On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

On macromodeling of nonlinear systems with time periodic coefficients

Some new techniques for reduced order (macro) modeling of nonlinear systems with time periodic coefficients are discussed in this paper. The dynamical evolution equations are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the new set of equations become time-invariant. The techniques presented here reduce the order of this transformed system and all original states are obtained via the appropriate transformations. This macromodel preserves the desired stability and bifurcation characteristics of the original large-scale system and due to relatively few states; it is suitable for simulation and controller design.In this work, methodologies based on linear and nonlinear projections as well as ‘time periodic invariant manifold’ idea are presented. The invariant manifold technique yields a ‘reducibility condition’ that determines when an accurate nonlinear order reduction is possible. A comparative study of these order reduction methods is also included. These techniques are compared by means of time traces and Poincaré maps. A numerical error analysis is also included and advantages and limitations are discussed by means of a practical example.     

System reduction using factor division algorithm and eigen spectrum analysis

A mixed method is proposed which combines the factor division algorithm with the eigen spectrum analysis for deriving reduced order models of high-order linear time invariant systems. Pole centroid and system stiffness of both original and reduced order systems remain same in this method. The proposed method guarantees stability of the reduced model if the original high-order system is stable and is comparable in quality with the other well known existing methods of order reduction. The method is illustrated by four numerical examples including one example of a multivariable system.     

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.     

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system.Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘reducibility conditions’. If these ‘reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover ‘resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.     

Problems with the estimation of stochastic differential equations using structural equations models

The present paper deals with the identification and maximum likelihood estimation of systems of linear stochastic differential equations using panel data. So we only have a sample of discrete observations over time of the relevant variables for each individual. A popular approach in the social sciences advocates the estimation of the “exact discrete model” after a reparameterization with LISREL or similar programs for structural equations models. The “exact discrete model” corresponds to the continuous time model in the sense that observations at equidistant points in time that are generated by the latter system also satisfy the former. In the LISREL approach the reparameterized discrete time model is estimated first without taking into account the nonlinear mapping from the continuous to the discrete time parameters. In a second step, using the inverse mapping, the fundamental system parameters of the continuous time system in which we are interested, are inferred. However, some severe problems arise with this “indirect approach”. First, an identification problem may arise in multiple equation systems, since the matrix exponential function denning some of the new parameters is in general not one‐to‐one, and hence the inverse mapping mentioned above does not exist. Second, usually some sort of approximation of the time paths of the exogenous variables is necessary before the structural parameters of the system can be estimated with discrete data. Two simple approximation methods are discussed. In both approximation methods the resulting new discrete time parameters are connected in a complicated way. So estimating the reparameterized discrete model by OLS without restrictions does not yield maximum likelihood estimates of the desired continuous time parameters as claimed by some authors. Third, a further limitation of estimating the reparameterized model with programs for structural equations models is that even simple restrictions on the original fundamental parameters of the continuous time system cannot be dealt with. This issue is also discussed in some detail. For these reasons the “indirect method” cannot be recommended. In many cases the approach leads to misleading inferences. We strongly advocate the direct estimation of the continuous time parameters. This approach is more involved, because the exact discrete model is nonlinear in the original parameters. A computer program by Hermann Singer that provides appropriate maximum likelihood estimates is described.     

Parallel nonlinear multisplitting methods

Summary Linear multisplitting methods are known as parallel iterative methods for solving a linear systemAx=b. We extend the idea of multisplittings to the problem of solving a nonlinear system of equationsF(x)=0. Our nonlinear multisplittings are based on several nonlinear splittings of the functionF. In a parallel computing environment, each processor would have to calculate the exact solution of an individual nonlinear system belonging to his nonlinear multisplitting and these solutions are combined to yield the next iterate. Although the individual systems are usually much less involved than the original system, the exact solutions will in general not be available. Therefore, we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as Newton's method. Several methods proposed in literature may be regarded as special nonlinear multisplitting methods. As an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants. One result is that the local convergence of these methods is determined by an induced linear multisplitting of the Jacobian ofF.Dedicated to the memory of Peter Henrici     

Nonlinear Model Order Reduction in Nanoelectronics: Combination of POD and TPWL

In this paper we demonstrate model order reduction of a nonlinear academic model of an inverter chain. Two reduction methods, which are suitable for nonlinear differential algebraic equation systems are combined, the trajectory piecewise linear approach and the proper orthogonal decomposition. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-level substructuring combined with model order reduction methods

In many fields of engineering problems linear time-invariant dynamical systems (LTI systems) play an outstanding role. They result for instance from discretizations of the unsteady heat equation and they are also used in optimal control problems. Often the order of LTI systems is a limiting factor, since it becomes easily very large. As a consequence these systems cannot be treated efficiently without model reduction algorithms. In this paper a new approach for the combination of model order reduction methods and recent multi-level substructuring (MLS) techniques is presented. Similar multi-level substructuring methods have already been applied successfully to huge eigenvalue problems up to several millions of degrees of freedom. However, the presented approach does not make use of a modal analysis like former algorithms. Instead the original system is decomposed in smaller LTI systems which are treated with recent model reduction methods. Furthermore, the error which is induced by this substructuring approach is analysed and numerical examples based on the Oberwolfach benchmark collection are given in this paper.     

基于Fuzzy推理的时变系统建模

提出一种基于Fuzzy推理的时变系统建模方法,其基本思想是:对时间维度进行分割,在每个较短的时间间隔内用时不变模型代替时变模型,将这些时不变模型组合在一起,最终获得一个整体非线性时变的微分方程模型.分别研究了输入输出型时变系统和状态空间型时变系统的模型建立方法,除了从理论上保证了所获得的模型对系统的逼近性,还从仿真实验验证了用该方法建立的模型对非线性时变系统有很好的逼近效果.     

Mechatronic Subsystem Studied of Approximate Method Modeled by Hypergraph

The transverse vibrating mechatronic subsystem is considered. Integral parts of this system are: a continuous beam with known boundary conditions and a transducer, extorted by harmonic voltage excitation, to be perfectly bonded to the beam surface. Findings this article are dynamical characteristics of the discussed mechatronic and mechanical system to model them by hypergraphs. Research limitation is that the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered, however for this kind of systems the approach is sufficient. Practical implications of this researches was that global approach is presented, that means in the domain of frequency spectrum analysis. The methods of analysis and obtained results can be base of design and investigation for this type of mechatronic systems. Originality of this paper is that the mechatronic system created from mechanical and electric subsystems with electromechanical bondage has been considered. This approach is different from those considered so far because is it relies on application approximate methods of analysis of mechatronic subsystem and modeling the one by hypergraph [1-7]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fault identification in nonlinear hybrid systems

The problem of fault identification in hybrid systems is investigated. It is assumed that the hybrid systems under consideration consist of a finite automaton, the set of nonlinear differential equations, and so-called mode activator that coordinates the action of these two parts. To solve the fault identification problem, sliding mode observers are used. The suggested approach for constructing sliding mode observers is based on the reduced order model of the original system. This allows to reduce complexity of sliding mode observers and relax the limitations imposed on the original system. Examples illustrate details of the solution.     

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

Gain-Scheduled Worst-Case Control on Nonlinear Stochastic Systems Subject to Actuator Saturation and Unknown Information

In this paper, we propose a method for designing continuous gain-scheduled worst-case controller for a class of stochastic nonlinear systems under actuator saturation and unknown information. The stochastic nonlinear system under study is governed by a finite-state Markov process, but with partially known jump rate from one mode to another. Initially, a gradient linearization procedure is applied to describe such nonlinear systems by several model-based linear systems. Next, by investigating a convex hull set, the actuator saturation is transferred into several linear controllers. Moreover, worst-case controllers are established for each linear model in terms of linear matrix inequalities. Finally, a continuous gain-scheduled approach is employed to design continuous nonlinear controllers for the whole nonlinear jump system. A numerical example is given to illustrate the effectiveness of the developed techniques.     

Weight Functions Method in Stability Study of Systems

A new method for systems stability analysis is presented. This method is called weight functions method and it replaces the problem of Liapunov function finding with a problem of finding a number of functions (weight functions) equal to the number of first order differential equations describing the system. It is known that there are not general methods for finding Liapunov functions. The weight functions method is simpler than the classical method since one function at a time has to found. Conditions of solution stability for linear and nonlinear systems and some examples are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A model order reduction technique for systems with nonlinear frequency dependent damping

Frequency domain solution of systems with frequency dependent damping is a computationally expensive endeavour especially when dealing with large order three-dimensional systems. A moment-matching based reduced order model is proposed in this work which is capable of handling nonlinear frequency dependent damping in second-order systems. In the proposed approach, local linear systems with frequency independent matrices are derived from the original system, and using the principles of the Rational Krylov approach, orthogonal basis vectors are computed from these local systems through the second-order Arnoldi procedure. The system is then projected on to the basis set to obtain a numerically efficient reduced order model, accurate in the entire frequency domain of interest. The proposed approach is also shown to be more accurate than the popular modal projection based multi-model approach of the same order. The proposed tool is applied to the problem of determining the frequency response of an idealised centrifugal compressor impeller with non-viscous (frequency dependent) damping.     

Reduced order controllers for distributed parameter systems: LQG balanced truncation and an adaptive approach

In this paper, two methods are reviewed and compared for designing reduced order controllers for distributed parameter systems. The first involves a reduction method known as LQG balanced truncation followed by MinMax control design and relies on the theory and properties of the distributed parameter system. The second is a neural network based adaptive output feedback synthesis approach, designed for the large scale discretized system and depends upon the relative degree of the regulated outputs. Both methods are applied to a problem concerning control of vibrations in a nonlinear structure with a bounded disturbance.     

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.