We first performed the simulation of the pendulum controled by the P(I)D law. The error, rate and control signals are then pictured below:

![0_image_0.png](0_image_0.png)

$$\begin{array}{l l}{{\mathrm{if}}}&{{\omega>0}}\\ {{\mathrm{if}}}&{{\omega<0}}\end{array}$$

![0_image_1.png](0_image_1.png)

![0_image_2.png](0_image_2.png)

If u,θ,ω are positive (resp. negative), then U,Ξ,Ω are given PST (
NGT) mnesor. We recall that multiplying two mnesors means choosing the more constrained of both (
PST × NGT λ = PST and 
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