Mathematical Formulation

Relative Coordinate

The report uses the relative displacement

q(t)=x(t)-z(t),

where x(t) is the platform position and z(t) is the base position.

Electromechanical Model

m\ddot{q}+c\dot{q}+kq=K_f i-m\ddot{z},

L\dot{i}+Ri+K_e\dot{q}=u.

The states are

x_1=q, \qquad x_2=\dot{q}, \qquad x_3=i.

For the nominal plant, the base disturbance is set to zero:

A=
\begin{bmatrix}
0 & 1 & 0 \\
-k/m & -c/m & K_f/m \\
0 & -K_e/L & -R/L
\end{bmatrix},
\qquad
B=
\begin{bmatrix}
0 \\ 0 \\ 1/L
\end{bmatrix},

C=
\begin{bmatrix}
1 & 0 & 0
\end{bmatrix},
\qquad
D=0.

Nominal Transfer Function

G(s)=\frac{K_f}
{Lm s^3+(Lc+Rm)s^2+(Lk+Rc+K_fK_e)s+Rk}.

The report applies the third-order Routh-Hurwitz criterion:

(Lc+Rm)(Lk+Rc+K_fK_e)>LmRk.

Reduced Design Model

For IMC-based PID tuning, the report uses

G_r(s)=\frac{1}{s^2+5s+100}.

The resulting ideal controller and filtered PID parameters reported in the PDF are

C(s)=50\left(1+\frac{1}{0.05s}+0.2s\right),

P = 50
I = 0.05
D = 0.2
N = 10

The supplied .slx file stores N = 100 in its PID Controller1 block. The report and model are preserved as submitted so that reviewers can inspect this difference directly.