Sliding Mode Control of FMII Systems: Handling Rice Fading Issues Under 2-D Frame
针对二维FMII系统在莱斯衰落信道下的滑模控制问题，构建了适用于二维系统的L阶莱斯衰落模型，设计了滑模函数和控制律，实现了闭环系统均方最终一致有界。
The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> th order Rice model can effectively describe imperfect transmission, including amplitude attenuation, time delays, and stochastic disturbance, and is a popular channel fading model in one-dimensional (1-D) systems. However, due to the state evolution feature of two-dimensional (2-D) systems, the previous state signals of 2-D systems are not as directly determinable as those of 1-D systems such that the construction of the conventional and obvious <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> th order Rice model become difficulty for 2-D systems. Motivated by these, this work will investigate the sliding mode control (SMC) problem of the 2-D Fornasini–Marchesini (FMII) system under Rice fading channels. First, the previous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -step state signals are defined according to the evolution feature of the FMII model. Afterwards, a new applicable <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> th-order Rice fading model for 2-D FMII systems is constructed, which not only has 2-D structural characteristics but also has similar properties to 1-D cases. A 2-D-type sliding function involving the predefined previous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> -step state signals and channel attenuation coefficients is designed, which exhibits a similar structural feature to the FMII model. A feasible SMC law is designed via the available fading states to achieve the ultimately uniformly boundedness in the mean square of the closed-loop system and the reachability of the sliding domain around the specified sliding surface. Furthermore, an optimization-solving algorithm is introduced to minimize the convergent bound. Finally, an example is applied to demonstrated the proposed results.