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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

It is shown in the paper how robustness can be guaranteed for consensus protocols with heterogeneous dynamics in a scalable and decentralized way i.e. by each agent satisfying a test that does not require knowledge of the entire network. Random graph examples illustrate that the proposed certificates are not conservative for classes of large scale networks, despite the heterogeneity of the dynamics, which is a distinctive feature of this work. The conditions hold for symmetric protocols and more conservative stability conditions are given for general nonsymmetric interconnections. Nonlinear extensions in an IQC framework are finally discussed.

H/sub /spl infin// 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.

We derive robust, decentralized and scalable stability conditions for networks of heterogeneous linear systems. The results are based on a graphical Nyquist type test which depends only on the frequency responses of an individual system and its nearest neighbours. If every system in the network satisfies such a condition, the network is guaranteed to be stable. The results are suitable for a broad class of networks, allowing arbitrary interconnection asymmetries and heterogeneous multi-input-multi-output (MIMO) systems.

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.

It is shown in the paper how by introducing interconnection symmetries by use of bidirectional communication, 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 a relaxed convex hull in the complex plane. 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. As a corollary, it is shown that in the presence of heterogeneous delays in the relative information (in which case stability is delay dependent) a simple scaling can guaranty stability on an arbitrary topology. The condition becomes necessary and sufficient when the delays are homogeneous. Time domain interpretations are finally discussed for certain formulations of the problem.

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 discussed in the paper how 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 the 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, thus implying important limitations in noise suppression capabilities. We finally consider the effects of noise in the case of nonlinear macroscopic models leading to bistabilities. 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 in a correct context.

It has recently been demonstrated within the mathematical biology community how linear noise approximations can be used to analyze intrinsic fluctuations in biological systems. By interpreting the fluctuation dissipation theorem in a noise filtering setting, we illustrate in the paper ways of exploiting conventional tools from robust control theory to reveal tradeoffs in gene regulatory networks. The control of plasmid copy numbers in bacteria is discussed.

We consider experimental (in)validation of a nominal discrete-time model by determining if observed input/output data is consistent with a system within a given /spl nu/-gap distance from the model (in light of given constraints on possible input and output noise). The general problem has been shown to be NP-hard; however we present linear matrix inequality (LMI) sufficient conditions for model consistency along with LMI necessary and sufficient conditions for invalidation with respect to a special class of input/output noise. A logical extension which considers time-varying perturbations of the nominal model is presented in parallel with the time-invariant /spl nu/-gap results.

This paper extends results on the approximation of uncertain systems in the /spl nu/-gap metric (1999), and those on the calculation of /spl nu/-gap distances for nonlinear systems (1998). We wish to be able to approximate nonlinear systems with complex uncertainty descriptions by linear systems with simpler uncertainty descriptions which are more amenable to robust control system design, and to evaluate the approximation errors in some suitable way. To this end, given two sets of systems P/sub 0/ and P/sub 1/, each pathwise connected in the graph topology and with a non-empty intersection, we define a directed distance /spl delta/~(P/sub 0/, P/sub 1/). This distance provides a bound on the worst case performance of any controller over the set P/sub 1/ in terms of its worst case performance over the set P/sub 0/. So, if both /spl delta/~(P/sub 0/, P/sub 1/) and /spl delta/~(P/sub 1/,P/sub 0/) are small, then the worst case performance of any feedback controller over each set is similar. If we regard P/sub 0/ as a (simpler) approximation of P/sub 1/, and if /spl delta/~(P/sub 0/, P/sub 1/) is small, then any controller designed to function well over the set P/sub 0/ is also guaranteed to work well over the set P/sub 1/ If, in addition, /spl delta/~(P/sub 1/,P/sub 0/) is small then any controller that fails to work well with some member of P/sub 0/ will also fail to work well with some member of P/sub 1/. That is, /spl delta/~(P/sub 1/,P/sub 0/) is a measure of how conservative the approximation is. The first main result is an appropriate generalization to the nonlinear case of the /spl nu/-gap metric, which reduces to the /spl nu/-gap metric when applied to LTI systems. We then give two different definitions of S for uncertain systems. The first is based on the nonlinear /spl nu/-gap, and hence is applicable to nonlinear systems, the second is defined as the supremum over frequency of a pointwise distance and so is only applicable to LTI systems. Both definitions have the desired properties, and both are equivalent to the /spl nu/-gap when applied to the distance between single systems. The second main result of this paper is that, for sets of LTI systems, the second distance is less than the first. So, it would appear that there is no general definition of the distance between sets of systems which specializes to the right (i.e., least conservative) notion in the LTI case.

The usual approach in robust control has been to start with an explicit representation of the plant uncertainty and then look for a way of guaranteeing the satisfaction of some performance objective with the real plant. Here we consider a complementary approach: if a controller satisfies a condition involving a nominal plant, then over what set of plants is it guaranteed to satisfy some closed-loop objective? This viewpoint has several benefits. It can provide less conservative results, and can provide insights into how closed-loop objectives should be chosen for robustness in the absence of hard bounds on the uncertainty.

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.

We present an LMI-based fixed-order controller design. We consider the robust stabilization problems for normalized coprime perturbation with the frequency weighting. We show that this problem can be converted to a one-block problem with coprime factorization representation and it can be solved with LMI optimization. This design method has the advantage that it can be extended to simultaneous control design for multiple plants. A design example is shown to demonstrate the design method.

Presents robust decentralized control design based on LMI optimization. The proposed design has the following features i) the diagonal plant model is not necessary, ii) a reduced-order controller can be obtained, and iii) simultaneous design for the multiple plants is possible.

Various alternative definitions for the nonlinear H2-, L2-, and $\nu$-gap metrics are studied. The concept of $\beta$-conjugacy and multiplicative homogeneity are introduced to relate the metrics to each other and to compare the stability margins of nonlinear feedback loops expressed in terms of the norms of complementary parallel projections. Left and right representations for the graph of a nonlinear system are studied. A new definition of "normalized" is introduced for left representations. Formulas for the gap metrics as the norm of the product of left and right representations are derived. The problem of controller synthesis for input-affine nonlinear systems to achieve norm bounds on the parallel projection operators is studied for input-affine nonlinear systems. The duality between the optimization of the two parallel projections is highlighted. State-space realizations for the normalized left and right representations are derived using nonlinear $H_\infty$ synthesis methods.

The gap metric and nu-gap metric have many appealing properties for assessing the uncertainty in a plant in a feedback configuration. This short paper addresses the question of relating these generalised stability margins to more traditional single-loop and multiloop robustness measures. In particular we show that when a system is controlled by a controller which provides a stability margin of /spl epsiv/ in the gap metric for the weighted plant then if the weighting matrices are diagonal the plant will have robust stability to easily interpreted simultaneous gain and phase variations at all the inputs and outputs.

We define a distance between two uncertain LTI systems1 which provides an upper bound on the difference in worst case robust performance when the same feedback controller is used with each system. We also suggest ways of using this measure to approximate complex uncertainty descriptions with simpler ones, with a view to simplifying any subsequent robust control design step.

This paper presents a design method for fixed-order robust controllers, based on coprirne factorization and LMI optimization, and considers the problem of local optima.

This paper presents a fixed structure H ∞ control design based on coprirne factorization and LMI optimization. With this design, we can design a reduced-order / decentralized controller for multiple-plants. We show the design methods and their examples.

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.