The power term 
, where 
, in the sliding surface equation (Eq. 12), 
, can lead to issues, and it may even be incorrect to follow the paper because it produces complex numbers when the error becomes negative.
, where , in the sliding surface equation (Eq. 12), , can lead to issues, and it may even be incorrect to follow the paper because it produces complex numbers when the error becomes negative.err = -1;
psi = 0.9;
pow = err^(psi)
While you can address the complex-valued problem by using the absolute value function, it no longer yields the correct control action. I understand this may seem absurd, as both the absolute value and surd functions are used together.
pow = abs(err)^(psi)
To rectify the sign issue, a radical change is needed for the term by multiplying it with the sign function. I refer to this as the Radical Sign function.
pow = abs(err)^(psi)*sign(err)
When it comes to the singularity issue, I believe it might be caused by the term 
when the error is zero, but the rate of change of error is non-zero. As demonstrated below, this term becomes infinitely large.
when the error is zero, but the rate of change of error is non-zero. As demonstrated below, this term becomes infinitely large.
err = 0;
pow = err^(psi - 1)
Given that you have opted for 
, which essentially makes the Radical Sign function nearly linear, it could be more advantageous to utilize the linear term 
directly. By doing so, you can evaluate whether the Simulink model runs smoothly without any complications when employing a linear sliding surface 
. This approach serves as your baseline for Higher-order Linear Sliding Mode Control.
, which essentially makes the Radical Sign function nearly linear, it could be more advantageous to utilize the linear term directly. By doing so, you can evaluate whether the Simulink model runs smoothly without any complications when employing a linear sliding surface . This approach serves as your baseline for Higher-order Linear Sliding Mode Control.To tap into the nonlinear capabilities of the Terminal Sliding Mode approach, it is recommended to select a value of ψ in proximity to zero rather than 0.9.
y = @(x, psi) abs(x).^(psi).*sign(x);
x = linspace(-1, 1, 20001); % range -1 ≤ x ≤ 1
plot(x, y(x, 0.9), 'linewidth', 1.5), hold on % when psi = 0.9
plot(x, y(x, 0.2), 'linewidth', 1.5), grid on % when psi = 0.2
title('Radical Sign Function')
xlabel('x'), ylabel('y')
legend('\psi=0.9', '\psi=0.2', 'fontsize', 16, 'location', 'NorthWest')
