Solution Of Nonlinear Matrix Equation Research Articles

On the Positive Definite Solutions of Nonlinear Matrix Equation

On the Positive Definite Solutions of Nonlinear Matrix Equation

Sub-optimal automatic generation control of interconnected power system using constrained feedback control strategy

This paper presents sub-optimal AGC regulator designs for a 2-area interconnected power system using constrained feedback control strategy. The power system with identical thermal plants consisting of non-reheat thermal turbines interconnected via parallel AC/DC links is considered for the investigations. The formulation of optimal AGC problem involves solution of non-linear matrix equations, which requires an appropriate computational technique to yield a solution with high speed and accuracy. The implementation of optimal AGC regulator requires monitoring of all the state variables of the system or state reconstruction, which may be undesirable from cost and complexity considerations. Due to these limitations of optimal AGC regulators, sub-optimal AGC regulators are designed based on the constrained feedback control strategy using the feedback of system states which are accessible and available for measurement. Different structures of state cost weighting matrix ‘Q’ are also considered in the study. The power system dynamic performance is analyzed based on time response plots achieved with the implementation of designed optimal and sub-optimal AGC regulators in the wake of 1% load disturbance in one of the areas.

Numerical solution of nonlinear matrix equations arising from Green’s function calculations in nano research

The Green’s function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form X+(C∗+iηD∗)X−1(C+iηD)=R+iηP, where R and P are Hermitian, P+λD∗+λ−1D is positive definite for all λ on the unit circle, and η→0+. For each fixed η>0, we show that the required solution is the unique stabilizing solution Xη. Then X∗=limη→0+Xη is a particular weakly stabilizing solution of the matrix equation X+C∗X−1C=R. In nano applications, the matrices R and C are dependent on a parameter, which is the system energy E. In practice one is mainly interested in those values of E for which the equation X+C∗X−1C=R has no stabilizing solutions or, equivalently, the quadratic matrix polynomial P(λ)=λ2C∗−λR+C has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute Xη efficiently even for very small values of η, thus providing good approximations to X∗. We also explain how the solution X∗ can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of P(λ) should be included in the computation. In some applications the matrices C,D,R,P have very special sparsity structures. We show how these special structures can be exploited to drastically reduce the complexity of the doubling algorithm for computing Xη.

On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation Xs+A∗X−t1A+B∗X−t2B=Q

Nonlinear matrix equation has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation are considered, where Q is a Hermitian positive definite matrix, A, B are nonsingular complex matrices, s is a positive number, and 0 < ti ≤ 1, i = 1, 2. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method.

On positive definite solutions of nonlinear matrix equation [formula omitted

In this paper, the nonlinear matrix equation X s - A ∗ X - t A = Q is investigated. Based on the fixed-point theory, the existence and the uniqueness of the positive definite solution are studied. An effective iterative method to obtain the unique positive definite solution is established given ‖ A ‖ · ‖ Q - 1 ‖ s + t s < s t . In addition, some computable estimates of the unique positive definite solution are derived. Finally, numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates.

A BMAP|G|1-analysis based on convolution calculus

Stochastic models of high-speed communication systems typically require the analysis of special Markov additive processes, known thus far as “batch Markovian arrival processes” (BMAPs). In most cases, the solution of nonlinear matrix equations of the form\(G = \sum\nolimits_{n = 0}^\infty {A_n G^n } \) or\(R = \sum\nolimits_{n = 0}^\infty {R^n A_n } \) is necessary, leading, in general, to severe numerical problems. Several efficient algorithms for that purpose have been reported in the literature. In this contribution, we present a new approach (the BMAP|G| 1-analysis), which is based on a convolution calculus for matrix sequences. The main results are the unified representation of Poisson, compound Poisson, and batch Markovian arrival processes by so-called convolutional exponential distributions, and the computation of the fundamental-period matrix G via so-called semi-convolutions. The resulting algorithms, in general, cannot outperform known algorithms as, for instance, those exploiting Horner schemes as proposed by D. M. Lucantoni and other authors, or the highly efficient cyclic reduction algorithm of D. Bini and B. Meini; nevertheless, they allow the direct computation of step matrices such as the “arrival matrices” An with\(A = \sum\nolimits_{n = 0}^\infty {A_n } \) and the component matrices Gn summing to G. Further, there is some hope that the proposed framework may lead to a unification of the theory in the sense that queues with arrival and/or service processes that are characterized by convolutional exponential distributions may possess formally similar properties, as known from the scalar case.

The Properties of an Output Feedback Scheme and Control System Design. Approach from Low-Order Model.

The optimal regulator is a key device in modern linear control theory. This paper deals with a problem of computation of the output feedback gain which minimizes the quadratic performance criterion. Numerous methods of computation of the output feedback have been presented in other reports, but they all require iterative calculation. However, use of such iterative algorithms requires solution of nonlinear matrix equations. Furthermore, the conditions for the existence of the output feedback gain which satisfies an algebraic equation are unclear. In this paper, a low-order model for computation of the feedback gain is proposed. This method is based on the second method of Lyapunov. The computational method for the output feedback gain and comparison of the Lyapunov functions of the original system and the low-order model are presented. This computational procedure does not involve use of any iterative algorithms. We have identified the conditions for the existence of the output feedback gain and the sufficient conditions for attainment of the optimal solution.

An efficient computational method for solving nonlinear matrix equation and its application in queuing analysis

The matrix analytic analysis of queues with complex arrival, vacation and service characteristics requires the solution of nonlinear matrix equation. The complexity and large dimensionality of the model require an efficient and smart algorithm for the solution. In this paper, we propose an efficient Adaptive Newton-Kantorovich (ANK) method for speeding up the algorithm solving the nonlinear matrix equation which is an inevitable step in the analysis of the queue with embedded Markov chain such as BMAP/SMSP/1/∞ queue or its discrete version. BMAP/SMSP/1/∞ is a queuing model with a Semi Markov Service time Process (SMSP) and a Batch Markovian Arrival Process (BMAP). The numerical result is presented for the discrete case of N-MMBP/D/1 queue which arises in analyzing traffic aspect of computer communication network, where MMBP is Markov Modulated Bermoulli Process. The comparisons of Adaptive Newton-Kantorovich (ANK) with Modified Newton-Kantorovich (MNK) show that ANK saves 30% of CPU time when the number of userN is 50.

Experimental results on matrix-analytical solution techniques–extensions and comparisons

The matrix-analytic analysis of many stochastic models requires the solution of nonlinear matrix equations. In this paper, several iterative algorithms for solving these equations are reviewed and insights gained on their convergence behavior are reported. An extension to these algorithms based on linear extrapolation is proposed with a view towards extending the computational feasibility of the matrix-analytic techniques to even larger classes of problems. Although this proposed extension is a minor addition to the existing algorithms, it has been observed to considerably improve their efficiency especially when the convergence rates are slow. This is illustrated through several examples, for which the CPU requirements of the reviewed algorithms are compared. Finally, the matrix-geometric approach is compared to the M/G/1 approach through specific examples and the conclusions are summarized

Performance of direct and iterative algorithms on an optical systolic processor

The frequency-multiplexed optical linear algebra processor (OLAP) is treated in detail with attention to its performance in the solution of systems of linear algebraic equations (LAEs). General guidelines suitable for most OLAPs, including digital-optical processors, are advanced concerning system and component error source models, guidelines for appropriate use of direct and iterative algorithms, the dominant error sources, and the effect of multiple simultaneous error sources. Specific results are advanced on the quantitative performance of both direct and iterative algorithms in the solution of systems of LAEs and in the solution of nonlinear matrix equations. Acoustic attenuation is found to dominate iterative algorithms and detector noise to dominate direct algorithms. The effect of multiple spatial errors is found to be additive. A theoretical expression for the amount of acoustic attenuation allowed is advanced and verified. Simulations and experimental data are included.

Computation of optimal output feedback gains for linear multivariable systems

For linear systems with quadratic performance indices, it is shown that the optimal output feedback gains can be computed using gradient techniques. Unlike previous algorithms, this approach avoids the solution of nonlinear matrix equations while appearing to ensure convergence. Computational results for a fourth-order system are presented.