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In this article, an adaptive quantized neural backstepping strategy is investigated for a class of nonlinear systems with input unmodeled dynamics and output constraints based on the small-gain method. A challenge lies in the considered input-quantized actuator possessing both unknown control gain and input unmodeled dynamics, and the application of the small-gain theorem when the system possesses input unmodeled dynamics and the output constraints. By the coordinate transformation of the state variables, the input unmodeled dynamics subsystem is transformed into a suitable form for applying the small-gain theorem. By a logarithmic one to one mapping, the time-varying output constraints are tackled. With these methods, the stability proof based on the small-gain theorem is completed. It is shown that all the signals are bounded, and the output signal is constrained within the preset range.

Summary In this article, the issue of adaptive finite‐time dynamic surface control (DSC) is discussed for a class of parameterized nonlinear systems with full state constraints. Using the property ...

In this study, the recursive three-step filter method introduced by Gillijns and Moor (2007) is proved strictly to be equivalent to the augment Kalman filter (AKF) when the covariance of process noise about the input approaches infinity. This result indicates that if the information about the exogenous input is unknown, the AKF can also be used to simultaneously obtain the state and exogenous input estimation. Numerical examples illustrate the authors' main results.

Summary In this paper, the issue of adaptive neural control is discussed for a class of stochastic nonstrict‐feedback constrained nonlinear systems with input and state unmodeled dynamics. A dynami...

In this paper, an adaptive quantized fuzzy dynamic surface control (DSC) stratergy is investigated for a class of nonlinear systems with input unmodeled dynamics and output constraints. A challenge lies in that the input-quantized actuator is considered to possess both unknown control gain and nonlinear input unmodeled dynamics. By quantized controller design and DSC technique, combining normalized signal, integral Lyapunov functions, Nussbaum functions and the adaptive laws, the obstacle caused by quantization and multiple uncertainties is effectively overcome. The designed novel quantizer has the advantages of both the existing uniform quantizer and hysteresis quantizer, which can avoid the chattering and reduce the quantization error no matter the control volume is large or small. By defining a group of transformation based on a logarithmic one to one mapping, time-varying output constraints is satisfied. It is shown that all the signals are bounded, and the output signal is constrained within the preset range.

This study proposes an adaptive quantised fuzzy control strategy for a class of non-linear systems with time-varying delay, state constraints, and input unmodelled dynamics. Design difficulties focus on the fact that the input-quantised actuator possesses both unknown control gain and non-linear input unmodelled dynamics, and all the states are required to satisfy state constraints. To remove these obstacles, integral barrier Lyapunov functions combined with Nussbaum functions are designed to deal with the unknown control gains and state constraints, and a quantised controller combined with the normalised signal is developed to tackle the quantised input and input unmodelled dynamics. To avoid the chattering and reduce the quantisation error in a wide scope of the control volume, a novel logarithmic uniform hysteresis quantiser is employed, which has both advantages of the existing uniform quantiser and hysteresis quantiser. Two simulation examples of practical control systems are conducted to demonstrate the effectiveness of the proposed control protocol.

Summary In this paper, we are concerned with the problem of adaptive output‐feedback tracking control for nonlinear systems with input quantization, unmodeled dynamics, and output constraints. A no...

Summary This paper focuses on the problem of adaptive control for a class of pure‐feedback nonlinear systems with full‐state time‐varying constraints and unmodeled dynamics. By introducing a one‐to...

Summary Stochastic adaptive dynamic surface control is presented for a class of uncertain multiple‐input–multiple‐output (MIMO) nonlinear systems with unmodeled dynamics and full state constraints ...

Summary This paper proposes an adaptive neural‐network control design for a class of output‐feedback nonlinear systems with input delay and unmodeled dynamics under the condition of an output const...

In this paper, the augment Kalman filter (AKF) approach is discussed under the infinity condition. It is proved that the AKF in infinity condition is equivalent to the minimum-variance unbiased (MV...

This paper solves the problem of adaptive neural dynamic surface control (DSC) for a class of full state constrained stochastic nonlinear systems with unmodeled dynamics. The concept of the state constraints in probability is first proposed and applied to the stability analysis of the system. The full state constrained stochastic nonlinear system is transformed to the system without state constraints through a nonlinear mapping. The unmodeled dynamics is dealt with by introducing a dynamic signal and the adaptive neural dynamic surface control method is explored for the transformed system. It is proved that all signals of the closed-loop system are bounded in probability and the error signals are semi-globally uniformly ultimately bounded(SGUUB) in mean square or the sense of four-moment. At the same time, the full state constraints are not violated in probability. The validity of the proposed control scheme is demonstrated through the simulation examples.

This paper is concerned with the adaptive control problem of a class of output feedback nonlinear systems with unmodeled dynamics and output constraint. Two dynamic surface control design approaches based on integral barrier Lyapunov function are proposed to design controller ensuring both desired tracking performance and constraint satisfaction. The radial basis function neural networks are utilized to approximate unknown nonlinear continuous functions. K-filters and dynamic signal are introduced to estimate the unmeasured states and deal with the dynamic uncertainties, respectively. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded, while the output constraint is never violated. Simulation results demonstrate the effectiveness of the proposed approaches.

In this paper, adaptive neural dynamic surface control (DSC) is developed using radial basis function neural networks (NNs) for a class of pure-feedback nonlinear systems with full state constraints and dynamic uncertainties. Based on a one-to-one nonlinear mapping, the pure-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. The dynamic uncertainties are dealt with using a dynamic signal. Using modified DSC and mean value theorem as well as Nussbaum function, two adaptive NN control schemes are proposed based on the transformed system. The designed control strategy removes the conditions that the upper bound of the control gain is known, and the lower bounds and upper bounds of the virtual control coefficients are known. It is shown that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the full state constraints are not violated. Two numerical examples are provided to illustrate the effectiveness of the proposed approach.

In this paper, the problem of adaptive neural network (NN) dynamic surface control (DSC) is discussed for a class of strict-feedback nonlinear systems with full state constraints and unmodeled dynamics. By introducing a one to one nonlinear mapping, the strict-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. Radial basis function (RBF) neural networks (NNs) are used to approximate unknown nonlinear continuous functions. Unmodeled dynamics is dealt with by introducing a dynamical signal. Using modified DSC and introducing integral-type Lyapunov function, adaptive NN DSC is developed. Using Young’s inequality, only one parameter is adjusted at each recursive step in the design. It is shown that all the signals in the closed-loop system are semi-global uniform ultimate boundedness (SGUUB), and the full state constraints are not violated. Simulation results are provided to verify the effectiveness of the proposed approach.

Summary In this paper, the problem of adaptive neural control is discussed for a class of strict-feedback time-varying delays nonlinear systems with full-state constraints and unmodeled dynamics, as well as distributed time-varying delays. The considered nonlinear system with full-state constraints is transformed into a nonlinear system without state constraints by introducing a one-to-one asymmetric nonlinear mapping. Based on modified backstepping design and using radial basis function neural networks to approximate the unknown smooth nonlinear function and using a dynamic signal to handle dynamic uncertainties, a novel adaptive backstepping control is developed for the transformed system without state constraints. The uncertain terms produced by state time delays and distributed time delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. All signals in the closed-loop system are proved to be semiglobally uniformly ultimately bounded. A numerical example is provided to illustrate the effectiveness of the proposed design scheme.

An adaptive dynamic surface control (DSC) method is proposed for a class of stochastic non-linear systems with input and state unmodelled dynamics in this paper. A normalisation signal is introduced to restrict the input unmodelled dynamics. By introducing a non-linear transformation, modified DSC is extended and applied to a class of more general stochastic non-linear systems. By theoretical analysis, it is shown that all the signals in the closed-loop system are bounded in probability. Two simulation examples further demonstrate the effectiveness of the proposed control approach.

In this technical note, $H_{\infty}$ control problem of continuous Markovian jump systems is investigated. A linear feedback control scheme, combined with a state observer design, is proposed in the form of linear matrix inequalities, which can ensure the systems' mean-square stability with $H_{\infty}$ performance. Then, a multi-step state transition conditional probability function is introduced for the continuous Markovian process, which is used to define the system's mean-square stability probability. Furthermore, the formulas for calculating the mean-square stability probability are derived for situations where the control force may not be strong enough to ensure the full stability. Simulation results are presented to show the effectiveness of the theoretical results.

This paper addresses the sliding mode control problem for uncertain MIMO linear Markovian jump systems. Firstly, by using the linear matrix inequality approach, sufficient conditions are proposed to guarantee the stochastically asymptotical stability of the system on the sliding surfaces. Secondly, an equivalent control based sliding mode control is proposed, such that the closed-loop system can be driven onto the desired sliding surfaces in a finite time. Finally, combining with multi-step state transition probability, the reaching and sliding probabilities are derived for situations where the control force may not be strong enough to ensure the fully asymptotical stability. Simulation results are presented to illustrate the effectiveness of the proposed design method.

In this paper, an adaptive neural output feedback control scheme is investigated for a class of stochastic nonlinear systems with unmeasured states and four kinds of uncertainties including uncertain nonlinear function, dynamic disturbance, input unmodeled dynamics, and stochastic inverse dynamics. The unmeasured states are estimated by K-filters, and stochastic inverse dynamics is dealt with by constructing a changing supply function. The considered input unmodeled dynamic subsystem possesses nonlinear feature, and a dynamic normalization signal is introduced to counteract the unstable effect produced by the input unmodeled dynamics. Combining dynamic surface control technique with stochastic input-to-state stability, small-gain condition, and Chebyshev's inequality, the designed robust adaptive controller can guarantee that all the 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 verify the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, centralized and decentralized adaptive fuzzy output feedback control schemes are investigated for a class of stochastic nonlinear interconnected large-scale systems with dynamic uncertainties and unmeasured states. Fuzzy systems are used to approximate the unknown nonlinear functions. Decentralized 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 (DSC) technique with backstepping design, the condition in which the approximation errors are assumed to be bounded is avoided. Using the defined compact set in the stability analysis, the unknown smooth interconnections and black box functions are effectively dealt with. Using It o ? formula and Chebyshev's inequality, it is shown that all the 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 demonstrate the effectiveness of the proposed approach.

This paper explores the probability problems on terminal sliding mode control of second-order nonlinear continuous Markovian jump systems. An equivalent control based terminal sliding mode control scheme is proposed that can guarantee the systems' finite time mean-square stability. By introducing a multi-step transition conditional probability, the novel reaching and sliding probabilities are derived for the situations where the control force is not strong enough. This indicates that the reaching and sliding probabilities are both monotonically bounded non-decreasing non-negative piecewise right-continuous functions of the control parameter. A numerical example is given to show the feasibility of the theoretical results.

This paper is concerned with the centralized and decentralized adaptive output feedback control problem for a class of uncertain interconnected nonlinear systems with unmodeled dynamics and unmeasured states. The known requirement of the upper bounds of dynamic disturbances and coupling terms in the considered system is avoided. Decentralized observers are constructed based on the K-filters to estimate the unmeasured states of each subsystem. In the design of adaptive output feedback dynamic surface control, the unknown nonlinear functions are directly approximated by neural networks, while the coupling terms being lumped together with those unknown continuous functions produced in deduction are approximated as a whole. Using the defined compact set in the stability analysis, some unknown nonlinear continuous functions are dealt with effectively. It is proved that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Simulation results are provided to illustrate the effectiveness of the proposed approach.

In this study, a robust adaptive control is proposed for a class of strict-feedback state delayed non-linear systems with unmodelled dynamics and distributed time-varying delays using radial basis function neural networks. Dynamic uncertainties are dealt with using separation technique and introducing a dynamic signal. The terms including state time-varying delay and distributed time-varying delay uncertainties are compensated for by constructing appropriate Lyapunov–Krasovskii functionals. Using Young's inequality, only one learning parameter need to be tuned online at each step of recursion. It is proved that the proposed design method is able to guarantee semi-global uniform ultimate boundedness of all signals in the closed-loop system. Simulation results demonstrate the effectiveness of the proposed approach.

In this paper, a novel adaptive control scheme is investigated based on the backstepping design for a class of stochastic nonlinear systems with unmodeled dynamics and time-varying state delays. The radial basis function neural networks are used to approximate the unknown nonlinear functions obtained by using Ito differential formula and Young׳s inequality. The unknown time-varying delays and the unmodeled dynamics are dealt with by constructing appropriate Lyapunov–Krasovskii functions and introducing available dynamic signal. It is proved that all signals in the closed-loop system are bounded in probability and the error signals are semi-globally uniformly ultimately bounded (SGUUB) in mean square or the sense of four-moment. Simulation results illustrate the effectiveness of the proposed design.

Summary In 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 o 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 & Sons, Ltd.

In this paper, adaptive neural tracking control is proposed based on radial basis function neural networks (RBFNNs) for a class of multi-input multi-output (MIMO) nonlinear systems with completely unknown control directions, unknown dynamic disturbances, unmodeled dynamics, and uncertainties with time-varying delay. Using the Nussbaum function properties, the unknown control directions are dealt with. By constructing appropriate Lyapunov-Krasovskii functionals, the unknown upper bound functions of the time-varying delay uncertainties are compensated. The proposed control scheme does not need to calculate the integral of the delayed state functions. Using Young's inequality and RBFNNs, the assumption of unmodeled dynamics is relaxed. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded.

In this article, two robust adaptive control schemes are investigated for a class of completely non-affine pure-feedback non-linear systems with input non-linearity and perturbed uncertainties using radial basis function neural networks (RBFNNs). Based on the dynamic surface control (DSC) technique and using the quadratic Lyapunov function, the explosion of complexity in the traditional backstepping design is avoided when the gain signs are known. In addition, the unknown virtual gain signs are dealt with using the Nussbaum functions. Using the mean value theorem and Young's inequality, only one learning parameter needs to be tuned online at each step of recursion. It is proved that the proposed design method is able to guarantee semi-global uniform ultimate boundedness (SGUUB) of all signals in the closed-loop system. Simulation results verify the effectiveness of the proposed approach.