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4 Further Engineering Mathematics Assignment 2021-22 Third Order State Space System (2021-22) The primitive structure of a third order system of ODE’s is defined by the block diagram of Fig. 1. Initial conditions are zero, and the input, 𝑢, is a unit step. Initially, the feedback parameter values are set at: 𝒂 = 𝟔. 𝟏𝟐𝟓; 𝒃 = 𝟖. 𝟕𝟓 The state equation is: 𝒙̇ = 𝑨𝒙 + 𝑩𝑢

a) Using the states, 𝑥1 , 𝑥2 , and 𝑥3 indicated in the block diagram, derive the system matrix, A, and the corresponding spectral matrix, S, and modal matrix, M. (15 Marks)

b) Hence, determine the unit step response vector, 𝒙 = [ 𝑥1 𝑥2 𝑥3 ]. (25 Marks) c) Check this result via Laplace transform analysis. (15 Marks)

d) Derive the controller canonical realisation of the state equations, and draw the corresponding block diagram (note, the states will differ from those of the original system). (10 Marks)

e) For the controller canonical realisation, determine a set of state feedback coefficients to give a unit step response, 𝑥1 , with the following specification: 1. Zero steady state error for the step input. 2. Percentage overshoot, 𝑃𝑂 ≈ 20% 3. Time to first peak, 𝑇𝑝 ≤ 5 sec (12 Marks)

) Investigate whether the set of parameter values derived in

e), above, are transferrable to the original block diagram, through appropriate selection of the ′𝑎′ and ′𝑏′ parameters. (10 Marks)

g) Now, the system output vector, 𝒚, is to be defined as follows: 𝒚 = 𝑪𝒙 where C is the row vector, 𝑪 = [1, 0, 2] Investigate how this modification may be applied to

(i) the controller canonical block diagram, and

(ii) the original primitive block diagram. (13 Marks)

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