Model Reduction Of Linear Systems Research Articles

Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H2-norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H2-norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H2-error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation

In this article we study balanced model reduction of linear control systems using the singular perturbation approximation. Balanced model reduction techniques have been successfully applied to systems with homogeneous initial conditions, with one of their most important features being a priori L2 and H∞ bounds for the approximation error. The main focus of this work is to derive an L2 error bound for the singular perturbation approximation for system with inhomogeneous initial conditions, extending related work on balanced truncation. This L2 error bound measures the difference between the input-output maps of the original and of the reduced initial value systems. The advantages and flexibility of this approach are demonstrated with a variety of numerical examples.

Optimal control of linear systems with balanced reduced-order models: Perturbation approximations

In this article we study balanced model reduction of linear systems for feedback control problems. Specifically, we focus on linear quadratic regulators with collocated inputs and outputs, and we consider perturbative approximations of the dynamics in the case that the Hankel singular values corresponding to the hardly controllable and observable states go to zero. To this end, we consider different perturbative scenarios that depend on how the negligible states scale with the small Hankel singular values, and derive the corresponding limit systems as well as approximate expressions for the optimal feedback controls. Our approach that is based on a formal asymptotic expansion of an algebraic Riccati equations associated with the Pontryagin maximum principle and that is validated numerically shows that model reduction based on open-loop balancing can also give good closed-loop performance.

Model Reduction for Linear Systems with Low-Rank Switching

We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.

Model reduction for linear and nonlinear magneto‐quasistatic equations

SummaryWe consider model reduction for magneto‐quasistatic field equations in the vector potential formulation. A finite element discretization of such equations leads to large‐scale differential‐algebraic equations of special structure. For model reduction of linear systems, we employ a balanced truncation approach, whereas nonlinear systems are reduced using a proper orthogonal decomposition method combined with a discrete empirical interpolation technique. We will exploit the special block structure of the underlying problem to improve the performance of the model reduction algorithms. Furthermore, we discuss an efficient evaluation of the Jacobi matrix required in nonlinear time integration of the reduced models. Copyright © 2017 John Wiley & Sons, Ltd.

A Frequency-Based Interpretation of Energy-Based Model Reduction of Linear Systems

Various modeling techniques have been proposed in order to make modeling a more systematic procedure and to facilitate the development of modeling software that provide users, even experienced users, more ease when developing good models for control and design. The author has previously developed an energy-based modeling metric called “element activity” that was implemented in the model order reduction algorithm (MORA). While MORA was originally developed for the reduction of nonlinear models, the purpose of this paper is to gain insight into this methodology when applied to the reduction of linear models. Toward this end, the steady-state response to sinusoidal inputs is considered. Element activity is calculated analytically for any given excitation frequency, and a series of reduced models, which depend on the excitation frequency content, are generated. The results show through a quarter-car vehicle model that MORA generates a series of models whose spectral radius increases, as successively lower activity elements are included. It is demonstrated that low activity elements are related with high-frequency dynamics.

Model Order Reduction for Magneto-Quasistatic Equations

We consider model reduction of Maxwell's equations arising in magneto-quasistatic problems. A finite element discretization of such equations leads to large-scale differential- algebraic equations of special structure. For model reduction of linear systems, we employ a balanced truncation approach, whereas nonlinear systems are reduced using a proper orthogonal decomposition method combined with a discrete empirical interpolation technique. We will exploit the special structure of the underlying problem to improve the performance of the model reduction algorithms.

Model Reduction of Linear Systems with Random Parameters

Abstract This paper is devoted to model reduction of linear time invariant (LTI) systems with random parameters. A novel framework is proposed to deal with this challenging problem. It consists in the combination of the generalized polynomial chaos (GPC) formalism and the truncated balanced realization (TBR) based method. The GPC formalism is known to be a powerful tool for the random uncertainty propagation and quantification while the TBR is efficient in model reduction of LTI systems. So, to couple these two methods helps to generate an efficient methodology to reduce linear models with random parameters. Two techniques exploiting this combination are proposed and tested on the model reduction of a mechanical system with random parameters. High efficiency is shown for both methods.

A method for model-reduction of non-linear thermal dynamics of multi-zone buildings

We propose a method for reducing the order of dynamic models of temperature and humidity in multi-zone buildings. Low-order models of building thermal dynamics are useful for model-based HVAC control techniques, especially to computationally intensive ones such as Model Predictive Control (MPC). Even a lumped parameter model for a multi-zone building, which is a set of non-linear coupled ordinary differential equations, can have large state-space dimension. Model reduction techniques are useful to simplify such models. Although there are a number of well-developed techniques for model reduction of linear systems, techniques available for non-linear systems are limited. The method we propose exploits the linear portion of the model to compute a transformation (by using balanced realization) and a specific sparsity pattern of the non-linear portion that building thermal models possess to obtain the reduced order model. Simulations show that the prediction of the zone temperatures and humidity ratios by the reduced model is quite close to that from the full-scale model, even when substantial reduction of model order is specified that reduces computation time by a factor of six or more.

Model Reduction of Linear Systems by PSO Technique based on ISE, Circumvented by Equality Constraints

Model Reduction of Linear Systems by PSO Technique based on ISE, Circumvented by Equality Constraints

Optimality of Balanced Proper Orthogonal Decomposition for Data Reconstruction

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.

Model Reduction of Linear Systems by Conventional and Evolutionary Techniques

Model Reduction of Linear Systems by Conventional and Evolutionary Techniques

Switched Systems Reduction Framework Based on Convex Combination of Generalized Gramians

A general method for model-order reduction of switched linear dynamical systems is presented. The proposed technique uses convex generalized gramian which is a convex combination of the generalized gramians. It is shown that different classical reduction methods can be developed into the generalized gramian framework for model reduction of linear systems and further for the reduction of switched systems by construction of the convex generalized gramian. Balanced reduction within specified frequency bound is taken as an example which is developed within this framework. In order to avoid numerical instability and also to increase the numerical efficiency, convex generalized gramian-based Petrov-Galerkin projection is constructed instead of the similarity transform approach for reduction. It is proven that the method preserves the stability of the original switched system at least for stabilizing switching signal and it is also less conservative than the method which is based on the common generalized gramian. Some discussions on the coefficient of the vertices of the convex variables are presented. The performance of the proposed method is illustrated by numerical examples.

Model reduction of interconnected linear systems

AbstractThe 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.

ℋ ∞ model reduction of linear systems with distributed delay

This paper is concerned with the problem of ℋ ∞ model reduction for linear distributed-delay systems. For a given stable system containing both discrete and distributed delays in the state, our attention is focused on the construction of reduced-order model of the same structure, which guarantees the corresponding error system to be asymptotically stable and has a prescribed H∞ norm error performance. This problem is solved through a delay-dependent approach. By defining a new Lyapunov functional and making use of the Newton-Leibniz formula, a delay-dependent bounded real lemma is first established in terms of linear matrix inequalities (LMIs). Based on this performance criterion and by employing the projection lemma, sufficient conditions are formulated for the existence of admissible reduced-order models. Owing to the presence of matrix inverse constraints, these conditions are intrinsically nonconvex. An algorithm based on the sequential linear programming matrix method is proposed for verifying these conditions, which can be readily implemented by using standard numerical software. Desired reduced-order models can be constructed if these conditions are satisfied. The effectiveness of the proposed model reduction method is illustrated via a numerical example.

Hankel norm approximation of linear systems with time-varying delay: continuous and discrete cases

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time cases are considered. A numerical example is provided to show the effectiveness of the proposed theory.

A Simplified Approach for Biased Model Reduction of Linear Systems in Special Canonical Form

A new method is proposed for the modelling of high-order linear continuous time systems represented in special canonical form. New algorithms are developed to generate low order biased models which retain both the initial time-moments and Markov parameters of the original high-order system to give better approximation in both transient and steady state parts of the time response. The procedure is computationally simple and direct and avoids the necessity of matrix inversions and multiplications of the original high order system matrices. The new method avoids the calculation of the time moments and Markov parameters of the original high order system beforehand unlike many of the available methods for the model reduction in canonical form. The proposed method is digital computer oriented and very useful for the analysis and design of high order systems in special canonical form. The method is illustrated through typical numerical examples.

System simplification using variable structure reduced models

A practical method of model reduction for linear systems is proposed. To minimize the order of the reduced plant model and to achieve a very good approximation of the original system, the reduced model of the original system is constructed from two submodels. One model emphasizes the transient behaviour, and the other gives a very good steady-state approximation. The submodels are estimated using the techniques of frequency response matching and dominant energy modes analysis. The main features of the proposed method are that the order of the reduced model can be minimized and the reduced model can give good approximations in both the transient and steady-state responses of the original system. Moreover, applications of the proposed reduced model to control system design can lead to good performances in the transient and steady-state periods. Some simulated examples arc given to demonstrate the effectiveness of the proposed method.

Optimal Hankel Norm Model Reductions and Wiener–Hopf Factorization I: The Canonical Case

We consider the problem for discrete time systems of approximating a given stable rational matrix function $K(z) = \sum\nolimits_{j = 1}^\infty {K_j z^{ - j} } $ of McMillan degree n by a function $\hat K(z) + H(z)$, where $\hat K$ has McMillan degree $l < n$ and H is antistable; we include the case $l = 0$ in which we then take $\hat K = 0$. The minimum possible $L^\infty $-norm (on the unit circle) of the error $\| {K - \hat K - H} \|_{L^\infty } $, or equivalently the minimum possible spectral norm of the induced Hankel matrix $\| {\mathcal{H}_{K - \hat K} } \|$, is known to be equal to the $(l + 1)$st singular value $\sigma _{l + 1} (K)$ of the Hankel matrix $\mathcal{H}_K = [K_{i + j - 1} ]_{i,j} $. Assume $\sigma _{l + 1} (K) < \sigma _l (K)$ and choose the number $\sigma $ to satisfy $\sigma _{l + 1} (K) < \sigma < \sigma _l (K)$; if $l = 0$, the condition is simply $\sigma _1 (K) = \| {\mathcal{H}_K } \| < \sigma $. We give an explicit linear fractional map parametrization of the class of all functions $\hat K + H$ as above which satisfy $\| {K - \hat K - H} \|_{L^\infty } \leqq \sigma $. The coefficients of the linear fractional map are completely determined by the matrices A, B, C in a realization $K(z) = C(zI - A)^{ - 1} B$ for $K(z)$ and the observability and controllability gramians for the discrete time system $(A,B,C)$. The analogous results for continuous time systems are derived by a linear fractional change of variable; in this way we recover some recent results of Glover. The basic idea is to use the Grassmannian approach of Ball and Helton to reduce the problem to one of spectral factorization; this in turn can be solved by the geometric factorization principle of Bart, Gohberg, Kaashoek and van Dooren. Known applications include sensitivity minimization in $H^\infty $ control theory and model reduction for linear systems.

Model reduction by power decomposition and frequency response matching

A mixed method of linear system model reduction, retaining the advantages of the power decomposition method and the frequency response matching method, is proposed. From the viewpoint of energy contributions to the system output, the dynamic modes with dominant energy contributions instead of those with dominant eigenvalues will be preserved. Having determined the denominator of the reduced model, the parameters of the numerator are found by using the frequency response matching technique.