In this article, we study the problem of Nash equilibrium (NE) seeking of N -player games with high-order integrator agents over jointly strongly connected networks. The same problem was studied recently over static, undirected, and connected networks. Since a jointly strongly connected network can be directed and disconnected at every time instant, our result strictly includes the existing results as special cases. Moreover, in addition to some analytic conditions on the payoff functions, the existing results also rely on the satisfaction of an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. By adopting a different approach, our result does not rely on the satisfaction of this inequality. Furthermore, our result can also be extended to the case where the additive disturbances are imposed on the input of each agent. Our design is illustrated by two numerical examples.

In this article, a fully event-triggered adaptive distributed control law is developed for solving the practical leader-following consensus problem (LFCP) of a group of uncertain Euler-Lagrange (EL) systems. An event-triggered output-based distributed observer (OBDO) is established first to estimate the state variable of the leader system over jointly connected switching networks (JCSNs). Then, the event-triggered adaptive control law is designed to exclude the Zeno phenomenon and to make the steady-state consensus error smaller than any specified value by adjusting some design parameters. Compared with the existing results, the proposed event-triggered control law only depends on the output of the leader system, works over JCSNs, and, moreover, is fully event-triggered so that it can be directly implemented in a digital platform. The overall design is illustrated using a group of two-link manipulators.

In this article, we study the leader-following practical attitude consensus problem of a group of multiple uncertain rigid spacecraft systems over jointly connected networks by a distributed event-triggered control law. We first establish a lemma that allows the problem to be converted to a distributed practical stabilization problem of a well-defined uncertain dynamical system. Then, we combine the adaptive distributed observer technique and the adaptive control technique to design an event-triggered adaptive control law and an event-triggered mechanism to solve our problem. The effectiveness of our design is illustrated by a numerical example.

The cooperative global robust output regulation problem for multi-agent systems is a generalization of the leader-following consensus problem. The problem has been studied for various multi-agent systems over connected static networks and for some special classes of nonlinear multi-agent systems over jointly connected switching networks. In this paper, we further consider the same problem for a class of heterogeneous lower triangular nonlinear multi-agent systems over jointly connected switching networks. This class of systems is quite general in that it contains inverse dynamics, is of any order, and its subsystems can have different relative degrees. We will integrate the adaptive distributed observer and the distributed internal model approach to come up with a recursive approach to deal with our problem. We will also apply our approach to a leader-following consensus problem for a group of hyperchaotic Lorenz systems.

In this paper, we study the leader-following consensus problem of multiple Euler-Lagrange (EL) systems subject to an uncertain leader system. We first establish an adaptive distributed observer for a neutrally stable linear leader system whose system matrix is not known exactly. Under standard assumptions, this adaptive distributed observer can estimate and pass the leader's state to each follower through the communication network of the system without knowing the leader's system matrix exactly. Under the additional assumption that the leader's state is persistently exciting, this adaptive distributed observer can also asymptotically learn the parameters of the leader's system matrix. On the basis of this adaptive distributed observer, we further synthesize an adaptive distributed control law to solve our problem via the certainty equivalence principle. Our result allows the leader-following consensus problem of multiple EL systems to be solved even if none of the followers knows the system matrix of the leader system exactly.

This paper studies the cooperative global robust output regulation problem for a class of heterogeneous second-order nonlinear uncertain multiagent systems with jointly connected switching networks. The main contributions consist of the following three aspects. First, we generalize the result of the adaptive distributed observer from undirected jointly connected switching networks to directed jointly connected switching networks. Second, by performing a new coordinate and input transformation, we convert our problem into the cooperative global robust stabilization problem of a more complex augmented system via the distributed internal model principle. Third, we solve the stabilization problem by a distributed state feedback control law. Our result is illustrated by the leader-following consensus problem for a group of Van der Pol oscillators.

This paper presents a discrete-time recurrent neural network approach to solving systems of linear equations with two features. First, the system of linear equations may not have a unique solution. Second, the system matrix is not known precisely, but a sequence of matrices that converges to the unknown system matrix exponentially is known. The problem is motivated from solving the output regulation problem for linear systems. Thus, an application of our main result leads to an online solution to the output regulation problem for linear systems.

In this paper, we study the cooperative global robust practical output regulation problem for a class of second-order uncertain nonlinear multiagent systems via a distributed event-triggered state feedback control strategy. Compared with the existing work, one of the main challenges is that we need to design two distributed internal models to learn both the desired steady-state state and steady-state input for each agent. Moreover, to obtain a directly implementable digital control law, the two distributed internal models of each agent only depend on the sampled states of the neighboring agents and itself. As a result, the resulting augmented system is more complicated, and the control law needs to be recursively designed. To overcome the difficulty, we propose a novel distributed event-triggered control law and a novel distributed event-triggered mechanism to deal with our problem. By adjusting a design parameter in the proposed event-triggered mechanism, we show that the Zeno behavior does not happen and the ultimate bound of the tracking error can be made arbitrarily small. Our design will be illustrated by two examples.

This paper studies the leader-following rendezvous and flocking problems with connectivity preservation for a class of second-order nonlinear multi-agent systems subject to both external disturbances and plant uncertainties. By combining the potential function method, the adaptive control technique and the distributed observer method, both the leader-following rendezvous and flocking problems are solved. Finally, one simulation example for a group of Van der Pol oscillators is given to illustrate our design.

This paper studies the problem of leader-following rendezvous with connectivity preservation for a linear multi-agent system where the leader system is a linear autonomous system and the follower system is a multiple single-integrator system. We propose a distributed state feedback control protocol that is able to maintain the connectivity of the system and, at the same time, achieve asymptotic tracking of all followers to the output of the leader system.

In this paper, the cooperative global robust output regulation problem for a class of strict feedback nonlinear uncertain multi-agent systems is studied. We first introduce a type of distributed internal model that converts the cooperative global robust output regulation problem into a global robust stabilization problem of a so-called augmented multi-agent system which is in block lower triangular form. We then further present a set of sufficient conditions under which the augmented multi-agent system is globally stabilizable by a distributed state feedback control law.

In this paper, we consider the cooperative output regulation of linear multi-agent systems under switching network. The problem can be viewed as a generalization of the leader-following consensus problem of multi-agent systems. Due to the limited information exchanges of different subsystems, the problem cannot be solved by the decentralized approach and is not allowed to be solved by the centralized control. By devising a distributed observer network, we can solve the problem by both dynamic state feedback control and dynamic measurement output feedback control. As an application of our main result, we show that a special case of our results leads to the solution of the leader-following consensus problem of linear multi-agent systems.

This paper addresses a global robust adaptive control problem for a class of uncertain nonlinear systems by output feedback control. The problem will be solved by the internal model design method. Our problem formulation includes the stabilization and synchronization problem of some typical nonlinear systems as special cases. As an illustration, our main result will be applied to solving a robust output synchronization problem of the generalized Lorenz system and an uncertain Harmonic system.

Receding horizon control (RHC), also known as model predictive control (MPC), is a suboptimal control scheme that solves a finite horizon open-loop optimal control problem in an infinite horizon context and yields a measured state feedback control law. A lot of efforts have been made to study the closed-loop stability, leading to various stability conditions involving constraints on either the terminal state, or the terminal cost, or the horizon size, or their different combinations. In this paper, we propose a modified RHC scheme, called adaptive terminal cost RHC (ATC-RHC). The control law generated by ATC-RHC algorithm converges to the solution of the infinite horizon optimal control problem. Moreover, it ensures the closed-loop system to be uniformly ultimately exponentially stable without imposing any constraints on the terminal state, the horizon size, or the terminal cost. Finally we show that when the horizon size is one, the underlying problems of ATC-RHC and heuristic dynamic programming (HDP) are the same. Thus, ATC-RHC can be implemented using HDP techniques without knowing the system matrix A.

In this paper, neural networks are used along with two-player policy iterations to solve for the feedback strategies of a continuous-time zero-sum game that appears in L2-gain optimal control, suboptimal Hinfincontrol, of nonlinear systems affine in input with the control policy having saturation constraints. The result is a closed-form representation, on a prescribed compact set chosen a priori, of the feedback strategies and the value function that solves the associated Hamilton-Jacobi-Isaacs (HJI) equation. The closed-loop stability, L2-gain disturbance attenuation of the neural network saturated control feedback strategy, and uniform convergence results are proven. Finally, this approach is applied to the rotational/translational actuator (RTAC) nonlinear benchmark problem under actuator saturation, offering guaranteed stability and disturbance attenuation.

The existing approaches to the discrete-time nonlinear output regulation problem rely on the offline solution of a set of mixed nonlinear functional equations known as discrete regulator equations. For complex nonlinear systems, it is difficult to solve the discrete regulator equations even approximately. Moreover, for systems with uncertainty, these approaches cannot offer a reliable solution. By combining the approximation capability of the feedforward neural networks (NNs) with an online parameter optimization mechanism, we develop an approach to solving the discrete nonlinear output regulation problem without solving the discrete regulator equations explicitly. The approach of this paper can be viewed as a discrete counterpart of our previous paper on approximately solving the continuous-time nonlinear output regulation problem.

The twenty-two papers in this special issue are devoted to neural networks for feedback control systems. Covers some of the following topics: reinforcement learning; applications; neurocontrol systems; discrete time systems; and network architectures and training methods.

The dynamic surface control (DSC) technique was developed recently by Swaroop et al. This technique simplified the backstepping design for the control of nonlinear systems in strict-feedback form by overcoming the problem of "explosion of complexity." It was later extended to adaptive backstepping design for nonlinear systems with linearly parameterized uncertainty. In this paper, by incorporating this design technique into a neural network based adaptive control design framework, we have developed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. Our development is able to eliminate the problem of "explosion of complexity" inherent in the existing method. In addition, a stability analysis is given which shows that our control law can guarantee the uniformly ultimate boundedness of the solution of the closed-loop system, and make the tracking error arbitrarily small.

The small gain theorem as established in [Z.P. Jiang and I. Mareels, 1997] and [Z.P. Jiang et al., 1994] has been given in terms of a quite general inter-connection of two nonlinear subsystems. This paper gives a more clear-cut version and a more straightforward proof for the small gain theorem for a simpler inter-connection of two nonlinear subsystems by using a different characterization of the ISS concept. Moreover, we have also incorporated some type of model uncertainties into the system description, thus making this special version directly applicable to the global robust stabilization and output regulation problems.

A feedback controller that solves the discrete-time nonlinear servomechanism problem relies on the solution of a set of nonlinear functional equations known as the discrete regulator equations. The exact solution of the discrete regulator equations is usually unavailable due to the nonlinearity of the system. The paper proposes to approximately solve the discrete regulator equations using a feedforward neural network. This approach leads to an effective way to practically solve the discrete nonlinear servomechanism problem. The approach has been illustrated using the well-known inverted pendulum on a cart system. The simulation shows that the control law designed by the proposed approach performs much better than the conventional linear control law.

The solution of discrete-time nonlinear servomechanism problem relies on the solvability of a set of nonlinear functional equations known as the discrete regulator equations. Due to the nonlinear nature, it is difficult to obtain the exact solution of the discrete regulator equations. This paper presents a feedforward neural network based approach to solving the discrete regulator equations. This approach leads to an effective way to practically solve the discrete nonlinear servomechanism problem. The approach has been illustrated using the well known inverted pendulum on a cart system. The simulation shows that the control law designed by the proposed approach performs much better than the linear control law does.

The solution of the nonlinear servomechanism problem relies on the solvability of a set of mixed nonlinear partial differential and algebraic equations known as the regulator equations. Due to the nonlinear nature, it is difficult to obtain the exact solution of the regulator equations. This paper proposes to solve the regulator equations based on a class of recurrent neural network, which has the features of a cellular neural network. This research not only represents a novel application of the neural networks to numerical mathematics, but also leads to an effective approach to approximately solving the nonlinear servomechanism problem. The resulting design method is illustrated by application to the well-known ball and beam system.