A model-order reduction method based on Krylov subspaces for mimo bilinear dynamical systems

In this paper, we present a Krylov subspace based projection method for reduced-order modeling of large scale bilinear multi-input multioutput (MIMO) systems. The reduced-order bilinear system is constructed in such a way that it can match a desired number of moments of multivariable transfer functions corresponding to the kernels of Volterra series representation of the original system. Numerical examples report the effectiveness of this method.     

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.     

A dynamical systems model of unorganized segregation

We consider Schelling’s bounded neighborhood model (BNM) of unorganized segregation, from the perspective of modern dynamical systems theory. We carry out a complete quantitative analysis of the system for linear tolerance schedules. We derive a fully predictive model and associate each term with a social meaning. We recover and generalize Schelling’s qualitative results.

For the case of unlimited population movement, we derive exact formulae for regions in parameter space where stable integrated population mixes can occur, and show how neighborhood tipping can be explained in terms of basins of attraction.

When population movement is limited, we derive exact criteria for the occurrence of new population mixes.

For nonlinear tolerance schedules, we illustrate our approach with numerical simulations.     

Regular bilinear systems

This paper deals with (weakly) regular and abstract bilinearsystems which allow some degree of unboundedness in the controland observation operators for infinite-dimensional bilinearsystems. By adaption of the notion of abstract linear systemsintroduced by Salamon (1989) and Weiss (1989a), we give a bilinearversion of their approach which works in our setting. The paperinvestigates also the relation between the shift-invariant bilinearoperator and an analytic function. This latter can be consideredas a transfer function for bilinear systems. The results thusobtained are used to study the problem of observer design forsuch systems.     

A dynamical systems model of intrauterine fetal growth

ABSTRACT

The underlying mechanisms for how maternal perinatal obesity and intrauterine environment influence foetal development are not well understood and thus require further understanding. In this paper, energy balance concepts are used to develop a comprehensive dynamical systems model for foetal growth that illustrates how maternal factors (energy intake and physical activity) influence foetal weight and related components (fat mass, fat-free mass, and placental volume) over time. The model is estimated from intensive measurements of foetal weight and placental volume obtained as part of Healthy Mom Zone (HMZ), a novel intervention for managing gestational weight gain in obese/overweight women. The overall result of the modelling procedure is a parsimonious system of equations that reliably predicts foetal weight gain and birth weight based on a sensible number of assessments. This model can inform clinical care recommendations as well as how adaptive interventions, such as HMZ, can influence foetal growth and birth outcomes.     

The reduction principle in stability theory of dynamical and semidynamical systems

The extended version of this paper will be published inAnnali di Matematica Pura ed Aplicata.     

On mean-square-stable bilinear systems

It has been shown in a previous paper that an infinite-dimensionalstochastic discrete bilinear system is mean-square-stable ifand only if the spectral radii of two transformations of Hilbert-spaceoperators are both less than one. The present paper investigatesconditions to be imposed on the model operators in order toensure that such spectral radii coincide. Several examples arepresented and the main result establishes the spectral radiusidentity for models with compact operators.     

Stability of bilinear time-delay systems

In this paper, the stability of the differential bilinear time-delaysystems is first studied. We consider time-varying bilineartime-delay systems with output feedback. The input or controlu(t)is not only a signal but also an input with output feedback.The analysis is given by using norm-transformation methods.     

Model reduction for large-scale dynamical systems via equality constrained least squares

In this paper, we present a new method of model reduction for large-scale dynamical systems, which belongs to the SVD-Krylov based method category. It is a two-sided projection where one side reflects the Krylov part and the other side reflects the SVD (observability gramian) part. The reduced model matches the first r+i Markov parameters of the full order model, and the remaining ones approximate in a least squares sense without being explicitly computed, where r is the order of the reduced system, and i is a nonnegative integer such that 1≤i<r. The reduced system minimizes a weighted ?₂ error. By the definition of a shift operator, the proposed approximation is also obtained by solving an equality constrained least squares problem. Moreover, the method is generalized for moment matching at arbitrary interpolation points. Several numerical examples verify the effectiveness of the approach.     

Nonlinearity tests for bilinear systems

The purpose of this paper is to develop nonlinearity tests for open-loop bilinear systems. Lagrange multiplier tests of linear systems against a bilinear alternative are proposed. A simulation study is performed to check the validity of the asymptotic null distributions of the test statistics and to investigate the power characteristics of the tests. Two recent nonlinearity tests in the time-series context are adapted to linear systems and compared with Lagrange multiplier tests. Simulation results show that the proposed Lagrange multiplier tests are more powerful than the other tests.     

Mean field limit of a dynamical model of polymer systems

This paper provides a mathematically rigorous foundation for self-consistent mean feld theory of the polymeric physics.We study a new model for dynamics of mono-polymer systems.Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces.Every two points on the same string or on two diferent strings also interact under a pairwise potential V.The dynamics of the system is described by a system of N coupled stochastic partial diferential equations(SPDEs).We show that the mean feld limit as N→∞of the system is a self-consistent McKean-Vlasov type equation,under suitable assumptions on the initial and boundary conditions and regularity of V.We also prove that both the SPDE system of the polymers and the mean feld limit equation are well-posed.     

Optimal feedback control of bilinear systems

Feedback synthesis of optimal constrained controls for single-input bilinear systems is considered. Quadratic cost functionals (with and without quadratic control penalization) are modified by the inclusion of additional nonnegative state penalizing functions in the respective cost integrands. The latter functions are chosen so as to regularize the problems, in the sense that feedback solutions of particularly simple form are obtained. Finite and infinite time horizon problem formulations are treated, and associated aspects of feedback stabilization of bilinear systems are discussed.     

The reduction principle for discrete dynamical and semidynamical systems in metric spaces

The main result of this paper is on decoupling by means of a topological transformation of a homeomorphism which admits invariant sets. A theorem of conjugacy of a given possible noninvertible mapping, which admits an invariant set, to a simpler mapping than the given one in terms of decoupling is also proved. The results are valid in an arbitrary complete metric space.This work has been completed with the financial support of Latvian Council of Science under Grants 90.224 and 93.809.     

Control of spectrum in bilinear systems

For a bilinear stationary control system, we obtain necessary and sufficient conditions for the solvability of the spectrum control problem for the case in which the coefficients have a special form.     

On systems of bilinear forms whose minimal division-free algorithms are all bilinear

In this paper we prove that for a certain class of systems of bilinear forms, all minimal division-free algorithms are essentially bilinear. This class includes systems for computing products in finite algebraic extension fields, and systems for computing the products of Toeplitz and Hankel matrices with vectors. Our results, together with the classification theorems of 10., 12., 169–180) completely describe all minimal division-free algorithms for computing these systems. We also prove, as an immediate consequence of our results, that the multiplicative complexity of the quaternion product over a real field is 8.