Linear Descriptor Research Articles

On prescribed-time functional observers of linear descriptor systems with unknown input

ABSTRACT This paper is concerned with prescribed-time functional unknown input observer (PTFUIO) design for linear descriptor systems. First, by constructing two identical asymptotic convergence functional observers, a new PTFUIO is developed which can achieve an exact estimation within an arbitrarily prescribed time, where both the estimation accuracy and the convergence time pre-definition are independent of the initial conditions of the original system. Also, the necessary and sufficient conditions for the existence of such the PTFUIO are given in terms of the original system matrices which are easy to examine. Finally, numerical simulations are given to illustrate the effectiveness of the proposed method.

The controllability of linear descriptor systems using direct methods

In this paper, the controllability of linear descriptor systems is investigated by using a direct approach. This approach is based on viewing the system as a whole. In general method, the system is first decomposed into two subsystems in the canonical form. Then, these subsystems are dealt with separately. However, in this paper, we developed the system generally not viewed separately. Then, we derived the structure of controllable subspaces for linear descriptor systems. From these results, we can state the definition of a controllable matrix and then we investigated the properties of the linear descriptor systems related to the controllability matrices by using geometric term.

Practical Descriptor Modeling of Traffic in Intersecting Metro Lines and Robust Controller Design for Train Synchronization at Transfer Stations

In this article, a discrete-event linear descriptor model for intersecting metro lines based on the deviations of train departure times from the nominal values is introduced. In intersecting metro lines, the delays are transmitted from one line to the other intersecting lines in transfer stations (TSs). Therefore, considering the effect of the TSs in traffic modeling leads to a more accurate model. The introduced model is an uncertain model, and we reformulate it as a linear uncertain descriptor model with additive uncertainty. A robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> controller is designed based on the bounded real lemma and linear matrix inequality to compensate the delays and regulate the time deviations in the presence of uncertainty. In addition to the aforementioned objective of the control system, another objective is synchronizing the trains at TSs of intersecting metro lines. The simulation results demonstrate the performance and effectiveness of the proposed automatic traffic regulation system using real data from two intersecting lines of the Tehran metro system.

Properties of Pattern Matrices With Applications to Structured Systems

In cases where we do not have the exact parameter values of a mathematical model, we often have at least some structural information, e.g., that some parameters are nonzero. Such information can be captured by so-called pattern matrices, whose symbolic entries are used to represent the available information about the corresponding parameters. In this letter, we focus on pattern matrices with three types of symbolic entries: those that represent zero, nonzero, and arbitrary parameters. We formally define and study addition and multiplication of such pattern matrices. The results are then used in the algebraic characterization of three strong structural properties. In particular, we provide sufficient conditions for controllability of linear descriptor systems, necessary and sufficient conditions for input-state observability, and sufficient conditions for output controllability of linear systems.

Two-dimensional descriptor systems

Linear descriptor systems are governed by dynamical equations subject to algebraic constraints. In the one-dimensional case, where the systems only depend on a single index, usually time, the Weierstrass canonical form splits up the state vector in two parts, a causal part, running forward in time, and a non-causal part, running backward. In this paper linear time-invariant autonomous descriptor systems in two-dimensions are discussed and the condition on the existence of a non-trivial solution is derived, together with an explicit formula for the output of such systems. It is shown that the output of the model can be related to a causal and a non-causal part in each of the dimensions of the model, running forward and backward in the various dimensions respectively. The results are obtained by requiring that the solutions, for states and outputs, which are defined on a two-dimensional grid, are path invariant and unique.

Unknown Input Observer Design for a Class of Linear Descriptor Systems

This paper addresses the problem of unknown inputs observer (UIO) design for a class of linear descriptor systems. The unknown inputs affect both state and output of the system. The basic idea of the proposed approach is based on the separation between dynamic and static relations in the descriptor model. Firstly, the method used to separate the differential part from the algebraic part is developed. Secondly, an observer design permitting the simultaneous estimation of the system state and the unknown inputs is proposed. The developed approach for the observer design is based on the synthesis of an augmented model which regroups the differential variables and unknown inputs. The exponential stability of the estimation error is studied using the Lyapunov theory and the stability condition is given in term of linear matrix inequality (LMI). Finally, to illustrate the efficiency of the proposed methodology, a heat exchanger pilot model is considered.

The study of the projective transformation under the bilinear strict equivalence

Abstract The study of linear time invariant descriptor systems has intimately been related to the study of matrix pencils. It is true that a large number of systems can be reduced to the study of differential (or difference) systems, $S\left ( {F,G} \right )$, $$\begin{align*} &amp; S\left({F,G}\right): F\dot{x}(t) = G{x}(t) \left(\text{or the dual, } F{x}(t) = G\dot{x}(t)\right), \end{align*}$$and $$\begin{align*} &amp; S\left({F,G}\right): Fx_{k+1} = Gx_k \left(\text{or the dual, } Fx_k=Gx_{k+1}\right)\!, F,G \in{\mathbb{C}^{m \times n}}, \end{align*}$$and their properties can be characterized by homogeneous matrix pencils, $sF - \hat{s}G$. Based on the fact that the study of the invariants for the projective equivalence class can be reduced to the study of the invariants of the matrices of set ${\mathbb{C}^{k \times 2}}$ (for $k \geqslant 3$ with all $2\times 2$-minors non-zero) under the extended Hermite equivalence, in the context of the bilinear strict equivalence relation, a novel projective transformation is analytically derived.

State feedback observer-based control design for linear descriptor systems with multiple time-varying delays

Abstract In this paper, we propose an linear matrix inequality (LMI)-based design method to observer-based control problem of linear descriptor systems with multiple time-varying delays. The delay function can be continuous and bounded but not necessarily differentiable. First, by introducing a new set of improved Lyapunov–Krasovskii functionals that avoid calculating the derivative of the delay function, we obtain new delay-dependent sufficient conditions for guaranteeing the system to be regular, impulse-free and asymptotically stable. Then, based on the derived stability conditions, we design state feedback controllers and observer gains via LMIs, which can be solved numerically in standard computational algorithms. A numerical example with simulation is given to demonstrate the efficiency and validity of the proposed deign.

H∞ control for linear descriptor systems with non-differentiable delays in both state and observation

ABSTRACT This paper deals with the problem of control for linear descriptor systems with time-variable delay. The delay function, which is non-differentiable, appears in both the state and observation. For stability, appropriate Lyapunov–Krasovskii functionals are constructed to eliminate the derivative calculating of the delay function and utilize smaller number of linear matrix inequality decision variables. The proposed functionals are employed to derive new delay-dependent sufficient conditions for admissibility of the unforced descriptor closed-loop system. For control, we design state feedback controllers which make the descriptor closed-loop system admissible with a maximum disturbance attenuation level. A numerical example with simulation is given to illustrate the effectiveness of the proposed method.

Hankel-norm approximation of large-scale descriptor systems

Hankel-norm approximation is a model reduction method for linear time-invariant systems, which provides the best approximation in the Hankel semi-norm. In this paper, the computation of the optimal Hankel-norm approximation is generalized to the case of linear time-invariant continuous-time descriptor systems. A new algebraic characterization of all-pass descriptor systems is developed and used to construct an efficient algorithm by refining the generalized balanced truncation square root method. For a wide practical usage, adaptations of the introduced algorithm towards stable computations and sparse systems are suggested, as well as an approach for a projection-free algorithm. To show the approximation behavior of the introduced method, numerical examples are presented.

Optimal Control for Discrete‐time Descriptor Noncausal Systems

AbstractIn this paper, optimal control problems governed by linear discrete‐time descriptor noncausal systems (with quadratic input variables) are investigated in order. A descriptor system assumed to be regular alone is called descriptor noncausal system. According to Bellman's principle of optimality in dynamic programming, a recurrence equation for simplifying the optimal control problems is derived. Then, employing the recurrence equation, a bang‐bang optimal control problem subject to a linear descriptor noncausal system and an optimal control problem subject to a descriptor noncausal system with quadratic input variables are both settled, and the optimal solutions are given through exact expressions. A numerical example is presented to illustrate the effectiveness of the results obtained concerning the bang‐bang optimal control problem.

State feedback stabilization of linear descriptor time-varying delay systems

In this paper, the augmented Lyapunov-Krasovskii function approach combining with singular value decomposition method is developed for stabilization of linear descriptor systems with time-varying delay. The delay function is non-differentiable, but continuous and bounded. By introducing a set of improved Lyapunov-Krasovskii functionals we propose delay-dependent sufficient conditions for admissibility of the system in terms of linear matrix inequalities. Then, based on the obtained stability results the problem of stabilization is solved via state feedback controllers, which guarantees that the descriptor closed-loop system is admissible. An numerical example with simulation is provided to show the effectiveness of the theoretical result.

Robust Stabilization and Weighted Damping of Bounded Disturbances in Descriptor Control Systems

We establish necessary and sufficient conditions for the existence of dynamic regulators guaranteeing the prescribed estimation of the weighted level of damping of bounded disturbances and the asymptotic stability of linear descriptor systems. An algorithm for the construction of these regulators in the problems of robust stabilization and generalized H∞-optimization is proposed for the descriptor systems with controlled and observed outputs. The main computational procedures of the algorithm are reduced to the solution of linear matrix inequalities with additional rank restrictions. The efficiency of the algorithm is demonstrated by an illustrative example of descriptor stabilization system with bounded disturbances.

General Solutions for Descriptor Systems of Coupled Generalized Sylvester Matrix Fractional Differential Equations via Canonical Forms

We investigate a descriptor system of coupled generalized Sylvester matrix fractional differential equations in both non-homogeneous and homogeneous cases. All fractional derivatives considered here are taken in Caputo’s sense. We explain a 4-step procedure to solve the descriptor system, consisting of vectorization, a matrix canonical form concerning ranks, and matrix partitioning. The procedure aims to reduce the descriptor system to a descriptor system of fractional differential equations. We also impose a condition on coefficient matrices, related to the symmetry of the solution for descriptor systems. It follows that an explicit form of its general solution is given in terms of matrix power series concerning Mittag–Leffler functions. The main system includes certain systems of coupled matrix/vector differential equations, and single matrix differential equations as special cases. In particular, we obtain an alternative procedure to solve linear continuous-time descriptor systems via a matrix canonical form.

Electronic parameters in cobalt-based perovskite-type oxides as descriptors for chemocatalytic reactions

Perovskite-type transition metal (TM) oxides are effective catalysts in oxidation and decomposition reactions. Yet, the effect of compositional variation on catalytic efficacy is not well understood. The present analysis of electronic characteristics of B-site substituted LaCoO3 derivatives via in situ X-ray absorption spectroscopy (XAS) establishes correlations of electronic parameters with reaction rates: TM t2g and eg orbital occupancy yield volcano-type or non-linear correlations with NO oxidation, CO oxidation and N2O decomposition rates. Covalent O 2p-TM 3d interaction, in ultra-high vacuum, is a linear descriptor for reaction rates in NO oxidation and CO oxidation, and for N2O decomposition rates in O2 presence. Covalency crucially determines the ability of the catalytically active sites to interact with surface species during the kinetically relevant step of the reaction. The nature of the kinetically relevant step and of surface species involved lead to the vast effect of XAS measurement conditions on the validity of correlations.

Iterative criterion of the descriptor system asymptotic steadiness

An iterative criterion for the asymptotic steadiness of a linear descriptor system is considered. The criterion is based on an iterative algorithm for computing a generalized matrix sign-function. As an example, the problem of analyzing the asymptotic steadiness of a large descriptor system is given. Keywords linear descriptor system; steadiness criterion; matrix sign-function; search algorithm

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.

On the Kronecker canonical form of a full row rank matrix pencil and applications

Abstract The Kronecker canonical form (KCF) of matrix pencils plays an important role in many fields such as systems control and differential–algebraic equations. In this article, we compute a finite and infinite Jordan chain and also a singular chain of vectors corresponding to a full row rank matrix pencil using an extended algorithm, first introduced by Jones (1999, Ph.D. Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, UK). The proposed method exploits these vectors forming the chains corresponding to the finite and infinite eigenvalues and to the right minimal indices of the pencil. This leads to the computation of two transformation matrices for obtaining under strict equivalence the KCF of the pencil. An application to the study of homogeneous linear rectangular descriptor systems is considered and closed form solutions are obtained in terms of these two transformation matrices. All the results are illustrated with an example.

Stability Analysis of Linear Polytopic Descriptor Systems Using a Novel Copositive Matrix Approach

In this paper, we propose a copositive matrix approach to study stability of linear descriptor systems with polytopic uncertainty. First, by using copositive matrix analysis, new stability conditions are established in terms of linear matrix inequalities ensuring the regularity, impulse free, and exponential stability of such systems. Then, we use the obtained results to provide stability conditions for the system with delay. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Positivity of continuous time irregular linear descriptor systems

Positivity of continuous time irregular linear descriptor systems