Frequency domain uncertainty and the graph topology

A new metric on linear, time-invariant systems is defined. This metric is no greater than the gap metric, and is in fact the smallest metric for which a certain robust stabilization result holds. Unlike other known metrics which induce the graph topology, it has a clear frequency response interpretation. This allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric     

Heterogeneity and scalability in group agreement protocols: Beyond small gain and passivity approaches

It is shown in this paper how by introducing interconnection symmetries as in bidirectional communication schemes, stability of consensus protocols on arbitrary topologies can be guaranteed in a decentralized and scalable way, despite the presence of higher order heterogeneous dynamics. The analysis is centred round the notion of an S-hull and other related convexifications in the complex plane, that can lead to decentralized certificates by means of their dual interpretation. In the case of linear dynamics the certificates derived include, as special cases, passivity and dissipativity approaches, though a wider class of dynamics is allowed by employing convexification arguments that exploit the interconnection structure. Examples are given of networks comprised of agents with heterogeneous higher order dynamics and also with non-identical input and communication delays. Special cases of the results recover small gain related delay independent stability, but can also lead to less conservative delay dependent conditions that are fully decentralized and can be necessary and sufficient. Time domain interpretations are finally discussed for certain formulations of the problem.     

The robustness of feedback systems with bounded complexitycontrollers

H_∞ methods for the analysis and design of robust feedback control systems have sometimes been criticized for an apparent conservatism. However, recent results have shown that they can provide the least conservative results possible under a particular set of assumptions. For example, a ball in the v-gap metric is the largest set of plants that can be guaranteed to be stabilized a priori by a controller known only to satisfy a bound on the induced norm of a particular closed-loop operator. Nevertheless, there are examples of uncertainty which, whilst large when measured by the v-gap metric, would be regarded as relatively benign by an experienced designer of control systems. The present paper examines the possibility that in arriving at this judgment, such a designer is implicitly using the knowledge that he will always choose the least complex controller necessary to do the job. It is shown that, given an appropriate bound on the complexity of the controller, significantly stronger a priori robustness results can be obtained     

Controller discretization: a gap metric framework for analysis and synthesis

Although techniques for directly synthesising sampled-data (SD) compensators are available in the literature, feedback controller design is perhaps best understood in a purely continuous-time setting. As such, a feedback controller is often designed in the continuous-time domain and then discretized for digital implementation. It is important for the discretization step involved to yield an SD approximation which captures the essential features of the original controller from the perspective of closed-loop behavior. In this note, a gap metric framework is developed for studying the controller discretization problem for linear time-invariant (LTI) plants and controllers. Importantly, knowledge of a gap metric distance between an LTI controller and an SD approximation permits explicit characterization of the possible difference in closed-loop performance, with any LTI plant for which the LTI controller is known to work well, accounting for intersample behavior. The central result of the new framework gives rise to an algorithm for computing a gap metric measure of the distance between an LTI controller and a given discretization, and a technique for synthesising a SD approximation which is optimal with respect to this metric.     

Scalable robust stability for nonsymmetric heterogeneous networks

Scalable decentralized stability certificates in networks i.e. decentralized stability guarantees for an arbitrary interconnection of heterogeneous dynamical systems, are often based on certain symmetry assumptions in the way the systems are interconnected. Such structure simplifies the mathematical analysis significantly, nevertheless a potential pitfall needs to be addressed: the stability proof might offer no robustness guarantees to deviations from protocol symmetry. This is, for example, the case for Internet congestion control stability results for arbitrary networks i.e. the stability proofs break down with an arbitrarily small deviation from protocol symmetry. We propose in the paper conditions that can guarantee scalable robust stability in a nonsymmetric interconnection setting for a class of networks that includes Internet congestion control models and consensus protocols. The certificates derived are decentralized and scale with the degree of nonsymmetry.     

Linear feedback systems and the graph topology

It is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed-loop stability and closed-loop performance are robust properties. In earlier works, similar topological results were obtained by exploiting the existence of particular coprime-factor system representations, not known to exist in general. By contrast, the development here does not rely on any specific system representations. Systems are simply characterized as subspaces of norm bounded input-output pairs, and the analysis hinges on the underlying geometric structure of the feedback stabilization problem. Unlike other work developed within such a framework, fundamental issues concerning the causality of feedback interconnections are discussed explicitly. The key result of this paper concerns the difference between linear feedback interconnections, with identical controllers, in terms of a performance/robustness related closed-loop mapping. Upper and lower bounds on the induced norm of this difference are derived, allowing for possibly infinite-dimensional input-output spaces and time-varying behavior. The bounds are both proportional to the gap metric distance between the plants, which clearly demonstrates the gap to be an appropriate measure of the difference between open-loop systems from the perspective of closed-loop behavior. To conclude, an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting     

Percutaneous transluminal angioplasty by a retrograde subintimal transpopliteal approach

We report three cases of femoropopliteal occlusive disease which were successfully recanalized subintimally and retrogradely via the popliteal artery. The merits and limitations of this approach are discussed.