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This paper investigates the observer design for a class of nonlinear discrete-time systems satisfying incremental quadratic constraints. A circle criterion based full-order observer is constructed by injecting output estimation error into the observer nonlinear terms. We also construct a reduced-order observer to estimate the unmeasured system state. The proposed observers guarantee exponential convergence of the estimation error to zero. The design of the proposed observers is reduced to solving a set of linear matrix inequalities. It is proved that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. Compared to some previous results in the literature, this work considers a larger class of nonlinearities and unifies some related observer designs for discrete-time nonlinear systems. Finally, a numerical example is given to illustrate the effectiveness of the proposed design.

In this paper, the problem positive edge consensus of systems described by linear dynamics on undirected nodal networks is studied. A new observer-based distributed positive edge consensus control protocol is first proposed, which includes the classical Luenberger observer-based protocol as a special case. Moreover, in virtue of line graph theory and topological transformation, we can map the edge topology to the line topology such that the states of each edge in the connected networks can be demonstrated to be consistent. Finally, theoretical analysis and a simulation example are given to verify the effectiveness of the proposed positive edge consensus protocol.

The dynamics of opinion formation with two types of individuals over social networks is analyzed in this paper. During each interaction, individuals pick up one of his neighbors with a probability depending on the neighbor's closeness and popularity and update their opinion based on the convex combination of the opinion of the interactive individuals. It is found that when the stubborn individuals have the same initial opinion, the whole population will definitely converge to a consensus, otherwise, opinion fluctuation can be be observed if regular individuals can reach at least two stubborn individuals with different opinion. Moreover, in the small world network larger average degree and rewiring probability can accelerate convergence speed to a large extent especially when they are relative small. The results may have some insight on the opinion dynamics in the real world.

For the logarithmic quantizer, under the condition that the input of the quantizer is a continuous random variable, some statistical properties of the quantization output, such as probability density function, characteristic function and moments are analyzed in detail. Then if the input contains unknown parameters, the moment estimate of the parameters can be obtained based on the quantization output observations. Since finite logarithmic quantizer is more feasible in practice, the statistical properties of the output for such quantizer are also analyzed. Two examples are given to illustrate the methods proposed in this paper.