Linear Singular Systems Research Articles

On the preconditioned MINRES method for solving singular linear systems

On the preconditioned MINRES method for solving singular linear systems

Consensus of Singular Linear Multiagent Systems Via Hybrid Control

In this article, the consensus problem of singular linear multiagent systems (SLMASs) is investigated. Based on the properties of Drazin inverse, the considered SLMAS is transformed into an equivalent differential-algebraic system. To address this problem, an event-triggered mechanism is adopted in the existing works. However, it is pointed out in this article that this mechanism is not so effective for this problem. The reason is that the discontinuity of event-triggered controller can cause the algebraic subsystem to malfunction at the triggering instants. To avoid this phenomenon, a hybrid control strategy is adopted. The main characteristic of this strategy is that it is organized by an event-triggered controller and an impulsive controller. To show the validity of theoretical results, an example is given.

Renormalized oscillation theory for singular linear Hamiltonian systems

Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C2n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ1,λ2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems.

State bounding estimation of positive singular discrete-time systems with unbounded time-varying delays

This paper investigates the problem of state bounding for linear positive singular discrete-time systems with unbounded time-varying delay. Our approach is based on the comparison principle and the spectral analysis of matrices. Using the proposed approach, we provide new sufficient conditions for the regularity, causality, positivity and the existence of the smallest ultimate upper bound of such systems. The conditions are given in terms of linear programming problems, which can be solved by LP optimal toolbox. Numerical examples with simulation are given to demonstrate the advantage and validity of the proposed theoretical results. The proposed method is the first trial in the state bounding problem of linear singular discrete-time systems with unbounded time-varying delays.

New Explicit Criteria for Finite-time Stability of Singular Linear Systems Using Time-dependent Lyapunov Functions

New Explicit Criteria for Finite-time Stability of Singular Linear Systems Using Time-dependent Lyapunov Functions

Sensitivity analysis for sectional stiffness of anisotropic beams: The direct and adjoint methods

This paper concerns with the determination of sectional stiffness matrix and its sensitivity for anisotropic beams with initial curvatures. The governing equations of a three-dimensional beam is formulated to a Hamiltonian system, and the reduction of the Hamiltonian matrix leads to a set of singular, linear equations of the sectional warping and compliance matrix. The left and right null spaces of the singular linear system are explicitly constructed, and the existence and uniqueness of sectional warping and compliance matrix are justified. The direct and adjoint methods for sensitivity analysis of sectional stiffness matrix are developed. The existence and uniqueness of solutions for singular equations resulting from the direct and adjoint methods are proved formally. Numerical examples are presented to validate the proposed methods. The predictions of sensitivities from the direct and adjoint methods are shown to agree well with results of the complex-step method.

Fault-Tolerant Consensus for Linear Singular Multi-Agent Systems with Dynamic Adaptive Event-Triggered Mechanism

Fault-Tolerant Consensus for Linear Singular Multi-Agent Systems with Dynamic Adaptive Event-Triggered Mechanism

Unknown input observer for linear singular systems with variable delay: the continuous and the discrete time cases

In this paper, we propose a new approach to design a functional observer for linear singular systems with unknown inputs. Variable-Time delay is present in both state and input vectors. The proposed observer is developed in time and frequency domains. The time domain observer design is obtained for both continuous and discrete time systems. The procedure design is based on the unbiasedness of the estimation error dynamic of the observer using Lyapunov functional. The problem is solved by means of linear matrix inequalities to find the optimal gain implemented in the functional observer design. Frequency domain procedure is derived from time domain results, where we propose a suitable co-prime matrix fractions description. The main interest is that the proposed algorithm estimates both a functional state and the unknown input independently from the considered delay in time and frequency domains. The proposed observer proves its effectiveness on a given numerical example.

Unknown input observer for linear singular systems with variable delay: the continuous and the discrete time cases

Unknown input observer for linear singular systems with variable delay: the continuous and the discrete time cases

A Novel Bounded Real Lemma for Discrete-Time Markov Jump Linear Singular Systems

This letter presents a bounded real lemma for discrete-time Markov jump linear singular systems (MJLSS). We show that a proper Linear Matrix Inequality (LMI) is sufficient for this class of systems to be bounded.

Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

Stability analysis of discrete-time Markov jump linear singular systems with partially known transition probabilities

In this paper, we present sufficient conditions for the stochastic stability (SS) of a Markov jump linear singular system (MJLSS) with partially known transition probabilities. To handle this problem, we give two different approaches: the first one is based on the Dynamics decomposition (DD) of the system and the second one on the Weierstrass decomposition (WD). We show the relationship between these two approaches and provide a numerical example. By using the MATLAB Econometrics Toolbox, a simulation is run in order to validate our results.

The modified Uzawa methods for solving singular linear systems

For nonsingular linear systems, Yun in 2013 studied three variants of the Uzawa method; see Yun (2013) [18]. These methods contain the Uzawa-AOR method and the Uzawa-SAOR method as special cases. On the basis of the Uzawa-AOR method and the Uzawa-SAOR method, Li in 2017 proposed two modified Uzawa methods by constructing the coefficient matrix of AOR method into a new form of the product of lower triangular matrix and upper triangular matrix; see Li (2017) [26]. In this paper, we present two modified Uzawa methods for solving singular linear systems. The semi-convergence of these methods is analyzed by using the techniques of singular value decomposition and Moore-Penrose inverse. The numerical results are used to verify the theoretical results.

Bounded real lemma for singular linear continuous-time fractional-order systems

This paper proposes novel necessary and sufficient strict linear matrix inequalities (LMIs), to characterize admissibility of singular fractional-order linear continuous-time systems with the fractional derivative of order α belonging to 1≤α<2. Then, the problem of the bounded real lemma corresponding to the H∞ norm computation is addressed involving additional variables. Necessary and sufficient conditions are established via a set of LMIs that can be effectively used to design H∞ controllers. Based on the corresponding bounded real lemma, a state feedback control with a prescribed H∞ performance index for the underlying systems is proposed. Finally, numerical examples are provided to show effectiveness of the given results.

Parameterization technique for solving variational formulation of index three linear time-varying singular control systems

This paper deals with finding an approximate solution to an index three time-varying linear differential algebraic control equations based on the theory of variational formulation. The critical points of the formulated variational functional subject to the differential- algebraic equation with the class of consistent initial conditions are shown to be the solution of the proposed index-three DAEs with control problem (singular control systems) and vise- versa. The given variational problem is also transformed from indirect method into direct method using generalized Ritz bases technique. From the numerical illustrations results, the efficiency and very good accuracy as well as the simplicity of the present approach are obtained.

New results on robust finite-time stability of singular large-scale complex systems with interconnected delays

In this paper, we provide an efficient approach based on combination of singular value decomposition (SVD) and Lyapunov function methods to finite-time stability of linear singular large-scale complex systems with interconnected delays. By representing the singular large-scale system as a differential-algebraic system and using Lyapunov function technique, we provide new delay-dependent conditions for the system to be regular, impulse-free and robustly finite-time stable. The conditions are presented in the form of a feasibility problem involving linear matrix inequalities (LMIs). Finally, a numerical example is presented to show the validity of the proposed results.

DLPNO-MP2 second derivatives for the computation of polarizabilities and NMR shieldings.

We present a derivation and efficient implementation of the formally complete analytic second derivatives for the domain-based local pair natural orbital second order Møller-Plesset perturbation theory (MP2) method, applicable to electric or magnetic field-response properties but not yet to harmonic frequencies. We also discuss the occurrence and avoidance of numerical instability issues related to singular linear equation systems and near linear dependences in the projected atomic orbital domains. A series of benchmark calculations on medium-sized systems is performed to assess the effect of the local approximation on calculated nuclear magnetic resonance shieldings and the static dipole polarizabilities. Relative deviations from the resolution of the identity-based MP2 (RI-MP2) reference for both properties are below 0.5% with the default truncation thresholds. For large systems, our implementation achieves quadratic effective scaling, is more efficient than RI-MP2 starting at 280 correlated electrons, and is never more than 5-20 times slower than the equivalent Hartree-Fock property calculation. The largest calculation performed here was on the vancomycin molecule with 176 atoms, 542 correlated electrons, and 4700 basis functions and took 3.3 days on 12 central processing unit cores.

The linear quadratic optimal control problem for discrete-time Markov jump linear singular systems

In this paper, we address the linear quadratic optimal control problem for discrete-time Markov jump linear singular systems. The results are obtained under conditions that bring some additional structure to the quite general and complex class of systems being considered. The approach involves base transformations and restricts the control actions into an appropriate subspace to ensure regularity of the closed-loop system. Both finite and infinite time horizon problems are considered. The results are evaluated by means of an example.

Semi-global bipartite consensus tracking of singular multi-agent systems with input saturation

This paper investigates the problem of bipartite consensus tracking for a class of linear singular multi-agent systems under a signed graph topology, where the control input of each agent is subject to saturation. By exploiting a parametric algebraic Riccati equation (ARE)-based low-gain feedback approach, a static state feedback control protocol and a dynamic observer-based output feedback control protocol are developed utilizing local information. It is shown that under the presented protocols, semi-global bipartite consensus tracking can be achieved if the underlying signed communication graph is structurally balanced and contains a spanning tree rooted at the leader. A numerical example is finally given to illustrate the effectiveness of the theoretical results.

Internally Positive Representation to Stability of Delayed Timescale-Type Differential- Difference Equation

This paper considers exponential stability for a class of timescale-type differential-difference equation with bounded time-varying delay. Based on time scale theory, internally positive representation technique, as well as the existing exponential results of positive differential-difference equation on time scale, criteria of exponential stability of the system under consideration are obtained and they are robust to time delay and time scale to some extent. The theoretical results are applied to study stability for a class of linear singular system, and they are validated via two numerical examples.