Optical Model Reduction and Robust Feedback Control for Aerodynamics

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

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.

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

Studies a class of balanced truncation algorithms applicable to relative/multiplicative model reduction. These algorithms seek to balance the controllability gramian of a given transfer function and the observability gramian of its right inverse. For this reason, the algorithms are referred to as inverse-weighted balanced truncation (IWBT) algorithms. It is shown that by using IWBT algorithms one can derive relative and multiplicative L/sub /spl infin// error bounds that are known to hold for other reduction algorithms. It is also shown that the balanced stochastic truncation (BST) method is actually one special, but an optimal version of the IWBT algorithms. As such, the authors' result also serves to establish the fact that the available error bounds pertaining to BST algorithms actually hold for IWBT algorithms.

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.

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

Historically, one of the basic issues in control systems design has been robustness: the ability of a controlled plant to withstand variations in or lack of knowledge of its dynamics. Even if the dynamics of a system are accurately known for purposes of implementation, it is often desirable to design a control system based on a simplified model. Consequently it is essential to be able to guarantee a reasonable performance not only for the nominal plant, but also for its neighbouring perturbations: this is the issue of robustness. Since the beginning of this decade major advances have been made in this area, notably using the H -approach; this term is meant to cover the solution of sensitivity reduction, approximation and model reduction, robustness and related control design problems using the mathematics of Hardy spaces and related areas in Harmonic Analysis. This book contains the proceedings of the NATO Advanced Research Workshop on Modelling, Robustness and Sensitivity Reduction in Control Systems, which was held at the University of Groningen, December 1986. Its aim was to explore the development of H -design techniques and its ramifications in Systems Theory in a unified and systematic way with the emphasis on recent advances and future directions in this fast developing area. In particular the following inter-related aspects were addressed: H -mathematical foundations, model approximation and robustness in control design, optimal sensitivity reduction, modelling and system identification and signal processing.

This paper provides three model reduction methods for linear time-invariant systems in the view of the Riemannian Newton method and the Jacobi-Davidson method. First, the computation of Hankel singular values is converted into the linear eigenproblem by the similarity transformation. The Riemannian Newton method is used to establish the model reduction method. Besides, we introduce the Jacobi-Davidson method with the block version for the linear eigenproblem and present the corresponding model reduction method, which can be seen as an acceleration of the former method. Both the resulting reduced systems can be equivalent to the reduced system originating from a balancing transformation. Then, the computation of Hankel singular values is transformed into the generalized eigenproblem. The Jacobi-Davidson method is employed to establish the model reduction method, which can also lead to the reduced system equivalent to that resulting from a balancing transformation. This method can also be regarded as an acceleration of a Riemannian Newton method. Moreover, the application for model reduction of nonlinear systems with inhomogeneous conditions is also investigated.

Balanced truncation is a standard method for model reduction, possessing many desirable properties such as preservation of model stability and a priori error bounds. Balanced truncation is conducted using controllability and observability Gramians. Generalized Gramians can be found by solving a set of linear matrix inequalities. In this paper, we show that these linear matrix inequalities can be extended so that the number of decision variables are at least doubled, leading to the concept of extended Gramians. Herein it is shown that the desirable properties of balanced truncation also hold with extended Gramians. The extended Gramians are especially useful for improving error bounds and for models possessing additional structural constraints, as is demonstrated by means of examples herein.

Building on our recently proposed model reduction methods for switched ordinary linear systems, we propose a comprehensive model reduction method for linear switched differential-algebraic equations (DAEs). In contrast to most other available model reduction methods for switched systems, we consider the switching signal as a given time-variance of the system. This allows us to exploit certain linear subspaces in the reduction process and also provide in general significantly smaller reduced models compared to methods which consider arbitrary switching signals. Model reduction for switched DAEs has some unique features that make a generalization of the available methods nontrivial; in particular, the presence of jumps and Dirac impulses in response to switches have to be carefully treated. Furthermore, due to the algebraic constraints, the reachability subspaces cannot be the full space, hence a straightforward application of balanced truncation is not possible (because the corresponding reachability Gramians will be structurally non-invertible). We resolve this problem by first applying an exact model reduction which reduces the switched DAE to a switched ordinary system with jumps and which carefully keeps track of the impulsive effects. As a second step, we then apply a midpoint balanced truncation approach to further reduce the switched system. In addition to the challenge to appropriately take into account the Dirac impulses, another novel challenge was the occurrence of input-dependent state-jumps. We propose to deal with input-dependent jumps by combining certain discrete-time reachability Gramians with continuous-time reachability Gramians. We provide corresponding Matlab implementations of the proposed algorithms and illustrate their effectiveness with some academic examples.

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.

In recent years, model order reduction has been recognized to be a powerful tool in analysis and simulation of integrated circuits. We consider balancing-related model reduction methods for differential-algebraic equations arising in circuit simulation. We show how positive real and bounded real balanced truncation can be used for passivity-preserving model reduction of circuit equations. These methods are based on balancing the solutions of projected Lur’e or Riccati matrix equations and admit computable error bounds. We also discuss efficient algorithms for solving such matrix equations that exploit the topological structure of circuit equations. Numerical experiments demonstrate the performance of the presented model reduction methods.

When time domain state-space identification algorithms are employed, identified models may need to be overparameterized due to system nonlinearity or measurement errors. After an initial oversized model is obtained, model reduction must be conducted. An important step in model reduction is to determine those significant modes Several commonly used model reduction methods such as the balanced realization (BR) technique select significant modes by minimizing the impulsive response error between the original model and reduced model. This study proposes an alternative method. The method is based on modal approximation. Two indices are devised to determine significant and insignificant modes. The parameter matrices expressed in the modal realization are reduced by eliminating the terms corresponding to the insignificant modes. Experimental modeling of a flexible manipulator system is used to demonstrate the effectiveness of the proposed method. The results using the BR technique are also given as a comparison.

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification---settings that require repeated model evaluations over different parameter values.

Model reduction methods are relevant when the computation time of a full convection–diffusion–reaction simulation based on detailed chemical reaction mechanisms is too large. In this article, we consider a model reduction approach based on optimization of trajectories and its applicability to realistic combustion models. As many model reduction methods, it identifies points on a slow invariant manifold based on time scale separation in the dynamics of the reaction system. The numerical approximation of points on the manifold is achieved by solving a semi-infinite optimization problem, where the dynamics enter the problem as constraints. The proof of existence of a solution for an arbitrarily chosen dimension of the reduced model (slow manifold) is extended to the case of realistic combustion models including thermochemistry by considering the properties of proper maps. The model reduction approach is finally applied to two models based on realistic reaction mechanisms: ozone decomposition as a small test case and syngas combustion as a test case including all features of a detailed combustion mechanism.