Linear Multivariable Systems Research Articles

Restarted Nonnegativity Preserving Tensor Splitting Methods via Relaxed Anderson Acceleration for Solving Multilinear Systems

Restarted Nonnegativity Preserving Tensor Splitting Methods via Relaxed Anderson Acceleration for Solving Multilinear Systems

Computation of tensors generalized inverses under M-product and applications

This paper introduces notions of the Drazin and core-EP inverses on tensors via M-product. We propose a few properties of the Drazin and core-EP inverses of tensors, as well as effective tensor-based algorithms for calculating these inverses. In addition, definitions of composite generalized inverses are presented in the framework of the M-product, including CMP, DMP, and MPD inverses of tensors. Tensor-based higher-order Gauss-Seidel and Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative methods for solving multilinear equations have been developed. Certain multilinear systems are solved using the Drazin inverse, core-EP inverse, and composite generalized inverses, such as CMP, DMP, and MPD inverse. A tensor M-product-based regularization technique is applied to solve the color image deblurring.

A new preconditioned Gauss-Seidel method for solving $${\mathcal {M}}$$-tensor multi-linear system

A new preconditioned Gauss-Seidel method for solving $${\mathcal {M}}$$-tensor multi-linear system

Bond graph realizations of state space linear time-invariant multivariable descriptions

Given a state space description of a linear time-invariant multivariable system, a similarity transformation is applied to get a controllability canonical form. This transformation assures a Bond Graph (BG) realization and avoids the use of active bonds. Active or signal bonds do not preserve the continuity of power flows and consequently, energy conservation properties are lost. The state matrix of the system in transformed coordinates contains an antisymmetric matrix associated with the storage field and the rest is associated with the dissipative field. Thus, in general, the dissipative field is implemented with a multiport element, and in order to give physical meaning, the storage field is implemented with one-port elements. The resulting BG model does not have zero-order causal paths, that is, algebraic loops between the dissipative elements. The results are applied to the BG realization of a linear robust multivariable controller that is designed for a two mass system modelled in BG.

Linear to multi-linear algebra and systems using tensors

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.

Optimal design of Luenberger reduced-order observer with low sensitivity for linear multivariable systems

For the linear multivariable systems, by combining both merits of orthogonal-function approach and evolutionary optimization, in this paper, a new method is presented for designing a Luenberger reduced-order observer to solve the low-sensitivity design issue for physical system parameter deviation and simultaneously to minimize a measurement of the quadratic performance for reducing state transient estimation error. Two given examples illustrate the effectiveness of the presented new low-sensitivity design approach on state estimation performance. From the given examples, it shows that the estimated state errors are not sensitive to system parameter deviation and have the asymptotical convergence property. Besides, the performances are apparently superior to those without considering low-sensitivity design means.

Filtered generalized iterative parameter identification for equation‐error autoregressive models based on the filtering identification idea

SummaryBy using the collected batch data and the iterative search, and based on the filtering identification idea, this article investigates and proposes a filtered multi‐innovation generalized projection‐based iterative identification method, a filtered generalized gradient‐based iterative identification method, a filtered generalized least squares‐based iterative identification method, a filtered multi‐innovation generalized gradient‐based iterative identification method and a filtered multi‐innovation generalized least squares‐based iterative identification method for equation‐error autoregressive systems described by the equation‐error autoregressive models. These filtered generalized iterative identification methods can be extended to other linear and nonlinear scalar and multivariable stochastic systems with colored noises.

Optimal experiment design for multivariable system identification using simultaneous excitation

Having an accurate model of a system is essential for many applications today, especially those related to advanced process control (APC). When executing an industrial delivery project of APC, often significant time is spent performing experiments on the real process to identify a model. By designing higher quality experiments the time needed on site carrying out experiments can be reduced, saving both resources and engineering efforts. This paper investigates optimal input design to minimize the time needed to identify a linear time-invariant multi-variable system fulfilling certain requirements on the model accuracy. This is an extension of previous work which focused on sequential excitation on one input signal at the time. Here, however, the effects of using combined simultaneous and sequential excitation is investigated. The resulting non-convex optimization problem is relaxed using binary decision variables. An important feature of the approach is that the experiment is carried out in closed-loop using a model predictive controller with zone constraints to further guarantee that the output constraints are not violated. Simulation results indicate that there are many cases where using combined simultaneous and sequential excitation outperforms the previous sequential approach.

A kernel-based approach to errors-in-variables identification of stable multivariable linear systems

A kernel-based approach to errors-in-variables identification of stable multivariable linear systems

A Scalable Optimization Approach for the Multilinear System Arising from Scattered Data Interpolation

A Scalable Optimization Approach for the Multilinear System Arising from Scattered Data Interpolation

Richardson Iterative Method for Solving Multi-Linear System with M-Tensor

In this paper, Richardson iterative method is employed to solve M-Equation. In order to guarantee the solution can be found, convergence theorems are established and confirmed numerically. The optimal α, which is a parameter of Richardson iterative method that can provide the best convergence rate, is also determined theoretically and numerically. Furthermore, a theorem establishing the range of initial vector for general splitting methods is extended from the range in past study. To further accelerate the convergence rate, Anderson accelerator and three preconditioners are incorporated into Richardson iterative method. Numerical results reveal that by including these accelerators, the convergence rates are enhanced. Finally, we show that Richardson iterative methods with optimal α perform better than the SOR type methods in past studies in terms of number of iterative steps and CPU time.

Improving the Gauss–Seidel iterative method for solving multi-linear systems with $$\mathcal {M}$$-tensors

Improving the Gauss–Seidel iterative method for solving multi-linear systems with $$\mathcal {M}$$-tensors

On proper and stable stabilising controllers for a class of linear multivariable systems

In this paper, we examine the problems of existence and computation of proper and stable stabilising compensators in the feedback path of a special class of linear time-invariant multivariable systems characterised by a strictly proper transfer function matrix. For the class of systems that are ‘square’, i.e. have the same number of inputs and outputs and have no zeros in the closed right half complex plane and the class of ‘non-square’ systems that include a ‘square’ subsystem that has no zeros in the closed right half complex plane we present a sufficient condition for the existence and computation of proper stabilising (pole placing) compensators which in certain cases that depend on the transfer function matrix of the system can be chosen to be stable.

Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification

Nonlinear (NL) and multilinear (ML) systems play a fundamental role in engineering and science. Over the last two decades, active research has been carried out on exploiting the intrinsically multilinear structure of input–output signals and/or models in order to develop more efficient identification algorithms. This has been achieved using the notion of tensors, which are the central objects in multilinear algebra, giving rise to tensor-based approaches. The aim of this paper is to review such approaches for modeling and identifying NL and ML systems using input–output data, with a reminder of the tensor operations and decompositions needed to render the presentation as self-contained as possible. In the case of NL systems, two families of models are considered: the Volterra models and block-oriented ones. Volterra models, frequently used in numerous fields of application, have the drawback to be characterized by a huge number of coefficients contained in the so-called Volterra kernels, making their identification difficult. In order to reduce this parametric complexity, we show how Volterra systems can be represented by expanding high-order kernels using the parallel factor (PARAFAC) decomposition or generalized orthogonal basis (GOB) functions, which leads to the so-called Volterra–PARAFAC, and Volterra–GOB models, respectively. The extended Kalman filter (EKF) is presented to estimate the parameters of a Volterra–PARAFAC model. Another approach to reduce the parametric complexity consists in using block-oriented models such as those of Wiener, Hammerstein and Wiener–Hammerstein. With the purpose of estimating the parameters of such models, we show how the Volterra kernels associated with these models can be written under the form of structured tensor decompositions. In the last part of the paper, the notion of tensor systems is introduced using the Einstein product of tensors. Discrete-time memoryless tensor-input tensor-output (TITO) systems are defined by means of a relation between an Nth-order tensor of input signals and a Pth-order tensor of output signals via a (P+N)th-order transfer tensor. Such systems generalize the standard memoryless multi-input multi-output (MIMO) system to the case where input and output data define tensors of order higher than two. The case of a TISO system is then considered assuming the system transfer is a rank-one Nth-order tensor viewed as a global multilinear impulse response (IR) whose parameters are estimated using the weighted least-squares (WLS) method. A closed-form solution is proposed for estimating each individual IR associated with each mode-n subsystem.

Iterative refinement method by higher-order singular value decomposition for solving multi-linear systems

In this paper, the higher-order singular value decomposition and the LU decomposition are applied to solve the general multi-linear systems, rather than special structure like M−tensor. Here, we transform the general tensor system into a form with a special structure. In addition, an iterative refinement method is constructed by the information of the coefficient tensor itself, the higher-order singular value, instead of the spectra of iterative tensor. Finally, the numerical experimental results are given to demonstrate the efficiency of the iterative refinement method.

Stabilization of the Pendubot: a polynomial matrix approach

Abstract This paper concerns the stabilization problem for an underactuated robot called the Pendubot. Relying on a computational algorithm which is based on various results of the ‘polynomial matrix approach’, we propose an output-feedback-based internally stabilizing controller to stabilize the Pendubot at the unstable vertical upright position. The algorithm utilizes results for the solution of polynomial matrix Diophantine equations required for the computation and parameterization of proper ‘denominator assigning’ and internally stabilizing controllers for linear time invariant multivariable systems and reduces the problem to that of the solution of a set of numerical linear equations. The controller presented uses only the measured output which consists of the angles of the two links and does not require knowledge of the angular velocities which are usually not directly measurable. Comparative simulations are carried out to verify the good performance of the proposed controller. Finally, experimental results are provided to demonstrate the validity and feasibility of the proposed method.

On the invertibility indices and the Morse normal form for linear multivariable systems

In this paper, by extending the controllability and Brunovsky indices for a matrix pair to a matrix quadruple, three kinds of indices, say, the invertibility indices, the left invertibility indices, and the right invertibility indices, for linear time-invariant systems are introduced and studied. Based on properties of these indices, a neat expression for the Morse lists of a linear system is given without transforming it into its Morse normal form, which is a deep and fundamental result in linear systems theory, and takes an important role in many analysis and design problems for linear systems. Furthermore, the Morse normal form with only algebraic equivalence transformation is also investigated and is simplified.

Optimized reduced order ARX modelling and robust control for multivariable uncertain systems

This paper deals with modelling and robust control of linear multivariable uncertain systems based on the new linear multivariable ARX (Auto-Regressive with eXogenous input)-Laguerre model and the loop-shaping design procedure with the relative gain array theory. In fact, to guarantee significant parameter number reduction of the multivariable ARX-Laguerre model compared with the classic multivariable ARX model, we propose the optimization of the Laguerre poles of the model using the genetic algorithm where the Fourier coefficients are identified using the recursive least square method. Later, the optimized multivariable ARX-Laguerre model is exploited to propose a robust control algorithm for linear multivariable uncertain systems. Indeed, we propose to combine the loop-shaping design procedure approach and the relative gain array theory to develop a simplified final robust controller guaranteeing reference tracking and robust stability against parametric uncertainties. This proposition aims for the simplification of the control’s scheme by adjusting the weighting matrix and the final robust controller calculated from the loop-shaping design procedure for a simpler and more experimentally applicable control scheme. The identification of Fourier coefficients and Laguerre poles as well as the robust multivariable control algorithm are experimentally validated on a laboratory coupled two-tank system showing good results in terms of modelling and attending desired control performances.

Filtered auxiliary model recursive generalized extended parameter estimation methods for Box–Jenkins systems by means of the filtering identification idea

AbstractFor equation‐error autoregressive moving average systems, that is, Box–Jenkins systems, this paper presents a filtered auxiliary model generalized extended stochastic gradient identification method, a filtered auxiliary model multi‐innovation generalized extended stochastic gradient identification method, a filtered auxiliary model recursive generalized extended gradient identification method, a filtered auxiliary model multi‐innovation recursive generalized extended gradient identification method, a filtered auxiliary model recursive generalized extended least squares identification method, and a filtered auxiliary model multi‐innovation recursive generalized extended least squares identification method by using the filtering identification idea and the auxiliary model identification idea. The proposed filtered auxiliary model recursive generalized extended identification methods can be generalized to other linear and nonlinear multivariable stochastic systems with colored noises.

A new preconditioner for Gauss–Seidel method for solving multi-linear systems

A new preconditioner for Gauss–Seidel method for solving multi-linear systems