Question

The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is \[ G(s) = \dfrac{k(s+z)^a}{s^b(s+p)^c}, \] where \(k\), \(z\), \(p\), \(a\), \(b\), and \(c\) are positive constants. The value of \((a+b+c)\) is ___________ (rounded off to the nearest integer).

Hint:

On an asymptotic magnitude Bode plot, each pole contributes a $-20$ dB/dec slope after its corner, and each zero contributes $+20$ dB/dec. The initial slope reveals any poles/zeros at the origin.

Answer 1

From the Bode plot: the low-frequency slope is $-20$ dB/dec. This comes only from the pole at the origin $\;s^b\;$, hence $b=1$.
At $\omega_1$ the slope changes from $-20$ to $0$ (i.e., $+20$ dB/dec), implying a zero of order $1$ at $z$ $\Rightarrow a=1$.
At $\omega_2$ the slope changes from $0$ to $-40$ (i.e., $-40$ dB/dec), implying a pole of order $2$ at $p$ $\Rightarrow c=2$.
Therefore, \[ a+b+c=1+1+2=4. \] \[ \boxed{4} \]