Abstract
In this study, we focus on designing a robust piecewise adaptive controller to globally asymptotically stabilize a semilinear parabolic distributed parameter systems (DPSs) with external disturbance, whose nonlinearities are bounded by unknown functions. Firstly, a robust piecewise adaptive control is designed against the unknown nonlinearity and the external disturbance. Then, by constructing an appropriate Lyapunov–Krasovskii functional candidate (LKFC) and using the Wiritinger’s inequality and a variant of the Agmon’s inequality, it is shown that the proposed robust piecewise adaptive controller not only ensures the globally asymptotic stability of the closed-loop system, but also guarantees a given performance. Finally, two simulation examples are given to verify the validity of the design method.
Highlights
In actual engineering applications, most physical models are widely distributed in space, continuously changing in time, and have the characteristics of spatiotemporal dynamics
They cannot be modeled by ordinary differential equations (ODEs), which are precisely determined by partial differential equations (PDEs)
At first, inspired by the fact that the main dynamics of the parabolic PDEs can be roughly described by a low-dimensional ODE systems, for linear parabolic PDEs, a design method of predictive boundary control and a design method of sampling data boundary control were introduced in [11] and [8], respectively
Summary
Most physical models are widely distributed in space, continuously changing in time, and have the characteristics of spatiotemporal dynamics. For intradomain control researches, the main idea in literatures [31,32,33] was that the order of the systems was reduced by using the Galerkin method to obtain the lowdimensional nonlinear ODEs firstly, and the appropriate controller was designed for the obtained low-dimensional nonlinear ODEs by using the existing fuzzy control technology. The existing literatures on the control problems of semilinear parabolic DPSs based on fuzzy PDEs models are all local stability results, which is one of the motivations of this study. 2. A robust piecewise adaptive controller designed in this study guarantees the globally asymptotic stability of the uncertain semilinear parabolic DPSs, which overcomes the defects of the existing semiglobal results.
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