Model Reduction Of Linear Negative Imaginary Systems

Establishing a proper mathematical model is the basis for simulating, analyzing and synthesizing of various engineering systems. With the development of engineering technology, adopting the common modeling techniques such as mechanism analysis often lead to high-order original system models, which make the consequent system simulation and system analysis very difficult due to the corresponding high numerical complexity. In this case model order reduction plays an important role. It consists in finding a reduced-order model preserving the properties of original system as well as approximating the input-output behaviors.In this thesis, we are dedicated to model order reduction of linear negative imaginary systems. Negative imaginary is very special property describing a class of “dissipative-like” characterization. Negative imaginary systems are generally arising from some flexible systems or circuit systems. For example, it has been revealed that lightly damped structures with collocated force actuators and position sensors are often of negative imaginary property. Although research on linear negative imaginary systems has received great attention in the recent several years, there exists no relevant result on the model order reduction of linear negative imaginary systems. On the other hand, model order reduction of linear negative imaginary systems is of great importance since many practical negative imaginary systems often lead to high-order models due to their complicate dynamic characterization. This motivates us to develop new methods and algorithms for solving the model reduction problems of linear negative imaginary systems. The major content of this thesis is as follows.Chapter 1 summarizes the main prevailing model order reduction methods for linear systems.Chapter 2 gives some basic definition and tools on the analysis of linear negative imaginary systems. Based on several examples and practical applications, the time-domain and frequency-domain properties of linear negative imaginary systems are illustrated.Chapter 3 investigates model order reduction of general linear negative imaginary systems in the framework of balanced truncation. Negative imaginary balanced truncation method is proposed based on negative imaginary lemma. By solving a couple of Lure?s equations or Riccati equations, a particular contragredient transformation of the original system can be obtained. Then the reduced system can be generated by truncating the transformed full system model. It is shown that the reduced system preserving negative imaginary property. Meanwhile, an upper bound of the approximation performance between the reduced system and full system is provided. Illustrating example are provided to show the effectives of the proposed method.Chapter 4 focuses on solving the model order reduction problems of second-order linear negative imaginary systems in the framework of balanced truncation theory. Second-order negative imaginary balanced truncation method is proposed by exploiting the system structure properties. Firstly an orthogonal congruent transformation matrix is computed by solving a Lyapunov equation. Then the reduced order model is generated by truncating the transformed original system model. It is validated that the reduced system not only preserving negative imaginary property but also preserving the second-order structure. Example are presented to show the effectives and advantages of the proposed approach.Chapter 5 investigates model order reduction of general linear negative imaginary systems based on moment matching theory. Special properties of the interpolation matrix with respect to the negative imaginary spectral zeros are derived at first, then the model order reduction problem can be recast as an interpolation problem. Example are carried out for illustration.Chapter 6 revisits model order reduction of second-order linear negative imaginary systems by adopting the moment matching approach. A second-order negative imaginary moment-matching model reduction algorithm is proposed. The algorithm generates an orthogonal basis spanning a class of second-order Krylov subspace, then the reduced system can be obtained by projecting the original system with respect to the orthogonal subspace. Example are included to show the effectives of the proposed approach.Finally, the results of the thesis are summarized and further research topics are pointed out.