The behavior of heterogeneous multi-agent systems is studied when the coupling matrices are possibly all different and/or singular, that is, its rank is less than the system dimension. Rank-deficient coupling allows exchange of limited state information, which is suitable for the study of multi-agent systems under output coupling. We present a coordinate change that transforms the heterogeneous multi-agent system into a singularly perturbed form. The slow dynamics is still a reduced-order multi-agent system consisting of a weighted average of the vector fields of all agents, and some sub-dynamics of agents. The weighted average is an emergent dynamics, which we call a blended dynamics. By analyzing or synthesizing the blended dynamics, one can predict or design the behavior of a heterogeneous multi-agent system when the coupling gain is sufficiently large. For this result, stability of the blended dynamics is required. Since stability of the individual agent is not asked, the stability of the blended dynamics is the outcome of trading off the stability among the agents. It can be seen that, under the stability of the blended dynamics, the initial conditions of the individual agents are forgotten as time goes on, and thus, the behavior of the synthesized multi-agent system is initialization-free and is suitable for plug-and-play operation. As a showcase, we apply the proposed tool to four application problems; distributed state estimation for linear systems, practical synchronization of heterogeneous Van der Pol oscillators, estimation of the number of nodes in a network, and a problem of distributed optimization.

In this paper, we present a scheme of fully distributed resilient state estimation for linear dynamical systems under sensor attacks. The proposed state observer consists of a network of local observers, where each of them utilizes local measurements and information transmitted from the neighbors. As a fully distributed scheme, it does not necessarily collect a majority of sensing data for the sake of attack identification, while the compromised sensors are eventually identified by the distributed network and excluded from the observers. For this, the overall network (not the individual local observer) is assumed to have redundant sensors and assumed to be connected. The proposed scheme is based on a novel design of a distributed median solver, which approximately recovers the median value of local estimates.

Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowlegde how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control -without any further assumptions -can achieve asymptotic tracking.

The Zeros o f the Riemann 261 Fig. 22, Plate 8, is a reproduction of the resulting spectrogram, in point-to-point coincidence with the diagram of the gap. The detonating pile o f azide crystals gives a continuous spectrum similar to fig. 20, the lead and calcium lines traversing the gap in the brass plate. The lead line is enormously broadened by the density of the vapour and pressure. The spectrum of the secondary flash is the continuous band at the top, inter rupted just over the lead line by the absorption of the vapour. This is not very striking in the enlargement, but it was verified by numerous repetitions of the experiment and was very clear on the original negatives. The spectrum below the surface on which the azide detonated is that of the vapour which was blown down over the front surface of the brass block, and is chiefly that of aluminium, the lines 3944 and 3961 and the oxide bands being very conspicuous. It will be of interest to repeat the experi ment of the lead azide hemisphere on the brass block in a high vacuum, as the " tamping " action of the atmosphere will be eliminated. The Zeros of the Riemann Zeta-FunctionBy E. C. Titchmarsh, F.R.S. {Received 20 July, 1936) In my previous paper* I described calculations which show that all the zeros of £ (s), where s = cs + it, between 0 and t = 390 lie line a = W ith the help of a Government Grant, these calculations have now been carried as far as t = 1468. The number of zeros point is 1041, and they all lie on the line < 7 = I have to thank Dr. L. J. Comrie for planning and supervising the calculations, which were carried out with Brunsviga, National, and Hollerith machines.The main results of the calculations are contained in a table giving the values of the function \/( t) = \e defined in the previous paper, for k -90, 90-5, 91, . . . to = 520*5. The corresponding range of ri s roughly 58 *6 to 233 *7. T table giving t, to five decimals, as a function of k, for the above values of