Linear Discrete-time Descriptor Systems Research Articles

An extension of Klamka’s method to positive descriptor discrete-time linear systems with bounded inputs

An extension of Klamka’s method to positive descriptor discrete-time linear systems with bounded inputs

Willems’ Fundamental Lemma for Linear Descriptor Systems and Its Use for Data-Driven Output-Feedback MPC

In this letter we investigate data-driven predictive control of discrete-time linear descriptor systems. Specifically, we give a tailored variant of Willems’ fundamental lemma, which shows that for descriptor systems the non-parametric modeling via a Hankel matrix requires less data compared to linear time-invariant systems without algebraic constraints. Moreover, we use this description to propose a data-driven framework for optimal control and predictive control of discrete-time linear descriptor systems. For the latter, we provide a sufficient stability condition for receding-horizon control before we illustrate our findings with an example.

Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm

<p style='text-indent:20px;'>The shuffle algorithm is applied to analysis of the fractional descriptor discrete-time linear systems. Using the shuffle algorithm the singularity of the fractional descriptor linear system is eliminated and the system is decomposed into dynamic and static parts. Procedures for computation of the solution and dynamic and static parts of the system are proposed. Sufficient conditions for the positivity of the fractional descriptor discrete-time linear systems are established.

Matrix inverse eigenvalue problem for stabilization of fractional descriptor discrete-time linear systems by forward and propositional output feedback

In this paper, stabilization of unstable fractional descriptor discrete-time linear system via forward and propositional output feedback is done to obtain satisfactory responses. To gain forward and propositional output feedback matrices, two standard linear systems need to exist. Assigning nonzero arbitrary eigenvalues to the first standard system and inverted the desired eigenvalues for standard descriptor system to the second one, desired eigenvalues are assigned by matrix inverse eigenvalue problem. Numerical examples are also presented to illustrate our method.

Index reduction for second order singular systems of difference equations

This paper is devoted to the analysis of linear second order discrete-time descriptor systems (or singular difference equations (SiDEs) with control). Following the algebraic approach proposed by Kunkel and Mehrmann for pencils of matrix valued functions, first we present a theoretical framework based on a procedure of reduction to analyze solvability of initial value problems for SiDEs, which is followed by the analysis of descriptor systems. We also describe methods to analyze structural properties related to the solvability analysis of these systems. Namely, two numerical algorithms for reduction to the so-called strangeness-free forms are presented. Two associated index notions are also introduced and discussed. This work extends and complements some recent results for high order continuous-time descriptor systems and first order discrete-time descriptor systems.

Set-based state estimation and fault diagnosis of linear discrete-time descriptor systems using constrained zonotopes

This paper presents new methods for set-valued state estimation and active fault diagnosis of linear descriptor systems. The algorithms are based on constrained zonotopes, a generalization of zonotopes capable of describing strongly asymmetric convex sets, while retaining the computational advantages of zonotopes. Additionally, unlike other set representations like intervals, zonotopes, ellipsoids, paralletopes, among others, linear static constraints on the state variables, typical of descriptor systems, can be directly incorporated in the mathematical description of constrained zonotopes. Therefore, the proposed methods lead to more accurate results in state estimation in comparison to existing methods based on the previous sets without requiring rank assumptions on the structure of the descriptor system and with a fair trade-off between accuracy and efficiency. These advantages are highlighted in two numerical examples.

The pointwise completeness and the pointwise degeneracy of fractional descriptor discrete-time linear systems

The pointwise completeness and the pointwise degeneracy of fractional descriptor discrete-time linear systems

Robust state-feedback control for discrete-time descriptor systems with norm-bounded parametric uncertainties

ABSTRACTThis paper deals with a state feedback control problem for linear time-invariant discrete-time descriptor systems with norm-bounded parametric uncertainties. To this end, bounded real lemma (BRL) is extended on the class of uncertain descriptor systems. The control design procedure based on the conditions of BRL for uncertain descriptor systems is proposed. Numerical examples are included to illustrate the effectiveness of the present result.

Fault detection and isolation for discrete-time descriptor systems based on H −/L ∞ observer and zonotopic residual evaluation

ABSTRACT This paper investigates actuator fault detection and isolation for discrete-time linear descriptor systems. We propose a novel fault detection and isolation scheme based on a bank of fault detection observers (FDOs). Under the assumption that the disturbances are unknown but bounded, analysis is used to attenuate the effects of disturbances. Meanwhile, the FDO is designed to be sensitive to the fault via considering the performance index in finite frequency domain. The design conditions of the proposed FDO are converted into solving a set of linear matrix inequalities. Furthermore, the residual evaluation based on a set-membership approach is proposed to reduce conservatism. Simulation results are provided to demonstrate the performance of the proposed method.

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.

Discrete-time linear descriptor system unknown input observer design: an auxiliary output-based approach

This paper develops a new unknown input observer (UIO) design method for a class of discrete-time linear descriptor systems when the observer matching condition is not satisfied. By equivalent transformations, the state and unknown input observations of the descriptor system are changed to the UIO design problem for a general linear dynamic system firstly. Then based on some further discussion on the concept of the system left invertibility, a new auxiliary output vector is constructed which allows us to remove the unknown inputs from the dynamic equation, and thereby a Luenberger-type observer can be constructed. One of the advantages of the present method is that it can work only under the single strongly detectable condition. Finally, a simulation example is given to verify the effectiveness and show the superiorities of our methods.

Robust anisotropy-based control of linear discrete-time descriptor systems with norm-bounded uncertainties

The paper deals with linear discrete-time time-invariant (LDTI) descriptor systems with norm-bounded parametric uncertainties. The input signal is supposed to be a “colored” noise with bounded mean anisotropy. Sufficient conditions of anisotropic norm boundedness for such class of systems are given. The control design procedure, based on these conditions, is proposed. Numerical example is given.

Eigenvalue assignment in fractional descriptor discrete-time linear systems

Abstract The problem of eigenvalue assignment in fractional descriptor discrete-time linear systems is considered. Necessary and sufficient conditions for the existence of a solution to the problem are established. A procedure for computation of the gain matrices is given and illustrated by a numerical example.

Drazin inverse matrix method for fractional descriptor discrete-time linear systems

Drazin inverse matrix method for fractional descriptor discrete-time linear systems

Design of an Optimal Preview Controller for a Class of Linear Discrete-Time Descriptor Systems

The preview control problem of a class of linear discrete-time descriptor systems is studied. Firstly, the descriptor system is decomposed into a normal system and an algebraic equation by the method of the constrained equivalent transformation. Secondly, by applying the first-order forward difference operator to the state equation, combined with the error equation, the error system is obtained. The tracking problem is transformed into the optimal preview control problem of the error system. Finally, the optimal controller of the error system is obtained by using the related results and the optimal preview controller of the original system is gained. In this paper, we propose a numerical simulation method for descriptor systems. The method does not depend on the restricted equivalent transformation.

Leader-following consensus for discrete-time descriptor multi-agent systems with observer-based protocols

In this paper, the leader-following consensus problem of discrete-time descriptor multi-agent systems is considered. Each agent’s dynamics is modelled by a discrete-time linear descriptor system, and the interaction topology among the agents is assumed to be directed. To solve the multi-agent consensus problem, three different architectures of connecting observers and controllers are used to construct the consensus protocols. A kind of modified generalized algebraic Riccati equation is provided to design the protocol’s gain matrices. Based on graph theory, matrix theory and the Lyapunov method, some sufficient convergence conditions are established to guarantee the descriptor multi-agent system achieves consensus. Finally, a simulation example is provided to illustrate the obtained results.

Minimum energy control of descriptor discrete-time linear systems by the use of Weierstrass-Kronecker decomposition

Abstract The minimum energy control problem for the descriptor discrete-time linear systems by the use of Weierstrass-Kronecker decomposition is formulated and solved. Necessary and sufficient conditions for the reachability of descriptor discrete-time linear systems are given. A procedure for computation of optimal input and a minimal value of the performance index is proposed and illustrated by a numerical example.

Drazin inverse matrix method for fractional descriptor discrete-time linear systems

Abstract The Drazin inverse of matrices is applied in order to find the solutions of the state equations of fractional descriptor discrete-time linear systems. The solution of the state equation is derived and the set of consistent initial conditions for a given set of admissible inputs is established. The proposed method is illustrated by a numerical example.

线性离散时间广义系统的最优预见控制

We studied the design of an optimal preview controller for linear discrete-time descriptor systems. We first improved the theorem on optimal regulator theory for linear discrete-time descriptor systems. Then, we applied the first-order forward difference operator for the status equation and the previewable reference signal. Thus, a descriptor augmented error system with a previewable reference signal was created, and the tracking problem was transformed into a regulator problem of the descriptor augmented error system. We then discussed the relationship between the stability and causal controllability of the transformed and original system. By applying the feature of causal controllability, we introduced pre-feedback to transform the problem into an equivalent regulator problem for the descriptor causal system. The problem was then further transformed into an optimal control problem for a standard system derived from the descriptor augmented error system by modifying the quadratic performance index. Finally, we derived the optimal preview controller with preview action from the theorem thus obtained by improving the optimal regulator theory. For the ease of application, we presented the steps for realizing the theory, and validated the results through numerical simulation.

Anisotropic norm computation for descriptor systems with nonzero-mean input signals

Linear stationary discrete-time descriptor systems with input sequences of random Gaussian nonzero-mean vectors with bounded mean anisotropy are under consideration. Conditions of anisotropic norm boundedness for such systems are given in terms of generalized discrete-time algebraic Riccati equations (GDARE) and linear matrix inequalities (LMI). On basis of these results, the algorithm of anisotropic norm computation using convex optimization techniques is developed. Numerical examples illustrate methods of anisotropic norm computation.