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An improved prescribed performance control using a backstepping technique and adaptive fuzzy is proposed for a strict feedback nonlinear dynamic system. A new virtual variable was defined to generate the virtual control that forces the tracking errors to fall within prescribed boundaries, and an adaptive fuzzy system was used to obtain required approximation performances. A strict feedback controller and adaptive laws for estimating the unknown non-linear function were designed to avoid a singularity problem and calculation of the explosive number of terms generated by the error transformations of conventional error constraint method and the recursive steps of traditional backstepping control. Lyapunov stability analysis confirmed the boundedness and convergence of the closed-loop system. The prescribed error constraint performance of the proposed control scheme was validated by applying it to control the position of a second-order non-linear system and a robot manipulator.

We introduce an adaptive dynamic surface control (ADSC) method tailored for high‐order strict‐feedback systems (SFSs) with input saturation, utilizing the fully actuated system (FAS) approach. We simplify the steps in designing the controller by combining the FAS approach with ADSC method to directly control each high‐order subsystem as a complete entity, without the need to transform it into first‐order systems. Smooth functions and Nussbaum functions are applied to solve the problem of input saturation. We use a sequence of low‐pass filters to calculate the higher‐order derivatives of the virtual control law. Lyapunov stability theory is used to demonstrate that all signals within the closed‐loop system become uniformly bounded, with the tracking error ultimately converging to a small vicinity around zero. We validated the efficiency of the proposed method of control through simulations on a flexible joint manipulator system. In contrast to the traditional first‐order system method, which requires four virtual control laws, the proposed method in this paper necessitates only two, resulting in a smaller initial value of the control input.

An event-triggered (ET) neural-networks (NNs) adaptive output-feedback control approach is proposed for a class of input-saturated strict-feedback nonlinear systems with external disturbance. Compared with overall existing event-triggered control strategies, which are free from control input nonlinearities and suffer from the explosion of complexity, the proposed event-triggered-based NNs controller is able to handle system input saturation along with completely avoiding the complexity explosion problem. First, by introducing alternative state variables, and by implementing a low-pass filter, the difficulty arising from the cascading of the input-saturated strict-feedback system has been avoided. Thus, the system is converted to the normal canonical system, for which the controller synthesis is much simpler without resort to traditional back-stepping approach. Then, an observer is adopted to estimate the unknown states of the newly derived canonical system based on strictly positive real theory. In the design procedure, the unknown nonlinear functions are approximated by NNs to design a baseline controller, for which an additional robust term is embedded to deal with the input saturation nonlinearity, unknown disturbances and approximation errors using only two adaptive parameters. The proposed ET adaptive NNs control scheme is shown to guarantee the convergence of the output of the system to a small neighborhood of the origin along with the boundedness of all signals in the closed loop. Finally, simulation examples are presented to show the effectiveness of the proposed controller.

This paper addresses the inverse optimal security control problems for a class of stochastic non-strict feedback nonlinear cyber-physical systems under sensor and actuator attacks. The concerned system model includes both stochastic disturbances and more general nonlinearity. First, to make the control design feasible, a linear state transformation is applied to the attacked system. Furthermore, in the process of backstepping design, based on the Nussbaum gain function formula, fuzzy logic system approximation method, and inverse optimal control theory, combing the available output signal, an output feedback inverse optimal controller is proposed. Specifically, the designed controller not only ensures that the system is secure under network attacks but also optimal in terms of the cost function. Finally, two physical examples are given to verify the effectiveness of the proposed control scheme in various network attacks.

This article studies the control ideas of the optimal backstepping technique, proposing an event-triggered optimal tracking control scheme for a class of strict-feedback nonlinear systems with non-affine and nonlinear faults. A simplified identifier-critic-actor framework is employed in the reinforcement learning algorithm to achieve optimal control. The identifier estimates the unknown dynamic functions, the critic evaluates the system performance, and the actor implements control actions, enabling modeling and control of anonymous systems for achieving optimal control performance. In this paper, a simplified reinforcement learning algorithm is designed by deriving update rules from the negative gradient of a simple positive function related to the Hamilton-Jacobi-Bellman equation, and it also releases the stringent persistent excitation condition. Then, a fault-tolerant control method is developed by applying filtered signals for controller design. Additionally, to address communication resource reduction, an event-triggered mechanism is employed for designing the actual controller. Finally, the proposed scheme’s feasibility is validated through theoretical analysis and simulation.

The predefined-time fuzzy adaptive output feedback control problem is considered for non-strict feedback stochastic nonlinear systems with state constraints. Since the controlled plant contains unknown nonlinear dynamics and unmeasured states, the unknown nonlinear dynamics are handled by using fuzzy approximation technique, and a fuzzy state observer is established to estimate unmeasured states. Then, under the frameworks of a predefined-time stability theory and backstepping control design technique, a new fuzzy adaptive output feedback control method is proposed. It is proved that the controlled system is semi-global practically predefined-time stable in probability by constructing suitable barrier Lyapunov functions. Finally, the spring–mass–damper system is given to confirm the effectiveness of the presented control method.

In this paper, the type of general nonholonomic systems proposed in part 1 and part 2 is generalized into a multi-vector form, for which the nonsingular condition is not assumed. First, by differentiating the first equation in the system, and through stability analysis, the stabilization of the proposed system is firstly converted into the stabilization of a sub-strict feedback system (sub-SFS), that is, a system in the form of a strict feedback system but with the gain matrices not satisfying the nonsingular conditions. Second, using the fully actuated system (FAS) approach, the formulated sub-SFS is further converted into a sub-FAS, for which the concepts of singular and feasible sets are defined. Finally, the problem is solved by designing a sub-stabilizing controller of the obtained sub-FAS, which drives all the system responses starting from a so-called region of exponential attraction to the origin exponentially, and the closed-loop response of the formulated sub-SFS controlled by the designed controller is also explicitly provided. Technically, an external parameter is introduced in the designed controller. In general, the designed controller contains an integral term. In the case that the introduced two sets of matrix functions in the proposed system are time-invariant, the controller turns out to be a continuous time-varying one, but when the external parameter is particularly chosen, the controller takes a simpler form but becomes a discontinuous time-invariant one.

A new prescribed-time state-feedback design is presented for stochastic nonlinear strict-feedback systems. Different from the existing stochastic prescribed-time design where scaling-free quartic Lyapunov functions or scaled quadratic Lyapunov functions are used, the design is based on new scaled quartic Lyapunov functions. The designed controller can ensure that the plant has an almost surely unique strong solution and the equilibrium at the origin of the plant is prescribed-time mean-square stable. After that, the authors redesign the controller to solve the prescribed-time inverse optimal mean-square stabilization problem. The merit of the design is that the order of the scaling function in the controller is reduced dramatically, which effectively reduces the control effort. Two simulation examples are given to illustrate the designs.

This paper investigates the event-triggered control problem for a class of strict-feedback uncertain nonlinear systems with time-varying state constraints. The main challenges of such work arise from two aspects: 1) The state is not constrained at all moments, so the constraints on the state are not fixed, resulting in switches between bounded and unbounded constraints. The existing research is more suitable for bounded constraints. 2) Due to the presence of time-varying constraints, we aim to address these scenarios with varying control performance, existing research addresses this solely via offline tuning of the controller and its parameters. To solve the above problems, we develop a new exponential nonlinear state-dependent function (ENSDF). The ENSDF transforms the constraint problem of the original state into the boundedness problem of the new state, independent of the boundedness of the constraint boundaries. Then, a new constraint-dependent switching event-triggered controller is developed. The switching event-triggered controller based on ENSDF can dynamically adjust the controller and its parameters online to adapt to time-varying constraints. It is theoretically shown that the state does not violate the constraints, all closed-loop signals are globally bounded, and the tracking error converges towards an adjustable compact set.

This article focuses on the prescribed-time tracking control problem for non-strict feedback nonlinear systems (NFNS) with unknown system uncertainties and external disturbances. In contrast to the previous work on finite-time and fixed-time stabilization control for a class of NFNS, the prescribed-time zero-error tracking control is realized. Firstly, the fuzzy logic system is used to handle unknown nonlinear functions. Secondly, the “complexity explosion” problem is solved in the control structure through a designed command filter and a new prescribed-time error compensation algorithm, which realizes the preset time compensation of the filter error signal to zero, and effectively eliminates the impact of the filter error signal on the system performance. Furthermore, a dynamic event-triggered mechanism (DETM) is proposed based on the co-design approach to effectively reduce the waste of communication resources and optimize network resource utilization. Meanwhile, based on the Lyapunov stability theory, it is proved that the system tracking errors will converge to zero within a prescribed time, which can be determined by the user in advance arbitrarily while avoiding the Zeno phenomenon. It is important to note that the preset time can be adjusted within the physical limits of the system, independent of the system initial conditions. Finally, the effectiveness of the proposed control method algorithm is verified by numerical simulation results and experimental studies.