Time-varying Descriptor Systems Research Articles

Solutions to the output regulation problem of time-varying descriptor systems

This paper studies the output regulation problem of time-varying descriptor systems and the problem of designing state feedback and dynamic measurement output feedback control laws which asymptotically achieves output regulation and disturbance rejection is considered. New regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. Based on the solution of the regulator equations, the state feedback and dynamic measurement output feedback control laws are designed to solve the output regulation problem. The work extends the existing results of output regulation problem for time-varying linear systems to the time-varying descriptor systems. Numerical examples are given to show the effectiveness of our methodology.

Set-membership estimation for linear time-varying descriptor systems

This paper considers the problem of set-membership estimation for discrete-time linear time-varying descriptor systems subject to unknown but bounded disturbance and noise. We propose a set-membership estimation method based on a descriptor system observer and a zonotopic estimator of the observer error bounds. The observer parameters are optimized in order to minimize the sizes of the zonotopes enclosing all admissible state trajectories. Finally, two simulation results are provided to demonstrate the effectiveness of the proposed method.

Interval Observer for Discrete Periodic Time-varying Descriptor Systems

This paper designs an interval observer for discrete linear periodic time-varying descriptor (LPTVD) systems. First, the discrete LPTVD system is transformed into a linear time-invariant (LTI) descriptor system by using a stacked form of periodic systems. Then, in order to reduce complexity, we further transform the equivalent LTI descriptor system into a minimal-order form. Finally, by using the obtained minimal-order implementation of the discrete LPTVD system, an interval observer is designed to estimate the system state interval at each step. At the end, an example is used to illustrate the effectiveness of the proposed results.

Admissibility analysis of discrete linear time-varying descriptor systems

This paper deals with the problem of admissibility analysis (i.e. regularity, causality and exponential stability) of discrete-time linear descriptor systems with uncertain time-varying parameters in the system state-space model matrices. The parameters enter affinely into all the matrices, and both their admissible values and variations are assumed to belong to given intervals. First, necessary and sufficient admissibility conditions for uncertainty-free discrete linear time-varying descriptor systems are presented. Next, strict linear matrix inequality conditions based on parameter-dependent Lyapunov functions are proposed to ensure robust admissibility of uncertain linear descriptor systems. Both the cases of Lyapunov functions with affine and quadratic dependence on the system uncertain parameters are considered. The robust admissibility analysis methods incorporate information on available bounds on both the admissible values and variation of the uncertain parameters. Numerical examples are presented to demonstrate the potentials of the proposed methods.

Robust H∞ state-feedback design for discrete-time descriptor systems

Abstract This paper deals with the design of robust H2 controllers for discrete-time descriptor systems with uncertain time-varying parameters in all state space matrices. It is assumed that the system matrices are affine functions of uncertain bounded parameters with bounded variations. Firstly, a novel version of the Bound Real Lemma for linear discrete time varying descriptor systems is presented. Then, the latter result is specialized for state-feedback design in terms of linear matrix inequality constraints in order to ensure the closed-loop admissibility while guaranteeing an upper-bound on the system H2-norm for all admissible uncertainty. Numerical examples are presented to illustrate the proposed control design technique.

New LMI conditions for admissibility analysis of time-varying descriptor systems

This paper presents new admissibility analysis conditions for discrete-time descriptor systems. Firstly, it proposes an extension, expressed as necessary and sufficient admissibility conditions, of admissibility criteria developed in the literature for linear time-invariant systems in order to incorporate the time-varying case. This extension exploits time-varying slack variables and uses the linear matrix inequality (LMI) formulation. Then, this extension is applied to the case of polytopic descriptor systems with bounded time-varying parameters. The use of sum of squares polynomials allows to eliminate the criteria parametric dependence and to provide new admissibility analysis conditions expressed as a LMIs feasibility problem. This new admissibility analysis result uses parameter-dependent slack variables and may be less conservative than the results already proposed in the literature as sustained by our numerical evaluation.

The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense

In this paper, we generalize the time-varying descriptor systems to the case of fractional order in matrix forms. Moreover, we present the general exact solutions of the linear singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense by using a new attractive method. Finally, two illustrated examples are also given to show our new approach.

Projection-based model reduction for time-varying descriptor systems: New results

We have presented a Krylov-based projection method for model reduction of linear time-varying descriptor systems in [13] which was based on earlier ideas in the work of J. Philips [17] and others. This contribution continues that work by presenting more details of linear time-varying descriptor systems and new results coming from real fields of application. The idea behind the proposed procedure is based on a multipoint rational approximation of the monodromy matrix of the corresponding differential-algebraic equation. This is realized by orthogonal projection onto a rational Krylov subspace. The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The proposed method works efficiently for macro-models, such as time varying circuit systems and models arising in network interconnection, on limited frequency ranges. Bode plots and step response are used to illustrate both the performance and accuracy of the reduced-order model.

Robust Finite-TimeH∞Control for Linear Time-Varying Descriptor Systems with Jumps

The finite-timeH∞control problem is addressed for uncertain time-varying descriptor system with finite jumps and time-varying norm-bounded disturbance. Firstly, a sufficient condition of finite-time boundedness for the abovementioned class of system is obtained. Then the result is extended to finite-timeH∞for the system. Based on the condition, state feedback controller is designed such that the closed-loop system is finite-time boundedness and satisfiesL2gain. The conditions are given in terms of differential linear matrix inequalities (DLMIs) and linear matrix inequalities (LMIs), and such conditions require the solution of a feasibility problem involving DLMIs and LMIs, which can be solved by using existing linear algorithms. Finally, a numerical example is given to illustrate the effectiveness of the method.

H∞Fault Detection for Linear Discrete Time-Varying Descriptor Systems with Missing Measurements

This paper deals with the problem ofH∞fault detection for a class of linear discrete time-varying descriptor systems with missing measurements, and the missing measurements are described by a Bernoulli random binary switching sequence. We first translate theH∞fault detection problem into an indefinite quadratic form problem. Then, a sufficient and necessary condition on the existence of the minimum is derived. Finally, an observer-basedH∞fault detection filter is obtained such that the minimum is positive and its parameter matrices are calculated recursively by solving a matrix differential equation. A numerical example is given to demonstrate the efficiency of the proposed method.

Input-output Finite-time Stability of Time-varying Descriptor Systems

Input-output Finite-time Stability of Time-varying Descriptor Systems

Sufficient Conditions of Asymptotic Stability of the Time-Varying Descriptor Systems

We discuss the time-varying descriptor systems. Firstly, a sufficient condition of asymptotic stability and impulse-free is derived based on Riccati equation. Secondly, we design a state feedback controller to make the close-loop system asymptotically stable and impulse-free. Finally, a numerical example demonstrates the proposed results.

Finite-Time Boundedness Control of Time-Varying Descriptor Systems

This paper mainly studies a control problem of finite-time boundedness of time-varying descriptor systems. Firstly, a sufficient and necessary condition of finite-time stability is given, then a sufficient condition of finite-time boundedness for time-varying descriptor systems is given. Secondly, we analyze the finite-time boundedness control problem and design the finite-time state feedback controller; the controller is given based on LMIs for time-varying descriptor systems and time-varying uncertain descriptor systems, respectively. Finally, a numerical example is given to prove the effectiveness of the method.

Fault diagnosis for linear time-varying descriptor systems

In this paper fault diagnosis is studied for linear time varying descriptor systems, which are the discrete time counterpart of the continuous time dynamic systems described by differential-algebraic equations. This class of systems includes and is broader than the well-known state space systems. The framework of descriptor systems is particularly useful for studying dynamic systems exhibiting time varying singularities. Actuator faults and sensor faults are respectively modeled as parametric changes in the state equation and in the output equation. The main result of this paper consists in extending an adaptive observer, initially designed for state space systems, to descriptor systems. Based on this result, fault diagnosis is performed by estimating the parameters characterizing actuator and sensor faults. Simulations examples are presented to illustrate the proposed method.

Hamiltonian (script capital H)]∞ filtering and smoothing for linear discrete time-varying descriptor systems with unknown inputs

This study is concerned with the finite horizon ℋ∞ filtering and smoothing problems for linear discrete time-varying descriptor (LDTVD) systems with unknown inputs (UI). Under the condition of Y-observability, the LDTVD system is transformed into a non-descriptor system. The design of ℋ∞ filter and smoother is equivalent to a positivity problem of a certain indefinite quadratic form. By relating this quadratic form to a Krein space state-space model, the Kalman filter theory and the innovation analysis technology are adopted to solve the formulated ℋ∞ estimation problem. A necessary and sufficient condition for solvability of the estimation problem is proposed, and the simultaneous state and UI estimator is obtained in terms of algebraic Riccati equations. Numerical examples are provided to illustrate the performance of the ℋ∞ filter and smoother.

Dilated LMI conditions for time-varying polytopic descriptor systems: the discrete-time case

This article addresses the admissibility analysis and state-feedback control synthesis problems of discrete-time polytopic descriptor systems with possibly time-varying parameters. First, we present a new necessary and sufficient strict linear matrix inequality (LMI) condition for the admissibility analysis of linear time-invariant (LTI) descriptor systems. Then, based on the concept of poly-quadratic admissibility, introduced in this article, we extend this result to the admissibility analysis of possibly time-varying parameter-dependent descriptor systems. This extension, which applies to both uncertain and measurable parameters, uses parameter-dependent Lyapunov functions and employs two slack variables. The extended conditions are also expressed as strict LMI conditions which are easily tractable numerically compared to the non-strict ones often encountered when dealing with descriptor systems. Note that we have proposed two separate admissibility analysis conditions: one directly exploitable for state estimation and the other for state-feedback control. The need for different admissibility analysis conditions for each synthesis problem is motivated by the fact that the duality between state estimation and state-feedback control, which hold in the case of measurable parameters, does not hold when dealing with uncertain ones. In our approach, we overcome the absence of such duality by considering a dilation only on the dynamical part of the descriptor system. The application of our analysis result to both robust state-feedback control and polytopic state-feedback control are also presented in this article. Our analysis results extend to descriptor systems, some existing results developed for regular systems.

Controllability and observability at infinity of linear time-varying descriptor systems

Controllability and observability at infinity of linear time-varying descriptor systems are considered. New characterisations of controllability and observability at infinity are given. Based on the definitions of controllability and observability at infinity, necessary and sufficient conditions for these properties are obtained and presented in terms of original system parameters. The present framework is shown to overcome several difficulties inherent in other treatments of descriptor systems.

Formula omitted] filtering for rectangular discrete-time descriptor systems

This paper deals with the H ∞ recursive estimation problem for general rectangular time-variant descriptor systems in discrete time. Riccati-equation based recursions for filtered and predicted estimates are developed based on a data fitting approach and game theory. In this approach, the nature determines a state sequence seeking to maximize the estimation cost, whereas the estimator tries to find an estimate that brings the estimation cost to a minimum. A solution exists for a specified γ -level if the resulting cost is positive. In order to present some computational alternatives to the H ∞ filters developed, they are rewritten in information form along with the respective array algorithms.

線形時変ディスクリプタシステムに対する安定条件と安定化法

線形時変ディスクリプタシステムに対する安定条件と安定化法

Projection‐Based Model Reduction for Time‐Varying Descriptor Systems Using Recycled Krylov Subspaces

AbstractWe will present a projection approach for model reduction of linear time–varying descriptor systems based on earlier ideas in the work of Philips and others. The idea behind the proposed procedure is based on a multipoint rational approximation of the monodromy matrix of the corresponding differential–algebraic equation. This is realized by orthogonal projection onto a rational Krylov subspace. The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The proposed method works efficiently for macromodels, such as time varying circuit systems and models arising in network interconnection, on limited frequency ranges. Bode plots and step response are used to illustrate the performance of the reduced order models. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)