Linear Multivariable Systems Research Articles

Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices

Upper and lower bounds for H-eigenvalues, Z-spectral radius and C-spectral radius of a third-order tensor are given by the minimax eigenvalue of symmetric matrices extracted from this given tensor. As applications, a sufficient condition for third-order nonsingular M-tensors and some valid sufficient conditions for the uniqueness and solvability of the solutions to multi-linear systems, tensor complementarity problems and non-homogeneous systems are proposed.

A two-step accelerated Levenberg–Marquardt method for solving multilinear systems in tensor-train format

Recently, several iterative algorithms have been proposed for the solution of multilinear system Axm−1=b in the sense that the tensor A of order m and dimension n is a structured one. In this paper, we continue to address this multilinear system in which the coefficient tensor A is an unstructured one given in tensor-train format, and present a new iterative method for it. The method we propose here is an accelerated version of the modified Levenberg–Marquardt method for nonlinear equations (Fan, 2012), which contains two-step line searches corresponding to the LM step and the approximate LM step equipped with a novel LM parameter. It is shown that this algorithm has cubic convergence under the local error bound condition which is weaker than nonsingularity, and the computational complexity of which is free from the so-called curse-of-dimensionality. The performed numerical examples show that our method is promising.

The tensor splitting methods for solving tensor absolute value equation

Recently, D.-D. Liu et al. (2018) presented the tensor splitting methods for solving multilinear systems and S.-Q. Du et al. (2018) generalized tensor absolute value equations. In this paper, we verify the existence of solutions of tensor absolute value equations and propose the tensor splitting methods for solving this class of equation. Furthermore, the convergence analysis of the tensor splitting method is also studied under suitable conditions. Finally, numerical examples show that our algorithm is an efficient iterative method.

A new preconditioner of the tensor splitting iterative method for solving multi-linear systems with $$\mathcal {M}$$-tensors

In this paper, we propose a new preconditioner of the tensor splitting iterative method for solving multi-linear systems with $$\mathcal {M}$$ -tensors. We theoretically show that the spectral radii of the preconditioner iterative tensor decrease as the parameters in the new preconditioners increase, if the preconditioned tensor is a strong $$\mathcal {M}$$ -tensor. Based on this, we give the comparison for spectral radii of preconditioned iterative tensors. Numerical examples are given to show our theoretical results and the efficiency of our new preconditioner. We also show the efficiency of our preconditioner used to solving higher order Markov chain.

The accelerated overrelaxation splitting method for solving symmetric tensor equations

This paper is concerned with solving the multilinear systems $${\mathscr {A}}\mathbf {x}^{m-1}=\mathbf {b}$$ whose coefficient tensors are mth-order and n-dimensional symmetric tensors. We first extend the accelerated overrelaxation (AOR) splitting method to solve the tensor equation. To improve the convergence, we develop a Newton-AOR (NAOR) method that hybridizes the Newton method and the accelerated overrelaxation scheme. Convergence analysis shows that the proposed methods converge under appropriate assumptions. Finally, some numerical examples are provided to show the effectiveness of the methods proposed.

A new preconditioned SOR method for solving multi-linear systems with an $${\mathcal {M}}$$-tensor

In this paper, we propose a new preconditioned SOR method for solving the multi-linear systems whose coefficient tensor is an $${\mathcal{M}}$$-tensor. The corresponding comparison for spectral radii of iterative tensors is given. Numerical examples demonstrate the efficiency of the proposed preconditioned methods.

Reverse-order law for core inverse of tensors

The notion of the core inverse of tensors with the Einstein product was introduced, very recently (Sahoo et al. Comp Appl Math 39:9, 2020). In this paper we establish a few sufficient and necessary conditions for reverse-order law of this inverse. Further, we discuss mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems via the Einstein product. The prowess of the inverse is demonstrated to solve the Poisson problem in the multilinear system framework.

Preconditioned Jacobi type method for solving multi-linear systems with [formula omitted]-tensors

The main purpose of this paper is devoted to solving the multi-linear systems by the preconditioned methods based on the tensor splittings. We consider a new preconditioner I+Gα, which is constructed by the elements at first column of the majorization matrix of A, and propose the preconditioned Jacobi type iterative tensors according to three different tensor splittings. Theoretically, we give convergence theorem and comparison theorem of the proposed tensor splitting iterative methods. Numerical results are reported to show the efficiency of the proposed splitting iterative methods.

Observer-Based Piecewise Multi-Linear Controller Designs for Nonlinear Systems Using Feedback and Observer Linearizations

This paper proposes observer-based piecewise multi-linear controllers for nonlinear systems using feedback and observer linearizations. The piecewise model is a nonlinear approximation and fully parametric. Feedback linearizations are applied to stabilize the piecewise multi-linear control system. Furthermore, observer linearizations are more conservative in modeling errors compared with feedback linearizations. In this paper, we propose robust observer designs for piecewise multi-linear systems. Moreover, we design piecewise multi-linear controllers that combine the robust observer with various performance such as a regulator and tracking controller. These design methods realize a separation principle that allows an observer and a regulator to be designed separately. Examples are demonstrated through computer simulation to confirm the feasibility of our proposals.

Method development for analysis of information components of the monitoring system with a limited delay

Methods of modeling and analysis of information components of a monitoring system are examined. They are based upon methods of the queueing theory. In particular, a limited waiting for service requests in a multi-linear system with a general queue is considered. Various types of failures have also been taken into consideration. Particular emphasis is made on deriving the analytic dependence of loss probabilities from factors that limit a waiting queue (i.e. number of positions available within a queue and maximum stay time in the given queue) in case of a simple stream of calls which are incoming for service, and the demonstrative law of channel occupancy distribution.

Existence and Uniqueness of Positive Solutions to Nonlinear Systems of Equations With M-Type Functions

This paper considers the existence and uniqueness of positive solutions to a class of nonlinear systems of equations, which has wide applications in fluid mechanics and electrical circuit theory. In this paper, we define a new class of functions, called M-type function, and then, we show some good properties of M-type function as well as the relationship between this kind of functions and other kinds of functions. In particular, by fully exploiting the properties of M-type function, we present that under a reasonable condition, the nonlinear system of equations with an M-type function, which can be seen as a generalization of the multilinear system with a nonsingular M-tensor, and a positive right-hand side vector has a unique positive solution. Meanwhile, for this class of problems, we give a preliminary and simple algorithm framework. What is more, under an additional assumption, we illustrate the feasibility of using Newton step when improving the original iteration algorithm to obtain better convergence. Numerical experiments show the stability and efficiency of the proposed algorithms.

Novel accelerated methods of tensor splitting iteration for solving multi-systems

Tensor splitting iteration method is a class of popular technique for solving multi-linear systems. In this paper, we present one kind of efficient alternating splitting iteration method, and further generalize accelerated overrelaxation method (AOR) and symmetric accelerated overrelaxation method (SAOR) from linear systems to multi-systems. Then, one type of preconditioned (alternating) tensor splitting method is also applied for solving multi-systems. Numerical experiments illustrate the efficiency of the provided methods.

A Passive Fault-Tolerant Switched Control Approach for Linear Multivariable Systems: Application to a Quadcopter Unmanned Aerial Vehicle Model

Abstract This paper proposes synthesis algorithms for the design of passive state- and output-feedback fault-tolerant controllers. Sufficient conditions for the existence and the construction of such fault-tolerant controllers are given in terms of linear matrix inequalities (LMIs) which can be solved efficiently. The state-feedback fault-tolerant controller consists of a family of state-feedback gains switched appropriately according to a stabilizing switching signal so that the closed-loop system satisfies a performance requirement expressed in terms of system L2 norm. Similarly, the output feedback controller consists of a family of full-order linear, time-invariant controllers switched according to a stabilizing signal that depends only on the controller states. Both approaches are passive in the sense that they do not rely on the detection and/or the estimation of the faults. The proposed approaches are tested on a nonlinear model of a quadcopter. Simulation results show that satisfactory stability, tracking, and disturbance rejection are maintained despite of time-varying actuator faults.

Robust output tracking of constrained perturbed linear systems via model predictive sliding mode control

SummaryA robustifying strategy for constrained linear multivariable systems is proposed. A combination of tracking model predictive control with output integral sliding mode techniques is used to completely reject bounded matched perturbations. It can be guaranteed that all constraints on inputs, states, and outputs are satisfied although only output information is used. Finally, real‐world experiments with an unstable plant are presented in order to demonstrate the validity and the effectiveness of the proposed approach.

Further study on tensor absolute value equations

In this paper, we consider the tensor absolute value equations (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of the TAVEs is an interesting generalization of matrix absolute value equations (AVEs), the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of the TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of the TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of the TAVEs for some special cases. To find a solution to the TAVEs, we employ the generalized Newton method and report some preliminary results.

An iterative algorithm based on strong [formula omitted]-tensors for identifying positive definiteness of irreducible homogeneous polynomial forms

Identifying the positive definiteness of an even-degree homogeneous polynomial form arises in various applications, such as evolutionary game dynamics, automatical control, magnetic resonance imaging, and spectral hypergraph theory. However, it is difficult to determine whether a given homogeneous polynomial is positive definite or not because the problem is NP-hard. In this paper, an iterative algorithm of identifying the positive definiteness of irreducible homogeneous polynomial forms is proposed by identifying strong H-tensors with weakly irreducibility. The validity of the iterative algorithm is proved theoretically. Experimental results on multilinear systems, high-order Markov chains and symmetric multi-player games are presented to illustrate the applications of the proposed methods.

Neural network approach for solving nonsingular multi-linear tensor systems

The main propose of this paper is to develop two neural network models for solving nonsingular multi-linear tensor system. Theoretical analysis shows that each of the neural network models ensures the convergence performance. For possible hardware implementation of the proposed neural network models, based on digital circuits, we adopt the Euler-type difference rule to discretize the corresponding Gradient neural network (GNN) models. The computer simulation results further substantiate that the models can solve a multi-linear system with nonsingular tensors.

Methods for Finding a Regularized Solution When Identifying Linear Multivariable Multiconnected Discrete Systems

Methods for Finding a Regularized Solution When Identifying Linear Multivariable Multiconnected Discrete Systems

Adaptive control of linear multivariable systems using dynamic regressor extension and mixing estimators: Removing the high-frequency gain assumptions

In this paper it is shown that, using the recently introduced dynamic regressor extension and mixing parameter estimation technique, it is possible to remove the key assumption of prior knowledge on the high frequency gain imposed in model reference adaptive control of linear time-invariant multivariable systems. For the case of scalar systems this is tantamount to removing the requirement of known sign of this coefficient. The proposed controller is not certainty equivalent in the sense that the parameters generated by the new estimator are not directly applied to the controller, but they are modified adding a shifting and a scaling factor. Under the weak assumption of interval excitation, parameter convergence in finite time of the modified parameters is achieved, ensuring good robustness of the proposed controller.

Visual analysis of impact factors of forest pests and diseases

In the field of forest pest and disease research, researchers have combined the experience and data accumulated over many years and conducted long-term and systematic observations of the research object; they have used regression analysis to determine the factors that affect the occurrence of pests and diseases. This traditional approach is time-consuming and highly dependent on expert experience. In this paper, we propose a multicombination multivariable linear regression model to quantitatively describe the multiple linear combinations of relationship between multiple independent variables and a single dependent variable. Based on this model and a data flow model combined with statistical principles and visualization techniques, we propose a multicombination multivariable linear regression visual analysis method to assist researchers in quickly assessing the correlations between the disease indexes of forest diseases and pests and the factors that may affect the pest and disease occurrences. Based on this approach, a multicombination multivariable linear regression visual analysis system was designed and implemented, and the cases of a given forest pest and disease data set were analyzed. It is shown that the multicombination multivariable linear regression visual analysis method can effectively assist researchers in quickly understanding pest and disease data, determining impact factors, and finding relevant laws.