The converse of the passivity and small-gain theorems for input-output maps

We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.Comment: 15 pages, 3 figure

Stabilization of Cascaded Two-Port Networked Systems Against Nonlinear Perturbations

A networked control system (NCS) consisting of cascaded two-port communication channels between the plant and controller is modeled and analyzed. Towards this end, the robust stability of a standard closed-loop system in the presence of conelike perturbations on the system graphs is investigated. The underlying geometric insights are then exploited to analyze the two-port NCS. It is shown that the robust stability of the two-port NCS can be guaranteed when the nonlinear uncertainties in the transmission matrices are sufficiently small in norm. The stability condition, given in the form of "arcsin" of the uncertainty bounds, is both necessary and sufficient.Comment: 8 pages, in preparation for journal submissio

Diagonal Lyapunov functions for positive linear time-varying systems

Stable positive linear time-invariant autonomous systems admit diagonal quadratic Lyapunov functions. Such a property is known to be useful in distributed and scalable control of positive systems. In this paper, it is established that the same holds for exponentially stable positive discrete-time and continuous-time linear time-varying systems

Robust stability conditions for feedback interconnections of distributed-parameter negative imaginary systems

Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems.Comment: Submitted to Automatica, A preliminary version of this paper appeared in the Proceedings of the 2015 European Control Conferenc

Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

In this paper, we propose a notion of dissipativity with dynamic supply rates for nonlinear differential input-state-output equations via the use of auxiliary systems. This extends the classical dissipativity with static supply rates and miscellaneous dynamic quadratic forms. The main results of this paper concern Lyapunov (asymptotic/exponential) stability analysis for nonlinear feedback dissipative systems that are characterised by dissipation inequalities with respect to compatible dynamic supply rates. Importantly, dissipativity conditions guaranteeing partial stability of the state of the feedback systems without concerning that of the state of the auxiliary systems are provided. They are shown to recover several existing results in the literature. Comparison with the input-output approach to feedback stability analysis based on integral quadratic constraints is also made

Steady-state analysis of networked epidemic models

Compartmental epidemic models with dynamics that evolve over a graph network have gained considerable importance in recent years but analysis of these models is in general difficult due to their complexity. In this paper, we develop two positive feedback frameworks that are applicable to the study of steady-state values in a wide range of compartmental epidemic models, including both group and networked processes. In the case of a group (resp. networked) model, we show that the convergence limit of the susceptible proportion of the population (resp. the susceptible proportion in at least one of the subgroups) is upper bounded by the reciprocal of the basic reproduction number (BRN) of the model. The BRN, when it is greater than unity, thus demonstrates the level of penetration into a subpopulation by the disease. Both non-strict and strict bounds on the convergence limits are derived and shown to correspond to substantially distinct scenarios in the epidemic processes, one in the presence of the endemic state and another without. Formulae for calculating the limits are provided in the latter case. We apply the developed framework to examining various group and networked epidemic models commonly seen in the literature to verify the validity of our conclusions

Connections between integral quadratic constraints and dissipativity

We show that a recent dissipativity approach to feedback stability analysis of potentially open-loop unstable systems, which encompasses the classical soft integral quadratic constraint (IQC) theorem, may be recovered by hard IQC theory. The latter is known to be subsumable by the more general soft IQC theory endowed with homotopies that are continuous in the gap topology. Additionally, we demonstrate how the aforementioned classical soft IQC theorem, initially introduced for the analysis of a feedback interconnection of a nonlinear component anda linear system, may be recast to analyse the stability of a feedback interconnection of two nonlinear systems. This generates a frequency-dependent (Q(ω), S(ω), R(ω))-dissipativity resul