Balanced Model Reduction Research Articles

Design of 3-D Noncausal Filters with Small Roundoff Noiseand No Overflow Oscillations

The contribution of this paper consists of two individual parts. First, an invertible mapping technique is presented for 3-D digital system design, and it is applied to approximate 3-D noncausal filters in the spatial domain. Secondly, an algorithm is proposed for obtaining a structure for 3-D IIR filters with small roundoff noise and no overflow oscillations. The design of noncausal filters can be carried out by three steps: 1), a given noncausal impulse response is transformed into the first octant using the proposed 3-D invertible mapping technique; 2), the transformed impulse response in the first octant is approximated by balanced model reduction of 3-D separable denominator systems;3) , the resultant 3-D IIR filter is transformed back to the original coordinates.

Dynamic analysis and reduced order modelling of flexible rotor-bearing systems

The application of the balanced model reduction techniques to complex rotor-bearing systems is investigated. It is demonstrated that the balanced model reduction techniques can be successfully applied to the actual rotor-bearing systems. Numerical examples are solved to demonstrate the validity and efficiency of the reduced order model in representing the dynamics of the actual rotor system.

Least-squares approximation of FIR by IIR digital filters

An algorithm is presented for the least-squares approximation of FIR filters by IIR filters. The algorithm is an iterative procedure where each iteration requires the solution of an overdetermined set of linear equations and some digital filtering operations. All calculations are performed with the numerator and denominator coefficients of the transfer functions. A conversion to state-space descriptions is not necessary. Examples show that the approximation error is as small as that of the IIR filters obtained with balanced model reduction. Moreover, the effects of numerical errors are negligible. Thus, our algorithm is applicable even in cases where the FIR filter length is large.

Balanced Model Reduction and Control of Rotor-Bearing Systems

An effective technique is applied to the suppression of vibrations in flexible rotor-bearing systems with small gyroscopic effects. A balanced linear-quadratic-Gaussian (LQG) controller design procedure is implemented. The size of the controller is reduced in two stages by using (i) a balanced model reduction, and (ii) an LQG balanced reduction. The condition for a gyroscopic matrix is developed that allows one to ignore the rotor gyroscopic effects in the process of the controller design, although they are included in the rotor dynamics. The approach is illustrated on a typical rotor-bearing system represented by a 48 degree-of-freedom finite element model.

Magnitude response peak detection and control using balanced model reduction and leakage to a target

Adaptive filters are often used in systems that need to adjust to unknown environments. Communication channels with frequency nulls, signals that lack energy in a frequency band, and transducers with a finite bandwidth present special problems since adaptive filters can develop a large gain at frequencies where excitation is lacking. Such magnitude response peaks can cause problems if unchecked. This paper suggests a procedure for detecting and controlling magnitude response peaks that uses a balanced model reduction technique to form a low-order IIR filter that approximates the performance of the filter. The poles are then studied to determine if magnitude response peaks are present. If a peak is detected, then to a is used to gradually reduce the peak with minimal effect on the equalizer's response at other frequencies. Several useful bounds on the equalizer frequency response magnitude are derived, and the frequency domain behavior of the leakage to a target algorithm is analyzed. A case study is provided.

Balanced model reduction of symmetric composite systems

In this paper, we investigate the structural properties of balanced realizations, and their truncation, of a class of systems with symmetric interconnected and identical structure. It is established that the balanced realizations of such symmetric composite systems can be constructed via the balanced realizations of two auxiliary systems of comparatively low dimensions. The corresponding truncated realizations, which also have a certain symmetric structure, are explicitly constructed. The extension of the results to generalized symmetric composite systems is also studied.

Identification of highly accurate low order state space models in the frequency domain

This paper discusses an integrated approach for high accuracy and low order model identification. The approach integrates subspace-based identification algorithms with model reduction and parameter estimation algorithms to generate highly accurate low order models. The specific identification algorithm presented in the paper is the integrated frequency domain observability range space extraction and least square parameter estimation algorithm (IFORSELS). IFORSELS is an iterative algorithm which integrates the frequency domain observability range space extraction (FORSE) algorithm, the balanced realization model reduction algorithm, and the logarithmic and additive least square parameter estimation algorithms in an iterative fashion. It is capable of achieving much higher modeling accuracy using a lower order model than that achieved using a higher order model by subspace-based algorithms, such as FORSE.

Bilinear transformation and generalized singular perturbation model reduction

A new general bilinear relationship is found between continuous and discrete generalized singular perturbation reduced-order models. This result is applied to the problem of deriving discrete analogs of continuous singular perturbation and direct truncation model reduction and leads to a new definition of discrete "Nyquist" model reduction. Also, "unit circle" bilinear transformations are used to relate several known facts about continuous and discrete balanced model reduction and incorporate them into a symmetrical, unified framework.

Balanced realisations and model reduction of periodically time-varying state-space digital filters

The authors discuss balanced realisations and model reduction of periodically time-varying (PTV) state-space digital filters. Controllability and observability Grammians of PTV state-space digital filters are discussed. It is extremely interesting to notice that although PTV state-space digital filters can be implemented by using a group of time-invariant coefficient sets, controllability and observability Grammians cannot be evaluated independently by using any one set of these time-invariant coefficients. Also, important physical interpretations of controllability and observability Grammians are considered. Based on these analysis results, balanced realisations of PTV state-space digital filters are defined and a synthesis method for balanced realisations is proposed. As one application of balanced realisations, a reduced-order model of a PTV state-space digital filter can be obtained by taking a subsystem of balanced realisation of the PTV state-space digital filter. Finally, a numerical example is given to illustrate the balanced model reduction procedure.

Model reduction of multidimensional and uncertain systems

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.

Model reduction with relative/multiplicative error bounds and relations to controller reduction

Balanced model reduction with a priori relative/multiplicative error bounds in L/sub /spl infin// norm is studied. Inverse-weighted balanced truncation (IWBT) is proposed that is a variant of the Enns' algorithm. It is shown that IWBT is equivalent to the balanced stochastic truncation (BST). This equivalence leads to the extension of an improved relative/multiplicative error bound for BST obtained by Wang and Safonov to general nonsquare transfer matrices. Natural relations between IWBT and observer-based controller reduction are investigated which motivate the simultaneous reduction of the plant and controller. Robust stability conditions for simultaneous reduction of the plant and the controller are established for the closed-loop system consisting of full-order plant and reduced-order controller. As a by-product, robust stability and stabilization problems are solved for general coprime factor plant descriptions with relative/multiplicative H/sub /spl infin// norm bounded uncertainties. Similar results are obtained for general coprime factors of the feedback controller involving relative/multiplicative H/sub /spl infin// norm bounded uncertainties. >

A note on the "Approximation of FIR by IIR digital filters: an algorithm based on balanced model reduction" [Comment and reply

In the present paper Rudko extends the algorithm presented by Beliczynski et al. (see ibid., vol.40, p.532, 1992), based on balanced order reduction, for the conversion of a narrow-band linear phase FIR filter to an almost linear phase IIR filter. It is shown that the algorithm can be used to reduce the order of wideband FIR filters. Beliczynski et al. reply to the Comment. >

Model reduction and process control of thermomechanical behaviour of non‐linear material deformation

AbstractThis paper describes a design procedure for metal forming processes by using the controllable subspace of the full system. The velocity profile of the moving die is designed using the reduced order system. The metal forming processes are simulated using non‐linear finite element methods based on the rigid viscoplastic flow formulation. The balanced model reduction technique is applied to reduce the full state space model to a reduced order model that retains the controllable subspace of the thermomechanical system. The linear quadratic regulator theory with output tracking is used as an off‐line design tool to design the die velocity schedule. The process design is carried out to maintain the strain rate of the critical portion of the billet at a desired value. The procedure for designing the process parameters is demonstrated using two case studies.

On “Subspace Methods” Identification and Stochastic Model Reduction

In this paper the problem of stochastic model identification from estimated covariances is considered. In this context, we analyze a class of popular subspace identification procedures in the theoretical framework of rational covariance extension and balanced model reduction, and we demonstrate that they are based on a hidden assumption which is not entirely natural. We also show that the balanced truncation procedure proposed by Desai and Pal preserves positivity but not balancing.

Balanced realization and model reduction of singular systems

The problem of balanced realization and model reduction of a singular system of the form E[xdot] = Ax + Bu, where E is a singular matrix, is considered. Using coordinate transformation, which can be computed by performing singular value decomposition of E, we derive our first approach to the balancing of singular systems. The second approach is based on standard form decomposition of singular systems to slow and fast subsystems and performing balanced realization on the decomposed model. In this sense model reduction can be established in two steps: first, by decomposing the singular system and second, by performing balancing transformation on the decomposed subsystems.

Design of 3‐D separable‐denominator digital filters using minimal decomposition and balanced realization

AbstractBased on a concept of minimal decomposition and balanced model reduction, a technique is developed for designing three‐dimensional (3‐D) separable‐denominator digital filters in the spatial domain. This is done by decomposing a desired 3‐D FIR filter into three parts and by representing it by a cascade connection of three 1‐D filters. The minimal decomposition treated here can be viewed as an extension of that in the 2‐D separable‐denominator case. A numerical example also is given to illustrate the utility of the proposed technique. The amount of calculation required here is relatively small and the stability of the resulting filter is always guaranteed.

Generalized moment method for the order reduction of multivariable systems

A generalized formulation of the generalized moment method is used to reduce a 10th-order power system model to 4th-order. The results are compared to those obtained by Bandekas and Papadopoulos who recently attempted to solve the same problem. Results obtained using balanced model reduction are also included.

A New Square-root Balancing-Free Stochastic Truncation Model Reduction Algorithm

The paper proposes a new computational approach with enhanced numerical robustness for computing reduced order models of continuous systems by using the balanced stochastic truncation model reduction method. The new approach circumvents the computation of possibly III-conditioned stochastic balancing transformations. Instead, well-conditioned projection matrices are determined for computing directly the state-space representations of the reduced order models. The projection matrices are computed in a numerically reliable way, by using exclusively the Cholesky (square-root) factors of systems Gramians. The proposed algorithm can handle both minimal and non-minimal systems.

Model reduction of 2-D systems with frequency error bounds

Some explicit frequency domain infinity -norm error bounds for 2-D model reduction of denominator separable systems are derived. They are based on 1-D balanced model reduction and Hankel approximation methods. The evaluation of these error bounds only involves the computation of Hankel singular values of two 1-D transfer matrices. >

Transient approximation of a bilinear two-area interconnected power system

A balancing model reduction scheme for high order bilinear systems, similar to the linear balanced reduction algorithm, is applied to power system modelling. An original 17th-order two-area interconnected bilinear power system is reduced, using both linear and bilinear balancing algorithms, to a 10th-order reduced model. Overall superiority of the bilinear reduction scheme over linear balancing is observed in the performed simulation study. The reduction scheme also leads to a very acceptable approximate response compared with that of the original system.