Identification/reduction to a balanced realization via the extendedimpulse response gramian

A new identification/reduction algorithm for linear, discrete time-invariant (LDTI) systems is proposed which is based on the extended impulse response gramian first defined here for LDTI systems. The reduction algorithm employs singular value decomposition to retain states corresponding to dominant singular values of these gramians. The proven properties of the reduced order models include convergence to a balanced realization with conditional controllability, observability, and asymptotic stability. A suboptimal property of the model (in minimizing an impulse response error norm) is also demonstrated. The proposed technique can handle impulse response data of deterministic or stochastic nature for system identification application     

Model reduction via time-interval balanced stochastic truncation for linear time invariant systems

In this article, a new method for model reduction of linear dynamical systems is presented. The proposed technique is from the family of gramian-based relative error model reduction methods. The method uses time-interval gramians in the reduction procedure rather than ordinary gramians and in such a way it improves the accuracy of the approximation within the time interval which is applied. It is proven that the reduced order model is stable when the proposed method applies to a stable system. The method uses a recently proposed inner–outer factorisation algorithm which enhances the numerical accuracy and efficiency. In order to avoid numerical instability and also to further increase the numerical efficiency, projector matrices are constructed instead of the similarity transform approach for reduction. The method is illustrated by a numerical example and finally it is applied to a practical CD player example. The numerical results show that the method is more accurate than ordinary balanced stochastic truncation.     

A note on the closed-form determination of the gramians of a stable time-invariant linear system

A method of determining the gramians of a stable time-invariant linear system based on the expansion of the input-state and state-output impulse responses of the system is given. The procedure presented is shown to be computationally more efficient than previous methods when working with symbols or exact rational numbers.     

Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems

This paper presents a framework for nonlinear systems analysis that is based upon controllability and observability covariance matrices. These matrices are introduced in the paper and it is shown that gramians for linear systems form special cases of the covariance matrices. The covariance matrices can be transformed via a balancing-like transformation and nonlinearity measures are defined based upon these transformed covariance matrices. Subsequently, the covariance matrices are used for reduction of the nonlinear model. It is shown that the model reduction procedure reduces to balanced model truncation for linear systems for impulse inputs. Furthermore, it is also shown that several model reduction procedures that were developed by other researchers, and assumed to be independent from one another, are related. The findings are illustrated with an example.     

Legendre orthogonal polynomials approximation of system gramians and its application to balanced truncation

The paper presents a procedure for computation of system gramians by using Legendre orthogonal polynomials approximation of the system state impulse and output responses. The proposed approach is trajectory based and relies on the system state and output trajectories snapshots selected either by experiment or computer simulation. It is defined in deterministic settings as opposed to similar approaches defined in stochastic settings by using the data covariance matrix. The advantage of using orthogonal series approximation for the gramians is to avoid solving the usual Lyapunov equations. The proposed method can be equally well applied to linear time-invariant as well as time-varying systems, and even to unstable systems, since the gramians approximation is performed on a finite interval of time. When the observation interval contains the whole energy of the system state impulse and output responses, the proposed method gives similar results as the gramians computed by solving Lyapunov equations. Several experiments are performed showing the good approximation properties of the presented method.     

Time-interval model reduction of bilinear systems

In this paper, a new method for model reduction of bilinear systems is presented. The proposed technique is from the family of gramian-based model reduction methods. The method uses time-interval generalised gramians in the reduction procedure rather than the ordinary generalised gramians and in such a way that it improves the accuracy of the approximation within the time-interval for which the method is applied. The time-interval generalised gramians are the solutions to the generalised time-interval Lyapunov equations. The conditions for these equations to be solvable are derived and an algorithm is proposed to solve these equations iteratively. The method is further illustrated with the help of two illustrative examples. The numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians.     

On gramians and balanced truncation of discrete-time bilinear systems

This paper obtains a balanced truncation model reduction method for discrete-time bilinear systems. After deriving the sufficient conditions for the discrete-time bilinear systems to lead to bounded outputs for some bounded inputs, we study the properties of the reachability and observability gramians. The gramians can be computed by solving two generalized Lyapunov equations. By balancing the reachability and observability gramians, a model reduction method is obtained. Some propositions of the reduced order models are given. A numerical example is given to illustrate the effectiveness of the proposed method.     

A Survey of Model Reduction by Balanced Truncation and Some New Results

Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey of balancing related model reduction methods and their corresponding error norms, and also introduce some new results. Five balancing methods are studied: (1) Lyapunov balancing, (2) stochastic balancing, (3) bounded real balancing, (4) positive real balancing and (5) frequency weighted balancing. For positive real balancing, we introduce a multiplicative-type error bound. Moreover, for a certain subclass of positive real systems, a modified positive-real balancing scheme with an absolute error bound is proposed. We also develop a new frequency-weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians. Two numerical examples are illustrated to verify the efficiency of the proposed methods.     

Model reduction based on limited‐time interval impulse response Gramians

In this paper, three new Gramians are introduced namely ‐ limited‐time interval impulse response Gramians (LTIRG), generalized limited‐time Gramians (GLTG) and generalized limited‐time impulse response Gramians (GLTIRG). GLTG and GLTIRG are applicable to both unstable systems and also to systems which have eigenvalues of opposite polarities and equal magnitude. The concept of these Gramians is utilized to develop model reduction algorithms for linear time‐invariant continuous‐time single‐input single‐output (SISO) systems. In the cases of GLTIRG and GLTG based model reduction, the standard time‐limited Gramians are generalized to be applied to unstable systems by transforming the original system into a new system which requires the solution of two Riccati equations. Two numerical examples are included to illustrate the proposed methods. The results are also compared with standard techniques.     

Factorization‐based frequency‐weighted optimal Hankel‐norm model reduction

In this paper, we present frequency‐weighted optimal Hankel‐norm model reduction algorithms for linear time‐invariant continuous‐time systems by representing an original higher‐order system into new fictitious systems. The new system representations are derived through factorization of the resulting sub‐matrices that are obtained after transformations. As the proposed approaches are factorization dependent, additional results with both approaches are included using another factorization of the fictitious input–output and weight matrices. The proposed algorithms generate stable reduced models with double‐sided weights and provide a substantial improvement in the weighted error. A numerical example is given to compare the efficacy of the proposed algorithms with the well‐known frequency‐weighted techniques.     

On Recursive identification of linear and non-linear systems: case of coloured noise

Recursive identification algorithms based on least square estimation procedure have been presented for both linear and a class of nonlinear systems. The linear system is represented by an impulse response model while the non-linear system is a cascade combination of a. linear part and u zero memory non-linear part. It has been shown that even in the presence of coloured noise, for the model selected, ordinary least square estimation gives unbiased estimates of the system parameters. Results are illustrated through suitable numerical examples.     

Structured low-rank approximation and its applications

Fitting data by a bounded complexity linear model is equivalent to low-rank approximation of a matrix constructed from the data. The data matrix being Hankel structured is equivalent to the existence of a linear time-invariant system that fits the data and the rank constraint is related to a bound on the model complexity. In the special case of fitting by a static model, the data matrix and its low-rank approximation are unstructured.We outline applications in system theory (approximate realization, model reduction, output error, and errors-in-variables identification), signal processing (harmonic retrieval, sum-of-damped exponentials, and finite impulse response modeling), and computer algebra (approximate common divisor). Algorithms based on heuristics and local optimization methods are presented. Generalizations of the low-rank approximation problem result from different approximation criteria (e.g., weighted norm) and constraints on the data matrix (e.g., nonnegativity). Related problems are rank minimization and structured pseudospectra.     

A time-domain weighted linear successive interference cancellation detector for rapidly time-varying OFDMA communication systems

In this work, a weighted linear SIC (WLSIC) multi-user detector that is suitable for large-dimension rapidly time-varying communication systems is introduced. The proposed scheme exhibits low computational complexity and can converge to either the decorrelator or the LMMSE detector for two distinct values of the weighting factor. It uses a weighting factor to determine which average BER level the linear SIC converges to. We study the convergence behavior of the proposed scheme and prove that the latter is convergent if the weighting factor is between 0 and 1. Moreover, we propose a reduced complexity version of the WLSIC multi-user detector in which the complexity is reduced by 15% compared to the original WLSIC structure. Finally, simulation results obtained by applying the proposed WLSIC multi-user detector to cancel the inter-carrier interference (ICI) in an uplink OFDMA system coincide well with our theoretical findings.     

Model reduction of discrete Markovian jump systems with time‐weighted H2 performance

This paper is concerned with the optimal time‐weighted H₂ model reduction problem for discrete Markovian jump linear systems (MJLSs). The purpose is to find a mean square stable MJLS of lower order such that the time‐weighted H₂ norm of the corresponding error system is minimized for a given mean square stable discrete MJLSs. The notation of time‐weighted H₂ norm of discrete MJLS is defined for the first time, and then a computational formula of this norm is given, which requires the solution of two sets of recursive discrete Markovian jump Lyapunov‐type linear matrix equations. Based on the time‐weighted H₂ norm formula, we propose a gradient flow method to solve the optimal time‐weighted H₂ model reduction problem. A necessary condition for minimality is derived, which generalizes the standard result for systems when Markov jumps and the time‐weighting term do not appear. Finally, numerical examples are used to illustrate the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Convolutional solutions of partial difference equations

A significant part of the theory of one-dimensional linear shift-invariant systems is based on the concept of weighting function (or impulse response): the output is the convolution of the weighting function with the input. This paper introduces the concept of linear translation-invariant systems and uses this notion in studying impulse response, z-transforms, and transfer functions for multidimensional systems.     

Routh approximations for reducing order of linear, time-invariant systems

A new method of approximating the transfer function of a high-order linear system by one of lower order is proposed. Called the "Routh approximation method" because it is based on an expansion that uses the Routh table of the original transfer function, the method has a number of useful properties: if the original transfer function is stable, then all approximants are stable; the sequence of approximants converge monotonically to the original in terms of "impulse response" energy; the approximants are partial Padé approximants in the sense that the firstkcoefficients of the power series expansions of thekth-order approximant and of the original are equal; the poles and zeros of the approximants move toward the poles and zeros of the original as the order of the approximation is increased. A numerical example is given for the calculation of the Routh approximants of a fourth-order transfer function and for illustration of some of the properties.     

Closed form ℒ 2/ℋ2-optimising of zeros for model reduction of linear continuous time systems

Closed form expressions of transfer function responses are applied in this article to model reduction of nth order continuous time systems with respect to the location of zeros for a given location of poles. Explicit closed form formulae are derived for the numerator coefficients in the transfer function of the reduced system that minimise the integrated square deviation from the original system with respect to the impulse response or higher order responses, i.e. effective ?₂ or ?₂-optimisation. The relative degree of the reduced model can be selected freely, e.g. as the original model's one, by selecting the number of numerator coefficients. Constraints are dealt with by introducing a vector of Lagrange multipliers corresponding to the response order. The formulae are also related to solutions of Lyapunov and Sylvester equations based on the companion matrices of the original and reduced systems. The formulae derived can be used to enhance the results obtained from other reduction techniques such as those based on balanced Grammian reduction and singular value decomposition for mid-sized systems. Two examples, demonstrating this, are presented. The formulae can also be used as a basis for a more general optimisation approach, where the optimisation with respect to the numerator coefficients or the zeros, resulting in the solution of a linear system of equations, is combined with non-linear optimisation with respect to the denominator coefficients or the poles.     

Contextual weighted representations and indexing models for the retrieval of HTML documents

The diffusion of the World Wide Web (WWW) and the consequent increase in the production and exchange of textual information demand the development of effective information retrieval systems. The HyperText Markup Language (HTML) constitues a common basis for generating documents over the internet and the intranets. By means of the HTML the author is allowed to organize the text into subparts delimited by special tags; these subparts are then visualized by the HTML browser in distinct ways, i.e. with distinct typographical formats. In this paper a model for indexing HTML documents is proposed which exploits the role of tags in encoding the importance of their delimited text. Central to our model is a method to compute the significance degree of a term in a document by weighting the term instances according to the tags in which they occur. The indexing model proposed is based on a contextual weighted representation of the document under consideration, by means of which a set of (normalized) numerical weights is assigned to the various tags containing the text. The weighted representation is contextual in the sense that the set of numerical weights assigned to the various tags and the respective text depend (other than on the tags themselves) on the particular document considered. By means of the contextual weighted representation our indexing model reflects not only the general syntactic structure of the HTML language but also the information conveyed by the particular way in which the author instantiates that general structure in the document under consideration. We discuss two different forms of contextual weighting: the first is based on a linear weighted representation and is closer to the standard model of universal (i.e. non contextual) weighting; the second is based on a more complex non linear weighted representation and has a number of novel and interesting features.     

Approximation of multivariable linear systems with impulse response and autocorrelation sequences

This paper considers the construction of approximants of multi-input-multi-output, discrete-time linear systems from the finite data of the impulse response and autocorrelation sequences. In the approximation of a multivariable linear system, it is common practice to use a finite portion of its impulse response sequence. This is formally equivalent to the Padé approximation technique, which may produce unstable approximants, even though the original system is stable. Mullis and Roberts proposed a new method, which yields stable approximants, in connection with approximation of digital filters. This is, however, restricted to the single-input-single-output case. This paper extends their method to the multi-input-multi-output case and shows a fast recursive algorithm to construct stable approximants of linear systems.