Balancing and model reduction for discrete-time nonlinear systems based on Hankel singular value analysis

This paper is concerned with balanced realization and model reduction for discrete-time nonlinear systems. Singular perturbation type balanced truncation method is proposed. In this procedure, the Hankel singular values and the related controllability and observability properties are preserved, which is a natural generalization of both the linear discrete-time case and the nonlinear continuous-time case.

Balanced Truncation of Networked Linear Passive Systems

This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order network model, which leads to a reduced-order system preserving the passivity of each subsystem. Moreover, it is proven that there exists a coordinate transformation to convert the resulting reduced-order model to a state-space model of Laplacian dynamics. Thus, the proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics, and it preserves the passivity of the subsystems and the synchronization of the network. Moreover, it allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example

Reduction of Second-Order Network Systems with Structure Preservation

This paper proposes a general framework for structure-preserving model reduction of a secondorder network system based on graph clustering. In this approach, vertex dynamics are captured by the transfer functions from inputs to individual states, and the dissimilarities of vertices are quantified by the H2-norms of the transfer function discrepancies. A greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters. Then, the reduced-order model is generated by the Petrov-Galerkin method, where the projection is formed by the characteristic matrix of the resulting network clustering. It is shown that the simplified system preserves an interconnection structure, i.e., it can be again interpreted as a second-order system evolving over a reduced graph. Furthermore, this paper generalizes the definition of network controllability Gramian to second-order network systems. Based on it, we develop an efficient method to compute H2-norms and derive the approximation error between the full-order and reduced-order models. Finally, the approach is illustrated by the example of a small-world network

Krasovskii's Passivity

In this paper we introduce a new notion of passivity which we call Krasovskii's passivity and provide a sufficient condition for a system to be Krasovskii's passive. Based on this condition, we investigate classes of port-Hamiltonian and gradient systems which are Krasovskii's passive. Moreover, we provide a new interconnection based control technique based on Krasovskii's passivity. Our proposed control technique can be used even in the case when it is not clear how to construct the standard passivity based controller, which is demonstrated by examples of a Boost converter and a parallel RLC circuit

Novel Gramians for linear semistable systems

In this paper, the notions of pseudo Gramians are introduced for linear time-invariant semistable systems, which allow multiple semisimple poles at the origin. The proposed Gramians are the generalizations of standard Gramian matrices defined for asymptotically stable systems, and they can be computed by a set of Lyapunov equations. Furthermore, it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo Gramians, and the controllability and observability energy functions are also characterized using the pseudo Gramians. Additionally, the H2-norm and H∞-norm of a semistable system are analyzed, and then the results are used for the model reduction of semistable systems. Finally, the effectiveness of the methods is illustrated by an example of a gene regulation network