A First Course in FUZZY and NEURAL CONTROL

22 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

Figure 2.4. Free-body diagrams of the cart and the pendulum

By summing the forces along the horizontal for the pendulum, we get the fol-
lowing equation of motion:

H=mx®+mL®θcosθ−mLθ ̇

2

sinθ

Substituting this equation into the equation of motion for the cart and collecting
terms gives

(M+m)®x+bx ̇+mL®θcosθ−mLθ ̇

2

sinθ=u (2.15)

This is thefirst of two equations needed for a mathematical model.
The second equation of motion is obtained by summing all the forces in the
vertical direction for the pendulum. Note that, as we pointed out earlier, we
only need to consider the horizontal motion of the cart; and as such, there is no
useful information we can obtain by summing the vertical forces for the cart.
By summing all the forces in the vertical direction acting on the pendulum, we
obtain
Vsinθ+Hcosθ−mgsinθ=mL®θ+mx®cosθ
In order to eliminate theHandVterms, we sum the moments around the
centroid of the pendulum to obtain

−VLsinθ−HLcosθ=I®θ

Substituting this in the previous equation and collecting terms yields

(mL^2 +I)®θ+mgLsinθ=−mLx®cosθ (2.16)

Equations 2.15 and 2.16 are the equations of motion describing the nonlinear
behavior of the inverted pendulum. Since our objective is to design a controller
for this nonlinear problem, it is necessaryfor us to linearize this set of equations.
Our goal is to linearize the equations for values ofθaroundπ,whereθ=πis
the vertical position of the pendulum. Consider values ofθ=π+φwhereφ