Single-neuron Double Power Super Twisting Controller Design:
Double power super twisting controller
The Double Power Super Twisting Sliding Algorithm (DPSTA) has been developed to improve the robustness of the Super Twisting algorithm by exploiting the properties of the double power sliding law to reduce chattering while maintaining fast convergence. The main idea behind DPSTA is to replace the switching function of the super-twisting algorithm with that of the Double Power Sliding Mode Reaching Law (DPSMRL). The basic form of this algorithm is as follows:
{█(&(dx_1)/dt=-k_1 ϕ_1 (x_1 )+x_2+φ_1 (x_1,t)@&(dx_2)/dt=-k_2 ϕ_2 (x_1 )+φ_2 (x_2,t) )┤
with:
{■(ϕ_1 (x)=|x|^(1/2) sign⁡(x)+λ|x|^(3/2) sign⁡(x)@ϕ_2 (x)=ϕ_1^2 (x)^'=2ϕ_1 (x)⋅ϕ_1 (x)^'=sign⁡(x)+4λ|x|sign⁡(x)+3/2 λ^2 |x|^2 sign⁡(x))□( );λ≥0┤
where k1 and k2 are sliding mode gain coefficients. 
If for constants δ_1≥0 and ξ_1=1/(ϕ_1^' (x_1 ) ) :
|φ_1 |≤δ_1 |ζ_1 |" and " φ_2=0;∀t≥0
The system converges to the equilibrium point x(0,0) in finite time if the sliding mode gains k1 and k2 satisfy the conditions :
{■(k_1>2δ_1@k_2>k_1  (5δ_1 k_1+4δ_1^2)/2(k_1-2δ_1 ) )┤
The proof of finite-time stability and the estimation of convergence time are well presented in our previous paper. 
As detailed in [1], the super twisted Double power controller is expressed as follows:
u(t)=-k_1 ((|x〖(t)|〗^(1/2)+λ|x(t)|^(3/2) )sign⁡(x(t)))-∫k_2 ((1+4λ|x(t)|+3/2 λ^2 |x(t)|^2 )sign⁡(x(t)))
Although the controller ensures system stability as long as the gains satisfy the stability conditions (13), these conditions allow a wide range of values, making it difficult to calculate the specific gain values k1 and k2. This poses challenges in the design of optimal controller parameters.
To address this problem, we designed an adaptive sliding-mode controller based on a single neuron, capable of adjusting the gain of the sliding-mode controller online. This control strategy combines the advantages of the single neuron control strategy, with its high adaptability and simple algorithm, with DPSTA control theory. The detailed method for designing this control algorithm is described below.
Single-Neuron Double Power Super Twisting Controller design:
Figure 1 shows the overall block diagram of the single neuron dual power torsion super controller (SN_DPSTC) approach proposed in this study.
 

Since the single neuron controller operates using a numerical control, it is essential to discretise the Double Power Super Twisting Sliding Algorithm described in (15). We used the direct finite difference method for this discretisation. This approach allowed us to transform the continuous controller into an incremental controller, adapted for a numerical environment. 
The discrimination of u(t) is given by
u(k)=k_1^' |s(k) |^(1/2)  sgn⁡(s(k))+ k_2^' |s(k)|^(3/2) sign(s(k))+〖  k〗_3^' ∑_(i=0)^k▒sign(s(i)) +〖  k〗_4^' ∑_(i=0)^k▒〖|s(i) |^   sgn⁡(s(i)) 〗+〖  k〗_5^' ∑_(i=0)^k▒〖|s(i)|^2 sgn⁡(s(i))〗
with s(k) the sliding variable and:
{■(k_1^'=k_1^ @k_2^'=k_1^  λ@■(k_3^'=k_2@k_4^'=4k_2 λ@k_5^'=3/2 k_2 λ^2 ))┤
The incremental controller is given by:
■(Δu(k)=&u(k)-u(k-1) )
             =k_1^' (|s(k)|^(1/2) sgn⁡(s(k))-|s(k-1) |^(1/2)  sgn⁡(s(k-1)) )
              + k_2^' (|s(k)|^(3/2) sign(s(k))-|s(k-1)|^(3/2) sign(s(k-1)))
             +〖  k〗_3^' sign(s(k))  + k_4^' |s(k) |^   sgn⁡(s(k))   +    k_5^' |s(k)|^2 sign(s(k))    
we define the set variables as:
{■(x_1 (k)=|s(k)|^(1/2) sgn⁡(s(k))-|s(k-1) |^(1/2)  sgn⁡(s(k-1))@x_2 (k)=|s(k)|^(3/2) sign(s(k))-|s(k-1)|^(3/2) sign(s(k-1))@■(x_3 (k)=sign(s(k))@x_4 (k)=|s(k) |^   sgn⁡(s(k))@x_5 (k)=|s(k)|^2 sign(s(k)) ))┤
These state variables are the inputs to our single norone, as shown in Figure 1. For each input variable xi(k), we assign a weighting factor i(k). 
Based on the input variables and weighting factors, the neuron calculates the incremental controller:
■(Δu(k)=&K∑_(i=1)^5▒〖〖〖 ω〗_i (k)  x〗_i (k) 〗)
where K is the neuron gain coefficient K>0
Therefore:
{■(k_1^'=K〖 ω〗_1 (k)@k_2^'=K〖 ω〗_2 (k)@■(k_3^'=K〖 ω〗_3 (k)@k_4^'=K〖 ω〗_4 (k)@k_5^'=K〖 ω〗_5 (k) ))┤
Adaptive learning was selected for the adjustment of the weight coefficients because it allows the weights to be modified dynamically in response to variations in the error. This approach ensures faster and more stable model convergence.
we define the objective function 𝐽 as a measure of the squared error J=(s(k)^2)/2:
Δω_i (k)=-η_i  ∂J/(∂ω_i )
Using improved Hebb supervised learning, we obtain:
Δω_i (k)=η_i s(k)u(k-1)x_i (k-1)

To prevent uncontrolled weight gain, the output is obtained after normalizing the weighting factors as follows:
■(Δu(k)=&K∑_(i=1)^5▒〖ω_i^' (k)  x_i (k)  〗)
■(ω_i^' (k)=&(ω_i^  (k))/(∑_(i=1)^5▒〖ω_i^  (k)   〗))
where: 
■(ω_i^  (k)=&ω_i^  (k-1)-)η_i s(k)u(k-1)x_i (k-1)
the super twisted Double power controller will be expressed as:
u(k)=u(k-1)+K∑_(i=1)^5▒〖ω_i^' (k)  x_i (k)  〗
 it is correct