Norm Model Reduction Research Articles

SSD – Software for Systems with Delays: Reproducible Examples and Benchmarks on Model Reduction and H2 Norm Computation

We present SSD – Software for Systems with Delays, a de novo MATLAB package for the analysis and model reduction of retarded time delay systems (RTDS). Underneath, our delay system object bridges RTDS representation and Linear Fractional Transformation (LFT) representation of MATLAB. This allows seamless use of many available visualizations of MATLAB. In addition, we implemented a set of key functionalities such as H2 norm and system gramian computations, balanced realization and reduction by direct integral definitions and utilizing sparse computation. As a theoretical contribution, we extend the frequency-limited balanced reduction to delay systems first time, propose a computational algorithm and give its implementation. We collected two sets of benchmark problems on H2 norm computation and model reduction. SSD is publicly available in GitHub∗. Our reproducible paper and two benchmark collections are shared as executable notebooks.∗https://github.com/gumussoysuat/ssd

Optimal Hankel norm model reduction for discrete-time descriptor systems

Optimal Hankel norm model reduction for dynamical systems is of great significance in model-based simulation and design. For the class of linear time-invariant systems, it is among the few optimal reduction methods for which a prior error bound between the original system and its approximation is known. However, for descriptor systems, this optimal approximation technique no longer applies. In this paper, we propose several definitions of the Hankel operator for dynamical discrete-time descriptor systems. We investigate the implications of these definitions for the problem of optimal model approximation of descriptor systems in the sense of the Hankel norm. Novel reduction algorithms are derived for this class of systems with and without preservation of the DAE-index. The performance of the proposed methods is illustrated by numerical examples.

Hilbert-Schmidt-Hankel norm model reduction for matrix second-order linear systems

This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.

H2 and H∞ optimal model reduction using genetic algorithms

The mathematical modeling of most physical systems, such as aerospace systems, heat processes, telecommunication systems, transmission lines and chemical reactors, results in complex high order models. The complexity of the models imposes a lot of difficulties in analysis, simulation and control designs. Several analytical model reduction techniques have been proposed in literature over the past few decades to reduce these difficulties. However, most of the optimal techniques follow computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Genetic Algorithm (GA) has proved to be an excellent optimization tool in the past few years. Therefore, the aim of this paper will be to use GA to solve H 2 and H ∞ norm model reduction problems, and help obtain globally optimized nominal models.

Global optimization for the -norm model reduction problem

A branch and bound algorithm is proposed to solve the -norm model reduction problem for continuous and discrete-time linear systems, with convergence to the global optimum in a finite time. The lower and upper bounds in the optimization procedure are described by linear matrix inequalities (LMI). Also proposed are two methods with which to reduce the convergence time of the branch and bound algorithm: the first one uses the Hankel singular values as a sufficient condition to stop the algorithm, providing to the method a fast convergence to the global optimum. The second one assumes that the reduced model is in the controllable or observable canonical form. The -norm of the error between the original model and the reduced model is considered. Examples illustrate the application of the proposed method.

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.

Design of optimal reduced order [formula omitted] filters

This paper shows that the design of a reduced order optimal H 2 filter is equivalent to a frequency weighted L 2 norm model reduction problem.

State space identification of stable systems

This paper studies a two stage approach to state space identification with certain optimality properties. The first stage is essentially a linear correlation stage via orthonormal filtering to reduce the problem to a setup with small equivalent noise. The second stage is an optimal Hankel norm model reduction step. This two stage algorithm is shown to be consistent for a large class of stable, possibly infinite dimensional, linear time-invariant systems under a very mild condition on the additive noise. The consistency analysis is based on a non-probabilistic framework for noise which allows the inclusion of large classes of non-ergodic and non-stationary noise, possibly having deterministic components, with maximum transparency. Furthermore, l and H convergence results are derived for a popular class of ortho1 normal basis expansions based on shift operators. These bases include, e.g. Laguerre bases.

Chebycheff approximation in system identification and model reduction

The aim of this paper is to show how Chebycheff approximation may be used in frequency weighted H ∞ system identification and model reduction. We show how these techniques may be combined with known results from Hankel norm model reduction theory to reduce the error between the plant and the low-order approximation. In identification we also apply an L ∞-norm variant of Levy's method (Levy, 1959).

Induced [formula omitted]2 norm model reduction of polytopic uncertain linear systems

In this paper we study an induced L 2 norm model reduction problem for polytopic uncertain linear systems. The polytopic system has its state-space data contained in a convex polytope, a situation which often arises. The problem we address is to approximate this type of system by a lower order polytopic uncertain linear system with a guaranteed induced L 2 norm error. A sufficient solvability condition is provided in terms of LMIs with one extra coupling rank constraint, which generally leads to a nonconvex feasibility problem. We use an improved computational scheme based on the alternating projection method to obtain solutions to the problem, but only local convergence is guaranteed. The reduced-order uncertain system with its associated polytope is given explicitly by the solution of the feasibility problem.

(J, J?)-Spectral factorization and conjugation for discrete-time descriptor system

This paper considers a (J, J?)-spectral factorization of a signed spectral matrix associated with a discrete-time improper transfer matrix by using the descriptor system representation and the generalized algebraic Riccati equation (GARE). Under the assumption that the improper transfer matrix is stable, a numerical algorithm for (J, J?)-spectral factorization is developed based on solutions of the generalized eigenvalue problem (GEP) and a related matrix equation. For an unstable transfer matrix, aJ-unitary conjugation method is applied to obtain a stable transfer matrix generating the same signed spectral matrix. Then a Hankel norm model reduction problem is briefly discussed. A simple example is also included to illustrate the numerical procedure.

Frequency-weighted ℒ/sub ∞/ norm and optimal Hankel norm model reduction

A new relative error model reduction method is proposed using frequency-weighted balanced realization, and explicit /spl Lscr//sub /spl infin// norm error bounds are also derived for the relative error and multiplicative error. The method only needs to solve two Lyapunov equations. It is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase. This paper also gives a complete solution to the frequency-weighted Hankel norm approximation with antistable weighting. These results are then applied to /spl Lscr//sub /spl infin// norm model reduction, and several numerically effective algorithms are proposed. It is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A duality theory for robust stabilization and model reduction

Fuhrmann (1991, 1993) developed a duality theory in the context of Hanel norm approximation and Nehari complementation problems. The class of functions involved were the scalar, antistable transfer functions. This work was extended, using normalized coprime factorizations, by Fuhrmann and Ober (1993a) to the class of all minimal transfer functions. In this paper we extend the scope of the duality theory significantly. The paper presents a unified approach to problems of Hankel norm approximation, model reduction, and robust control of rational multivariable transfer functions. The unification is achieved by considering two classes of transfer functions and corresponding normalized coprime factorization. Using the Youla-Kucera parametrization of all stabilizing controllers, we single out a unique controller by imposing a McMillan degree minimization restriction on the doubly coprime factorizations. With this controller we construct an associated stable transfer function which we call the characteristic function. Many problems on the original system can be reduced to the study of the characteristic function.

Characterization of the solution to the optimal H∞ model reduction problem

In this paper the H ∞ norm approximation of a given stable, proper, rational transfer function by a lower order stable proper rational transfer function is studied. The solution method is based on the authors observation that the H ∞ model reduction problem can be converted into a Hankel norm model reduction problem. A simple characterization of the solution to the optimal H ∞ model reduction problem is developed.

Parametric interpolation, H ∞-control and model reduction

Adamjan et al. (1971) established an important equivalence between the optimal Hankel norm model reduction problem (the Schur-Takagi problem) and a classical interpolation problem due to Ahiezer. In two later papers, Ball (1983) and Ball and Helton (1985) investigated a vector version of this interpolation problem, using the sophisticated Ball-Helton operator theory. A parametric approach to the work of Adamjan et al. (1971) was studied by Genin and Kung (1981) in the rational case. We demonstrate that a class of H ∞-optimal control problems and the Hankel norm model reduction problem can be recast as parametric vector interpolation problems (PVIPs), which we solve using the classical Schur construction and results on the inertia of matrices. The connection between parametric vector interpolation and model reduction provides an alternative approach to the PVIP using Glover's state-space formulae (Glover 1984). In both cases, special care needs to be taken at certain ‘non-regular’ values of a gain parameter, a difficulty not recognized in the non-parametric work (e.g. Delsarte et al. 1981).

L ∞ error bound for the phase matching approximation (the one-step-at-a-time Hankel norm model reduction version)

Parallel to what was done for the balanced model reduction version of the phase approximation problem (see Li and Jonckheere 1987), this paper provides an L ∞ bound on the phase approximation error when one-step-at-a-time optimal Hankel norm model reduction is used to reduce the stable projection of the phase function matrix. The one-step-at-a-time Hankel norm procedure allows for an L ∞ phase error bound twice as small as that of the balanced version.