Abstract: We consider model reduction of uncertain behavioural models. Machinery for gap-metric model reduction and multidimensional model reduction using linear matrix inequalities is extended to these behavioural models. The goal is a systematic method for reducing the complexity of uncertain components in hierarchically developed models which approximates the behavior of the full-order system. This paper focuses on component model reduction that preserves stability under interconnection. >

Abstract: We present model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation (LFT) on a repeated scalar uncertainty structure. These reduction methods can be interpreted either as doing state order reduction for multi-dimensionalsystems, or as uncertainty simplification in the case of uncertain systems, and are based on finding solutions to a pair of Linear Matrix Inequalities (LMIs). A related necessary and sufficient condition for the exact reducibility of stable uncertain systems is also presented.

TL;DR: Model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation on a repeated scalar uncertainty structure and a related necessary and sufficient condition for the exact reducibility of stable uncertain systems are presented.

Abstract: 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.

241 citations

Cites result from "Model reduction of behavioural syst..."

...These new results, which were first noted in [ 4 ], are based on machinery presented in [18]....

TL;DR: The reduction method proposed is applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 -induced stability constraint required in the standard nonfactored case.

Abstract: We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 -induced stability constraint required in the standard nonfactored case. Reduction error bounds in the l 2 -induced norm of the factorized mapping are given.

55 citations

Cites background from "Model reduction of behavioural syst..."

...In this case, contractive and expansive coprime factors realizations may be considered instead; see [8] for a discussion relating to this topic....

TL;DR: An approach is given for the model reduction of stabilizable and detectable systems, which requires the development and use of coprime factorizations for NSLPV models, and can be explicitly computed using semidefinite programming.

Abstract: This paper focuses on the model reduction of nonstationary linear parameter-varying (NSLPV) systems. We provide a generalization of the balanced truncation procedure for the model reduction of stable NSLPV systems, along with a priori error bounds. Then, for illustration purposes, this method is applied to reduce the model of a two-mass translational system. Furthermore, we give an approach for the model reduction of stabilizable and detectable systems, which requires the development and use of coprime factorizations for NSLPV models. For the general class of eventually periodic LPV systems, which includes periodic and finite horizon systems as special cases, our results can be explicitly computed using semidefinite programming

Abstract: The emphasis of this thesis is on the development of systematic methods for reducing the size and complexity of uncertain system models. Given a model for a large complex system, the objective of these methods is to find a simplified model which accurately describes the physical system, thus facilitating subsequent control design and analysis. Model reduction methods and realization theory are presented for uncertain systems represented by Linear Fractional Transformations (LFTs) on a block diagonal uncertainty structure. A complete generalization of balanced realizations, balanced Gramians and balanced truncation model reduction with guaranteed error bounds is given, which is based on computing solutions to a pair of Linear Matrix Inequalities (LMIs). A necessary and sufficient condition for exact reducibility of uncertain systems, the converse of minimality, is also presented. This condition further generalizes the role of controllability and observability Gramians, and is expressed in terms of singular solutions to the same LMIs. These reduction methods provide a systematic means for both uncertainty simplification and state order reduction in the case of uncertain systems, but also may be interpreted as state order reduction for multi-dimensional systems. LFTs also provide a convenient way of obtaining realizations for systems described by rational functions of several noncommuting indeterminates. Such functions arise naturally in robust control when studying systems with structured uncertainty, but also may be viewed as a particular type of description for a formal power series. This thesis establishes connections between minimal LFT realizations and minimal linear representations of formal power series, which have been studied extensively in a variety of disciplines, including nonlinear system realization theory. The result is a fairly complete development of minimal realization theory for LFT systems. General LMI problems and solutions are discussed with the aim of providing sufficient background and references for the construction of computational procedures to reduce uncertain systems. A simple algorithm for computing balanced reduced models of uncertain systems is presented, followed by a discussion of the application of this procedure to a pressurized water reactor for a nuclear power plant.

36 citations

Cites background or methods from "Model reduction of behavioural syst..."

...Alternatively, a coprime factorization approach could be used; an initial attempt at this is discussed in [4]....

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...These new results, which were first noted in [4], are based on technical machinery presented in [54] and [56]....

Abstract: Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of X_{c}, X_{\bar{o}} , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

Abstract: The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.

TL;DR: A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ.

Abstract: A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ. The μ-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods.

Abstract: A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >

1,186 citations

"Model reduction of behavioural syst..." refers background or methods in this paper

...For a more extensive treatment of these subjects, see references [ 27 ],[1], [11],[10] and [24]....

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...The most striking feature of the representation of dynamical systems in the behavioural framework, as proposed by Willems [ 27 ], is the fact that there are no explicit inputs and outputs....

[...]

...A dynamical system is defined in a behavioural framework as follows [ 27 ]....

TL;DR: A load dumping vehicle including a frame, a gas turbine engine supported by the frame and a dump body pivotally connected to theframe for movement relative to the frame between a load carrying position and adump position.

Abstract: Preliminaries.- Robust stabilization of uncertain systems.- Robust stabilization of normalized coprime factor plant descriptions.- Reduced order controller design.- A loop shaping design procedure.- Design examples.