2.3.2. Controller Design

In this paper, the Gaussian function is used as the hidden layer of the RBF−NN, and the output of the neural network can be expressed as

The expression of Gaussian function is

In the formula, cj is the center of the Gaussian function; σj represents the width of the Gaussian function. wj is the weight from the hidden layer to the output layer; b is the bias of the output neuron.

In this paper, a neural network is used to approximate the frictional disturbance torque. Therefore, the frictional torque can be expressed by the output of the neural network

where, W* is the ideal network weight of the neural network, ε is the error of the ideal neural network approximation, and |ε|≤εmax.

Take the estimated value of W* as W^ then the estimated value of Tf(x) can be expressed as

Then the estimated error of the network weight is W˜=W*−W^.

Take x1=θ, let the ideal angular position signal be θd, then the angular position error is e=θd−θ, and the sliding mode function of the EMA system is

Derivation of sliding mode function of the EMA system

Equation (18) is the designed sliding mode control law

where, η≥D+pεmax.

Based on the above equations, Equation (17) can be further simplified