Model order reduction of MIMO bilinear systems by multi-order Arnoldi method

doi:10.1016/j.sysconle.2016.04.005

Systems & Control Letters

Volume 94, August 2016, Pages 1-10

https://doi.org/10.1016/j.sysconle.2016.04.005 Get rights and content

Abstract

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples.

Introduction

Modeling of complex phenomena often leads to dynamical systems with high dimension. Those models are usually difficult to analyze and simulate and unsuitable for control design. Hence, model order reduction is called for. Model order reduction can be used to find a low-order model that approximates the behavior of the original high-order model, and facilitates both the controller design and the computationally efficient analysis of the system to induce desired behavior. Model order reduction of linear systems has been studied for several years, and there exists an effective theory. See [1], [2] and the references therein for details. Now, more and more attention is paid to model order reduction for nonlinear systems. Bilinear systems are a special class of nonlinear systems which arise in many sciences and engineering fields, such as nonlinear circuit, nuclear fission, heat transfer, fluid flow and thermal processes [3]. Bilinear systems can also be used to approximate the weakly nonlinear systems [4], [5].

Based on the techniques used for linear systems, several model order reduction methods for bilinear systems have been proposed in recent years. The SVD-based approaches to model order reduction for bilinear systems were studied in [6], [7], [8]. And the $H_{2}$ optimal model order reduction for the continuous and discrete bilinear systems was also developed in [9], [10], [11]. On the other hand, the interpolation-based Petrov–Galerkin projection techniques for model order reduction of bilinear systems were proposed in [4], [12], [13], [14], [15], [16], [17]. They include one-sided and two-sided projection methods. In addition, Laguerre-based model order reduction for bilinear systems was discussed in [18].

In [19], [20], model order reduction methods based on general orthogonal polynomials have been successfully used for linear and nonlinear systems. In this paper, we extend this method to bilinear systems, and present a time domain model order reduction method for MIMO bilinear systems. This approach expands the state variables in the space spanned by general orthogonal polynomials, then the expansion coefficient vectors are calculated by a recurrence formula. And a reduced system is obtained by a projection transformation, which is constructed by the multi-order Arnoldi procedure. The resulting reduced model can match the first finite expansion coefficients of the original system. The error estimate and the stability of the reduced system are discussed. Additionally, the two-sided projection methods for bilinear systems are also considered.

The remainder of this paper is organized as follows. In Section 2, we briefly introduce some preliminary properties of general orthogonal polynomials. In Section 3, a model order reduction method based on orthogonal polynomials for MIMO bilinear systems is produced. The main properties and stability of the reduced system are discussed. Furthermore, we also introduce the application of two-sided projection methods in this section. In Section 4, two numerical examples are given to indicate the efficiency of our approach. Conclusions are presented in Section 5.

Section snippets

Preliminary

General orthogonal polynomials, which can cover all kinds of orthogonal polynomials and nonorthogonal Taylor series, have been used by many researchers to approach control problems, such as system identification, optimal control, and model order reduction [19], [20], [21], [22], [23], [24], [25], [26]. In this section, we briefly introduce some important properties of orthogonal polynomials for model order reduction.

The orthogonal polynomials $ϕ_{i} (t)$ with the weight function $ω (t)$ over the

Model order reduction of bilinear systems

In this paper, we concentrate on the following time invariant multi-input multi-output (MIMO) bilinear system ${\begin{matrix} \dot{x} (t) = A x (t) + \sum_{i = 1}^{m} N_{i} x (t) u_{i} (t) + B u (t), \\ y (t) = C^{T} x (t), \end{matrix}$ with initial condition $x (0) = x_{0}$ , where $t$ is the time variable, $x (t) \in R^{n}$ is the state of the system, $n$ is the dimension of the state space. $u (t) \in R^{m}$ and $y (t) \in R^{p}$ are the input and output functions, and $u_{i} (t)$ is the $i$ th component of $u (t)$ . $A \in R^{n \times n}, B \in R^{n \times m}, C \in R^{n \times p}, N_{i} \in R^{n \times n}$ for $i = 1, 2, \dots, m$ , are constant matrices.

The goal of Petrov–Galerkin projection

Numerical examples

In this section, we present two numerical examples to illustrate the efficiency of our approach. We make a comparison with the Krylov subspace methods for model order reduction of bilinear systems in [12], [13], [14], [17], and also compare the performance of different orthogonal polynomials. Additionally, the input dependence of the presented method is discussed as well. All numerical experiments were run in MATLAB and we used ode15s to solve differential equations under investigation.

Example 1

In this

Conclusions

In this paper, we have considered the time domain model order reduction methods for MIMO bilinear systems based on general orthogonal polynomials. And a multi-order Arnoldi algorithm is presented for bilinear systems. The proposed algorithm uses multi-order Arnoldi procedure to construct a projection matrix. The resulting reduced model can match a desired number of expansion coefficients of the original system in the space spanned by orthogonal polynomials. And the freedom in choosing the left

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The conclusion (i) can be obtained immediately from Lemma 3.1. The proof of the conclusions (ii) and (iii) are similar to those in [19,23], and is omitted here. □
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^☆: This work was supported by the Natural Science Foundation of China (NSFC) under grant 11371287 and the International Science and Technology Cooperation Program of China under grant 2010DFA14700.

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Model order reduction of MIMO bilinear systems by multi-order Arnoldi method☆

Abstract

Introduction

Section snippets

Preliminary

Model order reduction of bilinear systems

Numerical examples

Conclusions

Systems Control Lett.

Automatica

Linear Algebra Appl.

Comput. Math. Appl.

Systems Control Lett.

J. Franklin Inst. B

Approximation of Large-Scale Dynamical Systems

Nonlinear Systems, Vol. II: Applications to Bilinear Control

Projection-based approaches for model reduction of weakly nonlinear, time-varying systems

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

Nonlinear Systems: Analysis, Stability, and Control

Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems

SIAM J. Control Optim.

Model Reduction of Bilinear System Using Balanced Singular Perturbation, Computer Applications for Security, Control and System Engineering

Balanced averaging of bilinear systems with applications to stochastic control

SIAM J. Control Optim.

Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems

Internat. J. Control

Interpolation-based H2-model reduction of bilinear control systems

SIAM J. Matrix Anal. Appl.

Interpolation Methods for the Model Reduction of Bilinear Systems

Interpolation-based $H_{2}$ -model reduction of bilinear control systems