Singular Perturbation Approximation Research Articles

Balanced model order reduction for linear random dynamical systems driven by Lévy noise

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Levy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

Model order reduction of commensurate linear discrete-time fractional-order systems

This paper presents an approach to model order reduction of linear time-invariant discrete-time commensurate fractional-order state space systems by means of the Balanced Truncation Approximation and Singular Perturbation Approximation methods. Mathematical formulas for computation of controllability and observability Gramians for the fractional-order systems are derived. This enables an extension of classical SVD-originated model order reduction algorithms to model reduction of discrete-time fractional-order state space systems. Simulation experiment illustrates the efficiency of the introduced methodology.

Type II Singular Perturbation Approximation for Linear Systems with Lévy Noise

When solving linear stochastic partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the...

Well-posedness of a model of nonhomogeneous compressible–incompressible fluids

We propose a model of a density-dependent compressible–incompressible fluid, which is intended as a simplified version of models based on mixture theory as, for instance, those arising in the study of biofilms, tumor growth and vasculogenesis. Though our model is, in some sense, close to the density-dependent incompressible Euler equations, it presents some differences that require a different approach from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to our model, using the Leray projector. Besides, we show the convergence of both a continuous version of the Chorin–Temam projection method, viewed as a singular perturbation approximation, and the artificial compressibility method.

A Simple Formula Describing Impedance Artifacts due to the Size and Surface Resistance of a Reference-Electrode Wire in a Thin-Film Cell

Reference electrodes are used in testing and designing thin-film cells in order to distinguish the effects of the positive and negative electrodes and determine the sources of significant resistance (or impedance), but the reference electrode introduces some distortion into the measurement due to non-uniformity of the current distribution. This often arises due to edge effects, the size and placement of the reference electrode, or both. Two common geometries for placing reference electrodes are internally, between the cathode and anode, and externally at a distance from the cathode and anode. Each design introduces some level of distortion, which must be clarified. This work focusses on internally-placed wire reference electrodes and elucidates the artifacts in cell impedance measurements as a function of wire size and wire surface resistance. Published simulations of impedance artifacts rely on computationally-intensive computer simulations, but we develop a simple formula to accurately approximate these effects. The formula is derived using a singular perturbation approximation to the impedance and then combining it with a simple equivalent circuit. Some comparisons with detailed numerical simulations show the accuracy of the resulting formula as a function of the diameter of the reference wire and its surface resistance.

EMPIRICAL MODELLING OF ACTIVATED SLUDGE PROCESS VIA SYSTEM IDENTIFICATION

Activated sludge process is an important stage in Wastewater Treatment Plant (WWTP). In this study, model of the activated sludge process from Bunus Regional Sewage Treatment Plant Kuala Lumpur, Malaysia is developed. This paper focuses on modelling and model reduction of the WWTP system. The model with best fits of higher than 80% and the order of less than 10 is selected. For modelling purposes, data obtained is stimulated and modelled using System Identification technique which employ linear model ARX. For model reduction purposes, the high order model is reduced using model order reduction (MOR) of a combination of Singular Perturbation Approximation (SPA) and Frequency Domain Gramian based Model Reduction (FDIG) method. From the modelling results obtained, the ARX model with best fit of 85.11% is selected. Meanwhile, for the MOR FD-SPA technique, a 9th order model is selected with reduction error between frequencies 0.05 rad/s and 1.4 rad/s.

Singular Perturbation Approximation of Linear Hyperbolic Systems of Balance Laws

This technical note deals with a class of linear hyperbolic systems of balance laws with multiple time scales. The scale of time constants is modeled by a perturbation parameter. This parameter is introduced in both dynamics and boundary conditions. The solution of the full system is approximated by that of the reduced subsystem when the perturbation parameter is small enough. Lyapunov technique is used to prove it. The main result is illustrated by an academic example. Moreover, the boundary control synthesis to a gas flow transport model is shown based on singular perturbation approach.

Partial Fraction Expansion Based Frequency Weighted Balanced Singular Perturbation Approximation Model Reduction Technique with Error Bounds

In this paper, a new frequency weighted partial fraction expansion based model reduction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation is incorporated into the algorithm. This technique results in stable reduced order models regardless if single sided or double sided weights are used. Error bounds are also derived for the proposed method. For minimization of the frequency weighted approximation error, free parameters are introduced into the algorithm. A numerical example is provided in order to validate the proposed algorithm.

Singular perturbation approximation by means of a [formula omitted] Lyapunov function for linear hyperbolic systems

A linear hyperbolic system of two conservation laws with two time scales is considered in this paper. The fast time scale is modeled by a small perturbation parameter. By formally setting the perturbation parameter to zero, the full system is decomposed into two subsystems, the reduced subsystem (representing the slow dynamics) and the boundary-layer subsystem (standing for the fast dynamics). The solution of the full system can be approximated by the solution of the reduced subsystem. This result is obtained by using a H2 Lyapunov function. The estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to H2 and satisfying suitable compatibility conditions. Moreover, for a particular subset of initial conditions, more precise estimates are obtained. The main result is illustrated by means of numerical simulations.

Dissipativity-Preserving Model Reduction for Large-Scale Distributed Control Systems

We propose a dissipativity-preserving structured model reduction method for distributed control systems. As a fundamental tool to develop structured model reduction, we first establish dissipativity-preserving model reduction for general linear systems on the basis of a singular perturbation approximation. To this end, by deriving a tractable expression of singular perturbation models, we characterize dissipativity preservation in terms of a projection-like transformation of storage functions, and we show that the resultant approximation error is relevant to the sum of neglected eigenvalues of an index matrix. Next, utilizing this dissipativity-preserving model reduction, we develop a structured controller reduction method for distributed control systems. The major significance of this method is to preserve the spatial distribution of dissipative controllers and to provide an a priori bound for the performance degradation of closed-loop systems in terms of the ${\cal H}_{2} $ -norm. The efficiency of the proposed method is verified through a numerical example of vibration suppression control for interconnected second-order systems.

Early Model-Based Design and Verification of Automotive Control System Software Implementations

Verification and validation (V&V) are essential stages in the design cycle of automotive controllers to remove the gap between the designed and implemented controller. In this paper, an early model-based methodology is proposed to reduce the V&V time and improve the robustness of the designed controllers. The application of the proposed methodology is demonstrated on a cold start emission control problem in a midsize passenger car. A nonlinear reduced order model-based controller based on singular perturbation approximation (SPA) is designed to reduce cold start hydrocarbon (HC) emissions from a spark ignition (SI) combustion engine. A model-based simulation platform is created to verify the controller robustness against sampling, quantization, and fixed-point arithmetic imprecision. In addition, the results from early model-based verification are used to identify and remove sources of errors causing propagation of numerical imprecision in the controller structure. Thus the structure of the controller is modified to avoid or to reduce the level of numerical noise in the controller design. The performance of the final modified controller is validated in real-time by testing the control algorithm on a real engine control unit. The validation results indicate the modified controller is 17–63% more robust to different implementation imprecision while it requires lower implementation cost. The proposed methodology from this paper is expected to reduce typical V&V efforts in the development of automotive controllers.

Singular Perturbation Approximation of Balanced Infinite-Dimensional Systems

This paper concerned with a model reduction of infinite dimensional systems by using the singular perturbation approximation. The system considered is that of the exponentially stable linear system with bounded and finite-rank input and output operators such that the balanced realization can be performed on the system. Furthermore, the singular perturbation method is applied to reduce the order of the balanced infinite dimensional systems. A reduced-order model can be obtained by setting to zero of derivative all states corresponding to smaller Hankel singular values. To show the effectiveness of the proposed method, numerical simulations are applied to the heat conduction.

Error bounds in the gap metric for dissipative balanced approximations

We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input–state–output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.

Singular Perturbation Approximations for a Class of Linear Quantum Systems

This technical note considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics. The technical note presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction.

Controller reduction by balanced approach with guaranteed closed-loop performance

This paper addresses an algorithmic approach for controller reduction problem using balanced singular perturbation approximation (BSPA) technique for unweighted cases. The controller design methods generally lead to a controller of order comparable to plant and sometimes unstable controllers. For reduction of such controllers, a controller decomposition algorithm in conjunction with BSPA technique is presented. It is shown by numerical examples that reduced order controller by balanced truncation fails for some cases to stabilize the closed-loop system whereas resulting closed-loop system with reduced order controller using BSPA is guaranteed to preserve the closed-loop system stability and H∞ performance.

Reducing order of large-scale systems by extended balanced singular perturbation approximation

In this paper, a simplified approach for model order reduction, based on balanced singular perturbation approximation (BSPA) technique, is explored which is applicable for linear time invariant, minimal/non-minimal continuous/discrete-time systems. An extension of BSPA technique is proposed for order reduction of unstable systems. For reduction of unstable system, a system decomposition algorithm using real Schur transformation is given to decompose the unstable system into stable and unstable parts. Reduced order model of original unstable system is obtained by reducing the decomposed stable part by BSPA technique and directly adding it to decomposed unstable part. The algorithm is illustrated with MATLAB-based computer-simulated results in both time and frequency domains.

Dynamics of actively regulated gene networks

A popular way of modelling gene regulatory networks is to design the models as if the genes were acting directly on each other. Genes are activated or inhibited by transcription factors which are direct gene products. The action of a transcription factor on a gene is modelled as a binary on–off dose-response function or a sigmoid around a certain threshold concentration. A transcription factor may act on several genes with equal or different thresholds. The combined effect of several transcription factors on a gene is generally assumed to obey Boolean-like composition rules. A mathematical challenge related to such network models is to analyse the behaviour in the narrow domains in phase space where at least one variable is close to one of its thresholds, called switching domains. In the present paper we analyse the motion in switching domains for models with steep sigmoidal response functions under the assumption that each transcription factor only regulates one gene at each of its thresholds. This biologically reasonable assumption allows us to establish rules that determine the passage through any switching domain and the sequence of domains through which the system passes. The rules can be effectively implemented in a sound automated analyser for such networks. The assumption simplifies certain mathematical and computational issues, but unfortunately may be at variance with one of the necessary conditions for the singular perturbation theory which our derivations are based on. We give sufficient conditions ensuring that, despite this, the difficulties raised by our assumption can be evaded and singular perturbation approximation safely applied.

Homogeneous groups and multiscale intensity models for multiname credit derivatives

AbstractWe discuss a computationally tractable approach to the valuation of multiname credit derivatives employing name grouping for dimension reduction, and singular perturbation approximations for model robustness. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Gramian-Based Model Reduction for Data-Sparse Systems

Model reduction is a common task within the simulation, control, and optimization of complex dynamical systems. For instance, in control and optimization problems for instationary partial differential equations (PDEs), the associated large-scale linear systems resulting from a (semi-) discretization have to be solved repeatedly. Model reduction is a more and more accepted technique to attack these problems in reasonable time. Here, we investigate model reduction for parabolic PDEs based on balanced truncation (BT). The system theoretic properties of BT and the related method of singular perturbation approximation (SPA) provide desirable features for the reduced-order system. The major computational task in BT is the solution of large-scale Lyapunov or Stein equations, depending on whether the semidiscretized or fully discretized system is considered. Therefore, the method is often considered to be of limited use for really large-scale applications, i.e., problems that require $\gg1$ GByte of storage, so that memory access becomes an issue. Here, we develop effective implementations of BT and SPA by exploiting the structure of the underlying PDE problem. This is achieved by a data-sparse approximation of the (semi-)discretized differential operator using the hierarchical matrix format. Furthermore, we review how to integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations and suggest an iterative scheme for solving Stein equations employing formatted arithmetic. These approaches are well suited for a class of practically relevant problems and allow the application of BT and SPA to systems coming from two-dimensional (2D) and three-dimensional (3D) finite element method (FEM) and boundary element method (BEM) discretizations. The resulting methods are most beneficial in the context of BEM discretizations for 3D problems where large-scale dense matrices define the (semi-)discretized control system. We also analyze the additional approximation error in the reduced-order model introduced by the hierarchical matrix approximation. The efficiency of the methods is demonstrated for linear, scalar convection-diffusion-reaction equations. The numerical results confirm that the model reduction/BT error and the hierarchical matrix approximation error can be balanced by choosing the tolerances and parameters involved appropriately.

Partial-Fraction Expansion Based Frequency Weighted Model Reduction Technique With Error Bounds

In this technical note, we present a new frequency weighted balancing related technique which is based on a parametrized combination of the unweighted balanced truncation technique [9] ,[8] and the partial fraction expansion technique [11] . The reduced-order models which are guaranteed to be stable in the case of double-sided weighting, are obtained by either direct truncation or singular perturbation approximation. Simple, elegant and easily computable a priori error bounds are also derived. Numerical examples and comparison with other well-known techniques show the effectiveness of the proposed technique.