Structure-preserving Model Reduction Research Articles

Structure-preserving model reduction based on multiorder Arnoldi for discrete-time second-order systems

This paper investigates the new structure-preserving model reduction methods based on multiorder Arnoldi for discrete-time second-order systems by using discrete Laguerre polynomials. The expansion coefficients of discrete-time second-order system and its equivalent first-order system satisfy the implicit recursive expressions in space spanned by discrete Laguerre polynomials, and then multiorder Arnoldi algorithm is applied to the implicit recursive expressions to produce the column-orthogonal matrices, where the orthogonal projection matrix for the equivalent first-order system is constructed by partitioning the resulting column-orthogonal matrix appropriately. The reduced-order systems can preserve the special structures of the original systems as well as match a desired number of expansion coefficients of the original outputs. The error estimate of output variables is also given for discrete-time second-order system. Two benchmark examples are simulated to illustrate the effectiveness of the proposed methods.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Canonical and noncanonical Hamiltonian operator inference

A method for the nonintrusive and structure-preserving model reduction of canonical and noncanonical Hamiltonian systems is presented. Based on the idea of operator inference, this technique is provably convergent and reduces to a straightforward linear solve given snapshot data and gray-box knowledge of the system Hamiltonian. Examples involving several hyperbolic partial differential equations show that the proposed method yields reduced models which, in addition to being accurate and stable with respect to the addition of basis modes, preserve conserved quantities well outside the range of their training data.

Structure-preserving model reduction for port-Hamiltonian systems based on separable nonlinear approximation ansatzes

We discuss structure-preserving model order reduction for port-Hamiltonian systems based on a nonlinear approximation ansatz which is linear with respect to a part of the state variables of the reduced-order model. In recent years, such nonlinear approximation ansatzes have gained more and more attention especially due to their effectiveness in the context of model reduction for transport-dominated systems which are challenging for classical linear model reduction techniques. We demonstrate that port-Hamiltonian reduced-order models can often be obtained by a residual minimization approach where a suitable weighted norm is used for the residual. Moreover, we discuss sufficient conditions for the resulting reduced-order models to be stable. Finally, the methodology is illustrated by means of two transport-dominated numerical test cases, where the ansatz functions are determined based on snapshot data of the full-order state.

Extended Balancing of Continuous LTI Systems: A Structure-Preserving Approach

In this article, we treat extended balancing for continuous-time linear time-invariant systems. We take a dissipativity perspective, thus, resulting in a characterization in terms of linear matrix inequalities. This perspective is useful for determining a priori error bounds. In addition, we address the problem of structure-preserving model reduction of the subclass of port-Hamiltonian systems. We establish sufficient conditions to ensure that the reduced-order model preserves a port-Hamiltonian structure. Moreover, we show that the use of extended Gramians can be exploited to get a small error bound and, possibly, to preserve a physical interpretation for the reduced-order model. We illustrate the results with a large-scale mechanical system example. Furthermore, we show how to interpret a reduced-order model of an electrical circuit again as a lower dimensional electrical circuit.

Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We demonstrate the effectiveness of all considered methods in terms of their accuracy and computational cost by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems. The results of the experiments are extensively discussed to derive guidelines for the choice of model reduction methods based on the problem setting.

On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks

This paper deals with the systematic development of structure-preserving and robust approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic Euler equations. Our approach is guided throughout by energy-based modeling concepts (port-Hamiltonian formalism, theory of Legendre transformation), which provide a convenient and general line of reasoning. Under mild assumptions on the approximation, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be shown. Our approach is not limited to conventional space discretization but also covers complexity reduction of the nonlinearities by inexact integration. Thus, it can serve as a basis for structure-preserving model reduction. Combined with an energy stable time integration, we numerically demonstrate the applicability and good stability properties of the approach using the Euler equations as an example.

Knowledge-Based Optimal Irrigation Scheduling of Agro-Hydrological Systems

Agricultural irrigation consumes about 70% of freshwater globally every year. To improve the water-use efficiency in agricultural irrigation is critical as we move toward water sustainability. An irrigation scheduler determines how much water to irrigate and when to irrigate for an agricultural field. To get a high-resolution irrigation-scheduling solution for a large-scale agricultural field is still an open research problem. In this work, we propose a knowledge-based optimal irrigation-scheduling approach for large-scale agricultural fields that are equipped with center pivot irrigation systems. The proposed scheduler is designed in the framework of model predictive control. The objective of the proposed scheduler is to maximize crop yield while minimizing irrigation water consumption and the associated electricity usage. First, we introduce a structure-preserving model reduction technique to significantly reduce the dimensionality of agro-hydrological systems. Then, based on the reduced model, an optimization-based scheduler is designed. In the design of the scheduler, knowledge from farmers is taken into account to further reduce the computational complexity of the scheduler. The proposed approach explicitly considers both the irrigation time and the irrigation amount as decision variables to keep the crop within the stress-free zone considering the weather uncertainty and heterogeneous soil types for large agricultural fields. The proposed approach is applied to three different scenarios with different soil types, crops, and weather uncertainty. The results show that in all the conditions, the scheduler is capable of keeping the crops stress-free, which results in maximum yield and, at the same time, minimizes water consumption and irrigation events.

A comparative study of reduced model based boundary control design for linear port Hamiltonian systems

A comparative study of passivity based boundary control design for a class of infinite dimensional port-Hamiltonian system using two different model reduction approaches is presented. The first approach is based on a direct low order structure preserving discretization while the second approach arise from the structure preserving model reduction of a high order discretzed model. Two passivity-based control techniques, namely control by interconnection and damping injection, are used to change the equilibrium point and the convergence rate of the closed-loop system. An Euler-Bernoulli beam example is used to illustrate the findings by means of discussion and numerical simulations.

Geometric Optimization for Structure-Preserving Model Reduction of Hamiltonian Systems

Classical model reduction methods disregard the special symplectic structure associated with Hamiltonian systems. A key challenge in projection-based approaches is to construct a sym-plectic basis that captures the essential system information. This necessitates the computation of a so-called proper symplectic decomposition (PSD) of a given sample data set. The PSD problem allows for a canonical formulation as an optimization problem on the symplectic Stiefel manifold. However, as with their Euclidean counterparts, symplectic projectors only depend on the underlying symplectic subspaces and not on the selected symplectic bases. This motivates to tackle the PSD problem as a Riemannian optimization problem on the symplectic Grassmann manifold, i.e., the matrix manifold of symplectic projectors. Initial investigations on this manifold feature in a recent preprint of the authors. In this work, we investigate the feasibility and performance of this approach on two academic numerical examples. More precisely, we calculate an optimized PSD for snapshot matrices that stem from solving the one-dimensional linear wave equation and the one-dimensional nonlinear Schrödinger equation.

Knowledge-based optimal irrigation scheduling of three-dimensional agro-hydrological systems

An agricultural irrigation scheduler determines how much to irrigate and when to irrigate for a field. The accurate and effective scheduler decision for large agricultural fields is still an open research problem. In this work, we address the high dimensionality of the agricultural field and propose a knowledge-based approach to provide optimal irrigation amount and irrigation time for three-dimensional agro-hydrological systems. First, we introduce a structure-preserving model reduction technique to decrease the dimension of the system. Then, based on the reduced model, an optimization-based scheduler is designed. In the design of the scheduler, empirical knowledge from farmers is considered to significantly reduce the computational complexity. The proposed scheduler is designed in the framework of model predictive control. The objective of the proposed scheduler is to maximize crop yield while minimizing irrigation water consumption and the associated electricity usage. The proposed approach is applied to a field to show the effectiveness and superiority of the proposed framework.

Dynamical reduced basis methods for Hamiltonian systems

We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.

Reduced Order LQG Control Design for Infinite Dimensional Port Hamiltonian Systems

This article proposes a method that combines linear quadratic Gaussian (LQG) control design and structure preserving model reduction for the reduced order control of infinite dimensional port Hamiltonian systems (IDPHS).For that purpose the weighting operators used in LQG control design are chosen such that the resulting dynamic controller is passive and the closed-loop system equivalent to control by interconnection. The method of Petrov-Galerkin is then used to approximate the balanced realization of the IDPHS by a finite dimensional port Hamiltonian system and to provide the associated reduced order LQG controller. The main advantages of the proposed method are that, first, both control and reduction are driven by closed-loop performances and that, second, due to the passivity properties of the controller the closed-loop stability is guaranteed when the finite dimensional controller is applied to the infinite dimensional system.

Structure preserving model reduction of second‐order time‐delay systems via approximate Gramians

In this study, a model-order reduction approach based on approximate Gramians for second-order time-delay systems is proposed. By treating the second-order time-delay system as a special case of the integro-differential system, it exhibits a connection with the time-delay Gramians of the corresponding first-order system. After applying the approximate Gramians and balanced truncation procedures, the resulting reduced system is expected to well approximate the original system. Besides, the e -embedding technique is applied to the case when the damping matrix of the second-order time-delay system is singular. Finally, numerical examples show the good performance in the time domain and the finite frequency range.

Clustering-Based Model Reduction of Laplacian Dynamics With Weakly Connected Topology

This article studies the structure-preserving model reduction of Laplacian dynamics, which represent weakly connected directed networks with diffusive couplings. The notion of clusterability is introduced to guarantee a bounded reduction error, and a clustering algorithm is then proposed to partition the nodes into clusters, such that the nodes in each cluster form a connected subgraph of the original network. Then, a reduced-order model, which is established using the generalized balanced form of the original network, preserves the weakly connection structure and consensus property. Finally, the effectiveness of the proposed approach is illustrated by a numerical example.

Structure-preserving model reduction for dynamical systems with a first integral

Since the expense of the numerical integration of large scale dynamical systems is often computationally prohibitive, model reduction methods, which approximate such systems by simpler and much lower order ones, are often employed to reduce the computational effort. In this paper, for dynamical systems with a first integral, new structure-preserving model reduction approaches are presented that yield reduced-order systems while preserving the first integral. We apply energy-preserving integrators to the reduced-order systems and show some numerical experiments that demonstrate the favourable behaviour of the proposed approaches.

Krylov projection methods for linear Hamiltonian systems

We study geometric properties of Krylov projection methods for large and sparse linear Hamiltonian systems. We consider in particular energy preservation. We discuss the connection to structure preserving model reduction. We illustrate the performance of the methods by applying them to Hamiltonian PDEs.

Clustering approach to model order reduction of power networks with distributed controllers

This paper considers the network structure preserving model reduction of power networks with distributed controllers. The studied system and controller are modeled as second-order and first-order ordinary differential equations, which are coupled to a closed-loop model for analyzing the dissimilarities of the power units. By transfer functions, we characterize the behavior of each node (generator or load) in the power network and define a novel notion of dissimilarity between two nodes by the mathcal {H}_{2}-norm of the transfer function deviation. Then, the reduction methodology is developed based on separately clustering the generators and loads according to their behavior dissimilarities. The characteristic matrix of the resulting clustering is adopted for the Galerkin projection to derive explicit reduced-order power models and controllers. Finally, we illustrate the proposed method by the IEEE 30-bus system example.

On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks

We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semidiscretization in space via a mixed finite element method and also during the further dimension reduction by structure- preserving Galerkin projection, which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models, and we discuss certain modifications needed to satisfy some algebraic compatibility conditions which are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our arguments are based on a careful analysis of the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests, and the necessity of the compatibility conditions is demonstrated by examples.

Model reduction of distributed nonstationary LPV systems

This paper is on the structure-preserving model reduction of distributed systems formed by heterogeneous, discrete-time, nonstationary linear parameter-varying subsystems interconnected over arbitrary directed graphs. The subsystems are formulated in a linear fractional transformation (LFT) framework, and a communication latency of one sampling period is considered. The balanced truncation method is extended to the class of systems of interest, and upper bounds on the ℓ2-induced norm of the resulting error system are derived. Balanced truncation suffers from conservatism since it only applies to stable systems which possess structured solutions to the generalized Lyapunov inequalities. The coprime factors reduction method is then provided as a partial remedy to this conservatism. An illustrative example is given to demonstrate the efficacy of the proposed approaches.