1. Show all of your workings and drawings clearly. 2. Submit your homework as a pdf. The homework can be typed, written digitally or via pen and paper. Do ensure that the orientation of your submitted homework is upright. 3. You may use numerical software such as MATLAB for this homework. However, you have to sketch the root locus manually and not generate it via MATLAB or other software. 1. The motions of a linear stage can be modelled as a mass that is connected to some springs and dampers (Fig. 1). To actuate the linear stage, we can apply an external force (f) on it to produce an effective displacement of x. It is given that K = 10 N/m, K = 5 N/m, C = 1 Ns/m, C = 2 Ns/m and m = 3 kg. Assuming zero initial conditions, derive the transfer function of this linear stage, and find its zero(es) and pole(s). C GH K M 3 K ww f Fig. 1. The model of a linear stage. (7 marks)