Minimum Information Loss Method For Model Reduction Based On Cross-Gramian Matrix (CGMIL)

Model reduction is one of the important topics in the field of control and system engineering. There is an ever-increasing need for improved accuracy in the description of dynamical systems, which leads to models of high complexity. The basic motivation for model reduction is the need for simplified models of dynamical systems, which capture the main features of the original complex models. The simplified model is then used in place of the original complex model for either simulation or control and so on. This dissertation mainly studied one kind model reduction method based on information theory.First, this dissertation introduced the present state of the art of model reduction and also introduced the basic mentality of some classics methods for model reduction. Second, it was the key innovative part that an improved algorithm of the minimum information loss (MIL) method, namely minimum information loss method for model reduction based on cross-Gramian matrix (denoted by CGMIL), was proposed by analyizing the MIL method as well as its extension (denoted by RMIL), and the application of CGMIL in a general multi-variable (MIMO) system was discussed.The main innovative work as well as contribution of this dissertation was as follows:(1) According to the information entropies of steady states of a LTI system and its dual system, the definitions of the controllability-information and the observability-information were standardized. From the perspective of the completeness of system information description, it was pointed out that there was a deficiency of MIL, namely that the information entropy of steady states only contains the controllability-information without reflecting the observability-information. Although RMIL considers the information completeness, it need a transformation from the original model to the output-normal of the second-order mode in order to deal with the observability- information in the dynamical part as the controllability-information. A new definition of cross-Gramian information was defined based on the information theoretic properties of the system cross-Gramian matrix. By analyizing the information desription of the system states, the physical meanings of the cross-Gramian information was clarified theoretically from entropic points of view For a linear asymptotic stable SISO system, a new improved method minimizing the loss of the system cross-Gramian information in the truncation was proposed.(2) The simulation of model reduction for both the model with 24 orders randomly produced by matlab and an international popular real benchmark with 598 orders was carried out by adopting MIL, CGMIL and balanced truncation (BT) algorithms. Respectively, from the perspectives of the time domain, frequency domain, the error of noise response as well as definite signal response and so on, the analysis and comparison among the MIL reduced-order model, the CGMIL reduced-order model and the BT reduced-order model was performed.The rationality and the validity of the CGMIL algorithm was illustrated.(3) The establishment of CGMIL was based on the cross-Gramian information. However, the key property that the square of the cross-Gramian matrix equals to the product of the controllability-matrix and controllability-matrix does not always hold for general MIMO system. The theoretical analysis indicated that the symmetric MIMO systems and the orthogonal symmetric MIMO systems possess this property. Therefore, the cross-Gramian information description method with clear physical meaning could be established and CGMIL could also be applied to the symmetric MIMO system and the orthogonal symmetric MIMO system. Based on the properties of symmetric MIMO systems, the preliminary idea about the application of CGMIL for a general MIMO system was proposed and this part of the work need for further studying.In summary, the cross-Gramian information is the mean values of the system controllability-information and observability-information. Therefore, the cross-Gramian information has good system information theoretic properties. Consequently, minimizing the loss of the system cross-Gramian information as the performance index of model reduction is reasonable. CGMIL does not need to carry on the transformation to the system so that it keeps the inherent distribution of system information, therefore, CGMIL avoids extra model truncation error. Compared to MIL, CGMIL can obtain much better approximation of the full order system. CGMIL method is situated between MIL and balanced truncation in performance of model reduction and has advantage among the model reduction methods based on the information theoretic methods.