New Approach For Stability Analysis Research Articles

Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information

This study presents a new approach for stability analysis of polynomial-fuzzy-model-based (PFMB) control system using membership function information. For the purpose of extracting the regional information of the membership functions, the operating domain is partitioned into several sub-domains. In each sub-domain, the boundaries of every single membership function overlap term and the numerical relation among all the membership function overlap terms are represented as a group of inequalities. Through the S-procedure, the regional membership function information is taken into account in the stability analysis to relax the stability conditions. The operating domain partition scheme naturally arises the motivation of constructing the PFMB control system with the sub-domain fuzzy controllers. Each polynomial fuzzy controller works in its corresponding sub-domain such that the compensation capability of controller is enhanced. The sum-of-squares (SOS) approach is proposed to obtain the stability conditions of the PFMB control system using the Lyapunov stability theory. The PFMB control system studied in this study has the feature that the number of fuzzy rules and the membership function shapes of the polynomial fuzzy controller can be designed independently from the polynomial fuzzy model. To verify the stability analysis result, a numerical example is given to demonstrate the validity of the proposed method.

New Approach for Stability Analysis of Interconnected Nonlinear Discrete-Time Systems Based on Vector Norms

The stability analysis of interconnected large-scale systems is generally characterized by some degree of difficulty, specifically when the interconnections gather a large number of nonlinear subsystems and the couplings evolve according to nonlinear functions in time. In this paper, a novel systematic analysis procedure of fully interconnected discrete-time nonlinear systems with nonlinear interconnections is presented. First, a new arrow form representation of the interconnected system state space model is generated. The obtained generalized thin arrow state matrix is developed by combining the instantaneous subsystems characteristic polynomials, as well as a set of arbitrary and freely parameters. Then, the stability analysis is achieved using the comparison principle and vector norms by translating the stability properties of the lower-dimensional comparison system into those of the considered interconnected system. Aside the proposed systematic formulation and simplicity over existing techniques based mainly on LMIs, it is shown that through an appropriate choice of the model parameters, the developed stability conditions are opportune to locate interesting estimation of the stability domains. Lastly, two numerical examples are included to show the effectiveness of the proposed approach.

Stability analysis of unstructured finite volume methods for linear shallow water flows using pseudospectra and singular value decomposition

The discretization of the shallow water system on unstructured grids can lead to spurious modes which usually can affect accuracy and/or cause stability problems. This paper introduces a new approach for stability analysis of unstructured linear finite volume schemes for linear shallow water equations with the Coriolis Effect using spectra, pseudospectra, and singular value decomposition. The discrete operator of the scheme is the principal parameter used in the analysis. It is shown that unstructured grids have a large influence on operator normality. In some cases the eigenvectors of the operator can be far from orthogonal, which leads to amplification of solutions and/or stability problems. Large amplifications of the solution can be observed, even for discrete operators which respect the condition of asymptotic stability, and in some cases even for Lax–Richtmyer stable methods. The pseudospectra are shown to be efficient for the verification of stability of finite volume methods for linear shallow water equations. In some cases, the singular value decomposition is employed for further analysis in order to provide more information about the existence of unstable modes. The results of the analysis can be helpful in choosing the type of mesh, the appropriate placements of the variables of the system on the grid, and the suitable discretization method which is stable for a wide range of modes.

Stability Analysis and Controller Design of a Wide-Area Time-Delay System Based on the Expectation Model Method

Stochastic time delay is a new challenge for communication networks in wide-area damping control (WADC) systems. Uncertainties introduced by stochastic time delays increase the difficulty of modeling, analyzing the stability, and designing the controller. In this paper, a new approach for stability analysis and controller design through expectation modeling is proposed to solve for the uncertainty in wide-area controlled power systems introduced by the effect of a stochastic time delay. In the proposed method, the expectation model that considers the detailed stochastic time delay distribution is derived to accurately model the effect of a communication network on the wide-area control performance. Through a matrix transformation of the expectation model, the eigenvalue of the WADC system that considers time delay can then be calculated; this eigenvalue can be used for analyzing the system stability. A novel expectation model-based controller design procedure is established, through which traditional methods can be applied in the controller design of a power system with a stochastic time delay. Finally, the effectiveness of the stability analysis and controller design methods are validated through simulations in a power system of four generators and two areas with an HVDC link, and a power system with eight generators and 36 buses.

A New Approach for Stability Analysis of Discrete Systems

ABSTRACTIn this paper, a new approach is presented to calculate the stability margin of the discrete systems. It is based on the algorithm for reduced conservatism of eigenvalues and the Gerschgorin circle theorem. The necessary and sufficient condition for conservatism of eigenvalues of matrix is stated and proved mathematically. Moreover, the proposed approach is illustrated with an example and it is compared with the other existing methods.

A New Approach For Stability Analysis Of Time‐Dependent Switched Continuous‐Time Linear Systems

AbstractIn this paper, a new approach for stability analysis of time‐dependent switched linear systems is proposed. System equivalence is the main idea in this new approach, which derives a switched discrete linear parameter‐varying system from the switched continuous‐time linear switched system with interval dwell time, and the stability properties of the two corresponding systems are proved to be equivalent. Then, by applying a quadratic Lyapunov function approach for the equivalent switched discrete system, the stability of the switched continuous‐time linear system can be established without checking any average dwell time condition. Finally the computation complexity is analyzed, and mode incidence matrix is introduced to reduce the computation cost.

Non-linear control of an uncertain hypersonic aircraft model using robust sum-of-squares method

In recent years, sum-of-squares (SOS) method has attracted increasing interest as a new approach for stability analysis and controller design of non-linear dynamic systems. In this study, the authors present a robust SOS/robust LMI method to design a non-linear controller for longitudinal dynamics of a hypersonic aircraft model with parametric uncertainties. To this end, the control design problem is first formulated as a robust SOS problem. Then, an LMI representation of the robust SOS problem is derived using a proposed algorithm and is solved via a stochastic ellipsoid method. As the simulation results show, the designed robust controller is capable of stabilising the aircraft and following pilot commands under up to 50% variation in aerodynamic coefficients.

A membership-function-dependent approach for stability analysis and controller synthesis of Takagi–Sugeno models

This paper presents a new approach for stability analysis and controller design of Takagi–Sugeno (TS) models. The analysis considers information derived from existing or induced order relations among the membership functions. Partitioning of the state-space and the use of piecewise Lyapunov functions (PWLF) arise naturally as a consequence of induced order relations. Conditions under the novel approach can be expressed as linear matrix inequalities (LMIs) facilitating the inclusion of performance design. Examples are provided to show the advantages over the classical quadratic approach.

A membership-function-dependent stability analysis of Takagi-Sugeno models

This paper presents a new approach for stability analysis of Takagi-Sugeno (TS) models. The analysis considers information derived from existing or induced order relations among the membership functions. Partitioning of the state-space and the use of piecewise Lyapunov functions (PWLF) arise naturally as a consequence of induced order relations. Conditions under the novel approach can be expressed as linear matrix inequalities (LMIs) so they can be efficiently solved. Examples are provided to show the advantages over the classical quadratic approach.

An approach for stability analysis of a single-bit high-order digital sigma-delta modulator

In this work we considered the stability of a single-bit high-order sigma-delta modulator under dc input. A new approach for stability analysis is proposed. A nonlinear circle map is suggested to model the dynamics of the modulator. An analogy between the dynamics of the sigma-delta modulator and the sinusoidal digital phase-locked loop (DPLL) is studied and an approximate fixed point solution is presented with stability criteria. Suggestions for designing stabilized high-order systems are presented.