Balanced Truncation Method Research Articles

Finite-frequency model order reduction of linear and bilinear systems via low-rank approximation

In this paper, we first investigate the finite-frequency model order reduction for linear systems based on low-rank Gramian approximations. An efficient algorithm for computing low-rank approximations of the finite-frequency and frequency-dependent Gramians based on Laguerre functions is proposed. The approach constructs the low-rank decomposition factors of the finite-frequency Gramians or frequency-dependent Gramians through a recursive formula of Laguerre functions expansion coefficient vectors and then combines the low-rank square root method and frequency-dependent balanced truncation method to obtain the reduced-order models. In this process, it avoids dealing with the matrix-valued functions and solving the related (generalized) Lyapunov matrix equations directly, making them computationally efficient. Furthermore, the above method is successfully extended to bilinear systems, and a corresponding efficient computation method for low-rank approximations of the finite-frequency Gramians of bilinear systems is derived. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms.

Optimal linear quadratic regulator-based reduced order model for hybrid AC/DC microgrid

The stability of hybrid microgrids represents an elementary requirement to ensure their security and reliability. This paper aims to propose a supplementary controller that can enhance the hybrid microgrid dynamic performance while preserving its stability. At first, the paper derives the small signal state-space model of a hybrid microgrid that includes inverter-based and conventional generation. This model is then reduced using the balanced truncation method. The reduced order model is utilized to design a supplementary linear quadratic output regulator known for its efficiency and high performance. A Kalman observer is employed to measure the states of the reduced order model to complete the control process. Finally, time domain simulations are used to validate the reduced order model and verify the performance of the proposed controller. The results revealed the accuracy of the model and the effectiveness of the controller.

A parallel power system linear model reduction method based on extended Krylov subspace

With the ever-increasing scale of power systems, stability analysis and control usually bear heavy storage and massive calculation burdens. In view of this, the model order reduction technique proves valuable by constructing a low-dimensional approximate model of the original system, which is crucial for efficiently handling large-scale systems. Balanced truncation (BT), a famous model reduction method, confronts practical limitations as it requires the systems to be stable and cannot deal with unstable models. Therefore, a parallel linear balanced truncation method for power systems based on extended Krylov subspace (EKS) is proposed in this work. Besides extending the BT method to unstable systems by α-shift, the key contribution also lies in strategies to enhance the convergence of the EKS method, whereupon the algorithm improvements include effective and efficient techniques for solving dual Lyapunov equations, and parallel acceleration of the singular value decomposition. The results of the simulation case verify that the proposed method can effectively improve the convergence of the EKS method by increasing α-shift, and the improvement work in this paper reduces the total time consumption of the BT method by about 26 %–33 % of the original, exhibiting better calculation efficiency. In addition, the case studies show that the simplified model still retains the time-domain and frequency-domain response characteristics of the original high-dimensional model.

Coprime Factor Model Reduction of Multidimensional Systems

In this work, the right coprime factor for balancing and truncating unstable systems is extended to parameter-varying multidimensional systems employing recent works on coprime factor model reduction of one-dimensional uncertain systems. Since the balanced truncation method cannot be applied directly to unstable systems, state feedback gains should be computed and incorporated in order to stabilize the given system and to be able to apply the balanced truncation technique via defining the so-called stable coprime factor, as coprime factorization overcomes the stability condition required for model reduction. Parameter-dependent state feedback and parameter-dependent Gramians are considered in this work yielding less conservative techniques. In addition, the Gramians are defined as block diagonal matrices, which are partitioned according to the structure of the multidimensional systems. The application to a simulation example demonstrates the applicability and validity of the proposed reduction approach leading to small error bounds between the full and the reduced models.

Model order reduction for discrete time-delay systems based on Laguerre function expansion

We investigate model order reduction for discrete time-delay systems via orthogonal polynomial expansion in the frequency domain. The transfer function of systems is expanded in the framework of Laguerre function basis. We show that Laguerre coefficients of the states satisfy a linear system and reduced models generated by projection methods can preserve some Laguerre coefficients of the original systems. We prove that the subspace spanned by Laguerre coefficients is exactly a high order Krylov subspace, thereby leading to an efficient computation of projection matrices and a more accurate coefficient-matching property in the two-sided framework. Further the Laguerre expansion of systems is employed to enable an approximate but fast execution of balanced truncation for discrete time-delay systems. Specifically, Gramians are approximated based on the derived Laguerre coefficients, which circumvents the main bottleneck of balanced truncation methods. Finally, two numerical examples are simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Dimension reduction based on approximate gramians via Laguerre polynomials for coupled systems

Abstract In this paper, we focus on the topic of model order reduction (MOR) for coupled systems. At first, an approximation via Laguerre polynomials expansions to controllability and observability gramians for such systems are presented, which provides a low-rank decomposition form whose factors are constructed from a recurrence formula instead of Lyapunov equations. Then, in combination of balanced truncation and dominant subspace projection method, a series of MOR algorithms are proposed that preserve the coupled structures. What’s more, some main properties of reduced-order models, such as stability preservation, are well discussed. Finally, three numerical simulations are provided to illustrate the effectiveness of our algorithms.

Integrating Physical and Data-Driven System Frequency Response Modelling for Wind-PV-Thermal Power Systems

This paper presents an integrated system frequency response (SFR) modelling method for wind-PV-thermal power systems (WPTPSs) by combining both physical model-based and data-driven modelling methods. The SFR physical model is built and simplified by the balanced truncation (BT) method. Based on the physical model, an improved radial basis function neural networks (RBFNNs) is then employed to establish an off-line SFR model using source data. Following the transfer learning principle, the transferred data from the source data set is determined by the maximum mean discrepancy (MMD) criterion. The RBFNN-based SFR model is then fine-tuned using both the transferred source data and target data. Finally, the fine-tuned RBFNNs is applied to investigate real-time SFR of WPTPSs. Simulation results confirm the effectiveness of the proposed SFR modelling strategy with an illustrative WPTPS.

From Data to Reduced-Order Models via Generalized Balanced Truncation

This paper proposes a data-driven model reduction approach on the basis of noisy data with a known noise model. Firstly, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin projection to all systems explaining the data characterized in a large-dimensional quadratic matrix inequality (QMI) can again be characterized in a lower-dimensional QMI. Next, we develop a data-driven generalized balanced truncation method that relies on two steps. First, we provide necessary and sufficient conditions such that systems explaining the data have common generalized Gramians. Second, these common generalized Gramians are used to construct matrices that allow to characterize a class of reduced-order models via generalized balanced truncation in terms of a lower-dimensional QMI by applying the data reduction concept. Additionally, we present alternative procedures to compute a priori and a posteriori upper bounds with respect to the true system generating the data. Finally, the proposed techniques are illustrated by means of application to an example of a system of a cart with a double-pendulum.

Order Reduction of Large-Scale Linear Dynamic Systems Using Balanced Truncation with Modified Cauer Continued Fraction

This article discusses a new technique to reduce the order of high-dimensional, continuous, linear time-invariant dynamic systems. The proposed approach utilizes the merits of the balanced truncation method, and modified Cauer continued fraction expansion. The reduced model denominator was retrieved via the concept of Lyapunov balancing, and state truncation and numerator coefficients were obtained by using a modified Cauer continued fraction. The proposed technique circumvents the steady-state gain discrepancy and steady-state error mismatch of the conventional balanced truncation method, and the instability issue of continued fraction expansion. The proposed method assures steady-state gain approximation, few moments, Markov parameters, stability, and other essential control features of a high-order system. The efficacy and applicability of the proposed technique are demonstrated using some typical test systems adopted from the literature. This method gives improved performance indices like minimum H∞ norm, H2 norm, a minimum ISE (integral squared error), IAE (integral absolute error), and retains approximately the full IRE (impulse response energy) of the existing system in the reduced model. The simulation findings of the proposed technique are compared with other current model reduction methods in recent literature using MATLAB.

Continuous Grasping Force Estimation With Surface EMG Based on Huxley-Type Musculoskeletal Model.

Continuous grasping force estimation based on electromyography (EMG) signals, is very useful in practical applications including prosthetic control and human force observation. However, implementing the practical grasping force estimation usually considers a trade-off between the computational precision and resources. Specifically, the estimation based on the Huxley-type muscle model reaches detailed approximation of physiological process at a cost of larger computational resources for solving nonlinear partial differential equations while the counterpart with a traditional Hill-type muscle model. In this article, we achieve the grasping force estimation based on a reduced Huxley-type musculoskeletal model with high accuracy yet low time delay. Leveraging on a balanced truncation method, we further reduce the dimensionality of the spectral method solution in the Huxley-type musculoskeletal model for the model simplification. In addition, we introduce the Kalman filter method to process the EMG signals obtained by an armband, yielding better real-time performances and accuracy compared to the signal treatment using the traditional EMG filter method. Moreover, we also implement an effective identification of the model parameters using a particle swarm method. Finally, we trained the model on the first day and made grasping force estimation experiments involved with three participants over the course of a month. We envision that this effective and practical method would further improve the practical applications in the field of grasping force estimation.

Co-rotational Reduced Order Kalman Filter Finite Element Method for the Real-time Implementation of Large-scale Structural Models

The real-time implementation of large-scale structural models without sacrificing their physical realism is challenging. This paper introduces a conceptual framework named the Co-rotational Reduced Order Kalman Filter Finite Element Method (CROKF-FEM) for the realistic linear elastic analysis of large-scale structures in real-time. The standard linear finite element method (FEM) is employed for high-precision deformation modeling, and element matrices are pre-computed in the space domain. To avoid the inflated elements of standard linear FEM due to large rotation, a co-rotational formulation is devised. Then, the system is discretized in the time domain using the Wilson- θ implicit integration scheme and a further state-space model is developed. Subsequently, a reduced-order model is constructed using the balanced truncation method. Finally, a Kalman filter algorithm is created on the reduced model for online estimation. This approach preserves all the benefits of traditional linear FEM with a similar deformation accuracy while achieving significant real-time performance.

Exhaustive modal analysis of large-scale power systems using model order reduction

This paper presents an efficient modal analysis methodology that computes all modes of any given large-scale power system in exhaustive manner using the model order reduction techniques. For this, a reduced order model is generated using the Balanced Truncation (BT) method for which the controllability and observability gramians are approximated using the low-rank Cholesky factors. This leads to a rapid identification of classes of coupled dynamic devices of the original system. Next, approximated oscillatory modes are computed for each class. Finally, the exact values of the oscillatory modes of the overall power system are determined by iterative computations (the Modified Arnoldi method) initialized to the approximated modes found at the first step. The proposed methodology is able to put into evidence all coupling modes of any given large-scale power system (containing power electronics or any other specific dynamic devices). No a priori knowledge about the pattern of oscillations is needed. The accuracy and efficiency of the proposed methodology are thoroughly validated on several power systems with different orders, including a large scale model of the interconnected European power system.

Comparative study on techniques of model order reduction using rational Krylov subspace method

We present Model Reduction (MR) techniques of the 1st order system and 2nd order dynamical systems using the Balanced Truncation (BT) method [1, 2] and Projection onto the dominant Eigenspace of the Gramian (PDEG) method [3]. We get help from the well-known Rational Krylov Subspace Method (RKSM) [4] for BT and PDEG. We also mention the Lyapunov equations [1] with their solutions known as the system gramians. Finally, a comparison is shown between BT and PDEG methods using numerical experiments.

Design and Application of Silicon on Insulator Based SiGe VTFET in IIR Filter by Balanced Truncation (BT) Method of Model Order Reduction

Design and Application of Silicon on Insulator Based SiGe VTFET in IIR Filter by Balanced Truncation (BT) Method of Model Order Reduction

Balanced truncation for discrete time-delay systems via the interpretation of system energy

In this paper, the balanced truncation method is investigated for discrete time-delay systems. We show that the energy associated with the system controllability and observability can be characterized via the delay Lyapunov matrices, similar to the case of continuous time-delay systems. Then, we balance the system via a coordinate transformation in order to retain the delay structure of systems naturally. In this way, the balanced truncation method is conducted to obtain structure-preserving reduced models. Further, we provide an efficient process to compute a low-rank approximation to delay Lyapunov matrices based on the equivalent expression of discrete time-delay systems, which enables an approximate but fast execution of the proposed method. The stability of reduced models is also discussed in the paper. Finally, numerical examples are simulated to verify the feasibility and efficiency of the proposed method.

Higher Order Modeling of Reactor Regulating System and Nonlinear Neural Model Predictive Controller Design for a Nuclear Power Generating Station

In the existing instrumentation and control system of an operating Pressurized Heavy Water Reactor (PHWR) based nuclear power plant, conventional controllers are used to control the reactor power. A new idea of Nonlinear Neural Model Predictive Controller (NNMPC) is introduced in this research work. The new 17th order nonlinear higher order model of Reactor Regulating System (RRS) is developed under different plant operating modes and various parametric conditions in Single Input Multi Output (SIMO) configuration with special emphasis on Helium Control Valve Dynamics (HCVD) and Coupled Nonlinear Iodine and Xenon Dynamics (CNIXD). The SIMO RRS model is developed based on first principle. The 17th order model is reduced to 9th order lower dynamic model using Balanced Truncation Method (BTM). The Reduced Order SIMO RRS (RO-SIMO-RRS) model is programmed, simulated and validated in SIMULINK environment. The plant Neural SIMO RRS (N-SIMO-RRS) model is developed using innovative data generated from RO-SIMO-RRS simulations. The plant neural N-SIMO-RRS model is optimized using Levenberg-Marquardt Algorithm (LMA). Using the identified N-SIMO-RRS model, the Nonlinear Neural Model Predictive Controller (NNMPC) is designed, trained, verified, validated, and finally optimized using the backtracking technique in the SIMULINK environment. The optimized results are obtained from designed closed loop RRS and found within the acceptable design limits. The performance of the proposed closed loop RRS is also tested in reference tracking mode with excellent fast tractability near the optimal target demanded power level.

Structure Preserving Uncertainty Modelling and Robustness Analysis for Spatially Distributed Dissipative Dynamical Systems

The paper deals with uncertainty modelling, robust stability and performance analysis of multi-input multi-output (MIMO) reduced order spatially distributed dissipative dynamical systems. While researching the topic of modern robust control of such systems, two key findings were discovered: (i) systematic modelling of the uncertainty and model order reduction (MOR) at the level of a subsystem gives both modelling freedom and the ability for obtaining less conservative uncertainties on the level of a subsystem; (ii) for a special class of interconnected dissipative dynamical systems, uncertainty conservatism at the subsystem level can be reduced—a novel, structure preserving algorithm employing subsystem partitioning and subsystem MOR by means of balanced truncation method (BTM) is used to obtain low-order robustly stable interconnected systems. Such systems are suitable for practical decentralized and distributed robust controller synthesis. Built upon a powerful framework of integral quadratic constraints (IQCs), this approach gives uncertainty modelling flexibility to perform robustness analysis of real world interconnected systems that are usually affected by multiple types of uncertainties at once. The proposed uncertainty modelling procedure and its practical application are presented on the numerical example. A spatially discretized vibration dynamical system comprised of a series of simply supported Euler beams mutually interconnected by springs and dampers is examined. Spatial discretization of the mathematical model is carried out using the finite element method (FEM).

A New Passivity Preserving Model Order Reduction Method: Conic Positive Real Balanced Truncation Method

This article is dedicated to model order reduction of linear time-invariant passive systems, i.e., positive real (PR) systems, based on the balanced truncation (BT) concept. The main feature of this article is that we have used the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , in model order reduction for the first time in the proposed method, which results in more accurate reduced-order models. It is worth noting that this hypothesis has been never made before in any other model order reduction techniques. Furthermore, new gramians are calculated corresponding to their associated new Riccati equations and Lur’e equations, which are in turn achieved by using Kalman–Yakubovic–Popov (KYP) lemma and linear matrix inequalities (LMI). Thereafter, a novel passivity preserving model order reduction algorithm, which takes the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , into account, is presented. It is demonstrated that the proposed method is a generalization of the positive real balanced truncation (PRBT) method, and it is perfectly capable of providing more accurate approximation error compared to PRBT. Finally, numerical examples are included to figure out the effectiveness of the presented method.

Parameter Optimization to Power Oscillation Damper (POD) Considering its Impact on the DFIG

The power oscillation damper (POD) in the doubly-fed induction generator (DFIG) is to damp the low- frequency oscillation (LFO) in the power systems. But the POD will introduce the LFO to the DFIG and intensify the oscillation of the latter. In this paper, open-loop subsystems of the DFIG with the POD and the rest part of power system are derived to describe the interaction between them. The balanced truncation method is used to reduce the transfer function of the DFIG and derive the analytical description to the oscillation of the DFIG. Considering the impact of the POD on the DFIG, the constraint to the DFIG is newly incorporated in the optimization model to the POD parameters, with the aim to alleviate the oscillation of both the DFIG and power system. The simulation verifies the accuracy of the reduced model of the DFIG, and validates the effectiveness of the improved optimization to the POD of the DFIG.

Phase Preserving Balanced Truncation for Order Reduction of Positive Real Systems

This paper presents a new passivity-preserving order reduction method for linear time-invariant passive systems, which are also called positive real (PR) systems, with the aid of the balanced truncation (BT) method. The proposed method stems from the conic positive real balanced truncation (CPRBT) method, which is a modification of the BT method for PR systems. CPRBT presents an algorithm in which the reduced models are obtained from some Riccati equations in which the phase angle of the transfer function has been taken into consideration. Although CPRBT is a powerful algorithm for obtaining accurate PR reduced-order models, it cannot guarantee that the phase diagram of the reduced model remains inside the same interval as that of the original full-order system. We aim to address such a problem by modifying CPRBT in the way that the phase angle of the reduced transfer function always remains inside the conic and homolographic phase interval of the original system. This is proven through some matrix manipulations, which has added mathematical value to the paper. Finally, in order to assess the efficacy of the proposed method, two numerical examples are simulated.