Second-order Nonlinear Systems Research Articles

Fully Distributed Containment Control for Second-Order Nonlinear Multi-Agent Systems With External Disturbances

Fully Distributed Containment Control for Second-Order Nonlinear Multi-Agent Systems With External Disturbances

Distributed cooperative control of autonomous multi-agent UAV systems using smooth control

This paper addresses the cooperative control problem of multiple unmanned aerial vehicles (multi-UAV) systems. First, a new distributed consensus algorithm for second-order nonlinear multi-agent systems (MAS) is formulated under the leader-following approach. The algorithm provides smooth input signals to the agents' control channels, which avoids the chattering effect generated by the conventional sliding mode-based control protocols. Second, a new formation control scheme is developed by integrating smooth distributed consensus control protocols into the geometric pattern model to achieve three-dimensional formation tracking. The Lyapunov theory is used to prove the stability and convergence of both distributed consensus and formation controllers. The effectiveness of the proposed algorithms is demonstrated through simulation results.

Adaptive Event-Triggered Consensus Control of a Class of Second-Order Nonlinear Multiagent Systems.

This paper addresses an adaptive event-triggered consensus control problem for a class of second-order nonlinear multiagent systems (MASs) in an undirected communication topology. A novel adaptive distributed event-triggered consensus control scheme is presented for the MAS with unknown functions based on the definition of an auxiliary state, and the coefficient of the triggered function can be regulated adaptively with dependence on the auxiliary state error to ensure not only the control performance but also the efficiency of the network interactions. Furthermore, two self-triggered algorithms are developed for two cases, known functions and unknown ones, by the current state and information at the previous event time instant instead of the requirement for continuous monitoring auxiliary state errors. In theory, the stability of the resulting closed-loop system is rigorously investigated, and it is proven that all signals in the closed-loop system are bounded and the Zeno behavior is ruled out. Finally, two simulation examples, both real-time and numerical ones, are provided to verify the theoretical claims.

Prescribed-time leader-following consensus for stochastic second-order multi-agent systems subject to actuator failures via sliding mode control strategy

Two control protocols based on the sliding mode control scheme are proposed to solve the prescribed-time leader-following consensus problem for second-order nonlinear multi-agent systems with actuator failures and stochastic noises under directed topology. First, a discontinuous controller protocol is proposed when the parameters related to the actuator failures can be acquired in advance. The controller can drive the system trajectories to reach the sliding manifold before a predefined time, and then ensure that the leader-following consensus is achieved in predefined time also. The upper bound of the reaching time and the upper bound of the setting time do not depend on initial conditions and system parameters, and can be arbitrarily specified offline. Next, when the prior knowledge of the actuator faults and nonlinearities cannot be obtained, a continuous control protocol is proposed, which can adaptively adjust the control gains to compensate for these unknown uncertainties. We theoretically prove that the prescribed-time leader-following consensus in mean square can be achieved under the designed control schemes. Finally, several examples are presented to illustrate the effectiveness of the proposed solutions.

Finite-Time Tracking Control for a Class of Uncertain Strict-Feedback Nonlinear Systems With State Constraints: A Smooth Control Approach.

This article is concerned with the finite-time tracking control problem for a class of strict-feedback nonlinear systems involving state constraints, unknown nonlinearities, and nonvanishing disturbances. Unlike the literature that mainly focuses on a C0 finite-time controller, in this article, a novel C1 smooth finite-time adaptive neural network (NN) controller is proposed by employing a smooth switch between the fractional and cubic form state feedback. The proposed controller not only avoids the singularity but also makes it possible to implement the dynamic surface control (DSC) technique. By applying the adaptive NN control technique, together with barrier Lyapunov functions (BLFs) and a generalized first-order filter including both linear and fractional terms, the desired fast finite-time control performance of the closed-loop nonlinear systems can be guaranteed, and meanwhile, the state constraints are never violated. Under the proposed control scheme, the tracking control problems of nonlinear systems with output constraint and full-state constraints are, respectively, discussed. It is explicitly shown that all the internal error signals are driven to converge into small regions in a finite time. Finally, the effectiveness of the control scheme is also confirmed by the applications to the control of a second-order nonlinear system and an uncertain ship autopilot.

Solvability of nonlinear problem for some second-order nonstrongly elliptic system

In this work, we investigate the second-order nonlinear elliptic system defined on an unbounded domain and with variable nonconstant coefficients. We establish the existence of the solution of the nonlinear problem for a second-order nonstrongly elliptic system by using a coercive estimate, which is obtained for the linear case. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Using the topological fixed-point approach and embedding theorems, the conditions for solvability of the semiperiodical nonlinear problem in a weighted Sobolev space are obtained.

Improved Function Augmented Sliding Mode Control of Uncertain Nonlinear Systems with Preassigned Settling Time

This paper investigates the robust tracking control of second-order uncertain nonlinear systems by adopting the function augmented sliding mode control approach. An improved version of this approach is proposed such that the exact information of the initial tracking error is not required when generating the desired trajectory of the tracking error. This evidently enlarges the application scope of the function augmented sliding mode control approach. Performance functions are introduced to form the performance envelopes for the sliding mode variables. A robust sliding mode controller is constructed such that the sliding mode variables are confined within their performance envelopes. This further guarantees that the tracking error converges to the given neighbourhood of zero within the preassigned settling time provided that proper control parameters are selected. An application example on the rendezvous control of spacecraft is also employed to illustrate the effectiveness of the proposed control approach.

Event-triggered-based adaptive super-twisting attitude tracking for RLV in reentry phase

An event-triggered control scheme is proposed for attitude tracking of Reusable Launch Vehicle (RLV) by combining adaptive and super-twisting techniques. Firstly, an adaptive event-triggered super-twisting control (AETSTC) strategy for multi-variable second-order nonlinear systems with unknown external disturbances is designed, which can adaptively adjust the control gains with respect to the unknown external disturbances and save computational cost or limited network resources while ensuring the desired control performance. Then, a formal proof of the stability of the closed-loop systems under the proposed method is derived based on Lyapunov stability theory, and the Zeno-free execution of triggering sequences is also guaranteed. Furthermore, the proposed AETSTC strategy is applied on attitude control of RLV without any knowledge of the bounds of external disturbances. Finally, simulation results on attitude tracking of RLV are provided to illustrate the efficiency and advantage of the developed algorithm.

Prescribed-time cluster lag consensus control for second-order non-linear leader-following multiagent systems

Prescribed-time Lag consensus, as a special case of prescribed-time cluster lag consensus, is first investigated. The task is to design a control protocol for each follower so that the multiagent system (MAS) achieves lag consensus in any specified time. To achieve this goal, we propose a new distributed controller, in which the control gains are designed as time-varying functions related to the pre-specified time. In addition, a state transformation is introduced to tackle the technical difficulty caused by time-varying functions of different powers in the theoretical proof process. Then, a solution for the cluster lag consensus problem of the MAS is provided, so that under the proposed control protocol, each subsystem composed of followers from the same group and the leader achieves lag consensus with a different lag time in the specified time. By using a state transformation, Graph theory and generalized Lyapunov stability theory, the validity of the designed schemes is verified theoretically and sufficient conditions for the two conclusions are given respectively. Finally, we give two simulation examples to show performances of the proposed solutions.

Adaptive proxy-based sliding mode control for a class of second-order nonlinear systems and its application to pneumatic muscle actuators

The human-centered robotic systems demand safe and robust controllers in many applications. This paper proposes an adaptive proxy-based sliding mode control approach for a class of typical second-order nonlinear systems. A new PID-type virtual coupling is designed between a virtual proxy and the physical object. Considering the unknown bound of lumped disturbances, an adaptation law is applied to online adjust the gain of a sign function which ensures the proxy to track the reference accurately. By using the Lyapunov theorem, the closed-loop system stability is proved. Both simulations and experiments are conducted to verify the proposed method based on a real-world pneumatic muscle actuator control platform. The results show that the proposed adaptive proxy-based sliding mode control approach presents better tracking accuracy, safety, and robustness than the conventional PID control and sliding mode control.

Distributed consensus tracking control of second‐order non‐linear multi‐agent systems with unmodelled dynamics

In this study, the consensus tracking control problem is considered for a class of second-order non-linear multi-agent systems with unmodelled dynamics. Radial basis function neural networks are introduced to approximate the unknown non-linearities and a dynamic signal is utilised to deal with the unmodelled dynamics. Meanwhile, appropriate Lyapunov functions are constructed and a new distributed adaptive control protocol is designed to guarantee that all the tracking error signals in the considered multi-agent systems are ultimately bounded, under which the distributed tracking consensus is reached for all undirected connected communication graphs. Finally, two demonstrative examples are given to illustrate the validity of the designed protocols.

Adaptive non‐singular fast terminal sliding mode control for multi‐agent systems with unknown non‐linear dynamics

This study considers the finite time consensus tracking problems for second-order non-linear multi-agent system with the unknown dynamics. Firstly, the radial basis function neural networks are utilised to approximate the lumped unknown dynamic model. And then, to achieve the finite-time consensus tracking, a distributed non-singular fast terminal sliding mode control strategy is designed which can speed up the convergence of the multi-agent system. Moreover, the consistency under the proposed control strategy is theoretically analysed and the upper bound of setting time is explicitly obtained. Finally, two simulation examples are provided.

Completely distributed neuro-learning consensus with position constraints and partially unknown control directions

In this paper, we propose a new neural network-based iterative learning leader-less consensus control algorithm for the second-order uncertain time-varying nonlinear multi-agent systems, where the position of each agent is constrained in an open interval which can be pre-selected and the control direction can be non-identical and partially unknown. With the help of a novel barrier composite energy function and multiple piece-wise Nussbaum functions, the designed completely distributed learning protocols can make the positions and velocities of the agents achieve the exact consensus on a finite time interval as the iteration number increases to infinity. Meanwhile, the position trajectories of the agents at each iteration do not exceed the constraints bounds. Besides, to testify the feasibility and utility of the presented algorithm, two simulation examples are given in the end.

Consensus of Heterogeneous Second-Order Nonlinear Uncertain Multiagent Systems Under Switching Networks

The consensus problem of heterogeneous second-order nonlinear uncertain multiagent systems under switching networks is addressed by a distributed control approach. The network communication of all systems is subject to arbitrarily bounded and nonuniform time-varying delays. Based on the internal model principle, a nonlinear distributed dynamic controller is developed to solve the problem. By means of Lyapunov analysis, consensus of heterogeneous second-order nonlinear multiagent systems is achieved in the presence of system uncertainties, disturbances, and uniformly connected topologies as well as arbitrarily bounded and nonuniform time-varying communication delays.

Finite-time consensus of second-order nonlinear multi-agent systems with impulsive effects

This paper addresses finite-time consensus of second-order nonlinear multi-agent systems with impulsive effects. A control protocol contains neighborhood and self state feedback without sign function is proposed for finite-time consensus. By employing Lyapunov stability theory, a new less conservative estimation of energy function is obtained, by solving which, it gets both finite-time consensus and exponential consensus criteria with or without impulsive effects. Moreover, three impulsive types: stability, divergence and no effects, are divided based on strengths of impulse and controller. Examples are provided to demonstrate the correctness of theoretical results and the effectiveness of the finite-time protocol.

Asymptotic Behavior of Reachable Setson Small Time Intervals

The geometric structure of small-time reachable sets plays an important role in control theory, in particular, in solving problems of local synthesis. In this paper, we consider the problem of approximate description of reachable sets on small time intervals for control-affine systems with integral quadratic constraints on the control. Using a time substitution, we replace such a set by the reachable set on a unit interval of a control system with a small parameter, which is the length of the time interval for the original system. The constraints on the control are given by a ball of small radius in the Hilbert space $$\mathbb{L}_{2}$$ . Under certain conditions imposed on the controllability Gramian of the linearized system, this reachable set turns out to be convex for sufficiently small values of the parameter. We show that in this case the shape of the reachable set in the state space is asymptotically close to an ellipsoid. The proof of this fact is based on the representation of the reachable set as the image of a Hilbert ball of small radius in $$\mathbb{L}_{2}$$ under a nonlinear mapping to $$\mathbb{R}^{n}$$ . In particular, this asymptotic representation holds for a fairly wide class of second-order nonlinear control systems with integral constraints. We give three examples of systems whose reachable sets demonstrate both the presence of the indicated asymptotic behavior and the absence of the latter if the necessary conditions are not satisfied.

THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero. The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension. The results of numerical simulation illustrate the theoretical conclusions of the paper.

A CENTER MANIFOLD THEORY-BASED APPROACH TO THE STABILITY ANALYSIS OF STATE FEEDBACK TAKAGI-SUGENO-KANG FUZZY CONTROL SYSTEMS

The aim of this paper is to propose a stability analysis approach based on the application of the center manifold theory and applied to state feedback Takagi-Sugeno-Kang fuzzy control systems. The approach is built upon a similar approach developed for Mamdani fuzzy controllers. It starts with a linearized mathematical model of the process that is accepted to belong to the family of single input second-order nonlinear systems which are linear with respect to the control signal. In addition, smooth right-hand terms of the state-space equations that model the processes are assumed. The paper includes the validation of the approach by application to stable state feedback Takagi-Sugeno-Kang fuzzy control system for the position control of an electro-hydraulic servo-system.

Global Adaptive Leader-Following Consensus for Second-Order Nonlinear Multiagent Systems With Switching Topologies

In this note, the leader-following consensus problem is considered for a class of second-order nonlinear multiagent systems, where the communication topology is switching. In view of that the Lipschitz constants of nonlinear terms are unknown, adaptive control strategy is adopted. Using the back-stepping technique, a new adaptive protocol is proposed. It is noted that the global information, including the eigenvalues of Laplacian matrix, is not used in the protocol design. It is proven that the practical leader-following consensus can be reached by the given control scheme. Finally, we present a numerical example to show the effectiveness of the proposed protocol.

Model-Free Iterative Learning Control With Nonrepetitive Trajectories for Second-Order MIMO Nonlinear Systems—Application to a Delta Robot

A model-free iterative learning control strategy (ILC) in nonrepetitive trajectories, applied to robotic manipulators is presented in this article. The development of this ILC controller is derived in general case for second-order nonlinear multi-input multi-output (MIMO) systems subjected to external disturbances. Unlike most of the ILC implementations in the literature, which considers both the resetting condition and periodic trajectories; the proposed ILC controller is model-free and works under the practical alignment condition and noncyclic desired trajectories. Furthermore, these trajectories may vary in magnitude and in the starting and terminal points. The Lyapunov stability approach is combined with a time scale transformation to synthesize the control law. The asymptotic convergence of both the position and velocity tracking errors are demonstrated along the iteration axis. Finally, experimental results in pick and place operations of a parallel Delta robot are presented and discussed. These results are very effective and clearly point out the high potential of the developed model-free ILC technique for industrial robotic applications.