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ELE2038 - CONTROL COURSEWORK

1. System Description

1.1. System. Consider the system shown in Figure 1.

Figure 1. System of a wooden ball on an inclined plane. The ball can be attracted downwards by an electromagnet, which is controlled by the voltage V .

A wooden ball of total mass m and radius r is placed on an inclined plane, which is at an angle φ with respect to the horizontal plane as shown in Figure 1. The ball can roll on the inclined plane without sliding. The ball is connected to an elastic spring of stiffness k and a linear damper with viscous damping coefficient b. Let x denote the distance of the centre of the ball from the wall. When x = d, the spring is at its natural length and no restoring force is applied. The ball is considered to be approximately isotropic.

At the centre of the ball there is a metal core of very small radius which can be attracted by an electromagnet (which is essentially an inductor). The centre of the electromagnet is positioned at x = δ > d. This inductor is connected in series with an Ohmic resistor of resistance R and a voltage V is applied to the circuit as shown in Figure 1. The nominal inductance of the inductor is L0; however, as the ball approaches at a distance y from the centre of the inductor, its inductance increases and is given by

                (1)

where L1 and α are given positive constants.

The electromagnet can exercise an attractive force to the metal core of the ball, whose magnitude is given by

                             (2)

where c is a positive constant, i is the current that runs through the circuit and y is the distance between the centre of the wooden ball and the centre of the electromagnet.

Lastly, the position of the ball, x, on the inclined plane can be measured with a sensor that can be modelled as a first-order system with time constant τm.

1.2. System parameters. It is given that m = 462 g, g = 9.81 m/s 2 , d = 42 cm, δ = 65 cm, r = 12.3 cm, R = 2.2 kΩ, L0 = 125 mH, L1 = 24.1 mH, α = 1.2 m−1 , c = 6.811 m3gA2s2 ,k = 1885 N/m, b = 10.4 Ns/m, φ = 41◦ , and τm = 30 ms.

2. Assignment

You are a team of engineers who need to design a controller for the above system. The controller should be able to control the system at set points close to x sp = 0.4 m. The closed-loop system is expected to (i) be BIBO-stable, (ii) have zero offset, (iii) be properly tuned to avoid oscillations of large amplitude during the operation of the controlled system, and (iv) reject disturbances. You may want to impose additional requirements (e.g., related to stability margins, sensitivity of the closed-loop system to various noise signals, and more).

You should prepare a technical report with the proposed solution, with all involved steps (e.g., derivation of system equations, linearisation, controller design, stability analysis, stability margins, etc). Prove theoretically — where appropriate — that the closed-loop system has the desired properties, and provide simulation results. Do not forget that the original system is a nonlinear system2 . You also need to discuss whether you believe that the proposed controller would work in practice.

Where necessary, you may use Python, MATLAB, or any other framework for your simulations. Do not include your code in the report. Instead, include a link to your code (e.g., a Python notebook or a GitHub repository).