Unknown Nonlinear Functions Research Articles

Dynamical analysis and event-triggered adaptive finite-time prescribed performance control of the FO coupled MEMS resonators

This paper explores the dynamic behaviors and control method of weakly coupled micro-electro-mechanical system (MEMS) resonators with fractional-order (FO) dynamics. In the dynamics analysis session, Lyapunov exponents, bifurcation theory, and phase diagrams are used to analyze the effects of system parameters, coupling stiffness, and FO on the complex dynamical behaviors such as periodicity, pseudo-periodicity, and chaos oscillations. To suppress chaotic oscillations, a dynamic event-triggered finite-time prescribed performance controller is proposed. The unknown nonlinear functions are approximated through the interval type-3 fuzzy system (IT3FS) with an adaptive law. A novel finite-time prescribed performance function (finite-time PPF) is constructed to establish constrained boundaries for tracking errors. Compared with the existing PPFs, the proposed approach allows for more flexible setting of the convergence boundary before reaching steady-state. Subsequently, the constrained tracking errors are mapped to an unconstrained form using a nonlinear transformation function, whether the error constraint is symmetric or asymmetric. To circumvent the “explosion of complexity” arising from backstepping design process, a tracking differentiator (TD) is utilized. Additionally, to alleviate strain on communication resources, an event-triggered mechanism is devised to update control signals. The trigger threshold of the control signals can be adaptively adjusted based on the values of the Lyapunov function and the dynamic auxiliary variables. This control method guarantees finite-time convergence for all signals of the system. Finally, extensive simulations are performed to verify the effectiveness of the proposed algorithm.

Indirect adaptive observer control (I-AOC) design for truck–trailer model based on T–S fuzzy system with unknown nonlinear function

Tracking is a crucial problem for nonlinear systems as it ensures stability and enables the system to accurately follow a desired reference signal. Using Takagi–Sugeno (T–S) fuzzy models, this paper addresses the problem of fuzzy observer and control design for a class of nonlinear systems. The Takagi–Sugeno (T–S) fuzzy models can represent nonlinear systems because it is a universal approximation. Firstly, the T–S fuzzy modeling is applied to get the dynamics of an observational system in order to estimate the unmeasurable states of an unknown nonlinear system. There are various kinds of nonlinear systems that can be modeled using T–S fuzzy systems by combining the input state variables linearly. Secondly, the T–S fuzzy systems can handle unknown states as well as parameters known to the indirect adaptive fuzzy observer. A simple feedback method is used to implement the proposed controller. As a result, the feedback linearization method allows for solving the singularity problem without using any additional algorithms. A fuzzy model representation of the observation system comprises parameters and a feedback gain. The Lyapunov function and Lipschitz conditions are used in constructing the adaptive law. This method is then illustrated by an illustrative example to prove its effectiveness with different kinds of nonlinear functions. A well-designed controller is effective and its performance index minimizes network utilization—this factor is particularly significant when applied to wireless communication systems.

Adaptive fixed-time fuzzy containment control for uncertain nonlinear multiagent systems with unmeasurable states

This paper addresses the adaptive fixed-time fuzzy containment control for uncertain nonlinear multiagent systems, where the states and nonlinear functions are not feasible for the controller design. To address the issue of unmeasurable states, a state observer is developed, and fuzzy logic systems are utilized to approximate unknown nonlinear functions. Under the framework of fixed-time Lyapunov function theory and cooperative control, an adaptive fixed-time fuzzy containment control protocol is derived via the adaptive backstepping and adding one power integrator method. The derived fixed-time containment controller can confirm that the closed-loop systems are practical fixed-time stable, which implies that all signals in the systems are bounded and all follower agents can converge to the convex hull formed by the leader agents within fixed-time in the presence of unmeasurable states and unknown nonlinear functions . Simulation examples are conducted to test the validity of the present control algorithm.

A signal smoothness-based approach to prescribed performance control of strict-feedback nonlinear systems with quantization

This paper investigates the prescribed performance control problems of strict-feedback nonlinear systems with state quantization, input quantization, unknown disturbances and unknown nonlinear functions. Since the quantized states are discontinuous, the differentiability of the stabilizing functions in the backstepping technique cannot be guaranteed. To this end, a smooth approximation of the quantized states is first obtained by introducing a class of functions. Based on this smooth approximation, a quantized control scheme is presented such that all the closed-loop signals are bounded with the prescribed performance bounds. It is shown that the unknown nonlinearities and the unknown disturbances are not estimated and the derivatives of the stabilizing functions are eliminated. Lastly, two examples are provided to illustrate the effectiveness of the presented method.

Position constrained adaptive time-varying formation control of autonomous surface vessels with uncertain dynamics

This paper investigates the adaptive time-varying formation control problem of autonomous surface vessels (ASVs) with position constraints, uncertain dynamics and external disturbances under directed communication topology. Different from the existing studies, the nonlinear error transformation function is combined with formation control to achieve position constraints of ASVs. Based on dynamic surface control and backstepping method, a virtual control law is designed for each vessel. Radial basis function neural networks are employed to approximate the unknown nonlinear continuous function caused by ASVs’ model uncertainties, and the minimum learning parameter (MLP) method is applied to reduce computational complexity. Then, a time-varying formation position constrained controller and an adaptive law are proposed for each ASV. By using the Lyapunov stability theory, it is proved that all signals of the closed-loop ASV system are semi-global uniformly ultimately bounded and the position constrained objective can be achieved. Finally, simulation analyses are given to verify the effectiveness of theoretical results.

Adaptive neural event-triggered consensus control for unknown nonlinear second-order delayed multi-agent systems

In this paper, we discuss event-triggered consensus control for a heterogeneous second-order multi-agent systems with unknown nonlinear functions and state delays (SODMASs). Firstly, a broad adaptive dynamic event-triggered mechanism (ETM), which includes many existing ETM, is proposed. Based on this ETM and the approximation property of neural networks, an adaptive event-triggered protocol with weight estimation of neural network and time-varying integral gain compensation is designed to reduce communication frequency and to eliminate the uncertainty of nonlinear function and state delays. Then, the consensus problem for the heterogeneous SODMASs under the protocol is solved successfully by selecting a suitable Lyapunov–Krasovskii functional. In addition to these, it is also proved that all agents can exclude Zeno behavior under the proposed event-triggered protocol. Finally, a numerical example is given to verify the effectiveness of the proposed consensus protocol algorithm.

Adaptive fault-tolerant control for ascent HSV with wing damage and function constraints on states

In this article, a fault-tolerant tracking control technique is developed for the ascent of hypersonic vehicles (HSV) experiencing wing damage and time-varying state constraints. The difficulty of control problem lies in handling state constraints associated with both state and time, while determining the unknown control directions of the multi-input multi-output (MIMO) nonlinear system. The presence of constraints in both state and time complicates the design of the control algorithm. This paper proposes the construction of a novel state transition function to establish a new unconstrained system, thereby eliminating the conservative feasibility conditions of Barrier Lyapunov Function (BLF) and integral BLF methods when dealing with time-varying constraints. The fusion of time-varying scaling functions and Nussbaum-type functions not only resolves the issue of unknown control directions in the MIMO nonlinear system but also enhances the transient and steady-state performance of the closed-loop system. Neural Networks are employed to approximate the unknown nonlinear functions. Using Lyapunov stability theory, it is proven that state constraints are not violated, and all signals in the closed-loop system are bounded. Simulation results demonstrate the merits of the designed fault-tolerant control (FTC) algorithm.

Fixed‐time neuroadaptive formation control for multiple QUAVs with external disturbance

AbstractThis paper studies the formation control of multiple quadrotor unmanned aerial vehicle systems (MQUAVSs) with external disturbance. A new adaptive fixed‐time cooperative control protocol is designed for MQUAVSs. A fixed‐time command filtered compensation control technology is presented to overcome the “explosion of complexity” issue, and a new fixed‐time error compensation signal is designed to compensate the filtering error, which improves the convergence speed of the system. Adaptive neural network technology is introduced to deal with unknown nonlinear functions in the system. A fixed‐time stability theorem is presented for MQUAVSs to ensure that MQUAVSs can reach the predetermined formation and the formation tracking errors converge to the neighborhood of the origin in a fixed time. Finally, the effectiveness of the proposed method is verified by the formation simulation of MQUAVs.

Command Filter-Based Adaptive Neural Control for Nonstrict-Feedback Nonlinear Systems with Prescribed Performance

In this paper, a new command filter-based adaptive NN control strategy is developed to address the prescribed tracking performance issue for a class of nonstrict-feedback nonlinear systems. Compared with the existing performance functions, a new performance function, the fixed-time performance function, which does not depend on the accurate initial value of the error signal and has the ability of fixed-time convergence, is proposed for the first time. A radial basis function neural network is introduced to identify unknown nonlinear functions, and the characteristic of Gaussian basis functions is utilized to overcome the difficulties of the nonstrict-feedback structure. Moreover, in contrast to the traditional Backstepping technique, a command filter-based adaptive control algorithm is constructed, which solves the “explosion of complexity” problem and relaxes the assumption on the reference signal. Additionally, it is guaranteed that the tracking error falls within a prescribed small neighborhood by the designed performance functions in fixed time, and the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB). The effectiveness of the proposed control scheme is verified by numerical simulation.

Adaptive expected-time tracking consensus control for multiagent systems with external disturbances and time delays

This paper studies the expected-time tracking control problem for multiagent systems with disturbances and time delays. The expected-time control scheme can ensure that control objectives can be completed within the desired time. In the other words, the output signal can track the reference signal in an expected time. Based on a special error and the adaptive technology, the expected-time tracking consensus control scheme is achieved. Moreover, by using the neural networks technique, the unknown nonlinear functions which often exist in the dynamic of the system are handled. According to the Lyapunov stability theorem and the Lyapunov–Krasovskii function, it is proved that the consensus errors are semi-globally uniformly ultimately bounded. At last, the feasibility of the control method is verified by a simulation example.

Adaptive predefined-time tracking control of uncertain permanent-magnet synchronous motor drive chaotic system

A predefined-time tracking control algorithm is formulated for the Permanent Magnet Synchronous Motor(PMSM) driver system subjected to chaotic oscillations and undeterministic parameters, which can determine the upper bound of settling-time by controller gains. Radial Basis Function Neural Networks(RBFNNs) are employed to approximate the unknown nonlinear functions as universal approximators. The innovative predefined-time tracking differentiators are constructed to estimate the derivatives of the virtual inputs, resolving “the explosion of complexity” in traditional backstepping method. Then, an adaptive predefined-time controller is proposed in the backstepping framework. Lyapunov’s theoretical analysis proves that all signals in the closed-loop systems are uniformly ultimately bounded and the output tracking errors can converge to a small neighborhood of the origin within the predefined-time. Finally, simulation studies are performed to demonstrate that the proposed control scheme can quash the chaotic behaviors of the PMSM drive system and ensure fast-tracking performance even under unknown parameters.

Asymptotic tracking control for a class of uncertain nonlinear systems with output constraint

In this paper, we investigate the asymptotic tracking control for a class of strict-feedback nonlinear systems with unknown nonlinear functions and output constraint. The nonlinear transformation function-based system transformation approach is employed to convert the initial constrained system into an unconstrained one. To achieve the objective of asymptotic tracking, the tracking error is scaled by an exponential proportional function. Furthermore, to address the challenge of repetitive derivation of virtual control laws that occurs in the traditional design process of high-order backstepping methods, a first-order filter is applied. The designed control scheme guarantees that the system output will asymptotically track a predefined reference tracking trajectory while ensuring that the output constraint is not violated. Moreover, it is shown that the filter error will converge asymptotically to the origin. The effectiveness of the developed control strategy is verified through simulation results.

Adaptive Neural Network Finite-Time Prescribed Performance Consensus Control for a Class of Second-Order Multi-Agent Systems

This research investigates the semi-global practical finite-time prescribed performance consensus control issue for a class of second-order multi-agent systems with unknown nonlinear functions. Unlike the previous finite-time control set by a finite-time performance function, we give finite-time control by constraining the terminal sliding manifold in a performance function. In addition, an adaptive neural network control scheme is designed, which simplifies the controller and avoids the chattering issue existing in traditional sliding mode control. Eventually, a novel adaptive finite-time prescribed performance consensus control strategy is designed, which ensures that all system variables are semi-globally practical finite-time stable and consensus errors of the multi-agent systems converge within the prescribed region in finite time. The effectiveness and practicality of the presented control strategy are evaluated by conducting simulation cases.

Neural network-based prescribed performance tracking control for a class of nonlinear systems with mismatched disturbances

This paper addresses the prescribed performance tracking control problem for a class of nonlinear systems subject to mismatched uncertainties. A novel disturbance observer-based control (DOBC) scheme is proposed through combining radial basis function neural network (RBFNN) and prescribed performance function (PPF). In contrast to traditional DOBC, the proposed approach employs RBFNN technology to approximate unknown nonlinear functions in the system, instead of treating them as part of lumped disturbances. A novel disturbance observer is developed to estimate disturbances characterised by a nonlinear exogenous system. To enhance control performance, a PPF that characterises both transient and steady-state behaviour is used for the transformation of tracking errors. It can be proved that all the states of the closed-loop system can be guaranteed to be uniformly ultimately bounded (UUB), and that the tracking error evolves within the prespecified boundaries. Theoretical results are validated and supported by two simulation examples.

Decentralized Output-Feedback Adaptive Event-Triggered Control for Interconnected Nonlinear Delay Systems with Actuator Failures

This paper investigates decentralized adaptive event-triggered fault-tolerant control for interconnected nonlinear delay systems with actuator failures. The actuator failures suffered include loss of effectiveness and bias faults. A control scheme based on the K-filter is proposed, which effectively compensates for the effects of unknown actuator failures. A hyperbolic tangent function and neural network are introduced to approximate the unknown interconnection function and nonlinear delay function. By introducing the dynamic surface control method, the “explosion of complexity” issue is addressed. Furthermore, our proposed controller can ensure that all states of the corresponding closed-loop system are semi-globally uniformly ultimately bounded and that the tracking error can converge to a small neighborhood of zero. Meanwhile, Zeno behavior can be effectively avoided. Finally, the validity of the proposed control scheme is verified using a simulation example.

Adaptive neural tracking control for flexible joint robot including hydraulic actuator dynamics with disturbance observer

AbstractIn this paper, a disturbance observer‐based adaptive neural backstepping integral sliding mode control (BISMC) is developed for a flexible joint robot (FJR) with the integration of an adjustable stiffness rotary actuator (ASRA). This system suffers from unknown system dynamics, external disturbance, and the influence of variable stiffness, which is a challenge for achieving precision tracking performance. Considering the lumped disturbances in FJR generated by the hydraulic system and the stiffness modulation of the ASRA, we investigate the structural dynamics nonlinear model of the FJR system, including hydraulic actuator dynamics. While other linear control strategies are applied for the FJR, the proposed controller uses BISMC, neural networks (NN), and nonlinear disturbance observers to deal with the disadvantages mentioned above. Radial basis function neural networks (RBFNN) are designed to tackle unknown nonlinear functions, and the disturbance observers are introduced to compensate for the influence of the variable stiffness, disturbance, and the approximation error caused by NN. Simulations and experiments are independently implemented to demonstrate the effectiveness and feasibility of the proposed controller. Results exhibit that the integral absolute error‐index is reduced by 20.4% when the proposed method is deployed for the experiment with a multistep trajectory.

Backstepping-Based Fuzzy Adaptive Stabilization of Reaction-Diffusion Equation With State Constraints.

This article proposes a stabilization scheme for a cascaded parabolic partial differential equation (PDE)-ordinary differential equation (ODE) system with state constraints. To begin, by employing the fuzzy-logic system (FLS) technique for the unknown nonlinear functions in the ODE subsystem, the influence of the nonlinearities is successfully eliminated. Then, the infinite and finite-dimensional backstepping methods are skillfully applied to the design of control schemes. Specifically, the infinite-dimensional backstepping transformation and its inverse are utilized to obtain a new PDE subsystem, which is convenient for control design. Next, based on the new target PDE-ODE cascade system, a novel first-order filter is subtly introduced into each step of the finite-dimensional backstepping-based controller design procedure to avoid the issue of "explosion of complexity." Furthermore, a novel barrier Lyapunov function (BLF) is constructed to guarantee that the states of the ODE subsystem do not violate the constraints. The controller designed in this article ensures that all signals in the closed-loop system are bounded and the states can converge to zero. Finally, a simulation example verifies that the proposed scheme can achieve satisfactory performance.

Command filter based input quantized adaptive tracking control for multi‐input and multi‐output non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints

SummaryThis paper addresses the problem of adaptive tracking control for multi‐input and multi‐output (MIMO) non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints. To tackle the interference of unmodeled dynamics, the dynamic signal generated by the auxiliary system is used. Hyperbolic tangent function is used as a nonlinear mapping tool to transform the constrained system into an unconstrained one. Hysteresis quantizer is introduced to mitigate the chattering phenomenon and quantization error in the quantization signal. The derivative of virtual signal can be approximated more efficiently by command filter. Furthermore, an error compensation mechanism is established to mitigate the error introduced by the command filter. Unknown nonlinear functions are approximated by radial basis function neural networks (RBFNNs). Stability analysis of the proposed controller is performed through the Lyapunov stability theory and the output tracking error can be constrained within a specified range. Finally, simulation results are presented to demonstrate the effectiveness of the proposed method.

Fuzzy functional observer‐based sliding mode control for T‐S fuzzy cyber‐physical systems subject to disturbances and deception attacks

AbstractThis paper is concerned with the fuzzy functional observer‐based sliding mode control for T‐S fuzzy cyber‐physical systems subject to disturbances and deception attacks. First, the deception attack is modeled by an exogenous system with state dependent nonlinearity. Second, a fuzzy learning functional observer with fuzzy logic systems learning unknown nonlinear function, whose parameters can be directly found, is designed to estimate the partly unavailable states, deception attacks and lumped disturbances simultaneously. Then, a fuzzy sliding mode control is designed to compensate the effect of deception attack and reject the disturbances. Furthermore, some sufficient conditions are given to guarantee that the closed‐loop fuzzy system is exponential convergent. Finally, a numerical example is given to illustrate the validity of the proposed method.

Event‐triggered adaptive neural‐network control of nonlinear MIMO systems

SummaryThis article investigates an adaptive neural networks (NNs) tracking control design issue for nonlinear multi‐input and multi‐output (MIMO) systems involving the sensor‐to‐controller event‐triggered mechanism (ETM). In the design, NNs are utilized to approximate the unknown nonlinear functions. A sensor‐to‐controller ETM is designed to save unnecessary transmission and communication resources. Subsequently, a first‐order filter technique is presented to solve the problem that the virtual control function is not differentiable. Furthermore, an event‐triggered adaptive NNs control strategy is presented by constructing Lyapunov functions and using adaptive backstepping recursive design. It is demonstrated that the presented scheme can ensure the whole closed‐loop signals are uniformly ultimately bounded without exhibiting the Zeno behavior. Finally, a numerical simulation example confirms the effectiveness of the presented adaptive event‐triggered control (ETC) approach.