Nonlinear Decoupling Control With PI $^{\lambda}$ D $^{\mu}$ Neural Network for MIMO Systems

In this brief, a fractional order proportional-integral-differential neural network (PIDNN) controller based on the beetle swarm optimization algorithm (BSO-PI
$^{\lambda}$
D
$^{\mu}$
NN) is proposed for multi-input multi-output (MIMO) systems with strong coupling. First, the fractional order PID operator is introduced to the hidden layer neurons of the neural network, where long memory characteristics of the fractional order neurons can improve the control accuracy and convergence speed. Second, a sufficient condition on the learning rate is established to ensure the stability of the controller by the Lyapunov theory. Third, the PI
$^{\lambda}$
D
$^{\mu}$
NN is initialized by the BSO algorithm to prevent weights from falling into local optima. The proposed fractional order PIDNN controller can eliminate the coupling between variables and achieve desirable control performance without specific system models. To the authors’ best knowledge, this is the first work that the fractional order PI
$^{\lambda}$
D
$^{\mu}$
neurons are employed in neural network. Two simulation examples verify the effectiveness and superiority of the proposed controller.