Prescribed-Time Fuzzy Control for Stochastic Nonlinear Systems with Hysteresis and Dynamic Uncertainties

This article studies the problem of performance‐guaranteed adaptive fuzzy optimal compensator control for stochastic affine nonlinear systems with dead‐zone and unknown nonlinear dynamics. First, by using the error transformation techniques, the original system is transformed into an equivalent unconstraint system. Then, a feedforward fuzzy compensator is constructed to offset the influence raised by unknown dead‐zone. By designing identifier‐critic‐actor construction reinforcement learning, an adaptive fuzzy optimal performance constraint compensate control algorithm is presented. The developed control scheme can ensure that all signals of the controlled system are uniformly ultimately bounded, and the output can track the reference signal with a prescribed accuracy. Finally, simulation results are given to illustrate the effectiveness of the developed control algorithm and theorem.

This article takes a fast finite-time control of stochastic nonlinear systems into account. The presence of unknown stochastic disturbance terms makes the traditional fast finite-time control approaches unavailable. To deal with this difficulty, by establishing an auxiliary function and using Jensen's inequality, in Lemma 6, a new criterion of fast finite-time stability is first established for the uncertain stochastic system. Based on the approximation ability of neural networks (NNs), an innovative fast finite-time strategy is put forward for stochastic nonlinear systems. Furthermore, by adopting the presented fast finite-time stability criterion, the stability of the stochastic systems is confirmed. Finally, two simulations are implemented to validate the feasibility of the presented NN control strategy.

In this article, the problem of event‐triggered adaptive asymptotic tracking control (ATC) for stochastic nonlinear systems with unknown control directions (UCDs) and full state constraints is concerned. It must be said that the controller design and system analysis is more complex and difficult since the existence of stochastic disturbances, UCDs and full state constraints simultaneously. By introducing the lower bound of the UCDs into the barrier Lyapunov functions, an event‐triggered adaptive MTN ATC scheme is proposed based on the boundary estimation method and a new event‐triggered control (ETC) strategy, which can achieve satisfactory asymptotic tracking performance and control performance of the system, while reduce the occupation of network resources. The simulation results not only verify the effectiveness of the proposed control scheme, but also present different tracking performances between three ETC strategies for comparison, further confirming the superiority of the proposed ETC strategy in achieving asymptotic tracking performance.

This paper is devoted to the adaptive finite-time prescribed performance control (FTPPC) for stochastic nonlinear systems with unknown virtual control coefficients (UVCCs), which are functions of system states. To eliminate the condition that the initial value of the performance function (PF) is bigger than the initial tracking error, a novel smooth shifting function, for the first time, is defined and embedded in FTPPC for the tracking error. New control laws are firstly proposed and employed to deal with UVCCs in the controller design, which are different from the Nussbaum gain technology in the existing papers. An adaptive FTPPC strategy is designed so that all of the signals in the closed-loop system are bounded in probability and the tracking error is restrained in a fixed bound after a preset finite time,even that the PF is smaller than the tracking error at the initial time instant.

On controllability of nonlinear stochastic systems

This paper investigates the problem of state feedback control for uncertain nonlinear stochastic systems with timedelay. A state feedback controller is designed so that the closed-loop system is exponentially stable in mean square for all admissible uncertainties. Sufficient conditions are obtained in guarantee that the involved system has robust stabilization in terms of matrix inequalities. A regrouped LMI method is adopted to solve these matrix inequalities. A numerical example is given to illustrate the effectiveness of the given method.

In this article, the problem of adaptive multi-dimensional Taylor network control for strict-feedback nonlinear stochastic systems with time-delay is investigated. To overcome the control degradation resulting from the delay terms, the appropriate integral-type Lyapunov–Krasovskii functions are introduced. A novel adaptive multi-dimensional Taylor network control scheme is provided via backstepping technique. The proposed adaptive multi-dimensional Taylor network controller can ensure that all signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood of the origin. Three simulation examples are given to demonstrate the effectiveness of the proposed control scheme.

This paper extends H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> control theory to a more general class of stochastic nonlinear systems with dynamic uncertainties. From the view of stochastic dissipation, we investigate the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> robust performance rule design for this general class of systems. Under some mild assumptions on the unmodeled dynamics, we establish the relationship between the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain of the dynamic uncertain stochastic nonlinear systems and the solution to a certain HJI inequality. Furthermore, for some systems satisfying some proper matching conditions, we discuss the methods of acquiring the control laws without solving HJI inequality.

Robust adaptive fuzzy output feedback control for stochastic nonlinear systems with unknown control direction

In this paper, a robust adaptive neural network (NN) backstepping output feedback control approach is proposed for a class of uncertain stochastic nonlinear systems with unknown nonlinear functions, unmodeled dynamics, dynamical uncertainties and without requiring the measurements of the states. The NNs are used to approximate the unknown nonlinear functions, and a filter observer is designed for estimating the unmeasured states. To solve the problem of the dynamical uncertainties, the changing supply function is incorporated into the backstepping recursive design technique, and a new robust adaptive NN output feedback control approach is constructed. It is mathematically proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing design parameters appropriately. The simulation example and comparison results further justify the effectiveness of the proposed approach.

The optimal control of nonlinear stochastic systems is considered in this paper. The central role played by the Fokker-Planck-Kolmogorov equation in the stochastic control problem is shown under the assumption of asymptotic stability. A computational approach for the problem is devised based on policy iteration/ successive approximations, and a finite dimensional approximation of the control parametrized diusion operator, i.e., the controlled Fokker-Planck operator. Several numerical examples are provided to show the ecacy of the proposed computational methodology. require the study of the time evolution of the Probability Density Function (PDF), p(t,x), corresponding to the state, x, of the relevant dynamic system. The pdf is given by the solution to the Fokker-Planck- Kolmogorov equation (FPE), which is a PDE in the pdf of the system, defined by the underlying dynamical system's parameters. In this paper, approximate solutions of the FPE are considered, and subsequently leveraged for the design of controllers for nonlinear stochastic dynamical systems. In the past few years, the authors have developed a generalized multi-resolution meshless FEM methodology, partition of unity FEM (PUFEM), utilizing the recently developed GLOMAP (Global Local Orthogonal MAPpings) methodology to provide the partition of unity functions 6,7 and the orthogonal local basis functions, for the solution of the FPE. The PUFEM is a Galerkin projection method and the solution is characterized in terms of an finite dimensional representation of the Fokker-Planck operator underlying the problem. The methodology is also highly amenable to parallelization. 8-12 Though the FPE is invaluable in quantifying the uncertainty evolution through nonlinear systems, per- haps its greatest benefit may be in the stochastic analysis, design and control of nonlinear systems. In the context of nonlinear stochastic control, Markov Decision Processes have long been one of the most widely used methods for discrete time stochastic control. However, the Dynamic Programming equations under- lying the MDPs suer from the curse of dimensionality. 13-15 Various approximate Dynamic Programming (ADP) methods have been proposed in the past several years for overcoming the curse of dimensionality, 15-19 and broadly can be categorized under the category of functional reinforcement learning. These methods are essentially model free method of approximating the optimal control policy in stochastic optimal control prob- lems. These methods generally fall under the category of value function approximation methods, 15 policy gradient/ approximation methods 17,18 and actor-critic methods. 16,19 These methods attempt to reduce the dimensionality of the DP problem through a compact parametrization of the value functions (with respect to

SummaryIn this paper, adaptive output feedback tracking control is developed for a class of stochastic nonlinear systems with dynamic uncertainties and unmeasured states. Neural networks are used to approximate the unknown nonlinear functions. K‐filters are designed to estimate the unmeasured states. An available dynamic signal is introduced to dominate the unmodeled dynamics. By combining dynamic surface control technique with backstepping, the condition in which the approximation error is assumed to be bounded is avoided. Using It ô formula and Chebyshev's inequality, it is shown that all signals in the closed‐loop system are bounded in probability, and the error signals are semi‐globally uniformly ultimately bounded in mean square or the sense of four‐moment. Simulation results are provided to illustrate the effectiveness of the proposed approach. Copyright © 2014 John Wiley &amp; Sons, Ltd.

This article investigates the issue of adaptive fixed-time control for stochastic nonlinear systems with unknown covariance noise. First, we derive a significant corollary regarding fixed-time stability in stochastic nonlinear systems. This corollary provides the minimum bound for the settling time, which is a crucial parameter in assessing the system’s performance and stability. The adaptive control strategy is used to compensate the detrimental effect caused by unknown covariance noise. Then, in order to ensure the fixed-time stability of the closed-loop system, an adaptive state-feedback controller is constructed by utilizing backstepping method and the adding a power integrator technique. Moreover, the settling time estimate provided by the proposed control law is less conservative, meaning it allows for a more accurate prediction of the actual settling time compared to more traditional approaches. Finally, the simulation results are presented to verify the correctness of our theoretical results proposed in this paper.

SummaryIn this paper, an adaptive output feedback dynamic surface control (DSC) strategy is proposed for strict‐feedback stochastic nonlinear systems with input quantization, prescribed performance and dynamic uncertainties. A new quantizer is used to process the input signal, which can avoid the chattering of the quantization signal and keep the upper bound of the quantization error constant. Radial basis functions are used to approximate unknown smooth functions, unmodeled dynamics are processed by dynamic signals, and unmeasurable states are estimated by high gain observer. Hyperbolic tangent functions are employed to handle prescribed performance. The second order command filter is used to replace the first order filter used in general DSC, and the compensation term is added in each step of DSC. By the Lyapunov stability analysis, all signals in the controlled system are semi‐globally uniformly ultimately bounded (SGUUB) in probability. Two examples further prove that the control scheme designed in this paper is reasonable and effective.

Predefined time fuzzy adaptive control for stochastic nonlinear systems with limited time interval output constraints