Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semilinear Parabolic PDE Systems

Abstract

This paper employs a Takagi-Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial L ∞ norm ∥ · ∥∞ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov-Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincaré-Wirtinger inequality in 1D spatial domain, as well as a vectorvalued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of ∥·∥∞, if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.

Multi-sensor information fusion detection system for fire robot through back propagation neural network

“…If too many hidden layer nodes are selected, the BPNN will spend too much time on learning, but it may not achieve the best learning results. If too few hidden layer codes are selected, the structure of the BPNN will be too simple to meet the requirements of better recognition, better robustness, and better anti-interference ability of the BPNN proposed in this study [ 16 ]. Therefore, based on previous studies, this study determines the number of hidden layers by the following equation.…”

Spatial domain decomposition approach to dynamic compensator design for linear space‐varying parabolic MIMO PDEs

“…The above results are only applicable for the singleinput-single-output case. For the case of parabolic MIMO PDEs, some results have been reported in [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] for feedback compensator design. In [13][14][15][16][17][18][19][20], static feedback compensator has been constructed by collocated pointwise observation [13][14][15] or collocated piecewise observation [16][17][18][19][20].…”

“…In [13][14][15][16][17][18][19][20], static feedback compensator has been constructed by collocated pointwise observation [13][14][15] or collocated piecewise observation [16][17][18][19][20]. The works [15,[21][22][23][24] have discussed static feedback compensator design for the case of non-collocated pointwise or piecewise observation. With the aid of observer-based control technique, observer-based dynamic compensators have been constructed in [25,26] for local piecewise control of parabolic MIMO PDEs with non-collocated pointwise observation and non-collocated piecewise observation, respectively.…”

“…With the aid of observer-based control technique, observer-based dynamic compensators have been constructed in [25,26] for local piecewise control of parabolic MIMO PDEs with non-collocated pointwise observation and non-collocated piecewise observation, respectively. Note that the results in [13][14][15][16][17][18][19][20][21][22][23][24][25][26] are developed under an assumption that the dimension of control input is the same as that of observation output. More recently, the observer-based dynamic feedback compensator design reported in [25,26] has been extended in [27][28][29] for a general case when the dimension of control input and observation output is different.…”

Boundary Control of a Vibrating String Subject to Input Saturation and Output Constraint

“…Boundary control has many advantages such as the nonintrusive sensing and actuation [1]- [4], the circumvention of the spillover effect generated from reduced-order approaches [5]- [8], and so on, thus it has always been viewed as an effective and practical solution to control infinite dimensional systems [9]- [21]. In recent decades, significant development of control of infinite dimensional flexible string systems has been achieved [22]- [31].…”