A useful identity in circuit theory

A useful identity expressing the derivative of an unknown variable x_k of X with respect to an entry in the coefficient matrix of a linear system AX = B is presented. If the derivatives of x_k with respect to each entry of A or their combinations are required, then the identity avoids the repeated solution of the linear equations, and may result in a symbolic solution provided A^-1 is known. A method to measure the sensitivity in an n-port linear system is proposed, and its relationships to Kron's method of tearing and Branin's formulae are also discussed.     

Hybrid Analysis of a Large-scale Network by Node-tearing

A new hybrid analysis technique for large-scale networks by node-tearing is presented. The interconnection function matrix is defined and the formulation for computing the transfer function matrix W, the general return difference matrix F_K(X) and the general null return difference matrix F̂_K(X) of a large-scale network is given. In the overall computational process, all the hybrid port parameters are derived from the indefinite-admittance matrix and/or the indefinite-impedance matrix of the component subnetworks.     

A new approach to the design of coupling networks

Mathematical results are derived, which enable one to find a vector of parameters k⁰ such that (P₁(s,k⁰)?H)∩(P₂(k⁰)=0), where P₁(s,k) is a polynomial in s and in the components of k,P₂(k) is a polynomial in the components of k, and H is the set of Hurwitz polynomials. The algorithm is based on an extension of the root locus technique to the multiparameter case. The design problem of coupling networks between a resistive generator and a passive load, under prescribed power gain characteristics, is translated into the above formulation. A numerical example is provided.     

H∞ control for network-based systems with time-varying delay and packet disordering

The H_∞ control problem is investigated in this paper for a class of networked control systems (NCS) with time-varying delay and packet disordering. A new model is proposed to describe the packet disordering phenomenon and then converted into a parameter-uncertain system with multi-step delay. Based on the obtained system model, a sufficient condition for robust stability of the NCS is derived. Furthermore, an optimization problem with linear matrix inequalities (LMIs) constraints is formulated to design the state feedback H_∞ controller such that the closed-loop NCS is robust stable and has an optimal H_∞ disturbance attenuation level. Finally, two illustrative examples are given to demonstrate the effectiveness of the proposed method.     

The inverse lure problem of optimal control

Given the linear system $\text{x}\text{}$ = Ax - bu, y = c^Tx, it is shown that, for a certain non-quadratic cost functional, the optimal control is given by u_opt(x) = h(c^Tx), where the function h(y) must satisfy the conditions ky²?h(y)y>0 for y≠0, h(0) = 0 and existence of h^-1 everywhere. The linear system considered must satisfy the Popov condition 1/k + (1 +?ωβ) G(?ω)>0 for all ω, G(s) being the y(s)/u(s) transfer function.     

Parameter identification of a class of time-varying systems via orthogonal shifted legendre polynomials

A method of using orthogonal shifted Legendre polynomials for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the repeated integration of the differential equation and the representations of $\text{∫}\text{0}\text{t}$s(τ) dτ = Ps(t) and ts(t) = Rs(t), where P and R are constant matrices and s(t) is a shifted Legendre vector whose elements are shifted Legendre polynomials. The differential input-output equation is converted into a set of overdetermined linear algebraic equations for a least squares solution. The results of simulation studies are included to illustrate the applicability of the method.     

Study on stability and stabilizability of discrete-time mean-field stochastic systems

This paper is to study the mean square stabilizability and regional stability of discrete-time mean-field stochastic systems. Firstly, a necessary and sufficient condition is presented via the spectrum of linear operator to illustrate the stabilizability of discrete-time mean-field stochastic systems. B(0, γ)-stabilizability is introduced and transformed into solving linear matrix inequalities (LMIs). Secondly, B_M-stability is characterized, especially, the stabilities of circular region, sector region and annulus regions are discussed extensively. Finally, as applications, it is shown that B(0, γ₁; γ₂)-stability has close relationship with the decay rate of the system state response and the Lyapunov exponent.     

Orthogonal nonnegative matrix tri-factorization for co-clustering: Multiplicative updates on Stiefel manifolds

Matrix factorization-based methods become popular in dyadic data analysis, where a fundamental problem, for example, is to perform document clustering or co-clustering words and documents given a term-document matrix. Nonnegative matrix tri-factorization (NMTF) emerges as a promising tool for co-clustering, seeking a 3-factor decomposition X≈USV^?$\mathit{X}\approx {\mathit{USV}}^{?}$ with all factor matrices restricted to be nonnegative, i.e., U?0,S?0,V?0.$\mathit{U}?0\text{,}\mathit{S}?0\text{,}\mathit{V}?0\text{.}$ In this paper we develop multiplicative updates for orthogonal NMTF where X≈USV^?$\mathit{X}\approx {\mathit{USV}}^{?}$ is pursued with orthogonality constraints, U^?U=I,${\mathit{U}}^{?}\mathit{U}=\mathit{I}\text{,}$ and V^?V=I${\mathit{V}}^{?}\mathit{V}=\mathit{I}$, exploiting true gradients on Stiefel manifolds. Experiments on various document data sets demonstrate that our method works well for document clustering and is useful in revealing polysemous words via co-clustering words and documents.     

The solution of boundary value problems by multiple Laplace transformations

By use of the multiple Laplace transform a partial differential equation and its associated boundary conditions characterizing a boundary value problem in n independent real variables can be transferred directly into an algebraic equation in n independent complex variables. This algebraic equation can be solved for the multiple transform of the solution of the boundary value problem. Multiple inversion of this transform then gives the desired solution. The general theory underlying such solution of boundary value problems in two and three independent variables is advanced in detail. Use of this theory is illustrated by solution of two specific problems.     

Walsh Operational Matrices for Fractional Calculus and Their Application to Distributed Systems

The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s²+ 1), e^-s and e^-√s etc. are established. Several distributed parameter problems are solved by the new approach.     

An event-triggered asynchronous H∞ filtering for singular Markov jump systems with redundant channels

In this paper, the problem of asynchronous H_∞ filtering for singular Markov jump systems with redundant channels under the event-triggered scheme is studied. In order to save the resource of bandwidth limited network and improve quality of data transmission, we utilize event-triggered scheme and employ redundant channels. The redundant channels are modeled as two mutually independent Bernoulli distributed random variables. To formulate the asynchronization phenomena between the system modes and the filter modes, the hidden Markov model is proposed so that the filtering error system has become a singular hidden Markov jump system. The criterion of regular, causal and stochastically stable with a certain H_∞ performance for the filtering error system has been obtained. The co-design of asynchronous filter and the event-triggered scheme is proposed in terms of a group of feasible linear matrix inequalities. Two examples are given to show the effectiveness of the proposed method.     

Robust state-feedback control design for active suspension system with time-varying input delay and wheelbase preview information

This paper develops a robust state-feedback controller for active suspension system with time-varying input delay and wheelbase preview information in the presence of the parameter uncertainties. By employing system augmentation technique, a multi-objective control optimization model is first established and then this controller design is converted to a static full-state feedback controller design with robust H_∞ and generalized H₂ performance, wherein the model-dependent control gain is evaluated by transforming the related nonlinear matrix inequalities into their corresponding linear matrix inequality forms based on Lyapunov theory, and then LMI (Linear-Matrix-Inequality) technique is applied to solve and obtain the desired controller. A numerical simulation case is finally provided to reveal the effectiveness and advantages of the proposed controller.     

The graph-theoretic field model— II. application of multi-terminal representations to field problems

This paper is a sequel to a paper entitled “The Graph-Theoretic Field Model—I: Modelling and Formulations” (1). Herein, the Theory of Multi-Terminal Representations is applied to the Graph-Theoretic Field Model to provide mathematical models of finite elements. The element models are obtained solely from the algebraic building blocks of the Graph-Theoretic Field Model, without recourse to any functional mathematics. The theory of Multi-Terminal Representations is developed for both linear and non-linear problems. Examples of the application of the theory to one- and two-dimensional field problems are presented from heat conduction and electrostatics.     

An LMI approach to robust H∞ filtering for uncertain systems with time-varying distributed delays

In this paper, the problem of robust H_∞ filtering for uncertain systems with time-varying distributed delays is considered. The uncertainties under discussion are time varying but norm bounded. Based on the Lyapunov stability theory, sufficient condition for the existence of full order H_∞ filters is proposed by linear matrix inequality (LMI) approach such that the filtering error system is asymptotically sable and satisfies a prescribed attenuation level of noise. A numerical example is given to demonstrate the availability of the proposed method.     

On rectangular systems of differential equations and their application to circuit theory

A system of differential equations A(d/dt) x = Bx+f, along with the initial condition x(0) = k, is considered where A and B are m x n matrices. Generalized inverses of the matrix A are used to derive algebraic conditions for the existence and uniqueness of a solution. An example is presented to illustrate application of the results to circuit theory.     

H∞ guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays

This paper investigates the H_∞ guaranteed cost control problem for mode-dependent time-delay jump systems with norm-bounded uncertain parameters. Both distributed delays and input delays appear in the system model. Based on a matrix inequality, a sufficient condition for the existence of robust H_∞ guaranteed cost controller is derived, which stabilizes the considered system and guarantees that both the H_∞ performance level and a cost function have upper bounds for all admissible uncertainties. By the cone complementary linearization approach, the desired state-feedback controller can be constructed. A numerical example is provided to show the effectiveness of the proposed theoretical results.     

H∞ filtering for linear neutral systems with mixed time-varying delays and nonlinear perturbations

In this paper, the problem of H_∞ filtering for neutral systems with mixed time-varying delays and nonlinear perturbations is investigated. Some new delay-dependent sufficient conditions are presented to ensure that the filtering error system is asymptotically stable with a prescribed level of H_∞ noise attenuation. In addition, the design procedures for the existence of such filter are presented in terms of a set of linear matrix inequalities (LMIs). Slack variables and convex combination technique are adopted to reduce the conservatism of obtained results. Finally, three numerical examples are given to illustrate the effectiveness of the proposed method.     

Modelling of linear discrete-time systems using modified cauer continued fraction

An algorithm, amenable for programming on a digital computer, has been presented for the modelling of linear discrete-time systems, as an alternative to the procedure of Shamash (1). The transformations inherent in the procedure are easily accomplished by the synthetic division technique. With the use of modified Cauer form of continued fraction (MCF), the new method matches a set of both the time-moments and Markov parameters of the system and of the model, as in the procedure of Parthasarathy and Singh (2), giving a better approximation to the system response at all times. A distinct feature of the proposed algorithm compared with the earlier methods of discrete system reduction (1),(2), is that a number of reduced-order models are generated simultaneously; this allows scope for better selection in choosing the right model for system analysis and design.     

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

By using a bilinear transformation and some linear algebraic techniques, new matrix bounds of the solution of the continuous algebraic Lyapunov equation (CALE) are derived in this paper. Comparing to existing works, these obtained matrix bounds are less restrictive and are easy to be calculated. A numerical example is also given to demonstrate the merits of the present results.