On Padé-type model order reduction of J-Hermitian linear dynamical systems

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.     

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.     

Interpolatory model reduction of parameterized bilinear dynamical systems

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.     

Moderate deviations for randomly perturbed dynamical systems

A Moderate Deviation Principle is established for random processes arising as small random perturbations of one-dimensional dynamical systems of the form X_n=f(X_n−1). Unlike in the Large Deviations Theory the resulting rate function is independent of the underlying noise distribution, and is always quadratic. This allows one to obtain explicit formulae for the asymptotics of probabilities of the process staying in a small tube around the deterministic system. Using these, explicit formulae for the asymptotics of exit times are obtained. Results are specified for the case when the dynamical system is periodic, and imply stability of such systems. Finally, results are applied to the model of density-dependent branching processes.     

Stochastic equilibria in von Neumann-Gale dynamical systems

This paper examines a class of random dynamical systems related to the classical von Neumann and Gale models of economic dynamics. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. We establish a general existence theorem for equilibrium, which holds under conditions analogous to the standard deterministic ones. Our results answer questions that remained open for more than three decades.

     

A memory-efficient model order reduction for time-delay systems

Michiels et al. (SIAM J. Matrix Anal. Appl. 32(4):1399–1421, 2011) proposed a Krylov-based model order reduction (MOR) method for time-delay systems. In this paper, we present an efficient process, which requires less memory consumption, to accomplish the model reduction. Memory efficiency is achieved by replacing the classical Arnoldi process in the MOR method with a two-level orthogonalization Arnoldi (TOAR) process. The resulting memory requirement is reduced from quadratic dependency of the reduced order to linear dependency. Besides, this TOAR process can also be applied to reduce the original delay system into a reduced-order delay system. Numerical experiments are given to illustrate the feasibility and effectiveness of our method.     

Stochastic stability in some chaotic dynamical systems

For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.With 1 Figure     

Perturbed linear stochastic dynamical systems

This paper is concerned with a class of stochastic differential equations which arises by adding a nonlinear term involving a small parameter δ to the drift coefficient of a linear stochastic system. First, a stochastic representation of the solutions of a certain class of Cauchy problems is studied. Second, a finite time expansion in powers of δ of the expectations of functions is established. Third, the corresponding ergodic expansions are sought with additional assumptions regarding the existence of a unique ergodic measure.     

Formulae of partial reduction for linear systems of first order operator equations

This paper deals with the reduction of non-homogeneous linear systems of first order operator equations with constant coefficients. An equivalent reduced system, consisting of higher order linear operator equations having only one variable and first order linear operator equations in two variables, is obtained by using the rational canonical form.     

Inversion of linear delay dynamical systems

We consider the problem of inversion of a dynamical system, that is, the problem of real-time reconstruction of the unknown input of the system on the basis of measurements of its output. We consider linear systems of functional-differential equations in the case of commensurable delays. We obtain necessary conditions for the invertibility of this class of systems and suggest an inversion algorithm that permits one to obtain an estimate of the unknown input signal with given accuracy.     

Minimal stabilizers for linear dynamical systems

Doklady Mathematics -     

Fuzzy linear dynamical systems over fields

We investigate the problem of formalization of ways of describing fuzzy systems. The concept of fuzzy linearity for the class of dynamical systems is analyzed. Also, we consider some properties of the fuzzy dynamical systems that are linear over fields. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 147–155, 2007.     

Stochastic semidiscretization method: Second moment stability analysis of linear stochastic periodic dynamical systems with delays

In this paper, an efficient numerical approach is presented, which allows the analysis of the moment dynamics, stability, and stationary behavior of linear periodic stochastic delay differential equations. The method leads to a high dimensional stochastic mapping with periodic statistical properties, from which the periodic first and second moment mappings are derived. The application of the method is demonstrated first through the analysis of the stochastic delay Mathieu equation. Then a practical case study, where the effect of spindle speed variation on the stability and the resulting surface quality of turning operations is investigated.     

Stability preserving model reduction for linearly coupled linear time-invariant systems

We develop a stability preserving model reduction method for linearly coupled linear time-invariant (LTI) systems. The method extends the work of Monshizadeh et al. for multi-agent systems with identical LTI agents. They propose using Bounded Real Balanced Truncation to preserve a sufficient condition for stability of the coupled system. Here, we extend this idea to arbitrary linearly coupled LTI systems using the sufficient condition for stability introduced by Reis and Stykel. The model reduction error bounds for this method also follow from results of Reis and Stykel, which allows the adaptive choice of reduced orders. We demonstrate the method on Reis's and Stykel's coupled string-beam example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-time stabilization of impulsive dynamical linear systems

Finite-time stabilization of a special class of hybrid systems, namely impulsive dynamical linear systems (IDLS), is tackled in this paper. IDLS exhibit jumps in the state trajectory which can be either time-driven (time-dependent IDLS) or subordinate to specific state values (state-dependent IDLS). Sufficient conditions for finite-time stabilization of IDLS are provided. Such results require solving feasibility problems which involve Differential–Difference Linear Matrix Inequalities (D/DLMIs), which can be numerically solved in an efficient way, as illustrated by the proposed examples.     

Balanced truncation model order reduction in limited time intervals for large systems

In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.