1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

Constrained Model Predictive Control

In this paper, an adaptive fuzzy controller is constructed to address the finite-time tracking control problem for a class of strict-feedback nonlinear systems, where the full state constraints are strictly required in the systems. Backstepping design with a tan-type barrier Lyapunov function is proposed. Meanwhile, fuzzy logic systems are used to approximate the unknown nonlinear functions. The addressed control scheme guarantees that the output is followed the reference signals within a bounded error, and all the signals in the closed-loop system are bounded. The simulation results demonstrate the validity of the proposed method.

Finite-Time Adaptive Fuzzy Control for Nonlinear Systems With Full State Constraints

This paper focuses on the practical output tracking control for a category of high-order uncertain nonlinear systems with full-state constraints. A high-order tan-type barrier Lyapunov function (BLF) is constructed to handle the full-state constraints of the control systems. By the BLF and combining a backstepping design technique, an adding a power integrator, and a fuzzy control, the proposed approach can control high-order uncertain nonlinear system with full-state constraints. A novel controller is designed to ensure that the tracking errors approach to an arbitrarily small neighborhood of zero, and the constraints on system states are not violated. The numerical example demonstrates effectiveness of the proposed control method.

Adaptive Fuzzy Control With High-Order Barrier Lyapunov Functions for High-Order Uncertain Nonlinear Systems With Full-State Constraints

In this brief, the practical finite-time consensus (FTC) problem is investigated for the second-order heterogeneous switched nonlinear multi-agent systems (MASs), where the subsystems and the switching signal for each agent are different. Mainly due to that agents’ dynamics are switched and the unknown nonlinearities in the systems are more general, the practical FTC problem of the MASs is rather difficult to be solved by existing methods. As such, a new protocol design framework for the FTC problem is developed. Then, a novel adaptive protocol is proposed for the switched nonlinear MASs based on the developed design framework and the neural network method. The sufficient conditions for the practical FTC of nonlinear MASs under arbitrary switching are given. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed control scheme.

Finite-Time Consensus of Second-Order Switched Nonlinear Multi-Agent Systems

This studies an adaptive control problem of pure-feedback non-linear systems with full state constraints. The mean value theorem is employed to deal with unknown non-linearities. A novel backstepping design is constructed via barrier Lyapunov function (BLF) combined with dynamic surface control (DSC). The BLF guarantees the full state constraints are not violated and all the closed-loop signals remain bounded. DSC solves the problems of restrictions on high order differentiability of stabilising functions and avoiding the complexity that arises due to the explosion of terms in backstepping design. It is shown that all the signals in the closed-loop system are ultimately bounded and the tracking error converges to an adjustable neighbourhood of the origin while the full state constraints remain unchanged. The performance of the BLF-based DSC is illustrated with two simulation examples.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-cta.2016.0333

Barrier Lyapunov functions-based dynamic surface control for pure-feedback systems with full state constraints

In this paper, an adaptive finite-time controller is considered for a class of strict-feedback nonlinear systems with parametric uncertainties and full state constraints. Novel tan-type barrier Lya...

Finite-time tracking control for strict-feedback nonlinear systems with full state constraints

Finite‐time stabilization for a class of switched stochastic nonlinear systems with dead‐zone input nonlinearities

This paper studies the problem of controller design for pure-feedback nonlinear systems with asymmetric time-varying full state constraints. The mean value theorem is employed to transform a pure-feedback system into a strict-feedback structure with non-affine terms. For the transformed system, a time-varying asymmetric Barrier Lyapunov Function (ABLF) with the error variables is employed to ensure the time-varying constraints satisfaction. By allowing the barriers to vary with the desired trajectory in time, the initial condition requirements are relaxed efficiently. The presented control scheme can guarantee that all signals in the closed-loop system are ultimately bounded. It is also proved that the tracking error converges to an adjustable neighborhood of the origin even in the presence of disturbance. The performance of the ABLF-based control are illustrated through two examples.

Barrier Lyapunov functions-based adaptive control for nonlinear pure-feedback systems with time-varying full state constraints

This article further discusses the problem of output-feedback regulation for more general stochastic nonlinear systems with stochastic integral input-to-state stable inverse dynamics, and focuses on solving the important and unsolved problem proposed in Yu and Xie (Yu, X., and Xie, X.J. (2010), ‘Output Feedback Regulation of Stochastic Nonlinear Systems with Stochastic iISS Inverse Dynamics’, IEEE Transactions on Automatic Control, 55, 304–320): How to weaken the conditions on nonlinearities in drift and diffusion vector fields? Under the weaker conditions, how to make full use of the known information of stochastic nonlinear systems to design an adaptive output-feedback controller such that all the closed-loop signals are almost surely bounded and the output is driven to zero almost surely?

Yuqiang Wu

Papers

Barrier Lyapunov functions-based dynamic surface control for pure-feedback systems with full state constraints

Finite-time tracking control for strict-feedback nonlinear systems with full state constraints

Finite‐time stabilization for a class of switched stochastic nonlinear systems with dead‐zone input nonlinearities

Barrier Lyapunov functions-based adaptive control for nonlinear pure-feedback systems with time-varying full state constraints

Further results on output-feedback regulation of stochastic nonlinear systems with SiISS inverse dynamics