Homework 6: Digital Systems Revision¶
Problems¶
Given that \(z = e^{sT} = e^{\sigma T} \angle \pm \omega T\), where \(T\) is the sampling period of a digital system, show that
a. points with frequencies of \(\pm\omega_s/2\) (where \(\omega_s\) is the sampling rate \(2\pi/T\)) in the s-domain map onto the negative real axis in the z-domain.
b. the imaginary axis in the s-domain maps onto the unit circle $\(\left|z\right| = 1\)$ in the z-domain
c. lines of constant \(\sigma\) in the s-domain map onto concentric circles centred at \(z = 0\) in the z-domain
d. lines of constant frequency in the s-domain map onto radial lines centred at \(z = 0\) in the z-domain.
Derive equations for curves of constant second-order damping ratio \(\zeta\) and natural frequency \(\omega_n\) in the z-domain. Comment on the effect of high sampling rates on the poles of a discrete system.
Determine the z-transform of the function \(f(t) = e^{-at}\), where \(a\) is real \(> 0\).
Given that $\(G(s) = \frac{s+4}{(s+1)(s+3)},\)\( determine \)G(z)\(. [*Hint*: determine \)g(t)$ first.]
A discrete transfer function \(G(z)\) is given by $\(G(z) = \frac{0.387z^2}{(z-1)(z^2 - 2.37z + 025)}.\)\( Determine the value of \)g(nT)\( as \)nT \rightarrow \infty$.
Use long division to find the inverse z-transform of $\(F(z)=\frac{\left(1-e^{-aT}\right)z}{z^2 - \left(1-e^{-aT}\right)z + e^{-aT}}\)$.
The system type of a unity gain feedback digital control system $\(\frac{C(z)}{R(z)} = \frac{G_o(z)}{1 + G_o(z)}\)\( is given by the number of poles at \)z = 1\(. Thus a Type 0 system has no poles at \)z = 1\(, a Type 1 system has one, and so on. Use the discrete version of the final value theorem \)\(\lim_{n\rightarrow \infty} x(nT) = \lim_{z \rightarrow 1}\left(1-z^{-1}\right)X(z)\)$ to derive formulae for the steady-state error of a digital control system to step, ramp and parabolic inputs.
(not examinable) Apply Jury’s test (MATLAB function
jury) to determine if the following characteristic equations have any roots outside the unit circle:a. \(z^2 + 0.25 = 0\)
b. \(z^3 - 1.1z^2 + 0.01z + 0.405 = 0\)
c. \(z^3 - 3.6z^2 + 4z - 1.6 = 0\).