A feedback control system has the following closed-loop transfer function: T(S)=4148 /S4 +26S3 + 309S2 + 896s + 4148
(a) Find the closed-loop poles and plot them on the S-plane (you can use Matlab's 'pole' and 'pzplot functions). What isare) the dominant pole(s) of T(s)? (The dominant pole(s) is(are) the pole(s) with the largest real part. For instance, if p1 = -3 and P23 1+2i, the dominant poles are pz and pz because Re(P2)=Re(p:)=-1>-3=Re(p:)).
(b) Plot the unit-step response of (1) (you can use Matlab's 'step' function). Right-click on the generated plot and select "Characteristics'. Write down the numerical values for:
(1) Peak amplitude and overshoot,
(ii) Settling time,
(iii) Rise time,
(iv) Final value.
(C) Consider the second-order approximation/simplification of T(s) given by the following closed-loop transfer function:
T2(S)= w(2)n/ $2 + 2wns +w^2n
where Wn and are the natural frequency and the damping ratio associated with the dominant poles (pdi and Pp2) of (1), respectively. Find wn and S by equating the coefficients on the LHS and RHS of the following equation:
S2 +2cwns + wn^2= (s - PD1)(s - PD2).
(d) Using formulas 5.13-5.17 in the book, calculate the performance indices of the second-order system (2) when subjected to a unit-step input. In other words, compute its:
(i) Response peak amplitude value and overshoot,
(ii) Settling time,
(iii) Rise time, and
(iv) Final value.
Compare these performance indices with the ones found in part (b) by using relative difference %. Do you think Tz(s) is a good approximation of T(s)?
(e) Plot the unit-step response of Tz(s) and graphically compare with the plot found in part (b).