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In this paper, we propose the problem of connectivity maintenance for a group of robots that are connected with wireless communication. The communication between the agents is modeled by a fading channel, and the exchange of information is possible at permissible distances. The distributed controller is designed to maintain the global connectivity of the network communication graph. On the other hand, obtaining the best communication quality between agents is the definition of good network connectivity. Therefore, the Laplacian matrix is defined as a weighted graph according to the communication parameters. Initially, the controller uses the supergradation algorithm to remake the network Laplacian matrix for having a bigger second eigenvalue. While each agent in the network only has access to the neighbor's information. The control algorithm uses a multi-agent estimator to find the eigenvector corresponding to the second small eigenvalue. The formation problem has been considered in the second step after the reference topology has been obtained. Because of the nonlinear dynamics of the agents, the sliding mode controller has been used for this purpose. This robust controller could be a suitable choice due to the modeling uncertainty and sensor measurement uncertainty. Finally, an example of multi-agent robots is provided to evaluate the algorithm.

The robust adaptive leader-follower formation control of uncertain unmanned surface vehicles (USVs) subject to stochastic environmental loads is investigated in this paper. The stochastic additive noises are included in the kinematics which stands for the un-modeled dynamics and uncertainty. The disturbances induced by waves, wind and ocean currents in the kinetics are also separated into deterministic and stochastic components. A comprehensive model including kinematics and kinetics of each USV agent is then derived as stochastic differential equations including standard Wiener processes. Thus, the problem formulation is much more challenging and practical since both the exogenous disturbances and kinematics states are defined by stochastic differential equations. In order to guarantee that all the tracking errors converge to a ball centered at the origin in probability, quartic Lyapunov functions synthesis, dynamic surface control (DSC) technique, the projection algorithm, and neural networks (NNs) are employed. Finally, the simulation experiments quantify the effectiveness of proposed approach.

The paper introduces a novel adaptive fuzzy fractional-order (FO) fast terminal sliding mode control procedure for a class of nonlinear systems in the presence of uncertainties and external disturbances. For this purpose, firstly, using fractional calculus, a new FO nonlinear sliding surface is proposed and then, the corresponding FO fast terminal sliding mode controller (FOFTSMC) is designed to satisfy the sliding condition in finite time. Next, to eliminate the chattering phenomenon, a fuzzy system is constructed to design a continuous switching control law. The finite-time stability of the proposed adaptive fuzzy FOFTSMC (AFFOFTSMC) is proved using the concept of Lyapunov stability theorem. Finally, to illustrate the effectiveness of the proposed AFFOFTSMC, three examples are given as case studies. The numerical simulation results confirm the superiority of the proposed controller, which are the better robust performance, faster convergence, finite-time stability of the closed-loop control system, and a chattering free control effort compared to other mentioned control methods.

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.

In this paper, a novel optimal control design method by discontinuous quadratic Lyapunov function and continuous quadratic Lyapunov function for 2-dimensional piecewise affine systems via semi-definite programming and LMI constraints is proposed.In fact, for designing optimal control we use from two different criteria. At the first, an upper bound for a quadratic cost function for a stable closed-system is obtained. Then after, considering a state-feedback control approach, not only sufficient conditions for the stability of the closed-loop system but also the upper bound of the cost function are obtained. The optimization problem is formulated as a semi-definite programming with bilinear constraints (BMI). Some variables in BMIs are searched by genetic algorithm, so the bilinear constraints are converted to linear constraints and the controller coefficients are calculated. The effectiveness of the proposed method is verified by numerical examples. The simulation results show that discontinuous quadratic lyapunov functions are more efficient that continuous quadratic lyapunov functions.

This paper investigates the stabilization problem of an autonomous Linear Time Invariant (LTI) switched system with interval uncertainty and unstable subsystems. It is proved that the system would be stable by using a common Lyapanov function whose derivative is negative and bounded by a quadratic function within activation regions of each subsystem. First, a sufficient condition for the stability of an Linear Time Invariant switched system with interval uncertainty, based on the convex analysis and interval set theoretical approach, is presented and proved. Moreover, conservatism in the stability robustness bound is obtained. Then, a switching control law is designed to shift the Linear Time Invariant switched system among subsystems to ensure the decrease of the Lyapanov function within the state space. Finally, in order to decrease the switching frequency and to avoid chattering, the switching law is modified. Two examples are included to demonstrate the effectiveness of the theoretical findings.