Solutions of Exercise 1.3

Solutions of Exercise 1.3#

../_images/ex5.png

Fig. 6 Step responses to consider (courtesy of Exercise Manual for Automatic Control)#

Pair each step response in the plot above with the correct transfer function from the list below:

  1. \(G_1(s) = \frac{2}{s^2 + s + 1}\)

  2. \(G_2(s) = \frac{1}{s^2 + s + 1}\)

  3. \(G_3(s) = \frac{1}{s^2 + 0.1s + 1}\)

  4. \(G_4(s) = \frac{1}{s^2 + 2s + 1}\)

Useful formulas to compute the damping ratio and natural frequency of complex-conjugate eigevalues are reported in the Useful formulas.

Solution#

Key Parameters for Second-Order Systems#

  1. \(G_1(s)\):

    • Poles: \(s = -0.5 \pm j\frac{\sqrt{3}}{2}\)

    • Static Gain: \(G_1(0) = 2\)

    • \(\omega_0 = 1\), \(\xi = 0.5\) (damped)

    • \(T_{99\%} = \frac{5}{0.5} = 10\) s

    • The corresponding plot is A

  2. \(G_2(s)\):

    • Poles: \(s = -0.5 \pm j\frac{\sqrt{3}}{2}\)

    • Static Gain: \(G_2(0) = 1\)

    • \(\omega_0 = 1\), \(\xi = 0.5\) (damped)

    • \(T_{99\%} = \frac{5}{0.5} = 10\) s

    • The corresponding plot is D

  3. \(G_3(s)\):

    • Poles: \(s = -0.05 \pm j\frac{\sqrt{3.99}}{2}\)

    • Static Gain: \(G_3(0) = 1\)

    • \(\omega_0 = 1\), \(\xi = 0.05\) (under-damped, almost purely imaginary)

    • \(T_{99\%} = \frac{5}{0.05} = 100\) s

    • The corresponding plot is B

  4. \(G_4(s)\):

    • Poles: \(s = -1\) (double pole)

    • Static Gain: \(G_4(0) = 1\)

    • \(\omega_0 = 1\), \(\xi = 1\) (real poles)

    • \(T_{99\%} = \frac{5}{1} = 5\)

    • The corresponding plot is C

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