Common Quadratic Lyapunov Function Research Articles

Stability of Wind Turbine Switching Control in an Integrated Wind Turbine and Rechargeable Battery System: A Common Quadratic Lyapunov Function Approach

The power generated by wind turbines varies due to variations in the wind speed. A pack of rechargeable batteries could be used as a reserve power source to alleviate the intermittency in the wind turbine power. An integrated wind turbine and battery storage system is constructed where the wind turbine is electrically connected to a rechargeable battery system. Such a system can operate in two modes depending on the wind speed, power demand, and battery limit. The switching conditions for the wind turbine to operate in multi-input, single-output and single-input, single-output control mode are discussed. Linearized approximations of the closed loop wind turbine system are derived in order to analyze the switching stability between control modes. Common quadratic Lyapunov function (CQLF) is established for both control modes to prove the system stability. Simulation results demonstrating system stability are also presented.

Bounds and invariant sets for a class of switching systems with delayed-state-dependent perturbations

We present a novel method to compute componentwise transient bounds, componentwise ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components. In this latter case, we also show how our componentwise method can be combined with standard techniques in order to derive bounds possibly tighter than those corresponding to either method applied individually.

Stability Analysis of MEMS Gyroscope Dynamic Systems

In this paper, the existence of a common quadratic Lyapunov function for stability analysis of MEMS Gyroscope dynamic systems has been studied then a new method based on stochastic stability of MEMS Gyroscope system has been proposed.

Adaptive H∞ Control of Piecewise‐Linear Systems with Parametric Uncertainties and External Disturbances

AbstractIn this paper, we study the adaptive H∞ control problem for piecewise linear systems with parametric uncertainties and external disturbances. Motivated by the vector projection technique and dissipation theory, we propose a piecewise H∞ controller and its associated projection‐type parameter adaptation law, which are capable of guaranteeing the system stability and minimizing the effect of disturbances. Moreover, the common quadratic Lyapunov function method is used such that all of the synthesis conditions are formulated as linear matrix inequities and thus can be solved efficiently. Finally, a numerical example is provided to illustrate the results.

A method for determining the non-existence of a common quadratic Lyapunov function for switched linear systems based on particle swarm optimisation

The existence of a common quadratic Lyapunov function (CQLF) for a switched linear system guarantees its global asymptotic stability. Even if progress in finding the conditions for the existence/non-existence of a CQLF is significant, especially in switched linear systems consisting of N second-order systems or two systems of order n, the general case of N systems of order n still remains open. In this article, a sufficient condition for the non-existence of a CQLF for N systems of order n is derived. Based on the condition, a new method for determining the non-existence of a CQLF, using particle swarm optimisation, was designed and is described. Examples illustrating the proposed method are introduced at the end of this article.

Stability of switched linear discrete-time descriptor systems: a new commutation condition

In this article, we study stability of switched linear discrete-time descriptor systems. Under the assumption that all subsystems are stable and there is no impulse occurring at the switching instants, we establish a new pairwise commutation condition under which the switched system is stable. It turns out that the condition is a natural and important extension to the existing commutation conditions, and it can be applied in discretised continuous-time switched systems in a straightforward manner. Finally, we also show that when the proposed commutation condition holds, there exists a common quadratic Lyapunov function for the subsystems.

Improved observer-based fault estimator design for a class of T-S fuzzy systems

For a class of Takagi–Sugeno (T-S) fuzzy nonlinear systems, in this article, an improved fuzzy observer-based fault estimator is designed to estimate the fault signals on-line based on common Lyapunov function. Attention is focused on the linear matrix inequality (LMI)-based design method with specified performance constraints and less conservatism, which is suitable for complex system models with more rules. The multiobjective and H ∞ optimisation theory is applied to cope with the constraints on the disc-regional poles assignment index and the robustness to disturbances. Some developed relaxing techniques are utilised to reduce the conservatism and the number of LMI constraints which were generated by the conventional design method with common quadratic Lyapunov function. Thus, the resulting estimator is not only less conservative, but can also guarantee the desired fault estimation performance on the rapidity as well as robustness to neglected modelling dynamics, and the robustness against external disturbances with satisfied estimation accuracy to faults signals. The conventional design is also presented for comparative study. All the results are formulated in the form of LMIs. Simulative examples demonstrate that our proposed method is less conservative and effective for fault estimation.

Controlled Hopwise Averaging and Its Convergence Rate

This technical note develops Ideal Controlled Hopwise Averaging (ICHA) and Controlled Hopwise Averaging (CHA), two asynchronous distributed averaging algorithms for wireless networks, which attempt to “make the most” out of each iteration by fully exploiting the broadcast nature of wireless medium and incorporating feedback control of when to initiate an iteration. The latter feature, enabled by a common quadratic Lyapunov function, is novel among existing schemes and represents a new way to apply Lyapunov stability theory. We also derive deterministic upper bounds on the exponential convergence rate of ICHA on general and specific graphs, express the bounds explicitly in terms of the graph invariants, and show that they outperform the stochastic convergence rate of Pairwise Averaging on some common graphs of opposing densities. Finally, we obtain upper bounds on the convergence rate of CHA and show that CHA is capable of closely mimicking the behavior of ICHA, while being practical.

Finding common quadratic Lyapunov functions for switched linear systems using particle swarm optimisation

It is undoubtedly important to be able to ensure the existence of a common quadratic Lyapunov function (CQLF) for a given switched system because this is proof of its asymptotic stability, but equally important is the ability to calculate it in order to obtain more specific information about the behaviour of the switched system under analysis. This article describes the development of a new methodology for calculating a CQLF based on particle swarm optimisation (PSO) once the existence of a CQLF has been assured. Several comparative analyses are presented to show the strengths and advantages of the proposed methodology.

Stability Analysis of PID Controlled Local Model Networks

This paper addresses closed-loop stability analysis of PID controlled local model networks. The proposed method allows to investigate exponential or asymptotic stability of the closed-loop system. For this purpose a common quadratic Lyapunov function is used as stability criterion. Due to the fact that the Lyapunov approach requires a state-space model a suitable closed-loop state-space system with integration of the controller parameters is introduced. An example demonstrates the effectiveness of the proposed method.

Common Lyapunov Function in the Control Problem for Linear Dynamical Systems

A problem of designing a bounded feedback control for a linear dynamical system is considered. A feedback control law is proposed which is bounded and steers any state in a neighborhood of the origin to the origin in a finite time. The construction is based on the notion of a common Lyapunov function. The existence of a common quadratic Lyapunov function for matrices of specific forms is proven. Some remarkable properties of this function are established.

Common Lyapunov Functions for Some Special Classes of Stable Systems

In this work we consider the problem of existence and determination of a common positive definite solution (common quadratic Lyapunov function) to a set of Lyapunov matrix inequalities. We investigate a parameterized matrix family and a compact (in particular finite) set of 2 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\times$</tex> </formula> 2 matrices.

Parameter-dependent Lyapunov functions for state-derivative feedback control in polytopic linear systems

In some practical problems, for instance, the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. Using Linear Matrix Inequalities (LMIs), and applying the reciprocal projection lemma in a Parameter-dependent Lyapunov Function (PDLF), this article proposes a method for the design of state-derivative feedback applied to uncertain linear systems. The control design aims the system stabilisation without and with decay rate restriction. When considering only the system stability, the proposed methodology becomes practically equivalent to the Common Quadratic Lyapunov Function (CQLF) technique. Otherwise, when the decay rate is taken in account, the proposed methodology is shown to be less conservative. Numerical examples illustrate its efficiency.

A commutation condition for stability analysis of switched linear descriptor systems

We study the stability analysis problem for switched linear descriptor systems. Assuming that all subsystems are stable and there is no impulse at the switching instants, we establish a new pairwise commutation condition under which the switched system is stable. We also show that when the proposed commutation condition holds, there exists a common quadratic Lyapunov function (CQLF) for the subsystems. These results are natural and significant extensions to the existing results for switched systems in the state space representation.

Feedback Stabilization of Switching Discrete-Time Systems via Lie-Algebraic Techniques

This technical note addresses the stabilization of switching discrete-time linear systems with control inputs under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems, which is ensured when the closed-loop component subsystem matrices are stable and generate a solvable Lie algebra. The present work develops an iterative algorithm that seeks the feedback maps required for stabilization based on the previous Lie-algebraic condition. The main theoretical contribution of the technical note is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution. The core of the proposed algorithm is a common eigenvector assignment procedure, which is executed at every iteration. We also show how the latter procedure can be numerically implemented and provide a key structural condition which, if satisfied, greatly simplifies the required computations.

Stability analysis for a class of switched linear systems

This paper investigates the stability of a class of switched linear systems, and proposes a number of new results on the stability analysis. A novel analysis method is developed by using the 2-norm technique, and then several stability results are obtained based on the new analysis method. It is shown that the main results obtained in this paper not only guarantee the stability of the systems under arbitrary switching, but also provide an algorithm to find the minimum dwell time (MDT) with which switches make the switched systems stable. Finally, several illustrative examples with numerical simulations are studied by using the results obtained in this paper. The study of examples shows that our analysis method works very well in analyzing the stability of some classes of switched linear systems and, moreover, it has some advantages over the common quadratic Lyapunov function method in some cases. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Existence of a common quadratic Lyapunov function for discrete switched linear systems with m stable subsystems

The problem on the existence of a common quadratic Lyapunov function (CQLF) for discrete switched linear systems with m stable subsystems is considered. System matrices are Schur stable. A necessary condition for the existence of a CQLF and a sufficient condition for the non-existence of a CQLF are derived, respectively. Numerical examples are presented to illustrate the results.

Componentwise ultimate bound and invariant set computation for switched linear systems

We present a novel ultimate bound and invariant set computation method for continuous-time switched linear systems with disturbances and arbitrary switching. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds and invariant sets in the form of polyhedral and/or mixed ellipsoidal/polyhedral sets, is completely systematic once the aforementioned transformation is obtained, and provides a new sufficient condition for practical stability. We show that the transformation required by our method can easily be found in the well-known case where the subsystem matrices generate a solvable Lie algebra, and we provide an algorithm to seek such transformation in the general case. An example comparing the bounds obtained by the proposed method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.

On the continuous-time Takagi–Sugeno fuzzy systems stability analysis

In this paper, a new approach for the stability analysis of continuous-time Takagi–Sugeno (T-S) fuzzy system is proposed. The universe set is divided to subregions, and piecewise quadratic Lyapunov function is then found for each of them. This class of Lyapunov function candidates is much richer than the common quadratic Lyapunov function. By exploiting the piecewise continuous Lyapunov function, we derive stability conditions that can be verified via convex optimization over linear matrix inequalities (LMIs) or bilinear matrix inequalities (BMIs). These conditions are shown to be less conservative than some quadratic stabilization conditions published recently in the literature. Since this method uses low numbers of LMIs or BMIs and less computation Lyapunov functions, it is highly applicable and has less computation. This approach is not dependent on the shape of fuzzy sets and also, stability of the system is guaranteed in the presence of state feedbacks. At first, in order to decrease length of computation (amount of LMIs (BMIs)), an approach is introduced based on properties of the T-S system with two-overlapped fuzzy sets. Some criterions are obtained for stability analysis, stability analysis in the presence of parametric uncertainties and a stability criterion is presented to provide a reasonable performance for the system. To demonstrate the new approach, an illustrative example is presented.

A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching

A unified approach to stability analysis of switched linear descriptor systems under arbitrary switchingWe establish a unified approach to stability analysis for switched linear descriptor systems under arbitrary switching in both continuous-time and discrete-time domains. The approach is based on common quadratic Lyapunov functions incorporated with linear matrix inequalities (LMIs). We show that if there is a common quadratic Lyapunov function for the stability of all subsystems, then the switched system is stable under arbitrary switching. The analysis results are natural extensions of the existing results for switched linear state space systems.