Reconsider the control system in Example 10.2.

a. Convert the transfer function \(G(s)=Y(s) / U(s)\) to a differential equation of \(y(t)\).

b. Using the differential equation obtained in Part (a) to represent the plant, build a Simulink block diagram to simulate regulation control, in which the reference signal \(R(s)\) is zero. Assume that the initial conditions are \(y(0)=0.1 \mathrm{~m}\) and \(\dot{y}(0)=0 \mathrm{~m} / \mathrm{s}\).

Data From Example 2:

Reconsider the cart position control system in Example 10.1. The transfer functions of the plant (combining the cart and the DC motor), the controller, and the sensor are
\[G(s)=\frac{3.778}{s^{2}+16.883 s}, \quad C(s)=85, \quad H(s)=1\]

Derive the closed-loop transfer functions \(Y(s) / R(s)\) and \(E(s) / R(s)\).

Data From Example 1:

Consider the electromechanical system described in Problem 3 of Problem Set 6.4. It consists of a cart that moves along a linear track and a DC motor that drives the cart. An encoder is included to measure the position of the cart. Assume that a controller is designed to control the position of the cart. Draw a block diagram for this feedback control system. Clearly label essential components and signals.