Glenn Vinnicombe

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. >

Fundamental limits on the suppression of molecular fluctuations

Negative feedback is common in biological processes and can increase a system’s stability to internal and external perturbations. But at the molecular level, control loops always involve signalling steps with finite rates for random births and deaths of individual molecules. Here we show, by developing mathematical tools that merge control and information theory with physical chemistry, that seemingly mild constraints on these rates place severe limits on the ability to suppress molecular fluctuations. Specifically, the minimum standard deviation in abundances decreases with the quartic root of the number of signalling events, making it extremely expensive to increase accuracy. Our results are formulated in terms of experimental observables, and existing data show that cells use brute force when noise suppression is essential; for example, regulatory genes are transcribed tens of thousands of times per cell cycle. The theory challenges conventional beliefs about biochemical accuracy and presents an approach to the rigorous analysis of poorly characterized biological systems.

ON THE STABILITY OF NETWORKS OPERATING TCP-LIKE CONGESTION CONTROL

We derive decentralized and scalable stability conditions for a fluid approximation of a class of Internet-like communications networks operating a modified form of TCP-like congestion control. The network consists of an arbitrary interconnection of sources and links with heterogeneous propagation delays. The model here allows for arbitrary concave utility functions and the presence of dynamics at both the sources and the links.

Robust control of plants with saturation nonlinearity based on coprime factor representations

We consider the robust control of plants with saturation nonlinearities from an input/output viewpoint. First, we present a parameterization for anti-windup control based on coprime factorizations of the controller. Second, we propose a synthesis method which exploits the freedom to choose a particular coprime factorization.

Scalable Decentralized Robust Stability Certificates for Networks of Interconnected Heterogeneous Dynamical Systems

We derive scalable decentralized conditions that can guarantee robust stability for networks of linearly interconnected, stable, linear time invariant dynamical systems. Unlike previous results of this kind, we allow for heterogeneous dynamics on arbitrary interconnection topologies (i.e., linear systems on arbitrary underlying graphs). Robust stability of the entire network is guaranteed by satisfying local rules that involve only an agent and its neighboring dynamics; each new agent will introduce only one additional condition of this kind and hence the stability certificates scale with the network size. An application of this theory is given, where robustness analysis of Internet congestion control protocols is carried out in the general case of dynamics at both users and resources without any global bounds on the dynamics

Noise in Gene Regulatory Networks

Life processes in single cells and at the molecular level are inherently stochastic. Quantifying the noise is, however, far from trivial, as a major contribution comes from intrinsic fluctuations, arising from the randomness in the times between discrete jumps. It is shown in this paper how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations. Such interpretation naturally exists in linear noise approximations, but it also provides an exact description of the jump process when the transition rates are linear. As an important example, it is shown in this paper how, by addressing the proximity of the underlying dynamics in an appropriate topology, a sequence of coupled birth-death processes, which can be relevant in gene expression, tends to a pure delay; this implies important limitations in noise suppression capabilities. Despite the exactness, in a linear regime, of the analysis of noise in conjunction with the network dynamics, we emphasize in this paper the importance of also analyzing dynamic behavior when transition rates are highly nonlinear; otherwise, steady-state solutions can be misinterpreted. The examples are taken from systems with macroscopic models leading to bistability. It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady-state solution of the master equation neither always imply one another nor do they necessarily lead to efficient switching behaviours: the underlying dynamics need to be taken into account. Finally, we explore some of these issues in relation to a model of the lac operation.

Synchronous long-term oscillations in a synthetic gene circuit

Synthetically engineered genetic circuits can perform a wide variety of tasks but are generally less accurate than natural systems. Here we revisit the first synthetic genetic oscillator, the repressilator, and modify it using principles from stochastic chemistry in single cells. Specifically, we sought to reduce error propagation and information losses, not by adding control loops, but by simply removing existing features. We show that this modification created highly regular and robust oscillations. Furthermore, some streamlined circuits kept 14 generation periods over a range of growth conditions and kept phase for hundreds of generations in single cells, allowing cells in flasks and colonies to oscillate synchronously without any coupling between them. Our results suggest that even the simplest synthetic genetic networks can achieve a precision that rivals natural systems, and emphasize the importance of noise analyses for circuit design in synthetic biology.

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.

Brief paper: 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.

Scalability in heterogeneous vehicle platoons

It is known that vehicle platoons exhibit string instability when each vehicle tries to maintain a fixed distance from its predecessor. This can be avoided if sufficiently strong coupling with the leader is employed. If instead each vehicle tracks the average distance form its neighbours, the interconnection can still be ill-conditioned in the sense that the response to disturbances is not uniformly bounded with the size of the platoon. We show in the paper that in a symmetric bidirectional scheme, arbitrarily weak coupling with the leader can make the platoon scalable. In addition, we show that despite the additional feedback in a bidirectional control law, the symmetry of the information flow enables the derivation of local conditions which, if satisfied, guarantee that an arbitrarily long heterogeneous interconnection is robustly stable.