Answer Happy

The dynamic equation of motion for an air levitation system is given as below with the schematics of the system shown in Fig. 4. The objective is to balance the height of the ball inside a tube via the air flow blown by a DC fan from the bottom, in an upward direction. The height of the ball z is measured with respect to a position sensor attached at the top of the tube. There is an adjustable nozzle allowing for the air to escape from the top of the tube. This means that at any constant input voltage, the steady state air flow speed and the ball position will depend on the diameter of the nozzle. The speed of the air flow is v, and the mass of the ball is m with the gravitational acceleration denoted as g. The parameter a is the product of the drag coefficient Ca, air density p, and the surface area of the ball exposed to the upward air flow divided by 2, a = CapA. The drag force is then given by fa= a(v, - 2)². m² = a(v₁ - 2)² - mg
Q3.1. [10] Assume a constant air flow of vy, is applied to bring the ball to a specific height (operating point) z. Derive the linearised state space model of the system at this operating point (zee). Note that this operating point is obtained at steady state, which is when the velocity and acceleration of the ball are zero z = z = 0. (Hint: Apply the operating/equilibrium point condition to find a required relation between g and input offset vye). Q3.2. [3] Find the eigenvalues of the linearised system, and then obtain the state transition matrix. Q3.3. [5] Assume the transfer function of the DC fan is modelled as a simple first-order system with a time constant r, a DC gain of Kar, input voltage v, and vy as it output. Combine the DC fan transfer function with the linearised models from Q3.1 and obtain a 3rd-order linear model with Av, as the input (small changes in the input voltage of the DC fan) and Az as the output (small changes in the position of the ball) Q3.4. [10] Derive a 3rd-order state space model by choosing the state variables as x₁ = Az, x₂ = Az, x₂ = Avy with Av, as the input and x, = Az as the output. Then use the transfer function from Q3.3 to find a different state space representation of this system using the same input and output.