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Discrete-time systems are a common tool in the modeling of processes in many application areas such as digital signal processing and population dynamics. Model reduction is an essential remedy to handle high-fidelity systems in practice. To benefit from the performance gained by using reduced-order models, the computation of these models itself must be done with a reasonable use of resources. In this paper, we consider the case of medium-scale dense discrete-time systems and compare the performance of different numerical methods for the implementation of two basic model reduction techniques. Therefore, we give an overview of the considered model reduction methods and of the techniques used in underlying implementations. The outlined methods are then compared with established implementations in several numerical examples in terms of accuracy and performance.

Dynamic models have proven to be helpful for determining the residual water content in combustible biomass. However, these models often require partial differential equations, which render simulations impracticable when several thousand particles need to be considered, such as in the drying of wood chips. Reduced-order models help to overcome this problem. We compare proper orthogonal decomposition (POD) based to balanced truncation based reduced-order models. Both reduced models are lean enough for an application to systems with many particles, but the model based on balanced truncation shows more accurate results.

Reducing the order of positive real and bounded real systems has received great interest over the years. Several methods have been published regarding these issues which are able to produce positive real or bounded real reduced models. However, none of them can guarantee that the reduced models are both positive real and bounded real at the same time. In this brief, a novel model reduction scheme is presented which focuses on preserving two possible properties of the system (positive realness and bounded realness) simultaneously. This is done by using one positive real Riccati equation and one bounded real Riccati equation in the reduction procedure. Thereafter, the gramians extracted from these Riccati equations are balanced based on the Balanced Truncation (BT) concept. Finally, a numerical example is provided to investigate the efficacy of the proposed method.

The objective of this article is the construction of a new model reduction method for the simplification of large-scale linear dynamical plants. In this method, the denominator of the lower order p...

In this article, we are exploring and implementing the new model order reduction (MOR) method for Large-Scale Linear Dynamic System (LSLDS) to achieve reduced order. These technologies are designed to better understand and explain LSLDS based on the Modified Balanced Truncation Method (BTM). This refers to continuous/discrete LTI structures that are minimal / non-minimum. This reduced method allows MOR to preserve complete parameters with reasonable accuracy. The approach is based on the maintenance of dominant system modes and a relatively small state truncation. As the reduced-order model (ROM) is derived from the retention of dominant modes, the reduction remains stable. The main demerit of the balanced truncation method is that the ROM, stable states, does not match the original structures. By modified BTM to narrow the deviations in the ROM transfer function matrix, a gain factor is added to adjust the steady-state values of the reduction system without altering the dynamic behaviour of the system. The proposed method has been successfully applied to a real-time single area power system with ease of extension to a discrete-time case and the results obtained show the efficacy of the method. Application model and the results obtained indicate the effectiveness of the methodology. The time response of the system has been demonstrated by the proposed method, which proves to be excellent match, effectiveness and superiority compared to the response of other approaches in the literature review of the original system.

We propose a model reduction approach for switched linear system based on a balanced truncation reduction method for linear time-varying systems. The key idea is to approximate the piecewise-constant coefficient matrices with continuous time-varying coefficients and then apply available balance truncation methods for (continuous) time-varying systems. The proposed method is illustrated with a low dimensional academic example.

This paper presents a new limited time interval impulse response Gramians (LTIRG) based model reduction method for single-input single-output (SISO) continuous-time systems. The proposed approach incorporates the time-interval Gramians with impulse response Gramians to preserve the input-output behaviour of the original system for the specified time interval. A numerical example is given to illustrate the proposed approach, and the results are compared with the standard techniques.

Network systems consist of subsystems and their interconnections, and provide a powerful framework for analysis, modeling and control of complex systems. However, subsystems may have high-dimensional dynamics, and the amount and nature of interconnections may also be of high complexity. Therefore, it is relevant to study reduction methods for network systems. An overview on reduction methods for both the topological (interconnection) structure of the network and the dynamics of the nodes, while preserving structural properties of the network, and taking a control systems perspective, is provided. First topological complexity reduction methods based on graph clustering and aggregation are reviewed, producing a reduced-order network model. Second, reduction of the nodal dynamics is considered by using extensions of classical methods, while preserving the stability and synchronization properties. Finally, a structure-preserving generalized balancing method for simplifying simultaneously the topological structure and the order of the nodal dynamics is treated., To be published in Annual Review of Control, Robotics, and Autonomous Systems

The paper presents a model reduction framework geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and engineering, the problem of approximating the global behavior of systems away from equilibrium is largely open. We address this problem in the framework of dominance theory. Classical balanced truncation is applied to systems that can be decomposed as the feedback interconnection of a large-scale linear system and a comparably low-dimensional nonlinear system . The method is illustrated on a classical feedback model of a biological oscillator.

The paper presents a model reduction framework geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and engineering, model reduction methods are not well developed for non-equilibrium behaviors. The proposed framework addresses this need by exploiting recent advances on dominance theory. Classical balanced truncation for linear time-invariant systems is used to develop a dominance-preserving model reduction method for Lure systems, i.e. systems that can be decomposed as the feedback interconnection of a linear system and a static nonlinearity. The method is illustrated by approximating the oscillatory behavior of a discretized heat flow control system., 6 pages, 6 figures, submitted to the IFAC World Congress 2020

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

This paper is on the coprime factors reduction of distributed systems formed by discrete-time, heterogeneous, nonstationary linear parameter-varying subsystems. The subsystems are represent...

We discuss balanced truncation model reduction of constrained mechanical systems. The incorporation of the algebraic constraints leads to linear differential-algebraic control systems of index 2 or 3. Then for the square-root method in balanced truncation, the solution of projected Lyapunov equations is required. In this paper we discuss how to avoid the explicit construction of the projectors and the formulation of a projection-avoiding balanced truncation method which is suitable for efficient numerical computations. We have implemented this method in the Matlab package M-M.E.S.S. and present some numerical results for illustration.

An effective nonlinear model reduction approach, empirical Gramians balanced reduction approach, is studied, to reduce the computation complexity in nonlinear power system model application. The realization procedure is: firstly, computing the empirical controllable and observable Gramians matrices of nonlinear power system model, secondly, by these two matrices, computing the balance transformation matrix to obtain the balanced system model of the original model, then, computing the controllable and observable matrices of the balanced system to obtain the diagonal Hankel singular matrix. Finally, deciding the lower-order subspace to obtain the reduced power system model. A 15-machine power system model is taken as an example to perform the reduction simulation analysis.

International audience; The paper proposes a model reduction algorithm for linear hybrid systems, i.e., hybrid systems with externally induced discrete events, with linear continuous subsystems, and linear reset maps. The model reduction algorithm is based on balanced truncation. Moreover, the paper also proves an analytical error bound for the difference between the input-output behaviors of the original and the reduced order model. This error bound is formulated in terms of singular values of the Gramians used for model reduction.

This paper discusses an efficient technique for the model order reduction (MOR) of second-order index 3 systems which are arising in different disciplines of science and technology. The paper focuses on the second-order-to-second-order reduction techniques of the underlying systems using balanced truncation (BT). The index 3 second-order system can be converted into a standard second-order system; however, this causes the system to lose its sparsity. The main contribution of this paper is to apply the MOR method by exploiting the sparsity of the original model. This approach is efficient as it saves memory and reduces computational cost drastically. Numerical results are shown to prove the reflectivity and efficiency of the techniques.

This paper is concerned with the model reduction (MR) problem for continuous-time phase-type semi-Markovian jump systems (PH-SMJSs) based on balanced truncation method. The time-varying delay in states is considered via Wirtinger-based integral inequality and the parameter uncertainties are assumed to be randomly occurring. By utilizing the generalized output and input dissipation inequalities, the generalized gramians are solved. And the balanced realizations of the original PH-SMJSs are yielded according to the abilities of the controllability and observability. Then the low-order models are found by state truncation technique. Furthermore, the balanced performances of the original PH-SMJSs are preserved in the low-order systems. A numerical example is conduced to clarify the effectiveness of the proposed balanced performance preserving MR method for SMJSs.

This paper extends the applicability of a balancing-based reduction method to power system models that are structured into a study area, a buffer area, and a linear parameter-varying (LPV) external area. The latter is represented by a network of linear time-invariant (LTI) and LPV models and forms the target of the reduction phase. Every LPV model is built to depict a region of the power system with a variable number of active generators. The various LTI and LPV models are simplified while accounting for the presence of their respective neighbors. The adequacy of the reduction process is verified on the IEEE 118-bus system model by comparing the dynamic behavior of the reduced model and the full nonlinear model for different sets of disconnected generators., info:eu-repo/semantics/published

Model Order Reduction (MOR) has become key enabler for the efficient simulation of large circuit models. MOR techniques based on Balanced Truncation (BT) offer very good error estimates and can provide compact models with any desired accuracy over the whole range of frequencies (from DC to infinity). However, in most applications the circuit is only intended to operate at specific frequency windows, which means that the reduced-order model can become unnecessarily large to achieve approximation over all frequencies. In this paper, we present a frequency-limited approach which, combined with low-rank factorized solution, can handle large input models and provably leads to reduced-order models that are either smaller or exhibit better accuracy than full-frequency BT.

Along the ideas of Curtain and Glover (in: Bart, Gohberg, Kaashoek (eds) Operator theory and systems, Birkhäuser, Boston, 1986), we extend the balanced truncation method for (infinite-dimensional) linear systems to arbitrary-dimensional bilinear and stochastic systems. In particular, we apply Hilbert space techniques used in many-body quantum mechanics to establish new fully explicit error bounds for the truncated system and prove convergence results. The functional analytic setting allows us to obtain mixed Hardy space error bounds for both finite-and infinite-dimensional systems, and it is then applied to the model reduction of stochastic evolution equations driven by Wiener noise.

In this paper a balanced truncation based model order reduction technique for discrete-time systems using limited-frequency controllability and observability Gramians has been proposed which preserves the stability after reduction. The efficacy of the proposed method has been established using a numerical example.

Model reduction of a system is an approximation of a higher-order system to a lower-order system while the dynamic behavior of the system is almost unchanged. In this paper, we will discuss model order reduction (MOR) strategies for unstable systems, in which the method based on the balanced truncation algorithm will be focused on. Since each MOR algorithm has its strengths and weakness, practical applications should be suitable for each specific requirement. Simulation results will demonstrate the correctness of the algorithms.

We consider the model reduction problem for linear time-invariant dynamical systems having nonzero (but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions, we develop a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response. This approach allows greater flexibility and offers better approximation properties than other comparable methods.

International audience; In this paper, a controller order reduction method for linear parameter-varying systems is presented.The proposed method is based on the frequency-weighted balanced truncation technique, whichhas the advantage to reduce the order in a specific frequency range. The approach is discussed and isproved to preserve the closed-loop stability with a guaranteed upper error bound. Effectiveness andperformance of the obtained reduced-order controller are investigated by applying it to an automotivesemi-active suspension control. The obtained simulation results showthat objectives such as theroad handling and the passenger comfort realised with the reduced-order controller are kept in thesame performance level as with the full-order controller.Moreover, a comparisonwith an other-orderreduction method is shown and confirms the advantage of the developed method.

This paper contains the hybrid method for reduction of higher order. The proposed method uses the balanced truncation of time domain and mode selection method. The balanced truncation used for the reduction of numerator coefficient and the denominator part is reduced by the mode selection method. The obtained result is compared with the result available in the literature by using the step response characteristics as settling time, rise time, peak, peak time, settling max, and settling min of the system and response indices. The step response characteristics are compared with a reason that the reduced order must retain the important characteristics of the original order.

This paper investigates the model reduction problem based on balanced truncation for uncertain continuous-time Markovian jump linear systems with time-varying delay in the states. Firstly, two generalized dissipation inequalities are proposed to find the positive matrices which could guarantee the stochastic stability of the original system. By simultaneously balancing these two positive matrices, we can obtain the balanced transformation matrix which balances the original system. Then the reduced order model can be derived by truncating the balanced model. In order to bound the reduced order error, we choose the reciprocally convex approach to deal with the delay term. Finally, an illustrative example is provided to show the effectiveness of the proposed new design techniques.

We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.

This paper deals with model reduction by balanced truncation of fault tolerant switched systems. The conditions, which guarantee the stability of fault tolerant switched system and the error between the states of switched system and reduced order switched system is in a certain range, are proposed. It shows that fault tolerant switched system can be reduced by the same truncated of original switched system without faults if fault matrices satisfy some conditions.

In this work, a novel technique of balanced truncation for model reduction of parameter-varying multidimensional systems in “Roesser Form” is proposed. The multidimensional system representation requires two operators defined on l 2 -space. The reduction is based on extended Gramians. Both, standard and improved error bounds for the model reduction problem are presented. The approach is demonstrated with the application to an illustrative example.

In this paper, a reduced order linear model is developed for the purpose of controller design (e.g. power system stabilizer (PSS) parameters tuning) in an unstable power system. This reduced order model is based on balanced truncation (BT) model reduction method. Generally, BT method cannot be applied directly to unstable systems. There are several techniques in the literature to extend the BT method to unstable systems. This paper utilizes three of model reduction techniques based on BT theory and evaluate their performance in power system applications. The proposed techniques is applied to 39-bus New England test system. The results of modal analysis and frequency response simulation indicate various aspects and features of these methods.

Offshore structures like jackets are exposed to various kinds of external loading conditions and examination of broader range of modes and natural frequencies are required for their dynamic analysis. In this study, four model reduction techniques — two modal truncation methods and two balanced truncation methods — are introduced and applied to a jacket structure. The reduction results are compared with those of the conventional Paz condensation methods. Four new methods show better fidelity, especially in high frequency ranges than the conventional methods.

Model reduction is a significant issue in dynamic system simulation and control, as a consequence of the unmanageable levels of storage and computational requirements for large-scale systems. In this paper, the concept of a balanced truncation approximation method is extended to large-scale systems with interval uncertainties to get the reduced-order model with uncertainties. In order to get the balanced system, the balancing transformation matrix is introduced by using the nominal system, and the reduced-order model with uncertainties is obtained by using balanced truncation. A major characteristic of this model reduction method is that the reduced-order model obtained in this way is also as uncertain as the original model. The closeness of the reduced-order model to the original model relies on the upper bounds of the ignored Hankel singular values. To compare the original model and the reduced-order model, a perturbation method is proposed to give the interval bounds of the responses of the original model and the reduced-order model. As applications of the proposed method, three numerical examples are given.

In this paper, we investigate model reduction for positive linear time-invariant systems in the H ∞ sense. First, the bounded real lemma is further developed to obtain a new characterization of the H ∞ performance, which does not include any product term between the Lyapunov matrix and the parameter of the reduced-order model. Then a new parameterization of a positive reduced-order model is proposed, and accordingly, an iterative algorithm is constructed, which makes use of the coarse reduced-order models resulting from the (generalized) balanced truncation as the initial value to search for a desired positive one. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed method.

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map., We have corrected a number of typos and inconsistencies. In addition, we added new results in Theorem 3

A lateral flight reduced-order controller for a flexible aircraft is presented. It is based on the reduced-order model of the flexible aircraft and theμ-analysis and synthesis toolbox. The first step deals with the order reduction of the full-model by using the balanced-truncation model reduction method. Then the reduced-orderHcontroller is achieved by using theμ-analysis and synthesis theory and applied to the reduced-order model and full-order model of the flexible aircraft. Finally, numerical results are presented and discussed. The simulation results show the efficiency of the reduced-order controller based on the reduced-order model since the close-loop system of the full-order model meets a set of realistic specifications in the frequency and time domains.

This paper introduces an interpolation framework for the weighted-H"2 model reduction problem. We obtain a new representation of the weighted-H"2 norm of SISO systems that provides new interpolatory first order necessary conditions for an optimal reduced-order model. The H"2 norm representation also provides an error expression that motivates a new weighted-H"2 model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach.

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.

As an alternative to the popular balanced truncation method, the cross Gramian matrix induces a class of balancing model reduction techniques. Besides the classical computation of the cross Gramian by a Sylvester matrix equation, an empirical cross Gramian can be computed based on simulated trajectories. This work assesses the cross Gramian and its empirical Gramian variant for state-space reduction on a procedural benchmark based on the cross Gramian itself.

In this paper we consider different model reduction techniques for systems with moving loads. Due to the time-dependency of the input and output matrices, the application of time-varying projection matrices for the reduction offers new degrees of freedom, which also come along with some challenges. This paper deals with both, simple methods for the reduction of particular linear time-varying systems, as well as with a more advanced technique considering the emerging time derivatives.

In this paper, stability and synchronization preserving model reduction schemes are developed for linear multi-agent systems. The multi-agent systems that are considered here are composed of general, yet identical linear subsystems, and the communication topology is assumed to be time-independent. First, under the assumption that the agents have stable internal dynamics and the network is stable, the dynamic order of the agents is reduced such that the corresponding reduced order network is again stable. Then, starting from a synchronized network where agents are allowed to have unstable dynamics, a reduced order model for the network which preserves synchronization is obtained. In addition, model reduction error bounds are established to compare the behavior of the original network to that of the reduced order model. The proposed results are illustrated through a numerical example. (C) 2012 Elsevier B.V. All rights reserved.

SUMMARY In this paper, we present a model order reduction approach for nonlinear circuit equations with a small number of nonlinear elements. This approach is based on the decoupling of linear and nonlinear subcircuits and reducing the linear part using balancing-related model reduction techniques. The efficiency and applicability of the proposed model reduction approach is demonstrated on numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.

A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138---1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples.

This comment explains why the passivity preserving frequency weighted model reduction technique of Heydari and Pedram is guaranteed to yield neither passive nor stable models in the case of double-sided weighting. It is further shown for this method that the passivity can be preserved in the single-sided weighting case only. An example is presented to show that their method may yield non-passive models for passive original systems in the case of double-sided weighting.

We discuss a model reduction for unstable infinite dimensional systems using balanced truncation of reciprocal systems. The systems considered are assumed to be exponentially stabilizable and detectable, with bounded and finite rank input and output operators. We decompose the systems into their stable and unstable parts and perform reciprocal transformation only on the stable part. Balanced truncation is carried out on the reciprocal systems. This yield is then translated using reciprocal transformation as the reduced-order model of the stable part. Finally, we added the unstable part to the reduction of the stable part as the overall reduced-order model for the systems. To verify the effectiveness of the proposed approach, numerical simulations are applied to the heat equation. The performance of the proposed reduction method is compared to the balanced truncation method.

This paper presents a generalization of the model reduction method proper orthogonal decomposition to systems of differential-algebraic equations of arbitrary index. It is known that proper orthogonal decomposition generalizes the method of empirical balanced truncation for linear time-invariant systems. This property will be preserved for differential-algebraic equation systems of arbitrary index. This is important for the application of proper orthogonal decomposition in control, where the input-output behaviour should be approximated accurately, which is a well-known property of balanced-truncated systems.

Weighted model reduction problems appear in many important applications such as controller reduction. The most common approach to this problem is the weighted balanced truncation method. Interpolatory approaches to weighted model reduction have been mostly limited to simply choosing interpolation points in the regions where the weights are dominant. In this paper, we extend the (unweighted) tangential interpolatory model reduction to the weighted case using the recent generalized interpolation framework of Beattie and Gugercin [2009]. We generate a reduced-order model such that the weighted reduced-order system tangentially interpolates the weighted full-order system at the provided interpolation data.

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.