Solutions to Exercises
All calculations assume $T = 0.1$ seconds.
Solution to Exercise 1
$s = -0.5 + j1$
$r = e^{-0.5 \cdot 0.1} = e^{-0.05} \approx 0.951$
$\theta = 1 \cdot 0.1 = 0.100$ rad
$z = 0.951 \cdot (\cos(0.1) + j \sin(0.1)) \approx 0.951 \cdot (0.9950 + j0.0998) \approx 0.946 + j0.095$
Solution to Exercise 2
$s = -0.3 + j0.8$
$r = e^{-0.3 \cdot 0.1} = e^{-0.03} \approx 0.970$
$\theta = 0.8 \cdot 0.1 = 0.080$ rad
$z = 0.970 \cdot (\cos(0.08) + j \sin(0.08)) \approx 0.970 \cdot (0.9968 + j0.0799) \approx 0.967 + j0.078$
Solution to Exercise 3
$s = +0.2 + j0.5$
$r = e^{+0.2 \cdot 0.1} = e^{0.02} \approx 1.020$
$\theta = 0.5 \cdot 0.1 = 0.050$ rad
$z = 1.020 \cdot (\cos(0.05) + j \sin(0.05)) \approx 1.020 \cdot (0.9988 + j0.0500) \approx 1.019 + j0.051$
The system is unstable because $r > 1$ (pole in right half-plane).
Solution to Exercise 4
$s = -0.6 + j0$
$r = e^{-0.6 \cdot 0.1} = e^{-0.06} \approx 0.942$
$\theta = 0 \cdot 0.1 = 0$ rad
$z = 0.942 + j0$
The system is stable because $r < 1$.
Solution to Exercise 5
$s = 0 + j1.2$
$r = e^{0 \cdot 0.1} = e^{0} = 1$
$\theta = 1.2 \cdot 0.1 = 0.120$ rad
$z = 1 \cdot (\cos(0.12) + j \sin(0.12)) \approx 0.9928 + j0.1197$
The pole lies on the unit circle (marginally stable).