Energy functions for dissipativity-based balancing of discrete-time nonlinear systems

Most of the energy functions used in nonlinear balancing theory can be expressed as storage functions in the framework of dissipativity theory. By defining a framework of discrete-time dissipative systems, this paper presents existence conditions for their discrete-time energy functions along with algorithms to find them based on dynamic optimization problems. Furthermore, the important case of the nonlinear discrete-time versions of the controllability and observability functions, its properties and algorithms to find them are presented. These algorithms are illustrated with linear and nonlinear examples.     

Fully distributed robust synchronization of networked Lur’e systems with incremental nonlinearities

This paper deals with robust synchronization problems for uncertain dynamical networks of diffusively interconnected identical Lur’e systems subject to incrementally passive nonlinearities and incrementally sector bounded nonlinearities, respectively, in a fully distributed fashion. Whereas in stabilization of one single Lur’e system the conditions of passivity and sector boundedness for the uncertain nonlinear function in the negative feedback loop are commonly assumed, in our context of networked Lur’e systems we adopt the stronger assumptions of incremental passivity and incremental sector boundedness. Throughout this paper the interconnection topologies among these Lur’e agents are assumed to be undirected and connected. We design robustly synchronizing protocols and subsequently implement these protocols in a fully distributed way by means of an adaptive control law that adjusts the coupling weights between neighboring agents. Both for the cases of incrementally passive as well as incrementally sector bounded nonlinearities we obtain sufficient conditions for the existence of fully distributed robustly synchronizing protocols. The state feedback matrices are computed by solving LMIs in terms of the matrices defining the individual agent dynamics. Numerical simulation examples illustrate our theoretical results.     

Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators

This paper investigates the differential eigenstructure of Hankel operators for nonlinear systems. First, it is proven that the variational system and the Hamiltonian extension with extended input and output spaces can be interpreted as the Ga/spl circ/teaux differential and its adjoint of a dynamical input-output system, respectively. Second, the Ga/spl circ/teaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear Hankel operator which is closely related to the Hankel norm of the original system. Third, a new characterization of the nonlinear extension of Hankel singular values are given based on the differential eigenstructure. Finally, a balancing procedure to obtain a new input-normal/output-diagonal realization is derived. The results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems.     

Fault detection method for nonlinear systems based on probabilistic neural network filtering

A fault detection method for nonlinear systems, which is based on Probabilistic Neural Network Filtering (PNNF), is presented. PNNF limits the maximum estimation error of the asymptotic Bayes optimal result and describes the tracking process with an expression. On the basis of these properties of PNNF and the statistical characteristics of the noise of the system, a fault threshold can be better calculated, especially for the tracking process corresponding to a strong disturbance. According to the threshold, the fault can be detected by evaluating the residuals. Also, for some special cases when a fault is potential but the system is in steady state, which causes the information for fault detection may be insufficient and a group of disturbances are artificially input with definite amplitudes, so that the result of detection can be enhanced by comparing the real with the expected tracking processes of the filter. Examples are given to demonstrate the method of fault detection based on PNNF.     

Balancing for nonlinear systems

We present a method of balancing for nonlinear systems which is an extension of balancing for linear systems in the sense that it is basd on the input and output energy of a system. It is a local result, but gives ‘broader’ results than we obtain by just linearizing the system. Furthermore, the relation with balancing of the linearization is dealt with. We propose to use the method as a tool for nonlinear model reduction and investigate some of the properties of the reduced system.     

Robust cooperative output regulation of heterogeneous Lur'e networks

In this paper, we study robust cooperative output regulation problems for a directed network of Lur'e systems that consist of a nominal linear dynamics with an unknown static nonlinearity around it through negative feedback. We assume that the linear part of each agent is identical, but the nonlinearities are allowed to be different for distinct agents. In this sense, the network is heterogeneous. As is common in the context of Lur'e systems, the unknown nonlinearities are assumed to be sector bounded within one given sector. The interconnection graph among these agents is assumed to contain a directed spanning tree. Similar to classical output regulation problems, there is a virtual exosystem generating a reference signal in which all the agents are required to track cooperatively. Our designed distributed dynamic state/output feedback protocol makes a copy of the reference signal at each agent asymptotically, and then the robust cooperative output regulation problem becomes a robust tracking problem that can be handled by each agent via local information. It turns out that our cooperative protocols are fully distributed. Sufficient conditions on the existence of output synchronization protocols are given along with some discussions on these conditions. Finally, two simulation examples illustrate our design. Copyright © 2016 John Wiley & Sons, Ltd.     

On Brayton and Moser's missing stability theorem

In the early 1960s, Brayton and Moser proved three theorems concerning the stability of nonlinear electrical circuits. The applicability of each theorem depends on three different conditions on the type of admissible nonlinearities in circuit. Roughly speaking, this means that the theorems apply to either circuits that contain purely linear resistors or conductors-combined with linear or nonlinear inductors and capacitors or to circuits that contain purely linear inductors and capacitors-combined with linear or nonlinear resistors and conductors. This brief note presents a generalization of Brayton and Moser's stability theorems that also includes the analysis of circuits that contain nonlinear resistors, conductors, inductors, and/or capacitors at the same time.     

Minimality and local state decompositions of a nonlinear state space realization using energy functions

In this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state-space realization of a formal power series is minimal. Specifically, it is shown that positivity of these functions, in addition to a stability requirement and a few technical conditions, implies minimality. Using the nonlinear analogue of the Kalman decomposition, connections are then established between minimality, singular value functions, balanced realizations, and various notions of reachability and observability for nonlinear systems.     

On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so-called singular value functions of a system describe the relative importance of each state component from an input–output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, distinct sets of singular value functions and balanced realizations are possible.     

Lagrangian modeling of switching electrical networks

In this paper, a general and systematic method is presented to model topologically complete electrical networks, with or without multiple or single switches, within the Euler–Lagrange framework. Apart from the physical insight that can be obtained in this way, the framework has proven to be useful for the application of passivity-based control techniques, which on a case by case basis already has shown to be useful for the control of power converters within the class of switching electrical networks. The switches are assumed to be ideal, and pulse-width modulation is taken into account. Magnetic coupling of inductive elements is also included in the framework.