Parameterized model order reduction for linear DAE systems via ε-embedding technique☆
Introduction
In some fields such as electrical circuits [1], [2], mechanical systems including micro-electro-mechanical systems [3], the singular perturbations of ODE systems [4], etc., the modelling of most systems in these fields involves the sets of the systems consisting of differential equations as well as algebraic equations. They result in DAE systems. These DAE systems are usually sparse and time-invariant, but they may be of high order. The algebraic equations part is also very important for the DAE system, especially when its dimension is very large. The index is an indicator of the computational complexity involved in numerically solving DAE systems. For DAE systems of index one, there are some methods used to get their numerical solutions, such as waveform relaxation [5], [6], [7], [8]. In general, the simulation of DAE systems of high index cannot be achieved within reasonable computing time due to the huge computational effort. Thus, it is advisable to construct a reduced-order system that can approximate the behavior of the original DAE system very well.
The purpose of MOR is to construct a system whose dimension is magnitude smaller than that of the original system, while it can approximate the input-output relation of the original system with high accuracy. It is known that the MOR methods for linear ODE systems have been well developed. However, the MOR methods for DAE systems are less than those for ODE systems. The linear systems can be reduced by using the Krylov subspace methods, such as the Arnoldi algorithm and Lanczos process, see [9], [10], [11], [12], [13]. Moment matching approximation based on Krylov subspace methods is one of the most useful MOR techniques for large-scale systems, see [14], [15]. Balanced truncation is another MOR approach that has been proved to be efficient for linear ODE systems, see [16], [17], [18], [19]. Moreover, for some other MOR methods such as POD methods and manifold-based methods one can refer to [20], [21], [22].
For the parameterized MOR, Benner studied and drew together the existing methods [23], e.g., the rational interpolation methods, balanced truncation methods and POD methods. Besides, the frequency-weighted balanced truncation technique was used to deal with the MOR of linear parametric-varying systems [24]. This method can preserve the closed-loop stability. Different from the above methods, the parameterized MOR method based on Taylor expansion and Laguerre polynomials was proposed [25]. According to a connection shown in this paper, an efficient algorithm based on the Krylov subspace was explored to obtain a reduced-order parametric system, where the calculation is independent on input variable.
For linear large-scale DAE systems, their MOR has received a lot of attention in recent years, see [26], [27], [28], [29], [30], [31], [32]. When we study the MOR of the DAE system, we often are inclined to reduce the order of the ODE equations part while keep the algebraic equations unchanged. But, in many cases, the order of the algebraic equations part is large. In [26], both the ODE equations part and the algebraic equations part were reduced, where the reduced-order system of the former was generated by POD and balancing methods, the latter was reduced by using the POD method. Utilizing the block-diagonal form, the DAE system with high index was decomposed into an ODE subsystem and a DAE subsystem to achieve the goal of MOR, where the ODE subsystem is reduced by the balanced truncation method and the reduced-order subsystem of the DAE subsystem is constructed by two Krylov subspace projection methods [27]. In [28], the balanced truncation MOR methods for coupled systems with DAE subsystems were explored. By individually applying the balanced truncation MOR method to the closed-loop system and the subsystems, two MOR methods for the coupled system were proposed. It should be noted that with the application of the ε-embedding technique, the DAE system can be turned into an ODE system [29], [30], [31], [32]. Thereby, the well-known MOR methods for ODE systems can be used to construct the reduced-order systems efficiently. But, the numerical ill-conditioning may be caused by the small parameter ε. Notice that after embedding the parameter ε, the original DAE system is converted into a system that shares the same form with the singularly perturbed system. Regarding the numerical efficiency of the singularly perturbed system, some strategies were given to avoid the ill-conditioning problem, e.g., see [33]. In [33], the authors discussed the optimal control problem for singularly perturbed linear systems with two-vector input systems. It shown that a reduced-order continuous algebraic Riccati equation (CARE) can be used to avoid possible numerical ill-conditioning in this setting.
In this paper, we explore the structure-preserving parameterized MOR method for DAE systems. By utilizing the ε-embedding technique, the DAE system is turned into an ODE system. After that, we view the resulting ODE system as a parametric system. Based on the parametric moments of the transfer function of the parametric ODE system, the column-orthonormal matrix is constructed to obtain the reduced-order parametric system. Theoretical analysis shows that the transfer functions of the original parametric ODE system and its reduced-order parametric system satisfy the parametric moment matching. Set and then we get the needed reduced-order DAE system, which can preserve the structure of the original DAE system. Furthermore, based on the Arnoldi algorithm, we give a new method to achieve the construction of the column-orthonormal matrix. Finally, the effectiveness of the proposed method is illustrated by two numerical examples.
In conclusion, the MOR process of our method is accomplished in three steps: First, turn the DAE system into a parametric ODE system by the ε-embedding technique; Second, reduce the resulting parametric ODE system and obtain the reduced-order parametric system by using the parametric moment matching method; Finally, set and get the needed reduced-order DAE system. Different from the existing MOR methods about DAE systems and parametric systems, e.g., [23], [24], [25], [26], [27], [31], [32], we combine the ε-embedding technique with the parametric moment matching method to construct reduced-order systems of DAE systems, where the frequency parameter s and the system parameter ε are considered at the same time. This provides a new way to deal with DAE systems. Making full use of the parametric moments and the Arnoldi algorithm, the column-orthonormal matrix is generated skillfully, which generates the structure-preserving reduced-order system. Moreover, the parametric reduced-order systems satisfy the parametric moment matching property. This guarantees that the reduced-order DAE system is a good approximation of the original DAE system. In fact, the good performances of the reduced-order systems have been verified by simulation results.
The outline of the paper is as follows. In Section 2, the ε-embedding technique is applied to the DAE system. Thereby, we get a parametric ODE system. The MOR of this parametric system is discussed via a structure-preserving parametric moment matching MOR method in Section 3. Additionally, the parametric moment matching for our method is analyzed in Section 3. In Section 4, we present the Arnoldi parameterized MOR method to construct the column-orthonormal matrix. In Section 5, two numerical examples are given to verify the effectiveness of our MOR method. Finally, some conclusions are given in Section 6.
Section snippets
Transformation via the ε-embedding technique
Consider a time-invariant linear DAE systemwhere is a singular matrix, is a nonsingular matrix, is the descriptor vector, is the input variable, is the output variable, and its initial condition is . Generally speaking, E does not have the special structure. It is known that if the matrix pencil (E, A) is regular, then it can be equivalently changed into the Weierstrass–Kronecker canonical form (PEQ,
The structure-preserving parametric moment matching MOR method
In general, ε in the system (2.3) can be chosen as a very small constant. In this case, the system (2.3) is a non-parametric ODE system. Thereby, we can obtain the corresponding reduced-order systems by utilizing the well-known non-parameterized MOR methods. In this section, we view ε as a varying parameter and thus we consider the parameterized MOR of the parametric system. First, we present the structure-preserving parametric moment matching MOR method. Then, we get a reduced-order parametric
The Arnoldi parameterized method for constructing the column-orthonormal matrix
In the above section, we discuss the MOR process by using the structure-preserving parametric moment matching method. To get the column-orthonormal matrix V, the classical Gram–Schmidt process can be applied to the space (3.4). However, this process may be not stable [36]. In this section, we use the Arnoldi algorithm to compute the column-orthonormal matrix V.
The space (3.4) can be rewritten aswhere the order of the
Numerical experiments
In this section, we apply the structure-preserving parametric moment matching MOR method to two numerical examples whose models can be described as linear DAE systems. In the first example, a linear RLC circuit is considered. The second example is a linear multi-input single-output DAE system with the order 980. We compare the structure-preserving parametric moment matching MOR method with the direct Arnoldi algorithm, so as to gauge the accuracy and efficiency of our method. All simulations
Conclusion
This paper presents the structure-preserving parametric moment matching MOR method for DAE systems. The DAE system is turned into a parametric ODE system by using the ε-embedding technique. Based on the parametric moments of the transfer function of the parametric ODE system, the column-orthonormal matrix is constructed to obtain the reduced-order parametric system. We have shown that the transfer functions of the original parametric ODE system and its reduced-order parametric system satisfy
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This work was supported by the Natural Science Foundation of China (NSFC) under grants 11871393 and 61663043, and the International Science and Technology Cooperation Program of Shaanxi Key Research & Development Plan under grant S2019-YF-GHZD-0003.