Matrix Differential Equation Research Articles

Outlier-resistant sequential filtering fusion for cyber-physical systems with quantized measurements under denial-of-service attacks

In this paper, an outlier-resistant sequential fusion problem is concerned for cyber-physical systems with quantized measurements under denial-of-service attacks. The multi-sensor measurements are quantized by a bank of logarithmic quantizers before entering into communication networks. The denial-of-service attack is, from the defenders' perspective, regarded to be randomly occurring and such an occurrence is governed by a Bernoulli-distributed sequence of certain probability distribution. To suppress effects from measurement outliers onto innovations, tailored saturation functions are dedicatedly introduced to filter structures at both local and fusion stages, thereby keeping satisfactory fusion performance. By finding solutions to a set of matrix difference equations, upper bounds are initially acquired on estimator error covariances, and associated estimator parameters are subsequently secured via minimizing these acquired bounds. Two examples are finally presented to showcase the applicability of this outlier-resistant sequential fusion algorithm.

Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations

Dynamical low-rank integrators for matrix differential equations recently attracted a lot of attention and have proven to be very efficient in various applications. In this paper, we propose a novel strategy for choosing the rank of the projector-splitting integrator of Lubich and Oseledets adaptively. It is based on a combination of error estimators for the local time-discretization error and for the low-rank error with the aim to balance both. This ensures that the convergence of the underlying time integrator is preserved. The adaptive algorithm works for projector-splitting integrator methods for first-order matrix differential equations and also for dynamical low-rank integrators for second-order equations, which use the projector-splitting integrator method in its substeps. Numerical experiments illustrate the performance of the new integrators.

Dynamical low-rank integrators for second-order matrix differential equations

In this paper, we construct and analyze a new dynamical low-rank integrator for second-order matrix differential equations. The method is based on a combination of the projector-splitting integrator introduced in Lubich and Oseledets (BIT 54(1):171–188, 2014. https://doi.org/10.1007/s10543-013-0454-0) and a Strang splitting. We also present a variant of the new integrator which is tailored to semilinear second-order problems.

Distributed estimation for multi‐agent systems with relative measurements and quantized communication: A feedback quadratic framework

SummaryIn this paper, the feedback quadratic distributed state estimation problem is investigated for a class of linear discrete time‐varying multi‐agent systems subject to non‐Gaussian noises. For the multi‐agent systems, each agent has access to not only the local measurements but the relative states to its adjacent agents as well. Due to the bandwidth constraints of the digital communication networks, the signals are quantized and then exchanged between two adjacent agents, where the probabilistic uniform quantizations are taken into account. The purpose of the addressed problem is to design a novel quadratic distributed estimator for each agent based on the local information and the predictions received from the neighbors. In particular, an output injection term is introduced to handle the unstable systems. By means of the state/measurement augmentation approach, the underlying system is transformed into an augmented one, which aggregates the original vector and its second‐order Kronecker power. Accordingly, a distributed estimator is constructed such that an upper bound is ensured for the estimation error covariance and the suboptimal estimator parameters, which minimize such an upper bound, are subsequently calculated in terms of the solutions to certain matrix difference equations. Finally, an illustrative example is provided to verify the effectiveness and superiority of the proposed quadratic estimation scheme.

Asymptotically Tracking Control of Structural Balance for Discrete-Time Links System Associated with External Stimulations and State Observer

Considering that the link states of some practical networks are not measurable, it is impossible to control the links system directly to achieve structural balance. In order to handle this problem, the state observer design and the structural balance tracking control for the links system with complex dynamical networks are investigated. The dynamical model of the links system is molded as the Riccati matrix difference equation, which contains a coupling term associated with external stimulations. For the discrete-time model, the pole assignment algorithm is adopted to calculate the gain matrix for the observer of the links system. It could guarantee that the estimation error of observer for links system is globally asymptotically stable. Further, combining the effective estimation information of observer and external stimulations, an asymptotically tracking controller is proposed, which could achieve the asymptotic tracking of the reference structure balance network to the links system. Finally, the validity of the proposed method is verified by the simulation example.

Dynamic matrix theory for resonance response of plasmonic metamaterial lattice

In this paper, a lattice dynamics method, named M-K matrix method, is proposed to investigate the near-field resonance response of a plasmonic metamaterial lattice under an oblique incident field with an arbitrary incident angle. By considering the electric, magnetic and field-dipole interactions, we construct a dissipative many-body Lagrange model for a reference lattice. A collective forced vibration equation, with the degree of freedom equals to the number of nanoparticles in a cell, is introduced to describe the lattice resonance under a polarized field. The resonance frequencies can be conveniently obtained from the poles of transfer function matrix. Based on this elegant matrix differential equation, one can calculate the amplitude-frequency and phase-frequency responses of plasmonic lattice, and analysis the normal modes from dispersion relations. The analytical results, which are from three examples: simple square lattice, binary chain and chessboard lattice, are perfectly matched with numerical simulations in a large frequency band, proving it to be an effective tool to calculate the dynamic response of plasmonic lattice.

On the Numerical Treatment of the Temporal Discontinuity Arising from a Time-Varying Point Mass Attachment on a Waveguide

A vibrating pylon, modeled as a waveguide, with an attached point mass that is time-varying poses a numerically challenging problem regarding the most efficient way for eigenvalue extraction. The reason is three-fold, starting with a heavy mass attachment that modifies the original eigenvalue problem for the stand-alone pylon, plus the fact that the point attachment results in a Dirac delta function in the mixed-type boundary conditions, and finally the eigenvalue problem becomes time-dependent and must be solved for a sequence of time steps until the time interval of interests is covered. An additional complication is that the eigenvalues are now complex quantities. Following the formulation of the eigenvalue problem as a system of first-order, time-dependent matrix differential equations, two eigenvalue extraction methods are implemented and critically examined, namely the Laguerre and the QR algorithms. The aim of the analysis is to identify the most efficient technique for interpreting time signals registered at a given pylon as a means for detecting damage, a procedure which finds application in structural health monitoring of civil engineering infrastructure.

Riccati matrix differential equation and the discrete order preserving property

In this paper we recall discrete order preserving property related to the discrete Riccati matrix equation. We present results obtained by applying this property to the solutions of the Riccati matrix differential equation.

The Ψ–Asymptotic Equivalence of the Lyapunov Matrix Differential Equations with Integral Term as Right Side and Modified Argument

Abstract Using the notion of strict h-contraction, existence results for Ψ-asymptotic equivalence of two pairs of (Lyapunov) matrix differential equations with integral term as right side and modified argument are given.

An Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

We consider the construction of a feedback according to the Kalman algorithm for a continuous nonlinear control system on a finite time interval with control constraints where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position. The solution of an auxiliary optimal control problem with a quadratic functional is used for this task by analogy with the SDRE approach. Because this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems where all coefficient matrices in differential equations and criteria can contain state variables, on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations with state-dependent coefficient matrices. Due to the nonlinearity of the system this issue significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy in comparison with the Kalman algorithm for linear-quadratic problems. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V.F. Krotov and developed by V.I. Gurman and allows one not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. This article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, which depend on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

Encoding-Decoding-Based Distributed Filtering for Time-Varying Saturated Systems With Constrained Bit Rate

In this brief, the encoding-decoding-based distributed filtering problem is studied for a class of time-varying saturated systems with constrained bit rate. The constrained bit rate model is adopted to reflect the bandwidth allocation rules of the digital channel. In order to improve the communication reliability, the encoding-decoding mechanism is used to transmit the data signal under the constrained bit rate. The purpose of this brief is to design a distributed filter based on the measurements after the encoding-decoding procedure such that an upper bound (UB) of the filtering error covariance (FEC) is recursively presented through solving the matrix difference equations. Then, the distributed filter gain is parameterized by minimizing the trace of the proposed UB matrix. Furthermore, the boundedness of the UB of the FEC is ensured. Finally, a numerical simulation is provided to demonstrate the validity of the developed distributed filtering strategy.

Discrete gain scheduling control approach to elliptical orbit rendezvous system with actuator saturation

AbstractThis paper studies the discrete gain scheduling control design problem of elliptical orbit spacecraft rendezvous system with actuator saturation. Due to the presence of actuator saturation, the dynamic performance of the spacecraft rendezvous system degrades significantly. In order to improve the dynamic performance of the system, a discrete gain scheduling control approach is adopted to construct a group of time‐invariant ellipsoidal invariant sets, which can be used to determine the switching points of the discrete gain scheduling control. By choosing some discrete parameter values, the discrete gain scheduling control is obtained from a solution of a periodic Riccati matrix differential equation. Under the control obtained, the dynamic performance of the system is much improved while accomplishing successfully the rendezvous mission of the spacecraft. Finally, a practical example is provided to show the effectiveness of the proposed control design approach.

Double tracking control for the directed complex dynamic network via the state observer of outgoing links

ABSTRACT From the perspective of large system, a directed complex dynamic network (DCDN) is regarded as being made up of the nodes subsystem (NS) and the links subsystem (LS), which are coupled to each other. Different from previous studies, which propose the dynamic model of LS with the matrix differential equations, this paper describes the dynamic behaviour of LS with the outgoing links vector at every node, by which the dynamic model of LS can be represented as the vector differential equation to form the outgoing links subsystem (OLS). Due to the fact that vectors have more flexible mathematical properties than matrices, this paper proposes the more convenient mathematic method to investigate the double tracking control problem of NS and OLS. Under the condition that the states of NS are available and the states of OLS are unavailable, the asymptotical state observer of OLS is designed, by which the tracking controllers of NS and OLS are synthesised to ensure achieving the double tracking goals. Finally, an example simulation for supporting the theoretical results is also provided.

The Existence and Uniqueness of Solution for Fractional Order Matrix Differential Equation

The Existence and Uniqueness of Solution for Fractional Order Matrix Differential Equation

RESULTS ON BUILDING FRACTIONAL MATRIX DIFFERENTIAL EQUATION SYSTEMS USING A CLASS OF BLOCK MATRICES

In this paper, some important objectives have been achieved, which are as follows: First, we present a method of the inverse for a class of non-singular block matrices and some associated properties. Also, the accuracy of a new method is verified with some illustrated examples by applying the MATLAB lines. Second, applying a class of block matrices, we give the exact solution for fractional matrix differential equation systems using the Laplace fractional transformation method. Finally, illustrative examples and individual cases are also presented and discussed to demonstrate our new approach.

A sweep-based low-rank method for the discrete ordinate transport equation

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via the “unconventional” basis update & Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

Recursive state estimation for stochastic nonlinear non-Gaussian systems using energy-harvesting sensors: A quadratic estimation approach

In this paper, the recursive quadratic state estimation problem is investigated for a class of stochastic nonlinear systems (SNSs) subject to non-Gaussian noises through energy-harvesting sensors. Stochastic nonlinearities are taken into account in both state and measurement equations. The energy-harvesting technique is employed to mitigate the phenomenon of energy depletion and supply energy for data transmissions between the sensors and the remote estimator. By augmenting the state/measurements and their respective second-order Kronecker powers, the original SNS is converted into a new nonlinear system that exploits more information about the non-Gaussian noises, and the resultant quadratic estimator can then be designed by developing an efficient recursive variance-minimization algorithm. Subsequently, an upper bound is proven to exist on the estimation error covariance matrix via solving certain matrix difference equations, and such an upper bound is minimized by resorting to the design of appropriate gain parameters of the quadratic estimator. Moreover, we analyze the monotonicity of the trace of the minimal upper bound with respect to the probability of measurement transmission. Finally, a simulation example is given to demonstrate the effectiveness of the proposed quadratic estimation algorithm.

Asymptotic switched composite adaptive control with application to robotic interaction tasks

AbstractThis article proposes a new switched adaptive control design for uncertain switched systems with composite (time‐driven and state‐dependent) switching and shows its applicability in switched impedance control. A composite switched adaptive control design, consisting of the direct switched adaptive control and the indirect switched adaptive control counterpart, is developed to improve the control performance. Specifically, a new stability condition for composite switching is proposed by making use of differential matrix equations and Sylvester matrix equations, which are a generalization of Lyapunov matrix equations. The design results in a time‐varying multiple Lyapunov function that is decreasing at the switching instants. From the theoretical point of view, the relevance of this work is the construction of the adaptive laws that guarantee asymptotic tracking error and asymptotic estimation for the direct and indirect switched adaptive control loops, respectively. From the practical point of view, the relevance of this work is validated in a new switched impedance control for the robot interaction with uncertain and discontinuous environments.

Continuous‐time optimal unbiased FIR filter for input‐delayed systems

ABSTRACTMost existing finite impulse response (FIR) filters are restricted to models without delays. This paper proposes an continuous‐time optimal unbiased FIR filter for input‐delayed systems (CTOUFFID). A new integral transformation relation was introduced to derive the FIR filter. By applying this relation, the CTOUFFID problem is represented as an optimal control problem with zero terminal state. The filter gain function is obtained by solving two coupled matrix differential equations using initial conditions. The paper also offers discussion on the horizon size and a few special cases. The main benefit of the proposed solution is that it provides the maximum likelihood estimate with no requirements for the initial values. Finally, an application with the ‐404 turbofan engine model is presented to demonstrate the highly robust nature of the proposed FIR filer against incomplete noise information and unspecified model uncertainties.

Exploiting a higher‐order scheme for matrix square root and its inverse simultaneously

In this work, an investigation on an iterative scheme to calculate the matrix square root and its inversion simultaneously is performed and further discussed via the concept of matrix sign function. Convergence properties are discussed under some conditions on the choice of the initial matrix as well as the input matrix . It is then attempted to propose an iterative method possessing higher convergence order, which is also stable. Extension of the proposed scheme to the th root of a matrix is also given. Ultimately, several tests including an application of the proposed iterative method to solve matrix differential equations are brought forward.