Descriptor Systems Research Articles

Robust Observer Design for Discrete Descriptor Systems with Packet Losses

This paper considers observer design for discrete-time descriptor systems with packet losses. By taking packet loss into consideration, the error dynamic of the proposed observer becomes a stochastic switched system. Consequently, the proposed observer is synthesized in a stochastic switched system framework. Sufficient conditions for the stochastic stability with a prescribed robust performance of the error dynamic system are derived and converted into linear matrix inequalities. Not only can the proposed observer deal with packet losses, but it also attenuates the effect of process disturbance and measurement noise. A numerical simulation of a truck-trailer is given to demonstrate the effectiveness of the proposed method.

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.

A fault estimation and fault-tolerant control based sliding mode observer for LPV descriptor systems with time delay

This paper considers the problem of fault-tolerant control (FTC) and fault reconstruction of actuator faults for linear parameter varying (LPV) descriptor systems with time delay. A polytopic sliding mode observer (PSMO) is synthesized to achieve simultaneous reconstruction of LPV polytopic descriptor system states and actuator faults. Exploiting the reconstructed actuator faults and state estimates, a fault-tolerant controller is designed to compensate the impact of actuator faults on system performance by stabilizing the closed-loop LPV delayed descriptor system. Besides, the controller and PSMO gains are obtained throughout the resolution of linear matrix inequalities (LMIs) using convex optimization techniques. The developed PSMO could force the output estimation error to converge to zero in a finite time when the actuators faults are bounded through the reinjection of the output estimation error via a nonlinear switching term. Simulation results applied to a given numerical system are presented to highlight the superiority and effectiveness of the proposed approach.

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.

L₂–l ∞ Control for Discrete-Time Descriptor Systems

The problem of $l_{2} - l_\infty $ controller design for a class of discrete-time descriptor systems is addressed in this paper. In other words, the controller is designed to ensure the admissibility and the prescribed $l_{2} - l_\infty $ performance for the descriptor control system. The $l_{2} - l_\infty $ performance analysis and the feedback controller design of the descriptor control system are studied by the matrix inequality technique. Novel design conditions of the feedback controller for the descriptor systems are introduced in terms of linear matrix inequality (LMI) representations. An RC circuit is applied to explain the effectiveness of the proposed $l_{2} - l_\infty $ control design method.

On the stability of port-Hamiltonian descriptor systems

We study stable differential-algebraic equations and stabilizable differential-algebraic systems. Besides characterizing the stability in terms of a generalized Lyapunov inequality, we show that these systems can always be rewritten as port-Hamiltonian systems on the subspace where the solutions evolve.

Structured Backward Errors for Eigenvalues of Linear Port-Hamiltonian Descriptor Systems

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum satisfies the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure there always exists a nearby port-Hamiltonian descriptor system with exactly that eigenstructure. We also derive bounds for how near this system is and show that the stability radius of the system plays a role in that bound.

A Nonlinear Observer for Robust Fault Reconstruction in One-Sided Lipschitz and Quadratically Inner-Bounded Nonlinear Descriptor Systems

This article introduces a nonlinear observer to reconstruct faults in a class of nonlinear descriptor systems affected by disturbances, in both the state and output equations. The fault enters the system through nonlinear functions in both its state and output equations. The original system is first transformed such that design freedom in its structure is easier to exploit, and then a nonlinear observer is designed based on the transformed system to reconstruct the fault. Linear matrix inequalities are used to determine the gains of the observer such that the effect of the disturbances on the fault reconstruction is bounded. Finally, three simulation examples are carried out to verify the effectiveness of the scheme.

Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady-state optimal controls. A key ingredient of the analysis of turnpike phenomena is ...

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, $S\left ( {F,G} \right )$, $$\begin{align*} & S\left({F,G}\right): F\dot{x}(t) = G{x}(t) \left(\text{or the dual, } F{x}(t) = G\dot{x}(t)\right), \end{align*}$$and $$\begin{align*} & S\left({F,G}\right): Fx_{k+1} = Gx_k \left(\text{or the dual, } Fx_k=Gx_{k+1}\right)\!, F,G \in{\mathbb{C}^{m \times n}}, \end{align*}$$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.

A Hybrid Control Approach to $H_{\infty }$ Problem of Nonlinear Descriptor Systems With Actuator Saturation

In this article, the <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> problem of nonlinear descriptor systems (NDSs) is investigated. For regular or nondescriptor systems, this problem can be solved via the event-triggered control law. But this control law may be ineffective for the same problem of descriptor systems. The reason is that the discontinuity of the event-triggered controller will cause sudden failure of the algegbraic part of descriptor systems at each triggering instant. From the perspective of mathematics, the algebraic equation will have no solution at each triggering instant. To address this problem, a novel hybrid control scheme is introduced. The main characteristic of this control scheme is that it consists of an event-triggered controller and an impulsive controller. It is shown that the impulsive controller plays an extremely important role in adjusting system states to satisfy the algebraic equation. Based on this hybrid control scheme, some improved results are proposed compared with the existing ones. Finally, an example about circuit systems is given to illustrate the effectiveness of the results.

Fuzzy Control and Filtering for Nonlinear Singularly Perturbed Markov Jump Systems.

This article addresses the H∞ control and filtering problems for Markov jump singularly perturbed systems approximated by Takagi-Sugeno fuzzy models. The underlying transition probabilities (TPs) are assumed to vary randomly in a finite set, which is characterized by a higher level TP matrix. The mode- and variation-dependent fuzzy static output-feedback controller (SOFC) and filter are designed, respectively, to fulfill the control and filtering purposes. To facilitate the fuzzy SOFC synthesis, the closed-loop system is transformed into a fuzzy piecewise-homogeneous Markov jump singularly perturbed descriptor system (MJSPDS) by descriptor representation. A rigorous proof of mean-square exponential admissibility for the resulting fuzzy MJSPDS is presented. The criterion ensuring the mean-square exponential stability of the fuzzy filtering error system is further formed based on similar procedures. By setting the specific forms of the related matrix variables, the solutions for the predesigned fuzzy SOFC and filter are furnished, respectively. Finally, feasibility and validities of the developed fuzzy control and filtering results are verified by two practical examples.

${\mathcal H}_{\infty }$ Functional Filtering for Linear Systems With Unknown Inputs

A new method for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal H}_{\infty }$</tex-math></inline-formula> functional filtering for systems with unknown inputs is presented. First, a functional filtering technique is presented, then we show that the dynamic of estimation error can be modeled in the descriptor system form, this formulation makes it possible to describe the dynamic estimation error free of the derivative of the disturbances. The filter parameters are obtained from the bounded real lemma of descriptor systems. The developed approach unifies the design of functional, reduced-order, and full-order filters. Necessary and sufficient conditions for the solvability of the problem are obtained in terms of a set of bilinear matrix inequalities. Under certain conditions these inequalities can be transformed to a set of linear matrix inequalities. Two numerical examples are presented to illustrate the approach described in this article, the first example concerns the estimation of the state of charge of the lithium-ion battery and the second example highlights the reduced nature of the proposed functional filter.

Discounted cost linear quadratic Gaussian control for descriptor systems

We consider the linear quadratic Gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon. Based on recent results from the deterministic framework, we characterise the feasibility of this problem using a linear matrix inequality. In particular, conditions for existence and uniqueness of optimal controls are derived, which are weaker compared to the standard approaches in the literature. We further show that also for the stochastic problem, the optimal control is given in terms of the stabilising solution of the Lur'e equation, which generalises the algebraic Riccati equation. We conclude our paper by examining an application of our theory to fluid dynamics problems.

Study of minimum entropy controller for descriptor systems

AbstractThis paper extends the results of minimum entropy control from the state‐space system to the descriptor system. We first give an explicit definition for entropy of descriptor systems, and then derive a formula for calculating the entropy. Next, we find the parameterization of all controllers for descriptor systems. Among these controllers, we find the one resulting in the minimum entropy. Finally, a numerical example is provided to illustrate the obtained results.

Asymptotic Estimation of State, Fault and Perturbation for Nonlinear Systems and Its Fault-tolerant Control Application

This paper addresses the challenge of robust simultaneous estimation of state and actuator faults for Lipschitz nonlinear systems with unknown perturbations acting on both the state dynamics and output measurements. The existing methods enhance the estimation robustness by suppressing the perturbations or decoupling them under satisfaction of the matching condition. This work considers a total elimination of the perturbation effects through perturbation reconstruction. An adaptive sliding mode unknown input observer (ASMUIO) is developed to realize simultaneous asymptotic estimation of the state, faults and perturbations. It leverages a descriptor system reformulation of the original system by regarding partial perturbations as virtual state and the rest part as virtual faults. The proposed ASMUIO has feasibility guarantee and is solved via a linear matrix inequality (LMI) setting. Based on the ASMUIO, an adaptive backstepping fault-tolerant control (FTC) is further designed to achieve good tracking performance. The design efficacy is illustrated through comparative simulations of the proposed method against representative methods in the literature.

Bipartite containment of descriptor multi‐agent systems via an observer‐based approach

This study considers the bipartite containment problem for a class of descriptor multi-agent systems (MASs) under a structurally balanced graph. A new bipartite containment control protocol acting on each follower is proposed on the basis of designing a singular observer, where relative output information is utilised. By employing the stability theory of descriptor systems and multiple Lyapunov function technique, some sufficient criteria are derived to guarantee observer-based bipartite containment of descriptor MASs. The feasibility of obtained theoretical results is finally verified via performing simulations.

Input-Output Admissibility Analysis of Continuous Descriptor System With Time-Varying Delay

This letter deals with the input-output admissibility analysis of the continuous descriptor system with a time-varying delay. A two-term approximation transformation-model has been used to convert the original system into two interconnected subsystems. Based on the scaled small gain theorem and a new Lyapunov-Krasovskii functional, a delay-dependent sufficient condition has been presented to ensure that the time-varying delay descriptor system is input-output stable, regular, and impulse free. The derived conditions are given in a set of linear matrix inequalities. Finally, an example has been used to test the merit and validity of the proposed method.

Tangential interpolatory projections for a class of second-order index-1 descriptor systems and application to Mechatronics

This paper studies the model order reduction of second-order index-1 descriptor systems using a tangential interpolation projection method based on the Iterative Rational Krylov Algorithm (IRKA). Our primary focus is to reduce the system into a second-order form so that the structure of the original system can be preserved. For this purpose, the IRKA based tangential interpolatory method is modified to deal with the second-order structure of the underlying descriptor system efficiently in an implicit way. The paper also shows that by exploiting the symmetric properties of the system the implementing computational costs can be reduced significantly. Theoretical results are verified for the model reduction of the piezo actuator based adaptive spindle support which is second-order index-1 differential-algebraic form. The efficiency and accuracy of the method are demonstrated by analyzing the numerical results.