Synthesis of dissipative systems using quadratic differentialforms: part II

For pt.1 see ibid., p.53-69 (2002). The authors discuss several important special cases of the problem solved in Part I. These are: disturbance attenuation and passivation, the full information case, the filtering problem, and the case that the to-be-controlled plant is given in input-state-output representation. An interesting aspect is the notion of full information, which we define in terms of the observability of the to-be-controlled variables from the control variables. When the system is given in state space form, we obtain conditions for the existence of a controller that renders a system dissipative in terms of two coupled algebraic Riccati inequalities. The controller turns out to be a feedback system with a transfer function that is proper, but, in general, not strictly proper. Another issue that we study in this paper is feedback implementability. We find conditions under which, in the context of synthesis of dissipative systems, a controlled behavior can implemented by a feedback controller     

Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

We illustrate procedures to identify a state-space representation of a lossless or dissipative system from a given noise-free trajectory; important special cases are passive systems and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; the state-space equations are then computed by solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order model.     

Zn in Athenian Black Gloss Ceramic Slips: A Trace Element Marker for Fabrication Technology

The black‐colored pottery slips produced in Athens from the 6th to 4th centuries B.C., had a consistent composition achieved through processing and refinement of raw clay. Little direct evidence has been established as to what were these refinement methods. To better understand how the slip material was prepared, the major and trace elemental compositions of 19 slips from different ceramic vessels and their corresponding bodies of Athenian red‐figure and black‐figure vases were determined using laser ablation inductively coupled plasma mass spectrometry (LA‐ICPMS). Notably higher Zn concentrations were found in the slips (271–1959 ppm) than in their corresponding body ceramics (<361 ppm). The Zn concentrations in the slips were also found to be above the natural background for typical clay (between 10 and 300 ppm) suggesting an unintentional anthropogenic enrichment of this metal. Based on the abnormally high Zn content of the slip, it is speculated that the clay was treated using vitriol (concentrated acid mine runoff which is rich in Zn), to induce flocculation and remove carbonate mineral phases from the raw material that, if present, would prevent the slip from vitrifying. This same signature of elevated levels of Zn with a corresponding Ce anomaly is also observed for black glosses produced in Corinthian and Etrurian (Italy) workshops indicating that these trace element signatures were imparted to the material by means of shared methods of manufacturing instead of being indicative of a single unique source for this material.     

Every storage function is a state function

It is shown that for linear dynamical systems with quadratic supply rates, a storage function can always be written as a quadratic function of the state of an associated linear dynamical system. This dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate.     

H∞ control in a behavioral context: the fullinformation case

The authors formulate the H_∞-control problem in a behavioral setting. Given a mathematical model, say a set of higher order differential equations together with some static equations, the vector of manifest variables is partitioned into yet to be controlled variables, unknown exogenous variables, and interconnection variables. The interconnection variables are available for interconnection, in the sense that they can be made to obey certain differential or static equations, to be specified by the designer. Such a system of differential equations and static equations is called a controller. The design problem that we consider is to find controllers such that the size of the to be controlled variables is less than a given tolerance, for all disturbances in the unit ball, and such that the interconnection is a stable system. We find necessary and sufficient conditions for the existence of suitable controllers, under the hypothesis that we have a full information problem. These conditions involve indefinite factorizations of polynomial matrices and a test on a given Pick matrix     

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.     

The Kalman-Yakubovich-Popov Lemma in a behavioural framework

The classical Kalman-Yakubovich-Popov Lemma provides a link between dissipativity of a system in state-space form and the solution to a linear matrix inequality. In this paper we derive the KYP Lemma for linear systems described by higher-order differential equations. The result is an LMI in terms of the original coefficients in which the dissipativity problem is posed. Subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices. This enables us to derive a new algorithm for polynomial spectral factorization in terms of an LMI in the coefficients of a polynomial matrix.     

Uniform synchronization in multi‐agent systems with switching topologies

This paper deals with uniform synchronization analysis of multi‐agent systems with switching topologies. The agents are assumed to have general, yet identical, linear dynamics. The underlying communication topology may switch arbitrarily within a finite set of admissible topologies. We establish conditions under which the network is uniformly synchronized meaning that synchronization is valid under all possible switching scenarios. The primary conditions established are in terms of a pair of Lyapunov strict inequalities. Following those conditions, small gain and passivity types of conditions are proposed under which uniform synchronization is guaranteed. The proposed results are also extended to the case of observer‐based protocols. Copyright © 2015 John Wiley & Sons, Ltd.     

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.