Model reduction of synchronized homogeneous Lur’e networks with incrementally sector-bounded nonlinearities

Abstract

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11 Reduced-order modeling of large-scale network systems

“…Reducing the dimension of each subsystem also results in a simplification of overall networks. To reduce the dynamics of vertices, balanced truncation based on generalized Gramian matrices is commonly used (see, e. g., [57,23,21]), in which preserving the synchronization property of the overall network is of particular interest. In this section, we review some recent results in the synchronization-preserving model reduction of large-scale network systems using the classic generalized balanced truncation.…”

“…We denoteG(s) as the transfer matrix of system (11.54), and the model reduction error is upper-bounded as 55) where σ i are the GHSVs computed using K −1 M and K m [57]. Inspired by the work [57] for linear networks, [18,23] consider dynamical networks of diffusively interconnected nonlinear Lur'e subsystems. The robust synchronization of the Lur'e network can be characterized by a linear matrix inequality (LMI).…”

“…The robust synchronization of the Lur'e network can be characterized by a linear matrix inequality (LMI). Different from [57,18], where the minimum and maximum solutions of the LMI are used as generalized Gramians, [23] suggests to only use the solution of the LMI with the minimal trace as a generalized controllability Gramian, while the observability counterpart is taken by the standard observability Gramian as the solution of the Lyapunov equation, which is less conservative than the LMI. Using such a pair, generalized balanced truncation is performed on the linear component of each Lur'e subsystem, and the resulting reduced-order network system is still guaranteed to have the robust synchronization property.…”

“…Relevant methods are developed using generalized Gramian matrices [34] that allow for more freedom to preserve some desired structures than the standard Gramians. Networked nonlinear robustly synchronized Lur'e-type systems are reduced in [23], which shows that performing model reduction on the linear component of each nonlinear subsystem allows for preserving the robust synchronization property of a Lur'e network. Techniques in [43,21] can reduce the complexity of network structures and subsystem dynamics simultaneously.…”

“…A network system can be simplified if the dimension of individual subsystems is reduced, which leads to the second research direction in reduced-order modeling of network systems; see, e. g., [69,57,23]. In this framework, the approximation is applied to each subsystem in a way that certain properties of the overall network, such as synchronization and stability, are preserved.…”