Structured model reduction for dynamical networked systems

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

This article is concerned with the problem of the Hankel-norm model reduction for stochastic discrete-time nonlinear systems in interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy framework. The IT2 T-S fuzzy model is an efficient model for describing uncertain nonlinear systems, and the model reduction is to simplify the high-order complex systems by reducing the order of the original system. The aim of this article is to reduce the order of the original stochastic discrete-time IT2 fuzzy system into lower order system without ignoring the influence of IT2 membership functions. First, the Hankel-norm performance of the stochastic discrete-time IT2 fuzzy model is analyzed. Then, based on the projection theorem and cone complementary linearization approach, a convex Hankel-norm-based model reduction approach subject to conditions in the form of linear matrix inequalities (LMIs) is obtained. A membership-functions-dependent (MFD) technique is applied to capture the information of IT2 membership functions and further reduce the conservativeness. A numerical example is presented to illustrate the effectiveness of the proposed results.

Moment matching model reduction for negative imaginary systems

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Computation of invariant sets via immersion for discrete-time nonlinear systems

A new algorithm based on model reduction using the Krylov subspace technique is proposed to compute the DC solution of large nonlinear circuits. The proposed method combines continuation methods with model reduction techniques. Thus it enables the application of the continuation methods to an equivalent reduced-order set of nonlinear equations instead of the original system. This results in a significant reduction in the computational expense as the size of the reduced equations is much less than that of the original system. The reduced order system is obtained by projecting the set of nonlinear equations, whose solution represents the DC operating point, into a subspace of a much lower dimension. It is also shown that both the reduced-order system and the original system share the first q derivatives w.r.t. the circuit variable used to parameterize the family of the solution trajectories generated by the continuation method.

In this paper a new balanced realization method for nonlinear systems is proposed which is based on singular value analysis of Hankel operators. The proposed method balances the relationship between the input-to-state behavior and the state-to-output behavior of nonlinear dynamical systems, whereas the existing results only balance the relationship among the coordinate axes of the state-space. This result is expected to be a basis for new model reduction and system identification of nonlinear systems.

In this paper, a new model-deviation compensation control based on the transformation of state for a general nonlinear system with uncertainty is introduced. By using the concepts and theorems of differential geometry, the original system can be converted into the simplified form. Firstly, the problem and the simplified nonlinear system are described. Secondly, the process of the transformation of state is given, including the existence conditions and the mathematical expressions. Then the model-deviation compensation control is developed. It has the state-error correction term and the model-deviation compensation term besides the equivalent control term. Finally, the simulation results verify the effectiveness of the proposed controller.

Nonlinear model reduction — method and CAE-tool development

The basic problem of order reduction of linear and nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via Time Periodic Center Manifold Theory. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.

Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

Multipoint interpolation of Volterra series and [formula omitted]-model reduction for a family of bilinear descriptor systems

Model order reduction (MOR) of nonlinear systems draws great attention in the past several decades. This paper presents a new MOR method in time domain for nonlinear dynamical systems. The new algorithm is based on the combination of the Taylor series expansion for the state variable x(t) and the trajectory piecewise polynomial technique. Firstly, the nonlinear system is approximated by a piecewise polynomial representation. Then, based on the Taylor series coefficients of x(t), we formulate the projection matrix V for the piecewise polynomial system and the compact model of the piecewise polynomial system is obtained in the following. Besides, error estimation and stability analysis are also presented in this paper. Finally, two nonlinear systems are tested to verify the effectiveness of the algorithm.

The problem of model reduction for the special class of continuous-time Metzlerian system is considered in this paper. The model reduction is performed under structural maintainability and stability preserving constraints. It is shown that model reduction based on aggregation method enables to maintain the structure of the original Metzlerian system while preserving its stability. Furthermore, we employ the frequency weighted balancing to Metzlerian systems and show that the reduced order models obtained by either truncation or singular perturbation approximation are guaranteed to be stable for the general case of double-sided weighting.

Realistic scenarios typically involve several interacting dependent variables, leading to nonlinearity in mathematical models. Nonlinear differential equations are a crucial mathematical phenomenon for describing real-world problems. On the other hand, fuzzy set theory is a valuable tool for modelling the uncertainty present in a system. So, nonlinear mathematical models under uncertainty need an explanatory study on a system of nonlinear fuzzy differential equations. In this paper, we analyse nonlinear fuzzy differential equations with possible applications. This paper advances the existing literature by introducing a definition of a fuzzy sequence and developing the existence and uniqueness theory for fuzzy solutions of nonlinear differential equations using the fixed-point theorem. The component-wise analysis of a fuzzy nonlinear system has been conducted. Furthermore, the analysed theory precedes an aptly fitted application of a fuzzy nonlinear dynamical system using a numerical scheme.