Descriptor Systems Research Articles

Stubborn State and Disturbance Observer Co‐Design for Nonlinear Descriptor Systems With δQC$$ \delta \mathrm{QC} $$ via a Dynamic Event‐Triggered Mechanism

ABSTRACTThis article studies the state and disturbance simultaneous estimation problem for a class of nonlinear descriptor systems with using a dynamic event‐triggered mechanism. refers to incremental quadratic constraints, which can provide a unified description of many types of common nonlinear functions. To reduce the negative impact of measurement outliers on the identification estimation, a stubborn state and disturbance observer co‐design scheme is proposed for the first time by embedding dynamic saturation output estimation errors. Meanwhile, a dynamic event‐triggered mechanism is introduced to avoid the need for continuously available output information, which can reduce the pressure on communication resources. By constructing a Lyapunov function, existence conditions of the stubborn state and disturbance observer are obtained in the form of a convex optimization problem so that the error estimation maintains an acceptable estimation performance. Finally, simulation examples illustrate the universality and stubbornness of the proposed observer.

Robust fusion filter for networked uncertain descriptor systems with colored noise and cyber-attacks

The robust fusion filtering problem of multi-sensor networked uncertain descriptor systems (NUDSs) with colored noise, uncertain noise variances and cyber-attacks is investigated. During data transmission in unreliable communication networks, the data can be maliciously attacked by attackers. In other words, the local filters (LFs) may receive false data or may not receive data because of the cyber-attacks. By adopting the singular value decomposition (SVD) method, the original NUDSs can be converted into two reduced-order subsystems with uncertain correlated fictitious white noises, and the cyber-attacks are transformed into the fictitious noises. Cross-covariance matrices between local filtering errors are derived. The robust LFs are obtained according to the minimax robust estimation principle. Under the linear unbiased minimum variance criterion, three weighted fusion algorithms are applied to fuse the LFs. For all allowable uncertainties of noise variances and cyber-attacks, the minimal upper bounds of covariance matrices of the local and distributed fusion filters are guaranteed. The proof of their robustness is established through the minimax estimation principle and Lyapunov equation method. Finally, the correctness and effectiveness of the proposed algorithms are verified by a circuit system example.

Event-Triggered Variable Structure Control of Nonlinear Switched Descriptor Systems Subjected to Matched/Mismatched Disturbances

Event-Triggered Variable Structure Control of Nonlinear Switched Descriptor Systems Subjected to Matched/Mismatched Disturbances

Feedback Control Design Strategy for Stabilization of Delayed Descriptor Fractional Neutral Systems with Order 0 < ϱ < 1 in the Presence of Time-Varying Parametric Uncertainty

Descriptor systems are more complex than normal systems, which are modeled by differential equations. This paper derives stability and stabilization criteria for uncertain fractional descriptor systems with neutral-type delay. Through the Lyapunov–Krasovskii functional approach, conditions subject to time-varying delay and parametric uncertainty are formulated as linear matrix inequalities. Based on the established criteria, static state- and output-feedback control laws are designed to ensure regularity and impulse-free properties, together with robust stability of the closed-loop system under permissible uncertainties. Numerical examples illustrate the effectiveness of the control methods and show that the results depend on the range of variation in the delays and on the fractional order, leading to stability analysis results that are less conservative than those reported in the literature.

Admissibility Analysis and Controller Design Improvement for T-S Fuzzy Descriptor Systems

In this paper, a stability analysis and the controller improvement of T-S fuzzy Descriptor system are studied. Firstly, by making full use of the related theory of fuzzy affiliation function and combining the design method of fuzzy Lyapunov function with the method of inequality deflation, a stability condition with wider admissibility and less system conservatism is proposed. The advantage of this method is that it is not necessary to ensure that each fuzzy subsystem is progressively stable. We also maximise the boundary of the derivatives of the affiliation function mined. Secondly, a PDC controller and a Non-PDC controller are designed, and the deflation conditions for the linear matrix inequalities of the two controllers are constructed. Finally, some arithmetic simulations and practical examples are given to demonstrate the effectiveness of the method studied in this paper, and the results obtained are less conservative and have larger feasible domains than previous methods.

Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

AbstractThe present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

On negative imaginariness and [formula omitted] control of descriptor systems with state–space symmetry

This paper studies negative imaginariness and the static output feedback H∞ control synthesis of descriptor systems based on the state–space symmetric model. First, under the assumption that the transfer function matrix is proper, necessary and sufficient conditions to determine negative imaginary properties are established for descriptor systems with state–space symmetry. Then, based on state–space symmetry, a sufficient condition is derived for the design of static output feedback controller, such that the resulting closed-loop system satisfies negative imaginary and H∞ performance. Finally, two circuit networks and a numerical example are provided to validate the main results of the paper.

Generic observability for port-Hamiltonian descriptor systems

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

Sensor fault estimation based on ellipsoid bundles for dynamic point-the-bit rotary steerable systems

The dynamic point-the-bit rotary steerable system (DPRSS) achieves high-performance trajectory control in oil and gas exploitation. However, the sensor faults significantly degrade the reliability of the DPRSS. In this paper, the problem of sensor fault estimation for the DPRSS is investigated. Firstly, the DPRSS is modeled by a descriptor system with unknown but bounded uncertainties, which include measurement noises, model perturbations, and unknown inputs. Next, by integrating the decoupling technique, a sensor fault estimator is put forward based on the ellipsoid bundles. Besides, order reduction and bound calculation methods are proposed to derive intervals of the states and faults. Then, with a defined minimization criterion of the ellipsoid bundle, a sufficient condition is presented to design the parameters of the estimator. Finally, numerical simulations and experiments on a DPRSS prototype are provided to demonstrate the effectiveness of the proposed fault estimation method.

Transfer matrices with positive coefficients of descriptor linear systems

Transfer matrices with positive coefficients of descriptor positive linear continuous-time systems are addressed. Two methods of checking of the positivity of descriptor linear systems are proposed. It is shown that if the positive descriptor system is asymptotically stable then all coefficients of its transfer matrix are positive.

Joint unknown input observer for descriptor system based on interval observer

In this paper, a novel joint unknown input observer (JUIO) is proposed for a class of descriptor systems. The unknown input (UI) to be estimated injects additively into both the state and output equations in a state space model. To the best of our knowledge, only a few contributions in existing work address this problem directly. To begin with, by introducing an auxiliary UI, the original system is transformed into a normal form in which the output is no longer affected by UI. In this way, the negative effect brought by the UI occurring in the output measurement is removed. An interval observer is developed to obtain upper and lower boundary estimates of the output of the reformulated system. After that, an algebraic relationship between the auxiliary UI and the states is established, and a UI reconstruction (UIR) method is developed. Based on the UIR, a JUIO comprising the UIR and a Luenberger-like state observer is developed to achieve asymptotic estimations of the UI and state simultaneously. Verifiable conditions for the existence of the proposed JUIO are given with respect to the original descriptor system. Finally, a simulation example is presented to verify the effectiveness of the proposed method.

A Novel Method for Stability and Stabilization of Interconnected Descriptor Systems

This paper addresses the issues of stability and stabilization in interconnected descriptor systems. The existing results based on different structures of the Lyapunov equation are not directly applicable to resolving time-varying problems. In this paper, based on the original literature, the stability of the system is re-described and extended, and new results are obtained, these results are extended to encompass time-varying singular systems exhibiting state-space symmetry, which is achieved by introducing a novel Lyapunov functional. The approach in this paper is grounded in the theory of trace inequalities for matrices, establishing a new sufficient condition for the admissibility of interconnected descriptor systems. In the last, an illustrative example demonstrates the feasibility of employing state feedback to render interconnected descriptor systems admissible, affirming the validity of our proposed method.

Design of finite‐time convergent functional observer for descriptor systems: Application to fault estimation

AbstractIn this paper, a finite‐time functional observer design for linear time‐invariant descriptor systems is presented. A generalized observer based on additional degrees of freedom is developed. This observer can be considered of a general structure since the existing proportional observer (PO) and proportional‐integral observer (PIO) constitute its particular cases. The main contribution is to generate a finite‐time convergent functional observer for descriptor systems by combining two generalized functional observers. The obtained results are applied to the simultaneous state, sensor, and actuator faults estimation. A numerical example is given to illustrate the approach. The obtained results show that the observer is robust in presence of uncertainties, compared to its particular cases.

Existence conditions for functional ODE observer design of descriptor systems revisited

This paper is devoted to the problem of designing functional observers for linear time-invariant (LTI) descriptor systems. The observers are realized by using state–space systems governed by ordinary differential equations (ODEs). Available existence results for functional ODE observers in the literature are extended by introducing new and milder sufficient conditions. These conditions are purely algebraic and provided directly in terms of the system coefficient matrices. The proposed observer has an order less than or equal to the dimension of the functional vector to be estimated. The observer parameter matrices are obtained by using simple matrix theory, and the design algorithm is illustrated by numerical examples.

Low-Rank Methods for Solving Discrete-Time Projected Lyapunov Equations

In this paper, we consider the numerical solution of large-scale discrete-time projected Lyapunov equations. We provide some reasonable extensions of the most frequently used low-rank iterative methods for linear matrix equations, such as the low-rank Smith method and the low-rank alternating-direction implicit (ADI) method. We also consider how to reduce complex arithmetic operations and storage when shift parameters are complex and propose a partially real version of the low-rank ADI method. Through two standard numerical examples from discrete-time descriptor systems, we will show that the proposed low-rank alternating-direction implicit method is efficient.

Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply

Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.

Robust Observer-Based Proportional Derivative Fuzzy Control Approach for Discrete-Time Nonlinear Descriptor Systems with Transient Response Requirements

This paper proposes an observer-based proportional Derivative (O-BPD) fuzzy controller for uncertain discrete-time nonlinear descriptor systems (NDSs). Representing NDSs with the Takagi–Sugeno fuzzy model (T-SFM), the proportional derivative (PD) feedback method can be utilized in the fuzzy controller design via the Parallel Distributed Compensation (PDC) concept, such that the noncausal problem and impulse behavior are avoided. A fuzzy observer is proposed to obtain unmeasured states to fulfill the PD fuzzy controller. Moreover, uncertainties and transient response performances are taken into account for the NDSs. Then, a stability analysis process and corresponding stability conditions are derived from the Lyapunov theory with the robust control method and the pole constraint. Different from existing research, the Singular Value Decomposition (SVD) and the projection lemma are utilized to transfer the stability conditions into the Linear Matrix Inequation (LMI) form. Because of this reason, the conservatism of the analysis process can be reduced by eliminating the restriction on the positive definite matrix in the Lyapunov function. By giving the proper center and radius parameters of the pole constraint in the O-BPD fuzzy controller design process, the expected transient responses can be obtained for different designers and different practical applications. Finally, the effectiveness and applicability of the proposed O-BPD fuzzy controller are demonstrated by two examples of the simulation.

Robust-optimal control of rotary inverted pendulum control through fuzzy descriptor-based techniques

Expanding upon the well-established Takagi–Sugeno (T–S) fuzzy model, the T–S fuzzy descriptor model emerges as a robust and flexible framework. This article introduces the development of optimal and robust-optimal controllers grounded in the principles of stability control and fuzzy descriptor systems. By transforming complicated inequalities into linear matrix inequalities (LMI), we establish the essential conditions for controller construction, as delineated in theorems. To substantiate the utility of these controllers, we employ the rotary inverted pendulum as a testbed. Through diverse simulation scenarios, these controllers, rooted in fuzzy descriptor systems, demonstrate their practicality and effectiveness in ensuring the stable control of inverted pendulum systems, even in the presence of uncertainties within the model. This study highlights the adaptability and robustness of fuzzy descriptor-based controllers, paving the way for advanced control strategies in complex and uncertain environments.

Controllability and Observability of the Descriptor Linear Systems Reduced to the Standard Ones by Feedbacks

Abstract In this paper, a new method for the reduction of the descriptor linear systems to standards ones is presented and verified. The method uses a state and/or state derivative feedback of output and output derivative feedback in order to transform the descriptor system into a standard one. The controllability and observability properties of the original descriptor as well as transformed standard systems are proved. Simple numerical examples illustrate the theorems introduced.

Effective numerical simulations of synchronous generator system

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.