Invasive weed optimization for model order reduction of linear MIMO systems

In this work, a model order reduction (MOR) technique for a linear multivariable system is proposed using invasive weed optimization (IWO). This technique is applied with the combined advantages of retaining the dominant poles and the error minimization. The state space matrices of the reduced order system are chosen such that the dominant eigenvalues of the full order system are unchanged. The other system parameters are chosen using the invasive weed optimization with objective function to minimize the mean squared errors between the outputs of the full order system and the outputs of the reduced order model when the inputs are unit step. The proposed algorithm has been applied successfully, a 10th order Multiple-Input–Multiple-Output (MIMO) linear model for a practical power system was reduced to a 3rd order and compared with recently published work.     

Two-sided model order reduction for two-directional difference system

This paper defines a two-directional difference system and constructs the projection matrix. Then the original system is projected into the smaller system, and we discuss its moment-matching properties. Next we define the dual system, and discuss the dual relation between the dual system and the original system. Then we can construct the projection matrix with the above mentioned dual relation, and project the dual system into the respectively smaller system, hence derive the moment-matching properties. Finally synthesizing the above two moment-matching properties we obtain the main results that the number of moments matched is twice as much as the number of the generating terms of the constructed projection subspace. We apply this result to the two-sided model order reduction for parameter time delay system, and obtain the result that the reduced system can preserve twice moments as the number of the generating terms of the constructed projection subspace. Finally we derive an algorithm to compute the basis of the subspace involved in the reduction process.     

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

Oscillation theorems for certain second order self-adjoint matrix differential systems

By means of generalized averaging pair technique and Riccati transformation method, oscillation criteria for self-adjoint differential matrix system of the form

     

Multiple periodic solutions for second order systems with changing sign potential

This paper deals with the multiplicity of solutions of a second order nonautonomous system. We extend a previous result of the author relaxing the assumptions on the sign of the potential.     

On second order degree of graphs

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v.In this paper we address the following question:What are the sufficient conditions for a graph to have a vertex v such that d2(v) ≥ d(v),where d(v) denotes the degree of v? Among other results,every graph of minimum degree exactly 2,except four graphs,is shown to have a vertex of second order degree as large as its own degree.Moreover,every K-4-free graph or every maximal planar graph is shown to have a vertex v such that d2(v) ≥ d(v).Other sufficient conditions on graphs for guaranteeing this property are also proved.     

Existence and decay of solutions of an abstract second order nonlinear problem

In this paper we consider the following nonlinear evolution problem with damping:

(∗)     

On sufficient second order optimality conditions in multiobjective optimization

A second order sufficient optimality criterion is presented for a multiobjective problem subject to a constraint given just as a set. To this aim, we first refine known necessary conditions in such a way that the sufficient ones differ by the replacement of inequalities by strict inequalities. Furthermore, we show that no relationship holds between this criterion and a sufficient multipliers rule, when the constraint is described by inequalities and equalities. Finally, improvements of this criterion for the unconstrained case are presented, stressing the differences with single-objective optimization     

Rotating periodic solutions for super-linear second order Hamiltonian systems

In this paper we consider a class of super-linear second order Hamiltonian systems. We use Morse theory to obtain the existence and multiplicity of rotating periodic solutions, which might be periodic, subharmonic or quasi-periodic ones.     

Order reduction and determination of dominant state variables of nonlinear systems

The method presented can simplify nonlinear system models by reducing the number of state equations. Starting from a special state space representation, the main idea is to take over all nonlinear terms into the reduced system and to renew all couplings of state variables, input variables and nonlinear functions. The steady state performance can be influenced by additional measures which are discussed in detail and which are illustrated by a technical example. A dominance analysis is introduced which helps choosing the system order and the dominant state variables. All computations are based on proven algorithms and most of them are free of iterations.     

On existence of oscillatory solutions of second order Emden-Fowler equations

We study the second order Emden-Fowler equation

(E)     

A connection between time domain model order reduction and moment matching for LTI systems

We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1 Y.L. Jiang and H.B. Chen, Time domain model order reduction of general orthogonal polynomials for linear input-output systems, IEEE Trans. Autom. Control 57 (2012), pp. 330–343. doi:10.1109/TAC.2011.2161839[Crossref], [Web of Science ®] , [Google Scholar]] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time-domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system.     

Notes on periodic solutions of subquadratic second order systems

Some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory.     

On the regularity of second order cone programs and an application to solving large scale problems

We prove the nonsingularity of the standard primal–dual system for second order cone programs assuming Slater’s condition, uniqueness and strict complementarity. This result is applied to the analysis of the augmented primal–dual method for solving linear programs over second order cones.     

Subharmonic solutions for nonautonomous sublinear second order Hamiltonian systems

Some existence theorems are obtained for subharmonic solutions of nonautonomous second order Hamiltonian systems by the minimax methods in critical point theory.     

Transformation of high order linear differential-algebraic systems to first order

We study general nonsquare linear systems of differential-algebraic systems of arbitrary order. We analyze the classical procedure of turning the system into a first order system and demonstrate that this approach may lead to different solvability results and smoothness requirements. We present several examples that demonstrate this phenomenon and then derive existence and uniqueness results for differential-algebraic systems of arbitrary order and index. We use these results to identify exactly those variables for which the order reduction to first order does not lead to extra smoothness requirements and demonstrate the effects of this new formulation with a numerical example.Dedicated to Richard S. Varga on the occasion of his 77th birthday.     

Existence of homoclinic solution for the second order Hamiltonian systems

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systems , ∀t∈R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t∈R, and W(t,x) satisfies the superquadratic condition W(t,x)/|x|²→+∞ as |x|→∞ uniformly in t, and need not satisfy the global Ambrosetti-Rabinowitz condition.