaya and Whale Optimizer Algorithm (WOA) are applied with the same objective function involving ITAE criterion. FOPI^λD^µ offers two additional tuning parameters unlike a nominal PID controller and hence the former gives more flexibility in controller design than the latter in terms of transient response. The FOPID controller provides a faster closed-loop output augmented with improved robust properties of the system. Despite inherent non-linearities and time variation in system parameters, FOPI^λD^µ controllers depict enhanced performance. Using the concept of conformal mapping, robust stability analysis of fractional order polynomials is done with uncertain interval structure using Vertex and Edge theorem. Based on the value set, this paper demonstrates numerical and graphical optimal robust stability analysis of a system with variations observed in five parameters, considering the minimum argument root of the polynomial of the aforementioned closed-loop system.

Stability comparison of different controllers for hydraulic turbine fractional order interval parameter time-delay system

This paper studies the stability comparison of four controllers for a hydraulic turbine fractional order interval parameter time-delay system. Considering the terms of parameter uncertainty and time-lag, the hydraulic turbine fractional order interval parameter time-delay system is established by using fractional calculus and identification technology. Through the edge theorem and D-decomposition method, the calculation process of controller parameter stability region is derived in every detail. Subsequently, the four different controllers including the PID, PID plus second order derivative (PID2D), fractional order PID (FOPID) and fractional order PID plus second order derivative (FOPID2D) controllers, are applied in the hydraulic turbine fractional order interval parameter time-delay system. According to the experimental results, the effect of the controller parameters kd , kd 2, α and λ on the stability region of the parameters kp and ki is analysed and the variable law of stability regions of the hydraulic turbine governing system is presented. Finally, the stability region of these four controllers is compared and the results validate the superiority of the FOPID2D controller over the other controllers.

On the asymptotic stability of linear difference equations with time-varying coefficients

<abstract><p>Issues of the asymptotic stability for linear difference equations with time-varying coefficients are discussed. It is shown that, in contrast to equations with constant coefficients, the condition of Schur stability of the characteristic polynomial for a linear difference equation with time-varying coefficients is neither necessary nor sufficient for the asymptotic stability of the difference equation. It is proved that the analog of Kharitonov's theorem on robust stability and the edge theorem do not hold for a difference equation if the coefficients of the equation are not constant.</p></abstract>

Single Grid-Current Sensor-Controlled Weak-Grid-Following Inverters: A State-and-Disturbance-Observer-Based Robust Control Scheme Achieving Grid-Synchronization and Disturbance Rejection

With the aim of reducing the number of sensors, this article investigates the single-sensor grid-following control for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> -filtered inverter system, and proposes an intuitive robust stability analysis method in the linear framework to handle power control and grid voltage variations in engineering practice. Specifically, through the use of a state-and-disturbance observer, the lumped-disturbance compensated suppression and nominal system full-state feedback technologies are equivalently realized. And the lumped-disturbance estimation carrying voltage phase information at the point of common coupling is further used as synchronization target of the phase-locked loop (PLL) to achieve grid-following control, and the resulting phase-shift is further corrected by accurate phase compensation. In addition, due to the nonlinear nature of the PLL and the power control and grid voltage variations, the system analysis is faced with the dilemma of complex nonlinear analysis or small-signal analysis without a fixed equilibrium point. In this case, a PLL small-signal model with bounded parameter perturbation characterizing power control and grid voltage variations is developed, and further incorporated into the output impedance modeling of the single-sensor grid-following controlled inverter. According to the edge theorem, the system stability in the studied uncertain region is analyzed via four edge output impedances, and appropriate control parameters are selected accordingly. Finally, the experimental results from a downscaled prototype are provided to verify the effectiveness of the proposed single-sensor grid-following control scheme.

A strengthening of the spectral chromatic critical edge theorem: Books and theta graphs

AbstractA graph is color‐critical if it contains an edge whose removal reduces its chromatic number. Let be the Turán graph with vertices and parts. Given a graph , let be the Turán number of . Simonovits' chromatic critical edge theorem states that if is color‐critical with , then there exists an such that and the Turán graph is the only extremal graph provided . Nikiforov proved a spectral chromatic critical edge theorem. It asserts that if is color‐critical and , then there exists an (which is exponential with ) such that and is the only extremal graph provided , where is the spectral radius of and . In addition, if is either a complete graph or an odd cycle, then is linear with . A book graph is a set of triangles sharing a common edge and a theta graph is a graph which consists of two vertices connected by three internally disjoint paths with length one, two, and . Notice that both and are color‐critical. In this article, we prove that if , then contains a book with unless . Similarly, we prove that if , then contains a theta graph with for odd and for even unless . Our results imply that in the spectral chromatic critical edge theorem is linear with for book graphs and theta graphs. Our result for book graphs can be viewed as a spectral version of an Erdős conjecture (1962) stating that every ‐vertex graph with contains a book graph with Moreover, our result for theta graphs yields that every graph with contains a cycle of length for each . This is related to an open question by Nikiforov (2008) which asks for the maximum such that every graph of large enough order with contains a cycle of length for every .

FOPID controller design for a perturbed PHWR system using enhanced crow search algorithm

The effectiveness of a control algorithm in the operation of a Pressurized Heavy Water Reactor (PHWR) relies upon its ability to counter the step back condition accurately. This paper presents an enhanced crow search algorithm for designing a fractional-order proportional integral derivative (FOPID) controller implemented in power control loop of a PHWR under step-back condition. The proposed controller is applied on eight different interval fractional transfer function models of PHWR derived using Edge theorem as well as eight step back conditions of PHWR. Simulation results reveal excellent reference tracking capability of the proposed controller. The effectiveness of the proposed controller has been proved by performing a comparison among different error performance indices with the existing controllers in the literature.

Construction of Polynomial Polytopes by Using Segment Lemma

A polynomial with all roots lying in the open left half plane of the complex plane is called Hurwitz stable. The convex hull of a finite number of polynomials of the same order is called a polynomial polytope. By the Edge theorem, a polynomial polytope with the invariant degree is Hurwitz stable if and only if all edges are Hurwitz stable. Due to the Edge Theorem, the segment stability criterions are of great importance. In this paper, we consider a construction of stable polytopes by using the Segment Lemma. It is constructed stable polytopes with nonzero volumes. The obtained results can be used, for example, in the stabilization problems of unstable transfer functions by lower-order controllers.

Robust Chatter Stability Prediction of the Milling Process considering Uncertain Machining Positions

Milling stability is a function of the tool point frequency response functions (FRFs), which vary with the movements of the moving parts within the whole machine tool work volume. The position-dependent tool point FRFs result in uncertain prediction of the stability lobe diagram (SLD) for chatter-free machining parameter selection. Taking the variations of modal parameters to represent the variations of tool point FRFs, this paper introduces the edge theorem to predict the robust milling chatter stability. The application of the edge theorem requires the minimum and maximum modal parameters within the machining space defined by the machining position and machining allowance information. Then, radial basis function artificial neural networks (RBFANNs) are used to predict the position-dependent modal parameters in X and Y directions based on the sample information of machining positions and related modal parameters at the tool point. Moreover, sample machining spaces are determined based on the aforementioned sample positions, and the trained RBFANNs are used to obtain corresponding sample extreme modal parameters. On this basis, RBFANNs for predicting the position and machining allowance-dependent extreme modal parameters can also be trained, and they are combined with the edge theorem and zero exclusion condition to calculate robust pairs of the spindle speed (n) and limiting axial cutting depth (aplim) and then plot the robust SLD (RSLD). A case study was performed on a real three-axial vertical machining center, and the plotted RSLD considering position variations was compared with the traditional SLD. Results of the chatter tests show that the RSLD can provide more reliable (ap, n) pairs to guarantee the milling stability, validating the feasibility of the proposed robust milling chatter stability prediction method.

On polytopes in Hurwitz region

We consider the stability region of monic polynomials in the coefficient space. Starting from specially chosen stable points we define polytopes, whose edges are either stable or belong to the stability boundary. Therefore by the Edge Theorem the interiors are stable. Stable polytopes, in the reverse directions which are directed to infinity have been studied in the work (Dzhafarov et al., 2019). Some applications of the obtained results to the synthesis of fixed-order stabilizing controllers are given.

Robust Performance in Parametric Control System Design With Application to Power Systems

This article deals with the problem of robust low-order controller design for systems with multimodel representations. Such models are frequently used for the description of nonlinear and complex industrial power systems over an operating profile. We first develop a generalized complex extension of convex direction based on the interpolation and edge theorems. Subsequently, for plant models with parametric-type uncertainty, new vertex results related to robustness margins and system performance are derived. Based on this, new tools to design robust low-order controllers for such systems are presented.

Stabilization of polytopes of plants by their vertices

SummaryThis paper aims to assess the stability of a polytope of linear systems by their vertices. These results are based on the Hermite‐Biehler and Edge theorems. A sufficient condition satisfying a constraint on the even (or odd) part of the closed‐loop characteristic polynomials associated with the stabilization of its vertices is proved. Finally, a constructive method to stabilize a polytope of plants with simultaneous and robust linear time‐invariant controllers is established.

Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality

The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To determine the root quality indices we propose to depict on the root plane not all edges of the interval parametric polytope (as the edge theorem says), but its particular vertex-edge route. In order to define this route we need to know the angle sequence at which the edge branches depart from any integrated pole on the allocation area. It is revealed that the edge branches can integrate into the route both fully or partially due to intersection with other branches. The conditions which determine the intersection of one-face edge images have been proven. It is shown that the root quality indices can be determined by its ends or by any other internal point depending on a type of edge branch. The conditions which allow determining the edge branch type have been identified. On the basis of these studies we developed the algorithm intended to construct a boundary vertex-edge route on the polytope with the interval parameters of the system. As an illustration of how the algorithm can be implemented, we determined and introduced the root indices reflecting the robust quality of the system used to stabilize the position of an underwater charging station for autonomous unmanned vehicles.

Robust evaluation of chatter stability for milling process with uncertainties based on optimal configuration of machining position and spindle speed

Chatter vibration in milling process is a major obstacle that limits the machining quality and productivity, which may be avoided by using stability lobe diagrams (SLDs). Many traditional models developed to predict chatter stability assume that dynamic parameters of the machine tool remain constant under operational conditions. However, these parameters such as natural frequencies, damping ratios, stiffness, and cutting force coefficients vary depending upon different aspects including spindle speed, tool wear, and machining position, reducing the accuracy of chatter prediction. In this study, a robust chatter prediction method based on conventional analytical milling stability models is presented by employing the Edge theorem and Zero Exclusion condition. In this method, optimal combinations of spindle speeds and machining positions are firstly researched to obtain higher critical depths of cut, based on the conventional stability model, modal fitting technique, Kriging model, and improved particle swarm optimization. At each combination, related nominal modal parameters and cutting force coefficients are identified, and their left and right worst-case deviations are also determined. Critical stable condition for each combination is detected by a graphic approach within the minimum and maximum bounds of uncertainties. Accordingly, a robust stability lobe diagram is obtained with related spindle speeds and critical cutting depths. The proposed method was verified by chatter tests on a real vertical machining center, demonstrating its reliability in chatter prediction compared to the conventional stability lobe diagram.

Resilient Transactive Control for Systems With High Wind Penetration Based on Cloud Computing

One of the main shortcomings, caused by high penetration of wind power, is intermittency of generation. For integrating high penetration of wind power, the frequency regulation and the transactive control systems are modified to be sufficiently resilient against fluctuations of wind power and malicious cyber threats. Here, a hierarchical state-space model is presented for the frequency regulation and the transactive control systems in a smart grid environment. To achieve a resilient control, a framework based on cloud computing is proposed for the communication network. Benefits and challenges of the cloud-based framework are also described in this paper. To optimize the operation of the proposed framework, control parameters are tuned robustly by using the edge theorem. Finally, the performance of proposed control system is evaluated in the case study section.

Robust optimization of machining parameters based on edge theorem

Traditional mathematic models for predicting milling stability assume that dynamic parameters of machine tools remain constant. However, these parameters such as natural frequencies and cutting force coefficients vary under operational state, reducing accuracies of the chatter prediction and related machining parameters optimization. In this study, the edge theorem and the zero exclusion condition are used to extend the traditional stability model for considering the effects of the uncertain parameters. Thus, robust combinations of the spindle speed and axial cutting depth are predicted. They are the inputs of the optimization model to obtain the maximum material remove rate MMR based on the particle swarm optimization method. The proposed machining parameters optimization method was applied on a real vertical machining center, and its effectiveness was validated by the chatter tests.

Design of coupling parameters for inducing amplitude death in Cartesian product networks of delayed coupled oscillators.

The present study investigates amplitude death in Cartesian product networks of two subnetworks, where each subnetwork has a different coupling delay. The property of the Cartesian product helps us to analyze the stability of amplitude death. Our analysis reveals that amplitude death can occur for long coupling delays if there is a suitable difference in the coupling delays in the two subnetworks. Furthermore, based on the edge theorem in robust control theory, we propose two design procedures of coupling parameters for inducing amplitude death in the Cartesian product networks. Our procedures do not require any information of topologies of the subnetworks. The validity of these procedures is numerically confirmed.

Large-Scale Power System Robust Stability Analysis Based on Value Set Approach

This paper presents a method for robust stability analysis of large-scale power systems based on the value set approach. The order of the system is first reduced by the dimension reduction and modal truncation to capture the oscillation modes of interest. The characteristic polynomial is then obtained by the diagonal expansion, which allows the structure of coefficient functions to be exploited. The Mikhailov plot is generated with ease using the edge theorem and mapping theorem, and by inspection of the plot, the uncertainty's impact becomes transparent. The performance of the proposed method is tested on a 547-machine 8647-bus model of the actual North China system. The results of several case studies are reported, and related works are reviewed for comparison.

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.

Application of Edge theorem for robust stability analysis of a power system with participating wind power plants in automatic generation control task

This study investigates the dynamic participation of wind power plants (WPPs) in the automatic generation control (AGC) task. The pre-specified model of wind farm in the AGC studies has been used. It is shown that operating of WPPs at the command mode results in a significant improvement in the frequency behaviour of power system due to their faster response. However, the WPPs may change their operation mode from the command mode to the maximum power point mode according to the wind speed conditions and load variations. This reduces the improvement in the frequency response. In this condition, the shortage in the wind power should be compensated by the conventional units. Thus, the share of different units becomes uncertain (but bounded) when the WPPs participate in the AGC task. To address this challenge, this study provides the robust stable frequency operation of power system under uncertain shares of different units in the AGC task using the Edge theorem. Simulations are performed on an IEEE 39-bus test system connected to some wind farms in different buses. Results confirm the robust stability of the underlying power system, and hence, the effective capability of the wind farms to participate in the AGC task.