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This paper proposes a method to compute solutions of optimal controls for dynamic systems in terms of radial basis function neural networks (RBFNN) with Gaussian neurons. The RBFNN is used to compute the value function from Hamilton–Jacobi–Bellman equation with the policy iteration. The concept of dominant system is introduced to create initial coefficients of the neural networks to stabilise unstable systems and guarantee the convergence of policy iteration. Model reduction and transfer learning techniques are introduced to improve robustness of RBFNN optimal control, and reduce computational time. Numerical and experimental results show that the resulting optimal control has excellent control performance in stabilisation and trajectory tracking, and much-improved robustness to disturbances and model uncertainties even when the system responses move to a domain of the state space that is larger than the domain where the neural networks are trained. [ABSTRACT FROM AUTHOR]

ABSTRACT This article presents a balancing‐based algorithm for reducing the complexity of structured discrete‐time linear time‐invariant (DT‐LTI) index‐2 descriptor systems. The proposed algorithm involves projecting the index‐2 system onto a hidden manifold, which converts it into a generalized system. However, this causes the system to lose its sparsity and become dense, which is impractical for large‐scale systems. To overcome this issue, the authors enhance the Smith‐based iterative method for solving discrete‐time algebraic Lyapunov equations, which allow for balanced truncation without explicitly forming the dense system. The proposed algorithm is shown to be efficient and robust through numerical simulations. [ABSTRACT FROM AUTHOR]

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems, we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics. [ABSTRACT FROM AUTHOR]

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems, we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics.

Abstract In the design process of large adaptive mirrors numerical simulations represent the first step to evaluate the system design compliance in terms of performance, stability and robustness. For the next generation of Extremely Large Telescopes increased system dimensions and bandwidths lead to the need of modeling not only the deformable mirror alone, but also all the system supporting structure or even the full telescope. The capability to perform the simulations with an acceptable amount of time and computational resources is highly dependent on finding appropriate methods to reduce the size of the resulting dynamic models. In this paper we present a framework developed together with the company Microgate to create a reduced order structural model of a large adaptive mirror as a preprocessing step to the control system simulations. The reduced dynamic model is then combined with the remaining system components allowing to simulate the full adaptive mirror in a computationally efficient way. We analyze the feasibility of our reduced models for Microgate’s prototype of the adaptive mirror of the Giant Magellan Telescope.

In the design process of large adaptive mirrors numerical simulations represent the first step to evaluate the system design compliance in terms of performance, stability and robustness. For the next generation of Extremely Large Telescopes increased system dimensions and bandwidths lead to the need of modeling not only the deformable mirror alone, but also all the system supporting structure or even the full telescope. The capability to perform the simulations with an acceptable amount of time and computational resources is highly dependent on finding appropriate methods to reduce the size of the resulting dynamic models. In this paper we present a framework developed together with the company Microgate to create a reduced order structural model of a large adaptive mirror as a preprocessing step to the control system simulations. The reduced dynamic model is then combined with the remaining system components allowing to simulate the full adaptive mirror in a computationally efficient way. We analyze the feasibility of our reduced models for Microgate's prototype of the adaptive mirror of the Giant Magellan Telescope. [ABSTRACT FROM AUTHOR]

We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples. [ABSTRACT FROM AUTHOR]

Modeling and simulating laser heating phenomena are crucial for optimizing manufacturing processes and ensuring high-quality final products. A major challenge in semiconductor manufacturing is achieving accurate, real-time temperature control during wafer heating. To reduce the computational burden of complex mathematical models, low-dimensional reduced models can be employed. In this paper, we develop a mathematical model for laser heating in silicon wafers. For model reduction, we use the balanced truncation method, considering both frequency-unrestricted and restricted cases. Additionally, the rational Krylov subspace method is applied to solve high-dimensional sparse matrix equations. To gain key physical insights, we use the COMSOL Multiphysics package. Finally, some numerical experiments are conducted using MATLAB to validate the proposed approach.

This paper proposes a novel algorithm, termed Mixed Riccati–Lyapunov Balanced Truncation (MRLBT), tailored for order reduction of Linear Time-Invariant Continuous-Time Descriptor Systems (LTI-CTD), commonly encountered in electrical and electronic circuit modeling. The MRLBT approach synergistically combines the advantages of balanced truncation (BT) and positive-real balanced truncation (PRBT) techniques while mitigating their limitations. Unlike BT, which preserves stability but not passivity, and PRBT, which retains passivity at the expense of larger reduction errors, MRLBT ensures the preservation of both stability and passivity inherent in the original system. Additionally, MRLBT achieves reduced computational complexity and minimized order reduction errors compared to PRBT. The proposed algorithm transforms the system into an equivalent Mixed Riccati–Lyapunov Balanced form, enabling the construction of a reduced-order model that retains the critical physical properties. Theoretical analysis and proofs are provided, establishing an upper bound on the global order reduction error. The efficacy of MRLBT is demonstrated through a numerical example involving an RLC ladder network, showcasing its superior performance over BT and PRBT in terms of reduced errors in the time and frequency domains. [ABSTRACT FROM AUTHOR]

This study presents a novel approach for the simplification and controller design of expansive dynamic linear time-invariant plants. This approach applies to both single input single output and multiple input multiple output models on a wide scale. The diminution approach is a straightforward method that guarantees the stability of the lower-order plant, given that the higher-order system is stable. Implementing the balanced truncation approach determines the denominator polynomial of the required system. The coefficients of the numerator polynomial are computed using a simple mathematical procedure, as outlined in the suggested scenario. The proposed method overcomes the constraints of the balanced truncation scheme while maintaining its crucial attributes, including stability, controllability, observability, passivity, etc. The strategy that has been developed guarantees the preservation of stability, time moments, Markov parameters, and other features of the higher-order plant in the reduced model. A two-input, two-output, single-machine infinite bus real-time power system model and a ninth-order system are utilized to test the accuracy and effectiveness of the proposed method. The findings of the suggested technique are compared against other popular algorithms. Furthermore, controllers are derived using the moment-matching process using the recommended lower-order plant for two real-time case studies. The controller’s design is shown, and its efficacy is confirmed using a real-time system described in the literature. [ABSTRACT FROM AUTHOR]

This paper considers model order reduction based on low-rank decomposition of the cross Gramian for linear systems. The proposed approach uses the low-rank factors of the cross Gramian to generate approximate balanced model for the large-scale system. Then, the reduced-order model is obtained by truncating the states corresponding to the small approximate Hankel singular values. The low-rank factors are directly constructed from the expansion coefficients of impulse responses in the space spanned by Legendre polynomials by solving block tridiagonal linear systems. In contrast to balanced truncation related approaches which require to solve Sylvester equation to compute the full cross Gramian, the proposed method needs to solve sparse linear equations, and only one singular value decomposition is applied to a low-dimensional matrix. The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, the reduction procedure is modified to alleviate the shortcoming, which may unexpectedly lead to unstable systems even though the original one is stable. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study. [ABSTRACT FROM AUTHOR]

To implement deep learning models on edge devices, model compression methods have been widely recognized as useful. However, it remains unclear which model compression methods are effective for Structured State Space Sequence (S4) models incorporating Diagonal State Space (DSS) layers, tailored for processing long-sequence data. In this paper, we propose to use the balanced truncation, a prevalent model reduction technique in control theory, applied specifically to DSS layers in pre-trained S4 model as a novel model compression method. Moreover, we propose using the reduced model parameters obtained by the balanced truncation as initial parameters of S4 models with DSS layers during the main training process. Numerical experiments demonstrate that our trained models combined with the balanced truncation surpass conventionally trained models with Skew-HiPPO initialization in accuracy, even with fewer parameters. Furthermore, our observations reveal a positive correlation: higher accuracy in the original model consistently leads to increased accuracy in models trained using our model compression method, suggesting that our approach effectively leverages the strengths of the original model.

This paper proposes a comprehensive model order reduction framework to enable fast power integrity verification at the system level. This approach is developed to compress models of complete power delivery networks of high-end multiprocessor systems, where electromagnetic models of board and package are connected through banks of per-core Fully Integrated Voltage Regulators to chip models and loads in a closed-loop configuration. Due to complexity in both dynamical behavior and number of signals to be monitored, a direct transient simulation at the system level is very challenging. We show that a careful topological formulation of the circuit equations leads to a global model format that enables a structured projection framework for the elimination of the redundant states. Within this framework, we present and compare two alternative approaches based on approximate interpolation and empirical balancing, here adapted for the application at hand. In both cases, the resulting system is proven to be unconditionally stable both in open and in closed-loop configuration. Transient simulation of the reduced system provides a speedup exceeding $100\times $ with respect to SPICE.

This paper presents a new method for the design of separable-denominator 2-D IIR filters with nearly linear phase in the passband. The design method is based on a balanced realization model reduction technique. The nearly linear-phase 2-D IIR filter is designed using 2-D model reduction from a linear-phase 2-D FIR filter, which serves as the initial filter. The structured controllability and observability Gramians Ps and Qs serve as the foundation for this technique. onal positive-definite matrices that satisfy 2-D Lyapunov equations. An efficient method is used to compute these Gramians by minimizing the traces of Ps and Qs under linear matrix inequality (LMI) constraints. The use of these Gramians ensures that the resulting 2-D IIR filter preserves stability and can be implemented using a separable-denominator 2-D filter with fewer coefficients than the original 2-D FIR filter. Numerical examples show that the proposed method compares favorably with existing techniques.

To ensure reliable and efficient operation of fuel cell systems, it is important to monitor them online. However, placing sensors inside the fuel cell is often challenging, so virtual sensing using an efficient state observer is used in this study. Detecting local internal phenomena, such as reactants’ starvation, membrane dryout/flooding, and nitrogen accumulation, requires knowledge of the spatial distribution of internal states. Lumped-parameter models are not suitable for this, as they use a single variable to describe parameters such as hydrogen concentration. Instead, a high-order distributed-parameter fuel cell model is used to predict the spatial profiles of various internal states. An observer algorithm is employed to correct the predicted quantities using a few measurements taken at the system boundary. This update step only considers dominant dynamics from a reduced model to adjust all system states accordingly, making it computationally efficient and robust. The observer algorithm’s performance was verified against a high-fidelity model through detailed simulations.

Balanced singular perturbation approximation (SPA) is a model order reduction method for linear time-invariant systems that guarantees asymptotic stability and for which there exists an a priori error bound. In that respect, it is similar to balanced truncation (BT). However, the reduced models obtained by SPA generally introduce better approximation in the lower frequency range and near steady-states, whereas BT is better suited for the higher frequency range. Even so, independently of the frequency range of interest, BT and its variants are more often applied in practice, since there exist more efficient algorithmic realizations thereof. In this paper, we aim at closing this practically relevant gap for SPA. We propose two novel and efficient algorithms that are adapted for different settings. First, we derive a low-rank implementation of SPA that is applicable in the large-scale setting. Second, a data-driven reinterpretation of the method is proposed that only requires input-output data and thus is realization-free. A main tool for our derivations is the reciprocal transformation, which induces a distinct view on implementing the method. While the reciprocal transformation and the characterization of SPA are not new, their significance for the practical algorithmic realization has been overlooked in the literature. Our proposed algorithms have well-established counterparts for BT and, as such, a comparable computational complexity. The numerical performance of the two novel implementations is tested for several numerical benchmarks, and comparisons to their counterparts for BT as well as to existing implementations of SPA are made. [ABSTRACT FROM AUTHOR]

The climate system can be numerically represented by a set of physically based dynamical equations whose solution requires substantial computational resources. This makes computationally efficient, low dimensional emulators that simulate trajectories of the underlying dynamical system an attractive alternative for model evaluation and diagnosis. We suggest that since such an emulator must adequately capture anomaly evolution, its construction should employ a grid search technique where maximum forecast skill determines the best reference model. In this study, we demonstrate this approach by testing different bases used to construct a Linear Inverse Model (LIM), a stochastically forced multivariate linear model that has often been used to represent the evolution of coarse‐grained climate anomalies in both models and observations. LIM state vectors are typically represented in a basis of the leading Empirical Orthogonal Functions (EOFs), but while dominant large‐scale climate variations often are captured by a subset of these statistical patterns, key precursor dynamics involving relatively small scales are not. An alternative approach is balanced truncation, where the dynamical system is transformed into its Hankel space, whose modes span both precursors and their subsequent responses. Constructing EOF‐ and Hankel‐based LIMs from monthly observed anomalous Pacific sea surface temperatures, both for the 150‐year observational record and a perfect model study using 600 years of LIM output, we find that no balanced truncation model of any dimension can outperform an EOF‐based LIM whose dimension is chosen to maximize independent skill. However, the dynamics of a high‐dimensional EOF‐based LIM can be efficiently reproduced by far fewer Hankel modes. Plain Language Summary: The climate system may be physically and dynamically represented by a set of partial differential equations, from which one can obtain various numerical models and typically require substantial computational resources to solve. It is therefore attractive to find a low order representation of the dynamical system that captures key climate phenomena. One such approach is the Linear Inverse Model (LIM), a multivariate linear empirical model that builds on the large scale climate variability, identified from principal component analysis, and infers the lagged response relationship between two temporally varying climate states. This approach, however, may not adequately identify the small scale precursor dynamics, the occurrences of which often leads to large scale phenomena. Alternatively, the balanced truncation approach is capable of capturing the precursor dynamics and the large‐scale response, by which we find the balanced truncation is more effective in approximating the dynamical system using a lower dimension than LIM. Our finding implies potential use of the balanced truncation approach in many numerical dynamical systems that may benefit from reducing the dimensions and improving the efficiency. Key Points: Low dimensional emulators of the climate system should capture predictable dynamics, which may be evaluated using forecast skill metricsObserved predictable dynamics of high‐dimensional linear inverse model (LIM) can be reproduced by lower dimensional balanced truncationHowever, no balanced truncation model of any dimension can outperform a LIM whose dimension is chosen to maximize independent skill [ABSTRACT FROM AUTHOR]

We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users' demand. Based upon the generalized multiscale simulation on the coarse scale, the balanced truncation is applied to solve its Lyapunov equations on the reduced scale for further savings while ensuring high accuracy. A θ-implicit scheme is utilized for the fully discretization process. Finally, numerical results validate the uniform stability and robustness of our proposed method. [ABSTRACT FROM AUTHOR]

Parallel manipulators are more energy-efficient and present higher speed/acceleration ratios when compared with serial manipulators. Moreover, reducing their components' inertia can further improve these performance criteria. However, this design alternative could induce vibrations due to its lighter components. In this case, model-based control design strategies can help the designer to meet high-performance requirements. These strategies require limited-sized models, while the methods for modeling system's flexibilities usually yield large-sized models. Therefore, strategies for reducing the size of these large-sized models should be employed. In this manuscript, a two-step model reduction strategy is proposed to derive suitable models for control design. This proposal preserves all rigid modes and selects the most relevant flexible ones. An Linear Quadratic Gaussian is designed for a flexible 3RRR parallel manipulator using a limited-sized model. This model-based control strategy is evaluated in a flexible multibody environment demonstrating the proposal's viability. [ABSTRACT FROM AUTHOR]

• A convex optimization problem is established to extract the symmetric system. • A modified version of the block Arnoldi process is investigated. • A mixed model reduction method based on the symmetrizer for regular linear systems is proposed. • The H 2 error bound between the original system and the reduced system is studied. • The mixed model reduction method for linear systems is discussed. In regard of the cross Gramian and the block Arnoldi process, this paper focuses on mixed model reduction for two classes of dynamical systems. By studying the correlation between the cross Gramian and the Hankel singular values, model reduction via the cross Gramian is investigated for symmetric regular linear systems. Considering the non-symmetric regular linear system, associated with the symmetrizer, a convex optimization problem is established to obtain a symmetric system. Then, the non-symmetric regular linear system is divided into two parts, in which one is determined by the symmetric system and another is a non-symmetric counterpart. Model reduction via the cross Gramian is employed to reduce the first part. According to the structure of the non-symmetric counterpart, a modified version of the block Arnoldi process is developed to reduce the non-symmetric counterpart. The H 2 error bound between the original system and the reduced system is discussed. This model reduction approach avoids partitioning the non-symmetric system into several single-input and single-output systems. Moreover, the proposed model reduction method can also be applied to the non-symmetric linear systems. Finally, the proposed model reduction approach is illustrated by a 2D convective thermal flow benchmark and an optimal cooling problem of steel profiles. [ABSTRACT FROM AUTHOR]

Accumulative error along with time impedes the application of many computational methods on high‐precision simulation; therefore, time‐interval methods emerge. In this paper, a new model reduction method of quadratic‐bilinear (QB) systems based on time‐interval Gramians (TIGs) is presented. The conditions for generalized Lyapunov equations whose solutions are exactly TIGs to be solvable are derived. Lyapunov stability and error bound are discussed to demonstrate the succession and advance of time‐interval balanced truncation (TIBT). The accuracy and robustness are improved, as is illustrated in numerical results. [ABSTRACT FROM AUTHOR]

This paper proposes a novel algorithm, termed Mixed Riccati–Lyapunov Balanced Truncation (MRLBT), tailored for order reduction of Linear Time-Invariant Continuous-Time Descriptor Systems (LTI-CTD), commonly encountered in electrical and electronic circuit modeling. The MRLBT approach synergistically combines the advantages of balanced truncation (BT) and positive-real balanced truncation (PRBT) techniques while mitigating their limitations. Unlike BT, which preserves stability but not passivity, and PRBT, which retains passivity at the expense of larger reduction errors, MRLBT ensures the preservation of both stability and passivity inherent in the original system. Additionally, MRLBT achieves reduced computational complexity and minimized order reduction errors compared to PRBT. The proposed algorithm transforms the system into an equivalent Mixed Riccati–Lyapunov Balanced form, enabling the construction of a reduced-order model that retains the critical physical properties. Theoretical analysis and proofs are provided, establishing an upper bound on the global order reduction error. The efficacy of MRLBT is demonstrated through a numerical example involving an RLC ladder network, showcasing its superior performance over BT and PRBT in terms of reduced errors in the time and frequency domains.

Abstract The climate system can be numerically represented by a set of physically based dynamical equations whose solution requires substantial computational resources. This makes computationally efficient, low dimensional emulators that simulate trajectories of the underlying dynamical system an attractive alternative for model evaluation and diagnosis. We suggest that since such an emulator must adequately capture anomaly evolution, its construction should employ a grid search technique where maximum forecast skill determines the best reference model. In this study, we demonstrate this approach by testing different bases used to construct a Linear Inverse Model (LIM), a stochastically forced multivariate linear model that has often been used to represent the evolution of coarse‐grained climate anomalies in both models and observations. LIM state vectors are typically represented in a basis of the leading Empirical Orthogonal Functions (EOFs), but while dominant large‐scale climate variations often are captured by a subset of these statistical patterns, key precursor dynamics involving relatively small scales are not. An alternative approach is balanced truncation, where the dynamical system is transformed into its Hankel space, whose modes span both precursors and their subsequent responses. Constructing EOF‐ and Hankel‐based LIMs from monthly observed anomalous Pacific sea surface temperatures, both for the 150‐year observational record and a perfect model study using 600 years of LIM output, we find that no balanced truncation model of any dimension can outperform an EOF‐based LIM whose dimension is chosen to maximize independent skill. However, the dynamics of a high‐dimensional EOF‐based LIM can be efficiently reproduced by far fewer Hankel modes.

Rapid prediction of the aeroacoustic response is a key component in the design of aircraft and turbomachinery. In this work, we propose a technique for highly accelerated prediction of aeroacoustic response using a data-driven model reduction approach based on the eigensystem realization algorithm (ERA). Specifically, we create and compare ERA reduced-order models (ROMs) based on the training data generated by solving the linearized and nonlinear Euler equations, and we use them to predict the aeroacoustic response of an airfoil in a purely predictive setting subject to different types of gust loading. Activating each input channel separately in the full-order model (FOM) to generate the Markov sequence for training makes it computationally challenging to use the ERA in aeroacoustics applications. We address this bottleneck by first proposing a multifidelity gappy POD method to reduce the computation cost on the FOM and ROM levels by querying the high-resolution FOM only for the input channels identified by gappy POD. Second, we use tangential interpolation at the ROM level to reduce the size of the Hankel matrix. The proposed methods enable application of the ERA for accurate online acoustic response prediction, and they reduce the offline computation cost of ROMs. [ABSTRACT FROM AUTHOR]

Data-driven reduced-order models often fail to make accurate forecasts of high-dimensional nonlinear dynamical systems that are sensitive along coordinates with low-variance because such coordinates are often truncated, e.g., by proper orthogonal decomposition, kernel principal component analysis, and autoencoders. Such systems are encountered frequently in shear-dominated fluid flows where nonnormality plays a significant role in the growth of disturbances. In order to address these issues, we employ ideas from active subspaces to find low-dimensional systems of coordinates for model reduction that balance adjoint-based information about the system's sensitivity with the variance of states along trajectories. The resulting method, which we refer to as covariance balancing reduction using adjoint snapshots (CoBRAS), is analogous to balanced truncation with state and adjoint-based gradient covariance matrices replacing the system Gramians and obeying the same key transformation laws. Here, the extracted coordinates are associated with an oblique projection that can be used to construct Petrov-Galerkin reduced-order models. We provide an efficient snapshot-based computational method analogous to balanced proper orthogonal decomposition. This also leads to the observation that the reduced coordinates can be computed relying on inner products of state and gradient samples alone, allowing us to find rich nonlinear coordinates by replacing the inner product with a kernel function. In these coordinates, reduced-order models can be learned using regression. We demonstrate these techniques and compare to a variety of other methods on a simple, yet challenging three-dimensional system and a nonlinear axisymmetric jet flow simulation with 105 state variables. [ABSTRACT FROM AUTHOR]

Electromigration (EM) has become one of the most significant challenges considering longterm reliability in integrated circuit design. The problem is caused by the large current density in circuit interconnections. However, in most cases, we are interested in the EM stress at specific points of the interconnect, such as vias, junctions and boundaries. As a result, Model Order Reduction (MOR) techniques can provide attractive methodologies to reduce the complexity of the original systems. System-theoretic techniques like Balanced Truncation (BT) offer very reliable bounds for the approximation error, compared to moment-matching methods. In this paper, we apply a computationally efficient Low-Rank BT procedure based on the extended Krylov subspace method, that can handle large-scale models and significantly accelerate the EM stress analysis for any general interconnect or power grid structure. For the experimental evaluation, we first consider a general structure interconnect tree to demonstrate the generalizability of the proposed method and then we employ our method on the industrial IBM power grid benchmarks where the its scalability is shown. For the latter, we demonstrate that our method can achieve a speedup up to 238 × over a standard transient analysis method and a speedup up to 15 × over COMSOL, while exhibiting negligible error. • Practical numerical approach that can be applied on general multi-segment structures. • Accurate results within a predefined error bound by using Balanced Truncation. • Efficient reduction by using low-rank EKS method for solving the Lyapunov equations. • Reusable reduced models for different electromigration stress scenaria. [ABSTRACT FROM AUTHOR]

• Reduced order model based on balanced truncation method is obtained for hybrid microgrid. • The reduced order model has captured the dominant oscillatory modes of the hybrid microgrid. • The reduced order model is utilized to design a supplementary LQR known for its efficiency and high performance. • The proposed optimal controller significantly improves the performance of the hybrid 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. [ABSTRACT FROM AUTHOR]

We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users’ demand. Based upon the generalized multiscale simulation on the coarse scale, the balanced truncation is applied to solve its Lyapunov equations on the reduced scale for further savings while ensuring high accuracy. A θ-implicit scheme is utilized for the fully discretization process. Finally, numerical results validate the uniform stability and robustness of our proposed method.

Background and Objectives: This paper proposes a new Model Order Reduction (MOR) method based on the Bilinear Balanced Truncation (BBT) approach. In the BBT method, solving the generalized Lyapunov equations is necessary to determine the bilinear system's controllability and observability Gramians. Since the bilinear systems are generally of high order, the computation of the Gramians of controllability and observability have huge computational volumes. In addition, the accuracy of reduced-order model obtained by BT is relatively low. In fact, the balanced truncation method is only available for local energy bands due to the use of type I Gramians. In this paper, BBT based on type II controllability and observability Gramians would be considered to fix these drawbacks. Methods: At first, a new iterative method is proposed for determining the proper order for the reduced-order bilinear model, which is related to the number of Hankel singular values of the bilinear system whose real parts are closest to origin and have the most significant amount of energy. Then, the problem of determining of type II controllability and observability Gramians of the high-order bilinear system have been formulated as a constrained optimization problem with some Linear Matrix Inequality (LMI) constraints for an intermediate middle-order system. Then, the achieved Gramians are applied to the BBT method to determine the reduced-order model of the bilinear system. Next, the steady state accuracy of the reduced model would be improved via employing a tuning factor. Results: Using the concept of type II Gramians and via the proposed method, the accuracy of the proposed bilinear BT method is increased. For validation of the proposed method, three high-order bilinear models are approximated. The achieved results are compared with some well-known MOR approaches such as bilinear BT, bilinear Proper Orthogonal Decomposition (POD) and Bilinear Iterative Rational Krylov subspace Algorithm (BIRKA) methods. Conclusion: According to the obtained results, the proposed MOR method is superior to classical bilinear MOR methods, but is almost equivalent to BIRKA. It is out-performance respecting to BIRKA is its guaranteed stability and convergence. [ABSTRACT FROM AUTHOR]

The high-speed train bogie is a typical underactuated mechanical system. It has been difficult to design controls to increase the stability against hunting motion with limited sensor measurements and actuation hardware. This paper develops a control design for the stability improvement of the bogie by making use of the balanced truncation reduced order model of the bogie. When the train travels at speed higher than the critical speed, the bogie has unstable modes. The balanced truncation model reduction (BTMR) is usually applied to stable systems. In this paper, we first apply the real Schur decomposition to separate the stable and unstable modes of the bogie. The BTMR is then carried out for the stable subsystem. An augmented X 0 -balanced truncation algorithm is considered to deal with the dependence of the BTMR on initial conditions. The reduced order subsystem is then combined with the unstable subsystem. A linear quadratic regulator control together with a Luenberger observer for the state estimation is designed for the bogie with non-zero initial conditions. Simulation results show that the control designed with the balanced truncation reduced order model performs very well in terms of the stability improvement by significantly increasing the critical train speed. [ABSTRACT FROM AUTHOR]

• Modeling and analysis of piezoelectric tonpilz transducer. • Frequency-limited balancing based model reduction. • Iterative methods for Lyapunov equations. • Low-rank Gramian factors. • Sparsity preservation Index-I descriptor system. Piezoelectric based Tonpilz Transducer is a mechatronics control device that works on high-power sound propagation to navigate or detect objects under the water. In order to analyze the design and performance, we generate a mathematical model of such apparatus by applying the finite element method, which yields a large-scale sparse index-I descriptor system. We also study low-rank Gramian factors based on Balanced Truncation to obtain the reduced-order model for efficient design on the limited frequency domain. The low-rank Gramian factors are obtained by solving a pair of frequency-limited Lyapunov equations. We have modified the low-rank Alternative Direction Implicit method to solve the Lyapunov equations for retaining sparsity and ensuring the preservation of the structure of the system matrices. Numerical analyses are conducted to validate the proposed techniques, and the reduced order physical model is regenerated for a comparative physical analysis with the full dimensional physical model. [ABSTRACT FROM AUTHOR]

The load-frequency control is a matter of interest in power system operation as it addresses the problem of controlling the system in response to the disturbances. A well-known solution to this problem is feedback stabilization through Linear Quadratic Regulator. The observability condition has to be satisfied for the implementation of LQR, but for certain cases of multi-source power systems this criteria fails. Also, to obtain the desired response, proper selection of the weighting matricesQand R for LQR controller is inevitable. On a trial and error based choice of these matrices, the classical optimal controller may fail to achieve an optimum set point tracking, due to the absence of an integral control action. In this paper, a balanced reduction technique is proposed to obtain a minimal realization of the unobservable system, which makes the optimal control design possible for the control of change in frequency. Thus the paper also applies PID output feedback controller designed through LQR. The impact of reduction for minimal realization and the LQR based PID controller in the frequency stabilization of the considered systems are highlighted in this paper through pertinent cases. [ABSTRACT FROM AUTHOR]

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.

Frequency limited model order reduction algorithm presented by Wang and Zilouchian for discrete-time systems provide unstable reduced-order models and also do not provide a priori error bound formula. Many stability-preserving model order reduction algorithms were presented; however, these methods produce significant approximation errors in the desired frequency interval. An improved algorithm of model order reduction for the discrete-time systems is presented. The proposed technique gives the stable reduced-order model and also provides less approximation error as compared with other algorithms and also provides the formula for the frequency response a priori error bound. Numerical examples provided at the end of the section show the efficacy of the proposed technique.

In this article, we analyze a structure-preserving model order reduction technique for deterministic and stochastic delay equations based on the balanced truncation method and provide a system theoretic interpretation. Transferring the framework of [6], we find error estimates for the difference between the dynamics of the full and reduced model. This analysis also yields new error bounds for bilinear systems and stochastic systems with multiplicative noise and non-zero initial states. [ABSTRACT FROM AUTHOR]

In this paper, based on low‐rank balanced truncation (BT), we present a splitting model reduction method for bilinear control systems with nonzero initial conditions. By analyzing Volterra series, the original bilinear control system is split up into two systems: one is a bilinear control system with zero initial condition, and the other is our constructed control system. Specifically, we prove that the Volterra series containing initial condition of the original system converges to the output of our constructed control system. Thus, we develop a new splitting model reduction method that involves reducing two systems independently by low‐rank BT method and then combining the reduced outputs into an aggregate output. A nonlinear RC circuit and a heat transfer model are taken as numerical examples to illustrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]