A counterexample to “Generalized eigenvalue-based stability tests for 2-D linear systems: Necessary and sufficient conditions” by Fu, P., Chen, J., and S.I. Niculescu

We present a counterexample contradicting the equivalence of statements (i)-(ii) and (iii)-(v) of Theorem 1 of Fu, Chen, and Niculescu (2006).     

Necessary stability conditions for linear delay systems

We present necessary conditions for the exponential stability of linear systems with multiple delays. They are expressed in terms of the delay Lyapunov matrix of the Lyapunov–Krasovskii functionals of complete type approach. New properties of independent interest, that establish connections of the system fundamental matrix with its Lyapunov matrix, are crucial elements of our proof. We illustrate our work with a number of examples.     

Necessary and sufficient conditions for the stability of linear parameter-dependent systems

In this paper we discuss the necessary and sufficient conditions for the Hurwitz stability of a linear system whose matrix depends continuously on several parameters. We also give an algorithm based on our derived condition to determine the stability of the system in a finite number of steps. Numerical examples are given to illustrate the results.     

Necessary and sufficient conditions of quadratic stability ofuncertain linear systems

The stability of linear systems subject to possibly fast time-varying uncertainties is analyzed. A necessary and sufficient condition of quadratic stability is derived. An uncertainty stability margin coefficient ρ is introduced to give a quantitative measure of the stability. It is proposed that the uncertain region be approximated by a convex hyperpolyhedron. In this case, the computation of ρ becomes a two-level optimization problem, in which the extremum of the inner level can be reached by one of the corners of the hyperpolyhedron     

Necessary and sufficient conditions for quadratic stability andstabilizability of uncertain linear time-varying systems

In this paper we consider linear time-varying systems subject to norm bounded, one block, structured uncertainties. First, given an unforced system, we find necessary and sufficient conditions for quadratic stability; then, considered a controlled system, we provide necessary and sufficient conditions for quadratic stabilizability both for the state and the output feedback case     

Stability analysis for linear systems with input backlash through sufficient LMI conditions

This paper addresses the problem of stability analysis for a given class of nonlinear systems resulting from the connection of a linear system with an isolated backlash operator. Constructive conditions based on LMIs to ensure closed-loop stability are proposed by using some suitable Lyapunov functionals with quadratic terms and Lure type terms, and generalized sector conditions. Additionally, the boundary of the associated set of all the admissible equilibrium points is precisely defined.     

Lyapunov stability tests for linear time-delay systems

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proofs are presented in details for the case of systems with multiple delays. The state of the art, ongoing research and potential extensions to other classes of delay systems are discussed.     

Output feedback stabilisation of single-input single-output linear systems with I/O network-induced delays. An eigenvalue-based approach

This work addresses the output feedback stabilisation problem for a class of linear single-input single-output systems subject to I/O network delays. More precisely, we are interested in the characterisation of the set of delay and gain parameters guaranteeing the stability of the closed-loop system. To perform such an analysis, we adopt an eigenvalue perturbation based approach. Various illustrative numerical examples complete the presentation.     

Necessary and sufficient conditions for robust oscillatory stability

The problem of robust oscillatory stability of uncertain systems is investigated in this article. For the uncertain systems, whose characteristic polynomial sets belong to the interval polynomial family or diamond polynomial family, sufficient and necessary conditions are given based on the stability and/or oscillation properties of some special extreme point polynomials. A systematic approach exploiting Yang's complete discrmination system is proposed to check the robust oscillatory stability of such uncertain systems. The proposed method is efficient in computation and can be easily implemented.     

Necessary and sufficient conditions for detectability of jumps in linear systems

Necessary and sufficient conditions for detectability of abrupt changes of unknown magnitude in inputs, outputs, and states of linear systems are derived. Results indicate the minimum number of sensors necessary for detection of different types of hypothesized jumps.     

Necessary and sufficient conditions for strong stability of distributed parameter systems

In this paper we propose definitions for strong stabilizability and strong detectability of infinite-dimensional control systems. We show that these definitions are appropriate by showing that they can be used to give necessary and sufficient conditions for the strong stability of a system in terms of its input-output stability. As an application, we discuss the strong stability of the feedback interconnection of two strongly stabilizable and strongly detectable systems.     

Necessary and sufficient conditions for componentwise stability of interval matrix systems

In the asymptotic stability (AS) analysis of interval matrix systems, some results are available that operate only as sufficient conditions, based on a unique test matrix, adequately built from the interval matrix. Our note reveals the complete role of this test matrix for fully characterizing the componentwise asymptotic stability (CWAS) of interval matrix systems. CWAS is a special type of AS which ensures the flow invariance of certain time-dependent sets with respect to the state-space trajectories. Hence, the sufficient conditions for AS get a new and deeper meaning by their reformulation as necessary and sufficient conditions with respect to the stronger property of CWAS.     

Necessary and sufficient stability condition of fractional-order interval linear systems

This paper establishes a necessary and sufficient stability condition of fractional-order interval linear systems. It is supposed that the system matrix A is an interval uncertain matrix and fractional commensurate order belongs to 1≤α<2. Using the existence condition of Hermitian P=P^∗ for a complex Lyapunov inequality, we prove that the fractional-order interval linear system is robust stable if and only if there exists Hermitian matrix P=P^∗ such that a certain type of complex Lyapunov inequality is satisfied for all vertex matrices. The results are directly extended to the robust stability condition of fractional-order interval polynomial systems.     

Algebraic theory for robust stability of interconnected systems: Necessary and sufficient conditions

We consider an interconnected system Somade of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functionsH. Given that thekth subsystem is perturbed fromG_{k} = N_{rk}D_{k}^{-1}totilde{G}_{k} = (N_{rk} + Delta N_{rk})(D_{k} + Delta D_{k})^{-1}and that the system SoisH-stable, we derive a computationally efficient necessary and sufficient condition for theH-stability, of the perturbed system. These fractional perturbations are more general than the conventional additive and multiplicative perturbations. The result is generalized to handle simultaneous perturbations of two or more subsystems.     

Necessary and sufficient conditions for the stability of polynomials with linear parameter dependencies

In this paper we show that Kharitonov's theorem will provide a necessary and sufficient condition for the stability of a particular family of polynomials where the coefficients are real and linearly dependent on a set of uncertain parameters. The main results are obtained by restricting the linear map from the parameter to the coefficient space to contain the Kharitonov polynomials. The results are applied to both continous-time and discrete-time polynomials and future extensions are discussed.     

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables. Received May 18, 2000     

Necessary and sufficient conditions for stability of LMS

Guo and Ljung (1995) established some general results on exponential stability of random linear equations, which can be applied directly to the performance analysis of a wide class of adaptive algorithms, including the basic LMS ones, without requiring stationarity, independency, and boundedness assumptions of the system signals. The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded, nonstationary, and non-φ-mixing signals. The results of this paper can be applied to a very large class of signals, including those generated from, e.g., a Gaussian process via a time-varying linear filter. As an application, several novel and extended results on convergence and the tracking performance of LMS are derived under various assumptions. Neither stationarity nor Markov-chain assumptions are necessarily required in the paper     

2-D unitary matrix pencil method for efficient direction of arrival estimation

In this study, we extended the one-dimensional (1-D) unitary matrix pencil method (UMP) [N. Yilmazer, J. Koh, T.K. Sarkar, Utilization of a unitary transform for efficient computation in the matrix pencil method to find the direction of arrival, IEEE Trans. Antennas Propagat. 54 (1) (2006) 175–181] to two-dimensional case, where 2-D matrix pencil (MP) method are used to find the 2-D poles corresponding to the direction of arrival (DOA), azimuth and elevation angles, of the far field sources impinging on antenna arrays. This technique uses MP method to compute the DOA of the signals using a very efficient computational procedure in which the complexity of the computation can be reduced significantly by using a unitary matrix transformation. This method applies the technique directly to the data without forming a covariance matrix. Using real computations through the unitary transformation for the 2-D matrix pencil method leads to a very efficient computational methodology for real time implementation on a DSP chip. The numerical simulation results are provided to see the performance of the method.