Linear Multivariable Systems Research Articles

DC Motor Speed Control through Parallel DC/DC Buck Converters

This article proposes a multivariable robust controller for the regulation of DC motor angular speed, and for the active current sharing in each parallel DC/DC buck converter. Considering that the parallel DC/DC buck converter is connected in cascade to the armature of the DC motor. Additionally, the controller rejects actively the internal and external disturbances, which is subjected the multivariable linear system. These disturbances are considered time-varying, but they are bounded. The control law is based on differential flatness and active disturbance rejection control with GPI observers. Computer simulations using Matlab/Simulink verified the effectiveness of the proposed controller.

Damage detection of a truss bridge utilizing a damage indicator from multivariate autoregressive model

This study proposes a damage indicator automatically derived from a set of multivariate autoregressive models estimated from ambient vibration of bridges. The damage indicator evaluates a stochastic distance between a set of reference data (healthy bridge data) and unknown test data. Statistical hypothesis testing based on a probability distribution of the damage indicator was applied for damage detection. A field experiment conducted on an actual steel truss bridge with truss members that were artificially severed was conducted to assess the damage detection efficacy of the proposed damage indicator. Experimentally obtained results showed that the proposed damage indicator enables detection of three damage patterns. The proposed damage indicator effectiveness was also assessed by comparison to the damage sensitive feature from a univariate autoregressive model using experimental data of the same bridge. This comparison also demonstrated the efficacy of the proposed damage indicator obtained from a multivariate linear system.

Self-terminating growth of platinum by electrochemical deposition

In this paper, an islanded microgrid is modelled as a linear multivariable dynamic system. Then, the multivariable analysis tools are employed. The generalized Nyquist diagram and the relative gain array are used respectively for the stability assessment and solving the paring problem among the inputs and outputs. Droop control dependency on the X/R ratio of the microgrid DERs is recognized and its type is proposed using the relative gain array concept. Robust stability, nominal performance and robust performance requirements are evaluated in order to a better understanding of the system dynamics. Finally, three different controllers including H∞, H2 and sequential proportional-integral-derivative controls are designed and compared.

Recursive Subspace Identification Algorithm using the Propagator Based Method

<p>Subspace model identification (SMI) method is the effective method in identifying dynamic state space linear multivariable systems and it can be obtained directly from the input and output data. Basically, subspace identifications are based on algorithms from numerical algebras which are the QR decomposition and Singular Value Decomposition (SVD). In industrial applications, it is essential to have online recursive subspace algorithms for model identification where the parameters can vary in time. However, because of the SVD computational complexity that involved in the algorithm, the classical SMI algorithms are not suitable for online application. Hence, it is essential to discover the alternative algorithms in order to apply the concept of subspace identification recursively. In this paper, the recursive subspace identification algorithm based on the propagator method which avoids the SVD computation is proposed. The output from Numerical Subspace State Space System Identification (N4SID) and Multivariable Output Error State Space (MOESP) methods are also included in this paper.</p>

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (ℳ) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (ℳ) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

A homotopy method for solving multilinear systems with M-tensors

Multilinear systems of equations arise in various applications, such as numerical partial differential equations, data mining, and tensor complementarity problems. In this paper, we propose a homotopy method for finding the unique positive solution to a multilinear system with a nonsingular M-tensor and a positive right side vector. We analyze the method and prove its convergence to the desired solution. We report some numerical results based on an implementation of the proposed method using a prediction–correction approach for path following.

An estimation approach for linear stochastic systems based on characteristic functions

This paper presents an alternative, characteristic function based approach for the Bayesian design of estimators for dynamic linear systems and linear detection problems. For a measurement update, the a posteriori characteristic function of the unnormalized conditional probability density function (ucpdf) of the state given the measurement history is obtained as a convolution of the a priori characteristic function of the ucpdf with the characteristic function of the measurement noise. It is shown that this convolution holds for a general measurement structure. Time propagation involves the product of the updated characteristic function of the ucpdf and the characteristic function of the process noise. Some estimation problems are found to be naturally tractable using only characteristic functions, such as the multivariable linear system with additive Cauchy measurement and process noise. It is shown that even the derivation of the Kalman filter algorithm has advantages when formulated using the characteristic function approach. Finally, in some instances the estimation problem can only be formulated in terms of characteristic functions. This is illustrated by a one-update scalar example for symmetric-α-stable distributions.

Geometric structure and properties of linear time invariant multivariable systems in the controller canonical form

In this paper, we analyse some fundamental structural properties of linear time-invariant multivariable systems in the controller canonical form and present a direct method for the computation of bases and associated friends for output-nulling, input-containing and reachability subspaces in terms of the parameters of the system and the invariant zero structure, both in the nondefective and in the defective case. Using this analysis, it is possible to express the solvability conditions of important control and estimation problems in terms of easily checkable conditions on the system matrices.

Polynomial matrix equivalences: system transformations and structural invariants

Abstract The present article is a survey on linear multivariable systems equivalences. We attempt a review of the most significant forms of system equivalence having as a starting point matrix transformations preserving certain aspects of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.

Further results on generalized inverses of tensors via the Einstein product

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.

Robust Multivariable Microgrid Control Synthesis and Analysis

In this paper, an islanded microgrid is modelled as a linear multivariable dynamic system. Then, the multivariable analysis tools are employed. The generalized Nyquist diagram and the relative gain array are used respectively for the stability assessment and solving the paring problem among the inputs and outputs. Droop control dependency on the X/R ratio of the microgrid DERs is recognized and its type is proposed using the relative gain array concept. Robust stability, nominal performance and robust performance requirements are evaluated in order to a better understanding of the system dynamics. Finally, three different controllers including H∞, H2 and sequential proportional-integral-derivative controls are designed and compared.

Semi-global output regulation for discrete-time singular linear systems with input saturation via composite nonlinear feedback control

The semi-global output regulation problem of multi-variable discrete-time singular linear systems with input saturation is investigated in this paper. A composite nonlinear feedback control law is constructed by using a low gain feedback technique for semi-global stabilisation of discrete-time singular linear systems with input saturation. The sufficient solvability conditions of the semi-global output regulation problem by composite nonlinear feedback control are established. When the composite nonlinear feedback control law is reduced to a linear control law, the solvability conditions are an exact discrete-time counterpart of the semi-global output regulation problem of continuous-time singular linear systems. With the extra control freedom of the nonlinear part in the composite nonlinear feedback control law, the transient performance of the closed-loop system can be improved by carefully choosing the linear feedback gain and the nonlinear feedback gain. The design procedure of the composite nonlinear feedback control law and the improvement of the transient performance are illustrated by a numerical example.

Proper representations of (multivariate) linear differential systems

A proper representation of a linear differential system is a representation with no singularity at infinity. It is shown that such a representation always exists. It turns out that for proper representations having minimal number of rows is equivalent to having minimal total row degree. One is led therefore to a natural definition of the notion of minimality. What is remarkable is that a minimal proper representation is uniquely determined up to premultiplication by a unimodular polynomial matrix of special form. This uniqueness result allows, in particular, to introduce important integer invariants.

Robust fuzzy stabilisation of interval plants

ABSTRACTThe paper deals with the problem of stabilisation of interval systems. To this end, by using Takagi–Sugeno fuzzy mechanism, a Mamdani-type PID-like fuzzy controller is modified and extended to develop a new PID-like Takagi–Sugeno fuzzy stabilising controller for the plant described by an interval system. Indeed, a PID-like Takagi–Sugeno fuzzy controller and an interval plant are considered in the forward path of a unity feedback system, and parameters in Takagi–Sugeno fuzzy controller are determined so that the stability of the closed-loop system is assured. The closed-loop system has a multilinear uncertainty structure. Therefore, based on the Zero Exclusion Condition for multilinear uncertain systems, a new theorem presenting sufficient conditions for the Takagi–Sugeno fuzzy controller to be robust stability guaranteed is also derived. An example is given to illustrate the application and the effectiveness of the proposed controller.

Solving Multi-linear Systems with $$\mathcal {M}$$ M -Tensors

This paper is concerned with solving some structured multi-linear systems, especially focusing on the equations whose coefficient tensors are $$\mathcal {M}$$M-tensors, or called $$\mathcal {M}$$M-equations for short. We prove that a nonsingular $$\mathcal {M}$$M-equation with a positive right-hand side always has a unique positive solution. Several iterative algorithms are proposed for solving multi-linear nonsingular $$\mathcal {M}$$M-equations, generalizing the classical iterative methods and the Newton method for linear systems. Furthermore, we apply the $$\mathcal {M}$$M-equations to some nonlinear differential equations and the inverse iteration for spectral radii of nonnegative tensors.

Standardizability of multivariable linear sampled-data systems with delay

Due to periodic sampling, linear sampled-data systems are a subclass of linear continuous periodic systems, even in the case, when all other elements are time invariant. Therefore, the well-established methods for LTI systems cannot be applied. Owing to the great practical importance of sampled-data control systems, various approaches for the rigorous description of those systems are known. Moreover, a lot of methods have been developed, that are able to yield rigorous solutions for numerous control and optimization problems. The application of those methods need representations in certain standard forms. Often, it is not clear, whether a given system can be transformed into a standard form or not. The paper considers this question for the standard sampled-data system with delay, for which numerous methods and tools are available, if the system belongs to the subclass with model structure. The paper provides necessary and sufficient conditions for a SD system with delay in standard structure to belong to this important subclass. The proof is given on basis of the parametric transfer matrix concept, which is at present the only one that can handle problems, where both difficulties -sampling and arbitrary time-delay - occur at the same time.

Application and Comparative Analysis of Various Classical and Soft Computing Techniques for Model Reduction of MIMO Systems

Model reduction techniques are simplification methods based on mathematical approaches employed to realize reduced models for the original high order systems. Some existing classical model reduction techniques for multivariable system are considered and compared for their performances. Interlacing property and coefficients matching (IPCM) method gives overall minimum integral square error (ISE), integral absolute error (IAE) and integral time absolute error (ITAE) values compared to other methods. Though the IPCM method is efficient, it may not guarantee for minimization of all objective functions simultaneously. In this paper, model reduction approach based on objectives like ISE, IAE and ITAE using multi-objective differential evolution (MODE) method is proposed for reducing the numerator and the denominator is reduced by interlacing property. MODE method minimizes the small, normal and large errors persisting for long time between original and reduced models. This multi-objective approach is applied for model reduction of 10th order multivariable linear time invariant power system model. Simulation results are demonstrated for single and multi-objective model reduction and compared with multi-objective particle swarm optimization (MOPSO) method to prove the validity of proposed MODE technique.

Moore–Penrose inverse of tensors via Einstein product

In this paper, we define the Moore–Penrose inverse of tensors with the Einstein product, and the explicit formulas of the Moore–Penrose inverse of some block tensors are obtained. The general solutions of some multilinear systems are given and we also give the minimum-norm least-square solution of some multilinear systems using the Moore–Penrose inverse of tensors.

Feedback Decoupling of Linear Multivariable Systems

In this paper, the decoupling problem of linear multivariable systems using a static state feedback is considered. Two main contributions related to this problem are presented. The first one is a complete characterization of the decoupled closed-loop structure of a decouplable square linear system, along with the necessary and sufficient conditions for the internal stability of the decoupled closed-loop system. The second contribution is a result presenting necessary and sufficient conditions for a right invertible linear system to be decouplable with a desirable infinite structure using nonregular state feedback. These conditions are stated in terms of the row image of two real matrices, which are obtained using the extended interactor of the system and the desirable infinity structure of the closed-loop system. A procedure is provided to find a realizable full column rank compensator that solves the problem. Given a system satisfying these conditions, it is shown how to obtain a non regular static state feedback that decouples the system.

H ∞-approach to controller synthesis under parametric uncertainty and polyharmonic external disturbances

We consider the robust stabilization problem for linear multivariable systems whose physical parameters may deviate from computed (nominal) in some known bounds, and the control object is subject to non-measurable polyharmonic external disturbances (with unknown amplitudes and frequencies) bounded in power. We pose the problem of synthesizing a controller that guarantees robust stability of the closed-loop system and additionally ensures given errors with respect to controlled variables in the established nominal mode. The solution of this problem is based on the technique of opening the object---controller system with respect to varying object parameters and can be reduced to a standard H?-optimization procedure, while the necessary accuracy is achieved by choosing the weight matrix for controlled object variables. We show the solution for a well-known benchmark problem.