Robust Stability Analysis Method Research Articles

Reduction of Conservativeness of Robust PID Control System Design Method: An Approach Using the Mapping Theorem and the Cremer–Leonhard–Mikhailov Criterion

This paper gives a novel method to design less‐conservative robust PID control systems. The partial model matching method is a design method for PID control systems. That is a method to design a control system that matches a reference model as closely as possible. This method does not guarantee robust stability of a PID control system because it is assumed that there are no uncertainties on the coefficients of the lower‐order terms of the plant. Therefore, in our previous work, a robust design method based on the partial model matching method was proposed. However, the drawback of this conventional method is that the results of stability analysis are conservative. In this paper, we aim to reduce the conservativeness caused by the conventional method. To achieve this goal, we use a robust stability analysis method based on the Cremer–Leonhard–Mikhailov criterion and the mapping theorem. Moreover, we also propose a new method to update the value of a design parameter. A numerical example shown in this paper implies that the conservativeness is reduced by the proposed method. © 2023 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

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 robust stability and performance analysis method for multi‐actuator real‐time hybrid simulation

A robust stability and performance analysis method for multi‐actuator real‐time hybrid simulation

Novel robust stability analysis method for uncertain systems with interval time-varying delay

Novel robust stability analysis method for uncertain systems with interval time-varying delay

Nonlinear Fractional Active Disturbance Rejection Speed Control for Stabilization of Hydraulic Turbine Regulating Systems With Mechanical Delay

An improved fractional-order nonlinear active disturbance rejection speed control (FO-NADRC) method is proposed for hydraulic turbine regulating systems (HTRSs) with a mechanical delay. First, the mathematical model of an HTRS with a mechanical time delay is established. Based on the principle of coordinate transformation, the state space equation of an HTRS with a time delay is transformed into the controllable normative mathematical model. Second, a new nonlinear function is proposed for the extended state observer ESO) that improves the observation accuracy and suppresses the high-frequency oscillation of the HTRS. Third, a new fractional-order state error feedback law (FO-SEFL) is proposed by introducing double adjustable parameters. Fourth, according to the improved ESO and the FO-SEFL, a novel FO-NADRC is designed for the HTRS. Furthermore, the Popov-Lyapunov robust stability analysis method is used to analyze the stability of the hydraulic turbine regulating control systems with mechanical delay. Finally, numerical simulation experiments demonstrate the effectiveness and superiority of the proposed control scheme.

Novel Robust Stability Criteria of Uncertain Systems with Interval Time-Varying Delay Based on Time-Delay Segmentation Method and Multiple Integrals Functional

Interval time-varying delay is common in control process, e.g., automatic robot control system, and its stability analysis is of great significance to ensure the reliable control of industrial processes. In order to improve the conservation of the existing robust stability analysis method, this paper considers a class of linear systems with norm-bounded uncertainty and interval time-varying delay as the research object. Less conservative robust stability criterion is put forward based on augmented Lyapunov-Krasovskii (L-K) functional method and reciprocally convex combination. Firstly, the delay interval is partitioned into multiple equidistant subintervals, and a new Lyapunov-Krasovskii functional comprising quadruple-integral term is introduced for each subinterval. Secondly, a novel delay-dependent stability criterion in terms of linear matrix inequalities (LMIs) is given by less conservative Wirtinger-based integral inequality approach. Three numerical comparative examples are given to verify the superiority of the proposed approach in reducing the conservation of conclusion. For the first example about closed-loop control systems with interval time-varying delays, the proposed robust stability criterion could get MADB (Maximum Allowable Delay Bound) about 0.3 more than the best results in the previous literature; and, for two other uncertain systems with interval time-varying delays, the MADB results obtained by the proposed method are better than those in the previous literature by about 0.045 and 0.054, respectively. All the example results obtained in this paper clearly show that our approach is better than other existing methods.

Robust stability analysis of incommensurate fractional-order systems with time-varying interval uncertainties

This paper focuses on the stability analysis of the incommensurate fractional-order systems describing by the pseudo-state-space form in the presence of interval uncertainties. By using the generalized small-gain theorem and the properties of M-matrices, a condition is achieved for robust finite-gain L2 stability of these systems. In addition, satisfying this condition ensures the asymptotic stability of uncertain fractional-order systems, based on the Caputo definition. Unlike the methods which are based on the eigenvalue argument checking, the proposed robust stability analysis method can be used for systems with irrational fractional orders and time-varying uncertainties. Compared with the Lyapunov based methods, the conservatism due to the lack of relation between stability condition and fractional-order of the system does not occur in the proposed method. An illustrative example is presented to confirm this method does not lead to conservatism generated by reformulation of the interval uncertainty, which is the basis of the commonly used methods in the literature. Finally, the suggested method is employed for robust stability checking of the Sallen–Key filter including fractional capacitors with irrational orders.

Real-time hybrid substructuring of a base isolated building considering robust stability and performance analysis

This paper demonstrates a real-time hybrid substructuring (RTHS) shake table test to evaluate the seismic performance of a base isolated building. Since RTHS involves a feedback loop in the test implementation, the frequency dependent magnitude and inherent time delay of the actuator dynamics can introduce inaccuracy and instability. The paper presents a robust stability and performance analysis method for the RTHS test. The robust stability method involves casting the actuator dynamics as a multiplicative uncertainty and applying the small gain theorem to derive the sufficient conditions for robust stability and performance. The attractive feature of this robust stability and performance analysis method is that it accommodates linearized modeled or measured frequency response functions for both the physical substructure and actuator dynamics. Significant experimental research has been conducted on base isolators and dampers toward developing high fidelity numerical models. Shake table testing, where the building superstructure is tested while the isolation layer is numerically modeled, can allow for a range of isolation strategies to be examined for a single shake table experiment. Further, recent concerns in base isolation for long period, long duration earthquakes necessitate adding damping at the isolation layer, which can allow higher frequency energy to be transmitted into the superstructure and can result in damage to structural and nonstructural components that can be difficult to numerically model and accurately predict. As such, physical testing of the superstructure while numerically modeling the isolation layer may be desired. The RTHS approach has been previously proposed for base isolated buildings, however, to date it has not been conducted on a base isolated structure isolated at the ground level and where the isolation layer itself is numerically simulated. This configuration provides multiple challenges in the RTHS stability associated with higher physical substructure frequencies and a low numerical to physical mass ratio. This paper demonstrates a base isolated RTHS test and the robust stability and performance analysis necessary to ensure the stability and accuracy. The tests consist of a scaled idealized 4-story superstructure building model placed directly onto a shake table and the isolation layer simulated in MATLAB/Simulink using a dSpace real-time controller.

Simplified computation of robust stability region for DOB-based control system with parametric uncertainty

Motivated by the conservatism and complexity of the existing robust stability analysis method for disturbance observer (DOB)-based control system, a robust stability analysis method, addressing parametric uncertainty, is proposed for different relative order processes with time delay. By Bounded Phase Condition, a simplified Generalised Kharitonov Theory is proposed for stability analysis of inner loop, replacing the redundant stability verification of generalised Kharitonov segment polynomial by stabilising the end points of the segment polynomial. Then, based on the simplified theorem and the stability equation method, all the feasible low pass filter (LPF) parameters that stabilised the inner-loop can be obtained. It is the first time that all the feasible LPF parameters are given for different relative order. Furthermore, the proposed method is extended to ensure the stability of outer loop and obtain the admissible range of controller parameter. Simulation studies present how to obtain the stability region and demonstrate the validity of the proposed method.

Prediction of robust stability boundaries for milling operations with extended multi-frequency solution and structured singular values

Reliable prediction of machine tool chatter is an essential problem in efficiency-oriented machine tool centers, since it requires the precise characterization of the dynamics of the machine-tool-workpiece system and the cutting force characteristics. Due to imperfect measurements, noise, uncertain and varying operational conditions, the mathematical models provide a deficient representation of the system. This leads to the need for the adaptation of robust stability analysis methods, which guarantee stability against bounded uncertainties and perturbations. In this paper, a frequency-domain approach is presented to calculate the robust stability boundaries of chatter-free machining parameters for milling operations. The idea is based on the concept of the stability radius and structured singular values, which is combined with the extended multi frequency solution. The proposed method is tested in a real case study.

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.

A Modeling Methodology for Robust Stability Analysis of Nonlinear Electrical Power Systems Under Parameter Uncertainties

This paper develops a modeling method for robust stability analysis of nonlinear electrical power systems over a range of operating points and under parameter uncertainties. Standard methods can guarantee stability under nominal conditions, but do not take into account any uncertainties of the model. In this study, stability is assessed by using structured singular value (SSV) analysis, also known as $\mu$ analysis. This method provides a measure of stability robustness of linear systems against all considered sources of structured uncertainties. The aim of this study is to apply the SSV method for robust small-signal analysis of nonlinear systems over a range of operating points and parameter variations. To that end, a modeling methodology is developed to represent any such system with an equivalent linear model that contains all system variability, in addition to being suitable for $\mu$ analysis. The method employs symbolic linearization around an arbitrary operating point. Furthermore, in order to reduce conservativeness in the stability assessment of the nonlinear system, the approach takes into account dependences of operating points on parameter variations. The methodology is verified through $\mu$ analysis of the equivalent linear model of a 4-kW permanent magnet machine drive, which successfully predicts the destabilizing torque over a range of different operating points and under parameter variations. Further, the predictions from $\mu$ analysis are validated against experimental results.

A systematic method for analyzing robust stability of interval neural networks with time-delays based on stability criteria

This paper presents a systematic method for analyzing the robust stability of a class of interval neural networks with uncertain parameters and time delays. The neural networks are affected by uncertain parameters whose values are time-invariant and unknown, but bounded in given compact sets. Several new sufficient conditions for the global asymptotic/exponential robust stability of the interval delayed neural networks are derived. The results can be casted as linear matrix inequalities (LMIs), which are shown to be generalizations of some existing conditions. Compared with most existing results, the presented conditions are less conservative and easier to check. Two illustrative numerical examples are given to substantiate the effectiveness and applicability of the proposed robust stability analysis method.

A new method for robust Schur stability analysis

This article is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power κ of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by κ. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of κ needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be introduced, yielding linear matrix inequalities optimisation problems of improved convergence.

IMPROVED ROBUST STABILITY CRITERION FOR UNCERTAIN CELLULAR NEURAL NETWORKS WITH TIME-VARYING DELAY

In this paper, we propose a new robust stability analysis method for uncertain cellular neural networks with time-varying delay. The proposed stability criterion is based on the Lyapunov function with sector bounded nonlinear function. The sufficient condition for the stability is derived in terms of LMI (linear matrix inequality). Numerical examples show the effectiveness of the proposed method.

A novel delay-dependent criterion for delayed neural networks of neutral type

This Letter considers a robust stability analysis method for delayed neural networks of neutral type. By constructing a new Lyapunov functional, a novel delay-dependent criterion for the stability is derived in terms of LMIs (linear matrix inequalities). A less conservative stability criterion is derived by using nonlinear properties of the activation function of the neural networks. Two numerical examples are illustrated to show the effectiveness of the proposed method.

Synthesis of Robust Performance Controllers Based on Discrete-Time Noncausal Linear Periodically Time-Varying Scaling

This paper is concerned with the discrete-time noncausal linear periodically time-varying (LPTV) scaling technique. It is defined through the discrete-time lifting, and it has been shown that even static noncausal LPTV scaling induces some frequency-dependent scaling when it is interpreted in the context of lifting-free treatment. It has been also shown that this feature together with the fact that the uncertainties come to be dealt with in their lifted forms leads to an effective method for robust stability analysis. This paper shows that noncausal LPTV scaling is effective also in controller design. More precisely, we study the periodic robust performance controller synthesis problem, and discuss how the properties of noncausal LPTV scaling lead to the performance improvement when the period of the controller is increased.

Bounded-Impedance Absolute Stability of Bilateral Teleoperation Control Systems.

Available passivity-based robust stability methods for bilateral teleoperation control systems are generally conservative, as they consider an unbounded range of dynamics for the class of passive operators and environments in the complex plane. In this paper, we introduce a powerful 3D geometrical robust stability analysis method based on the notions of wave variables and scattering parameters. The methodology, which was originally a 2D graphical method used in microwave systems for single-frequency analysis [1], is further developed in this paper for teleoperation and haptic systems. The proposed method provides both mathematical and visual aids to determine bounds or regions on the complex frequency response of the passive environment impedance parameters for which a potentially unstable system connected to any passive operator is stable, and vice-versa. Furthermore, the method allows for the design of bilateral controllers when such bounds are known, or can even be utilized when the environment dynamics are active. The geometrical test can also be replaced by an equivalent mathematical condition, which can easily be checked via a new stability parameter. The proposed method results in less conservative guaranteed stability conditions compared to the Llewellyn's criterion; thus, promising a better compromise between stability and performance. The new method is numerically evaluated for two bilateral control architectures.

Stability Analysis of Implicit Polynomial Systems

This technical note develops a robust stability analysis method for a class of implicit polynomial systems subject to uncertain constant parameters. A polynomial (with respect to the state and uncertain parameters) Lyapunov function is computed to assess the robust local stability of the system and to provide a robust estimate of the domain of attraction. The results are given in terms of linear matrix inequalities that are affinely dependent on the system state and uncertain parameters. Numerical examples illustrate the effectiveness of the developed approach.

Improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay

An innovative method for robust stability analysis of dynamic systems with interval time-varying delay and uncertainties of structural and polytopic types is proposed. The innovation of the method includes employment of a tighter integral inequality and construction of an appropriate type of Lyapunov function. The stability criteria derived from this method significantly improves and simplifies the existing stability conditions obtained through employing the technique with the least number of Linear matrix inequality variables. The simplicity of the stability criteria results from the fact that neither any useful terms are ignored in dealing with cross-product terms nor any superfluous free weighting matrices are introduced in the theoretical derivation. Numerical examples are given to demonstrate the effectiveness of the proposed method.