Balanced Stochastic Truncation Research Articles

Relative error-based time-limited [formula omitted] model order reduction via oblique projection

In time-limited model order reduction, a reduced-order approximation of the original high-order model is obtained that accurately approximates the original model within the desired limited time interval. Accuracy outside that time interval is not that important. The error incurred when a reduced-order model is used as a surrogate for the original model can be quantified in absolute or relative terms to access the performance of the model reduction algorithm. The relative error is generally more meaningful than an absolute error because if the original and reduced systems’ responses are of small magnitude, the absolute error is small in magnitude as well. However, this does not necessarily mean that the reduced model is accurate. The relative error in such scenarios is useful and meaningful as it quantifies percentage error irrespective of the magnitude of the system’s response. In this paper, the necessary conditions for a local optimum of the time-limited H2 norm of the relative error system are derived. Inspired by these conditions, an oblique projection algorithm is proposed that ensures small H2-norm relative error within the desired time interval. Unlike the existing relative error-based model reduction algorithms, the proposed algorithm does not require solutions of large-scale Lyapunov and Riccati equations. The proposed algorithm is compared with time-limited balanced truncation, time-limited balanced stochastic truncation, and time-limited iterative Rational Krylov algorithm. Numerical results confirm the superiority of the proposed algorithm over these existing algorithms.

Dynamic Thermal Estimation Methodology for High-Performance 3-D MPSoC

In 3-D integrated circuits, accurate runtime sensing of on-chip temperature is required to establish dynamic thermal management instruction sets. Placement restrictions and excessive runtime thermal variations, however, compromise the performance and reliability of the sensor readings. Within this framework, a novel methodology for thermal estimation based on unscented Kalman filter, augmented only with a limited number of temperature sensors at a few selected locations, is proposed. In addition, we extend discontinuous Galerkin finite-element method to include coupling mechanism between neighboring grid cells for accurate thermal profile estimation and introduce a balanced stochastic truncation to find a low-dimensional but accurate approximation of the thermal network over the whole frequency domain. As the experimental results show, the runtime thermal estimation method reduces temperature estimation errors by an order of magnitude.

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 Counterexample to “ Positive Realness Preserving Model Reduction With ${\cal H}_{\infty}$ Norm Error Bounds”

We provide a counterexample to the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in “Positive realness preserving model reduction with H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm error bounds,” IEEE Trans. Circuits Syst, I, Fundam. Theory Appl., vol. 42, no. 1, pp. 23-29 (1995). The proof of the error bound is based on a lemma from an earlier paper, “A tighter relative-error bound for balanced stochastic truncation,” Syst. Control Lett., vol. 14, no. 4, 307-317 (1990), which we also demonstrate is false by our counterexample. The main result of this paper was already known in the literature to be false. We state a correct H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric.

Frequency-domain generelaized singular peruturbation method for relative error model order reduction

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in this paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.

Two Algorithms for Fast and Accurate Passivity-Preserving Model Order Reduction

This paper presents two recently developed algorithms for efficient model order reduction. Both algorithms enable the fast solution of continuous-time algebraic Riccati equations (CAREs) that constitute the bottleneck in the passivity-preserving balanced stochastic truncation (BST). The first algorithm is a Smith-method-based Newton algorithm, called Newton/Smith CARE, that exploits low-rank matrices commonly found in physical system modeling. The second algorithm is a project-and-balance scheme that utilizes dominant eigenspace projection, followed by a simultaneous solution of a pair of dual CAREs through completely separating the stable and unstable invariant subspaces of a Hamiltonian matrix. The algorithms can be applied individually or together. Numerical examples show the proposed algorithms offer significant computational savings and better accuracy in reduced-order models over those from conventional schemes.

Modeling of normalized coprime factors with ν-metric uncertainty

Modeling of uncertain systems with normalized coprime factor description is investigated, where the experimental data is given by a finite set of frequency response measurement samples of the open loop plant that is linear, time-invariant, and possibly infinite-dimensional. The objective is not only to identify the nominal model but also to quantify the modeling error with sup-norm bounds in frequency domain. The uncertainty to be identified and quantified is chosen as the /spl nu/-metric, proposed by Vinnicombe (1993), because of its compatibility with H/sub /spl infin//-based robust control. An algorithm is developed to model the normalized coprime factors of the given plant using techniques of discrete Fourier analysis and balanced stochastic truncation and is shown to be robust in the presence of the worst case noise. Upper bounds are derived for the associated modeling error based on the minimum a priori information of the underlying model set and of the noise level in the measurement data. A simulation example is used to illustrate the effectiveness of the proposed algorithm.

Multiplicative approximation of transfer functions with frequency weighting

An algorithm for transfer function order reduction is presented, which generalizes the balanced stochastic truncation algorithm to allow for input weighting. An example illustrates use of the algorithm to secure smaller dB error in selected frequency bands through the introduction of the weighting.

Positive realness preserving model reduction with H/sub /spl infin norm error bounds

Many physical systems that occur in applications are naturally passive, for example, mechanical systems with dual sensors and actuators, and electrical circuits with passive components. Taking advantage of this property, many controller schemes have been proposed with the property that the controller is strictly positive real. Due to design and implementation considerations, the plant or the controller may need to be approximated by a lower-order system. It is highly desirable for the reduced-order system to also possess the positive realness property to guarantee that the resulting closed-loop system remains stable. Motivated by this problem, this paper considers the general model-reduction problem for a positive real system under the constraint that the reduced system is also positive real. We present a solution based on the balanced stochastic truncation. When the higher-order system is strictly positive real, we derive an H/sub /spl infin norm bound on the approximation error. We also consider alternate approaches of approximating the spectral factors with associated H/sub /spl infin norm error bounds. An example is included to show the efficacy of this method and comparison with other approaches.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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. >

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 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.

Relative-error H/sub infinity / identification from autocorrelation data-a stochastic realization method

A variant of the balanced stochastic truncation (BST) method for approximated realization of power spectrum matrices is shown to form the basis for an identification procedure that is well-suited to the task of determining relative-error-bounded approximate plant models for use in control design from input-output cross correlation data. Central to the theory is a novel L/sub infinity /-norm bound on the relative-error between an exact realization of the data and BST approximate realization. >

Multiplicative-error bound for balanced stochastic truncation model reduction

The multiplicative error for the balanced stochastic truncation model reduction method is discussed and an H/sub infinity / norm error bound is presented. The result, together with a previous result of the authors (Syst. Contr. Lett., vol.14, p.307-14, 1990), establishes that the same error bound applies to the relative error and the multiplicative error.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Relative-error bound for discrete balanced stochastic truncation

Fundamental differences between discrete-time balanced stochastic truncation (BST) and its continuous-time counterpart are shown to preclude a straightforward application of bilinear mappings for directly generalizing existing continuous-time error bounds. Nevertheless, discrete-time BST is shown to satisfy precisely the same Lk error bound as continuous-time BST, although the discrete-time derivations are somewhat more complex.

A tighter relative-error bound for balanced stochastic truncation

The properties of balanced stochastic truncation (BST) model reduction for stable multi-input multi-output systems are reviewed and analyzed. A relative-error bound for BST is obtained as 2∑ i=r+1σ i/(1−σ i . This error bound is tighter than the relative-error bound previously derived by Green.

Model reduction for robust control: A schur relative error method

AbstractA numerically robust relative error method for state‐space model order reduction is described. Our algorithm is based on Desai's balanced stochastic truncation technique for which Green has obtained an L∞ relative error bound. However, unlike previous methods, our Schur method completely circumvents the numerically delicate initial step of obtaining a minimal balanced stochastic realization of the power spectrum matrix G(s)GT(−s).

A relative error bound for balanced stochastic truncation

Recently, several methods have appeared for the approximation of (power) spectra, notably balanced stochastic truncation (BST). It is shown that BST satisfies a relative error bound approximately twice the bound for the relative error method (REM) proper. This offers a quantitative basis for the observation that BST and REM produce similar reduced-order models. Balanced stochastic truncation can therefore be interpreted as providing a computationally simple algorithm for relative error approximation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Balanced stochastic realizations

The balanced stochastic realization is introduced as a balanced solution to the continuous time positive real equations and the dual positive real equations. The structure of the associated balanced spectral factors is derived, the main result being a product decomposition of the spectral factors. The properties of the stochastic model reduction technique of balanced stochastic truncation are considered. In particular it is shown that balanced stochastic truncation preserves the right half plane zeros of the spectral factors, leading to a transformation approach to the approximation of non-minimum-phase stochastic systems.