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# EE 350 Control Systems Assignments

## Assignment 1: Modeling of a dynamic system (cont'd...)

1.4 The figure 1.4 shows a simplified version of a car or motorcycle suspension system. The input $u$ is the displacement of the road with respect to a reference (you may say unevenness of the road) and $y$ is the displacement of the body of the car. Obtain the transfer function $Y(s)/U(s)$ of the system.

Fig 1.4 Car suspension system

1.5 The figure 1.5 shows a inverted pendulum system. Assume that the mass of the inverted pendulum is $m$ and is evenly distributed along the length of the rod. Input of the system is $u$ and output is $\theta$. Assuming that the angle $\theta$ is small, derive mathematical models of the system in the forms of differential equations, transfer functions, and state-space equations.

Fig 1.5 Inverted pendulum on a car

1.6 Obtain a state-space representation of the mechanical system shown in figure 1.6, where $u_1$ and $u_2$ are the inputs and $y_1$ and $y_2$ are the outputs.

Fig 1.6 Mechanical spring mass damper system