Truncated model reduction methods for linear time-invariant systems via eigenvalue computation

This chapter proposes a clustered model reduction method for interconnected linear systems evolving over bidirectional networks. This model reduction method belongs to a kind of structured model reduction methods, where network clustering, namely clustering of subsystems, is performed according to a notion of uncontrollability of local states. We refer to this notion of uncontrollability as cluster reducibility, which can be captured by a coordinate transformation called positive tridiagonal transformation in an algebraic manner. In this chapter, it is shown that the aggregation of the reducible clusters retains the stability of the original system as well as an interconnection topology among clustered subsystems. Furthermore, an \(\fancyscript{H}_{\infty }\)-error bound is derived for the state discrepancy due to the cluster aggregation. The efficiency of the clustered model reduction is shown through an example of large-scale complex networks.KeywordsClustered model reductionInterconnected linear SystemsLarge-scale complex networks

The problem of model reduction of linear systems with certain interconnection structure is considered in this paper. To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken. This problem is important and timely because of the recent focus on complex networked systems in control engineering. Two different model reduction methods are introduced and compared in this paper. Both methods are extensions to the well‐known balanced truncation method. Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities. Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived. The so‐called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input–output map for the entire interconnected system. It is shown how these structured Hankel singular values can be used to select an approximation order. Finally, the two methods are applied to a model of a mechanical device. Copyright © 2008 John Wiley & Sons, Ltd.

Riemannian optimal model reduction of linear port-Hamiltonian systems

Considering the speed information as a prior constraint, constrained cubature kalman filter (CCKF) tracking filtering method based on model reduction (MR) method and perfect measurement (PM) method is studied and compared for the tracking problem of fixed single observer passive positioning using only angle pair constant speed targets. The simulation results show that the fixed single observer passive positioning system can track the moving target with constant speed by measuring angle only under the condition of speed prior information, and the perfect measurement method is more robust to the initial state of the filter than the model reduction method.

The very general problem of model reduction of nonlinear systems was made tractable by focusing on the very large subclass consisting of linear subsystems connected by nonlinear interfaces. Such problems constitute a large part of the nonlinear structural problems encountered in addressing the Sandia missions. A synthesis approach to this class of problems was developed consisting of: detailed modeling of the interface mechanics; collapsing the interface simulation results into simple nonlinear interface models; constructing system models by assembling model approximations of the linear subsystems and the nonlinear interface models. These system models, though nonlinear, would have very few degrees of freedom. A paradigm problem, that of machine tool vibration, was selected for application of the reduction approach outlined above. Research results achieved along the way as well as the overall modeling of a specific machine tool have been very encouraging. In order to confirm the interface models resulting from simulation, it was necessary to develop techniques to deduce interface mechanics from experimental data collected from the overall nonlinear structure. A program to develop such techniques was also pursued with good success.

Finite element models (FEMs) are extensively used in the design optimization of utility scale steam turbines. As an example, by simulating multiple startup scenarios of steam power plants, engineers can obtain turbine designs that minimize material utilization, and at the same time, avoid the damaging effects of large thermal stresses or rubs between rotating and stationary parts. Unfortunately, FEMs are computationally expensive and only a limited amount of simulations can be afforded to get the final design. For this reason, numerous model reduction techniques have been developed to reduce the size of the original model without a significant loss of accuracy. When the models are nonlinear, as is the case for steam turbine FEMs, model reduction techniques are relatively scarce and their effectiveness becomes application dependent. Although there is an abundant literature on model reduction for nonlinear systems, many of these techniques become impractical when applied to a realistic industrial problem. This paper focuses on a class of nonlinear FEM characteristic of thermo-elastic problems with large temperature excursions. A brief overview of popular model reduction techniques is presented along with a detailed description of the computational challenges faced when applying them to a realistic problem. The main contribution of this work is a set of modifications to existing methods to increase their computational efficiency. The methodology is demonstrated on a steam turbine model, achieving a model size reduction by four orders of magnitude with only 4% loss of accuracy with respect to the full order FEMs.

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

This paper extends some methods in model order reduction (MOR) of nonlinear models by trajectory piecewise-linear models based on output weighting (TPWLOW). The devised methods generalize the balanced-truncation method as the core for the proposed algorithms. These algorithms preserve the stability in some mentioned conditions. An error bound of reduction is calculated for these algorithms. In addition, a Krylov–TBR combinational MOR is also casted which benefits from the advantages of both balanced-truncation and Krylov subspace methods. The results show that in balanced-truncation-based algorithms, the efficiency of the first proposed algorithm fairly exceeds the second one. In addition, it is shown that the Krylov–TBR algorithm can reach the same efficiency despite remarkable lower order, provided that the designer can find a stable subspace. The robustness of TPWLOW-reduced models to linearization point movement is also evaluated, which shows the less sensitivity of the proposed methods in comparison with formerly introduced trajectory piecewise-linear models.

Model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure. These methods involve a complete generalization of balanced realizations, balanced Gramians, and balanced truncation model reduction with guaranteed error bounds, based on solutions to a pair of linear matrix inequalities which generalize Lyapunov equations. The resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems.

Reduced Order Controller Design for Discrete Time Systems

Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms

Model reduction of discrete time-delay systems based on Charlier polynomials and high-order Krylov subspaces

Large scale finite element models are routinely used in design and optimization for complex engineering systems. However, the high model order prevents efficient exploration of the design space. Many model reduction methods have been proposed in the literature on approximating the high dimensional model with a lower order model. These methods typically replace a fine scale model with a coarser scale model in schemes such as coarse graining, macro-modeling, domain decomposition and homogenization. This paper takes a systems perspective by stating the model reduction objective in terms of the approximation of the mapping between specified input and output variables. Methods from linear systems theory, including balance truncation and optimal Hankel norm approximation, are reviewed and compared with the standard modal truncation. For high order systems, computational load, numerical stability, and memory storage become key considerations. We discuss several computationally more attractive iterative schemes that generate the approximate gramian matrices needed in the model reduction procedure. A numerical example is also included to illustrate the model reduction algorithms discussed in the paper. We envision that these systems oriented model reduction methods complementing the existing methods to produce low order models suitable for design, optimization, and control.

1 Methods for Model Reduction.- 2 Multiplicative Approximation.- 3 Low Order Controller Design.- 4 Model and Controller Reduction Based On Coprime Factorizations.

Weighted model reduction for nonlinear systems and computation method