Exact model matching of generalized state space systems

The problem of exact model matching for generalized state space (GSS) systems via pure proportional state and output feedback is studied. The following two major issues are resolved here for the first time: The necessary and sufficient conditions for the problem to have a solution and the general analytical expressions for the exact modelmatching controller matrices. The important case of left invertible systems is treated separately wherein simple solutions are established for the above two major issues and results on structural properties of the closed-loop system are reported. Known results on model matching of regular systems are derived as a special case of the GSS systems results, thus unifying the solution of the exact model-matching problem of regular and singular systems.The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.     

Balanced truncation for linear switched systems

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.     

Reachability and Observability Concepts for Stochastic Systems

We define observability and reachability and introduce the corresponding Gramians for a Levy driven linear system like Benner, Damm in [3] which focused on the case of Wiener noise. We additionally show that the sets of observable and reachable states are characterized by these Gramians. This is analogous to deterministic systems, where observability and reachability concepts are described in Sections 4.2.1 and 4.2.2 in Antoulas [1]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Balancing Transformations for Infinite-Dimensional Control Systems

In order to facilitate model reduction by balanced truncation, we introduce state space transformations that can be used to construct an ℓ₂-balanced realization of a regular, linear input-ouput map with nuclear Hankel-operator directly from the system generators of an arbitrary, given realization. These balancing transformations are based on factors of the Gramians and, for infinite-dimensional systems, they are usually unbounded operators. Subsequently the ℓ₂-balanced realization can be truncated in a non-trivial way to obtain an approximating, finite-dimensional model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Empirical Gramian-based spatial basis functions for model reduction of nonlinear distributed parameter systems

Correct selection of spatial basis functions is crucial for model reduction for nonlinear distributed parameter systems in engineering applications. To construct appropriate reduced models, modelling accuracy and computational costs must be balanced. In this paper, empirical Gramian-based spatial basis functions were proposed for model reduction of nonlinear distributed parameter systems. Empirical Gramians can be computed by generalizing linear Gramians onto nonlinear systems, which results in calculations that only require standard matrix operations. Associated model reduction is described under the framework of Galerkin projection. In this study, two numerical examples were used to evaluate the efficacy of the proposed approach. Lower-order reduced models were achieved with the required modelling accuracy compared to linear Gramian-based combined spatial basis function- and spectral eigenfunction-based methods.     

Balanced Truncation Model Reduction of Large-Scale Dense Systems on Parallel Computers

Model reduction is an area of fundamental importance in many modeling and control applications. In this paper we analyze the use of parallel computing in model reduction methods based on balanced truncation of large-scale dense systems. The methods require the computation of the Gramians of a linear-time invariant system. Using a sign function-based solver for computing full-rank factors of the Gramians yields some favorable computational aspects in the subsequent computation of the reduced-order model, particularly for non-minimal systems. As sign function-based computations only require efficient implementations of basic linear algebra operations readily available, e.g., in the BLAS, LAPACK, and ScaLAPACK, good performance of the resulting algorithms on parallel computers is to be expected. Our experimental results on a PC cluster show the performance and scalability of the parallel implementation.     

On evaluating the performance of GSS: Furthering the debate

Following criticism from proponents of workstation group support systems about the lack of validity of ‘wide-band’ group support systems (GSS), a recent article argued that it was inappropriate to attempt to establish the level of experimental validity for such GSS that is demanded by researchers working with 'narrow-band' GSS. It was argued that ‘wide-band’ and workstation GSS are very different and this means that it is unlikely that the same measures of success would be appropriate. This paper agrees with this view but not with its extension, that the overall validation requirement is different; theoretical validity is a necessary feature of a full evaluation but it is not sufficient by itself - experimental validity must also be sought. This paper argues against the view that since experimental validity cannot be obtained, theoretical validity will suffice. A twin-track research approach involving theoretical and experimental validity is proposed that can unite researchers investigating (almost) all forms of GSS, modelled on the PIMS programme and supported by one of the research frameworks that already exist.     

Nonlinear Dynamics of Marine Systems in Random Waves

The dynamics of ships or offshore structures is influenced by several different effects, some of which have a distinctly nonlinear characteristic. Even though in many situations the motion can sufficiently be described by linear models, nonlinear phenomena play a crucial role in the investigation of some more critical operating conditions: Large amplitude motions, sudden jumps in the dynamical behavior and sensitivity to the initial conditions are likely to occur under some circumstances. The response of floating systems such as moored buoys and barges in regular waves can be approximated by analytical or numerical techniques. These analyses reveal the characteristics of different periodic motions. In order to determine how these responses change under a more general forcing, the motion of floating structures under the influence of random disturbances is described by probability distributions. Different mathematical tools can efficiently be applied to models with few degrees of freedom. The localized statistical linearization used here is also promising for larger systems. Modelling aspects of offshore structures and random waves are discussed as well as the determination of probability distributions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of Signature Symmetric Balanced Realizations

A new numerical scheme for computing balancing coordinate transformations for signature symmetric realizations in linear systems theory is presented. The method is closely related to the Jacobi method for diagonalizing symmetric matrices. Here the minimization of the sum of traces of the Gramians by orthogonal and hyperbolic Jacobi-type rotations is considered. Local quadratic convergence of the algorithm is shown.     

Generalized shift-splitting iteration method for a class of two-by-two linear systems

In this paper, we propose a generalized-shift splitting (GSS) iteration method for solving a broad class of two-by-two linear systems. The convergence theory of the GSS iteration method is established and the spectral properties of the corresponding preconditioned matrix of some special choices for the parameter matrices are analyzed. In addition, the optimal choice of the iterative parameter is discussed. Numerical experiments are used to verify the validity of the theoretical results and the effectiveness of the new method.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

On periodic solutions of nonlinear second order differential systems

The paper deals with the second order differential systems with periodic boundary conditions. In the first part of the paper four different methods are employed to find the unique solution of the linear systems. One of the methods is a shooting type which converts the periodic boundary value problem into its equivalent initial value problem. In the second part of the paper several sufficient conditions are provided for the existence and uniqueness for the nonlinear systems. The technique developed for the linear systems to convert into its equivalent initial value problem is used in an iterative way for the nonlinear systems. It is shown that the iterations converge quadratically.     

Interval Iterative Methods Under Partial Ordering (Ⅱ)

Many types of nonlinear systems can be solved by using ordered iterative methods. These systems are discussed in [2] in a unified form for five different initial conditions. This paper is a continuation of [2]. Under arbitrary initial conditions, some iterative methods are given, and several theorems for the existence and uniqueness of the solution and convergence of the methods are proved.     

Bases of topological epi-reflections

In the first part of this paper a characterization of epi-reflective subcategories of the category TOP of topological spaces in terms of initial topologies is given. This characterization enables us to associate to each epi-reflective subcategory of TOP, regular systems of cogenerators. After having examined some properties of these regular systems, the second part of the paper considers an A-closure operator associated to every epi-reflective subcategory A of TOP which for each regular system of cogenerators of A determines an epi-reflective subcategory of A.     

Invariant surfaces of two-dimensional periodic systems with bifurcating rest points in the first approximation

Two-dimensional, time-periodic systems of differential equations with a small positive parameter whose first-approximation systems are conservative, depend on the parameter, and have one, two, or three rest points are considered. Explicit conditions on the coefficients under which the initial system has one or several two-dimensional invariant surfaces homeomorphic to the torus for all sufficiently small parameter values are obtained, and formulas for these surfaces are presented. As an example, a class of systems with three two-periodic invariant surfaces is constructed. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

On the solvability of initial boundary-value problems for a class of operator-differential equations of third order

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability.     

On the Stability of Solitary Water Waves with a Point Vortex

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS) , but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state-dependent. As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized Korteweg–de Vries (KdV) and Benjamin-Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss , who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion and obtain a new instability result for fractional KdV. © 2020 Wiley Periodicals, Inc.     

Nonlinear martingale problems involving singular integrals

We study a class of nonlinear martingale problems in one dimension, that involve a singular integral of the density in the drift term, and are related to systems of particles with singular interactions. First, we prove existence and uniqueness of regular solutions of the associated nonlinear evolution equation. Then, we establish a suitable framework and conditions where the martingale problem is well posed. This extends the results of Bonami et al. (J. Funct. Anal. 165 (1999) 390) to a wide class of coefficients and initial conditions. Finally, we obtain our solution of the martingale problem as the chaotic limit of some systems of particles interacting through regular approximating kernels.     

Irreducibly inconsistent systems of linear inequalities

Irreducibly inconsistent systems of linear inequalities are considered from the point of view of applying simplex-like methods for investigation.We give necessary and sufficient conditions, for a system to be irreducibly inconsistent, which can be found by applying a simplex algorithm. Next we consider general inconsistent systems. The simplex method can be used to identify irreducibly inconsistent subsystem. Such subsystems can give structural insight in the inconsistency and are used in trying to make the overall system consistent. This can be done in an easy way e.g. by right-hand side manipulations. Finally the problem of finding a ‘cheapest’ way of doing this can be formulated and solved as an extended linear program.