Q1. The transfer function of a system is given by:

1.5791 × 10⁶��� + 1.9844 × 10¹⁰

���(���) =

���3 + 1.2604 × 104���2 + 2.0529 × 106��� + 1.9844 × 1010

i. Compute the state-space representation of this three-state system – show all the steps.

ii. Ascertain whether this three-state, state-space representation obtained in (i) is a Minimal Realization, by performing the controllability and observability tests.

iii. If it is Minimal, go to (iv). If not, find the Minimal Realization of this system.

iv. Confirm Minimal Realization by comparing the poles and zeros of the transfer function ���(���) given, the three-state state-space realization obtained in (ii) and the minimal Realization (if different) as obtained in (iii) → show pole-zero cancellation/s.

v. Generate a Bode Plot for ���(���) and its Minimum Realization on the same figure for frequencies ranging between 1 Hz and 10 KHz → Figure 1

a. Top subfigure Frequency (Hz) vs Magnitude (dB)

b. Bottom subfigure Frequency (Hz) vs Phase (deg)

c. ���(���) → dashed red; Minimum Realization → solid blue

d. Line thickness → 1.5

vi. Plot the Pole-Zero maps for ���(���) and its Minimal Realization on the same figure and reaffirm that the minimal realization has identical dynamics → Figure 2

vii. Generate a reference input → ���(���)  for a length of 5 s,  which is a sine wave  of amplitude 1, phase 0 and frequency equal to the resonant frequency of the system.

viii. Simulate and plot the output ���(���), of the system ���(���), for the input ���(���) designed in (vii) → Figure 3

a. Top subfigure → ���(���) between 2.8 s and 2.85 s → dashed red

b. Middle subfigure → ���(���) from (viii) between 2.8 s and 2.85 s → solid blue

ix. Find pole locations such that the output ���_���������(���) of this ‘regulated system’ (after relocating the pole/s of the Minimal Realization of ���(���) as obtained in (iii)) to the input ���(���) as generated in (vii), has an amplitude of ±10 units.

x. Compute the Regulator Gain Matrix ��� (using any method you know) to effect this pole-placement – show all the steps.

xi. Superimpose the frequency response of the regulated system ���_���������(���) in Figure

1 → solid black

xii. Simulate the output, ���_���������(���) of the regulated system ���_���������(���), for the input ���(���) designed in (vii). Plot ���_���������(���) as the Bottom subfigure of Figure 3 between 2.8 s and 2.85 s → solid black

Q2. A closed-loop control scheme is shown below in Figure Q2.

Figure Q2.

If the state matrices for each block are given by:

Plant ⟹ [���, ���, ���, ���]

���_��� ⟹ [���_���, ���_���, ���_���, ���_���]

���_��� ⟹ [���_���, ���_���, ���_���, ���_���]

i. Derive the overall closed-loop state-space representation that connects reference input ��� to overall output ���. Show all the steps.

ii. Consider the individual blocks are the following transfer functions:

��������������� =

6.448 × 10⁶

���2 + 90��� + 1.9344 × 107

5.5 × 10⁷

������ = ���2 + 1.18 × 104��� + 4.5 × 107

1200

���_��� = ���

Substitute the relevant state-space matrices for each of the block in the parametric

closed-loop state-space representation you obtained in Q2i and obtain the full state-space representation for this closed-loop system. Use MATLAB to do this. Include the code used in the Appendix.

iii. Compute the s-domain closed-loop transfer function of the overall system given in

���(���)

Figure Q2 directly using the given transfer functions in Q2ii →

���(���)

iv. Demonstrate that the dynamics of the overall closed-loop state-space representation that you obtained in Q2.ii match those of the closed-loop transfer function you obtained in Q2. iii.

v. Explain what the two controllers ���_��� and ���_��� doing? Use (any number of) appropriate plots and analyses to support your explanation / reasoning and provide the 3 dB bandwidth of the overall closed-loop system.