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Root locus of Second Order System

The locations of the poles of $H(s)$ in the s-plane determines the system behavior. Here we consider the pole locations and the corresponding system response as $\zeta$ changes from $-\infty$ to $\infty$. We assume $N(s)=\omega_n^2$ below.

\begin{displaymath}H(j\omega)=\frac{\omega_n}{s-p_1}\frac{\omega_n}{s-p_2}
=\frac{\omega_n}{2\sqrt{\zeta^2-1}}[\frac{1}{s-p_1}-\frac{1}{s-p_2}] \end{displaymath}

The inverse Laplace transform of $H(s)$ is the impulse response $h(t)$ of this system

\begin{displaymath}h(t)=\frac{\omega_n}{2\sqrt{\zeta^2-1}}[e^{p_1t}-e^{p_2t}]u(t)
=M\;[e^{p_1t}-e^{p_2t}]u(t) \end{displaymath}

where

\begin{displaymath}M\stackrel{\triangle}{=}\frac{\omega_n}{2\sqrt{\zeta^2-1}}
=\frac{\omega_n}{2j\sqrt{1-\zeta^2}} \end{displaymath}

This second order system can also be considered as the product (i.e., cascade) of two first order systems:

\begin{displaymath}H(s)=\frac{\omega_n}{s-p_1} \cdot \frac{\omega_n}{s-p_2}=H_1(s) \cdot H_2(s) \end{displaymath}

and in the Bode diagram of this system, both the magnitude and the phase plot of $H(j\omega)$ can be obtained by adding the corresponding plots of $H_1(s)$ and $H_2(s)$. (For example, the sum of two curves with 20 dB/decade drop at high frequency is a curve with 40 dB/decade drop.)

The frequency response $H(j\omega)$ at any frequency $\omega$ is determined by the product of the two vectors $v_1=j\omega-p_1$ and $v_2=j\omega-p_2$ in the denominator of $H(j\omega)$, and the behavior of the system as the frequency changes can be qualitatively determined by observing the locations of the poles in the s-plane, which vary as a function of the parameter $\zeta$. We first list the two poles for different values of $\zeta$:
$\zeta$ $p_1$ $p_2$ comments
$\zeta=-\infty$ $\infty$ $0$  
$-\infty<\zeta<-1$ $(-\zeta+\sqrt{\zeta^2-1})\omega_n$ $(-\zeta-\sqrt{\zeta^2-1})\omega_n$ real poles $0<p_2<p_1$
$\zeta=-1$ $\omega_n$ $\omega_n$ repeated real poles $0<p_1=p_2$
$ -1<\zeta<0$ $(-\zeta+j\sqrt{1-\zeta^2})\omega_n$ $(-\zeta-j\sqrt{1-\zeta^2})\omega_n$ complex conjugate pair $Re[p_1]=Re[p_2]>0$
$\zeta=0$ $j\omega_n$ $-j\omega_n$ imaginary poles
$ 0<\zeta<1$ $(-\zeta+j\sqrt{1-\zeta^2})\omega_n$ $(-\zeta-j\sqrt{1-\zeta^2})\omega_n$ complex conjugate pair $Re[p_1]=Re[p_2]<0$
$\zeta=1$ $-\omega_n$ $-\omega_n$ repeated real poles $p_1=p_2<0$
$ 1<\zeta<\infty$ $(-\zeta+\sqrt{\zeta^2-1})\omega_n$ $(-\zeta-\sqrt{\zeta^2-1})\omega_n$ real poles $p_2<p_1<0$
$\zeta=\infty$ $0$ $-\infty$  

Note 1: When $\vert\zeta\vert<1$, the poles are a complex conjugate pair located along a circle with center at the origin and radius equal to $\omega_n$. The phase angles of the poles are $\pm cos^{-1}\zeta$. In other words, different $\omega_n$ values are represented by concentric circles of different radius, and different $\zeta$ values are represented by rays from the origin of different angles.

Note 2: When $\zeta$ increases from $-\infty$ to $\infty$, each of the two poles moves in the s-plane along a path called the root locus.

Motion of $p_1$:

\begin{displaymath}\infty \stackrel{\mbox{leftward}}{\Longrightarrow}
\omega_n ...
...ga_n \stackrel{\mbox{rightward}}{\Longrightarrow} \mbox{origin}\end{displaymath}

Motion of $p_2$:

\begin{displaymath}\mbox{origin} \stackrel{\mbox{rightward}}{\Longrightarrow}
...
... -\omega_n \stackrel{\mbox{leftward}}{\Longrightarrow} -\infty \end{displaymath}

Here we discuss the pole locations corresponding to various $\zeta$ values, and the corresponding system behaviors:

Summarizing the above, we can trace the movement of the two roots (root locus) in the s-plane when $\zeta$ changes its value in the range $-\infty \sim \infty$, and find the corresponding system behaviors, as shown in the table below ( $M=\omega_n/2\sqrt{\zeta^2-1}=\omega_n/2j\sqrt{1-\zeta^2}$).

$\zeta$ $H(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$ $h(t)=M(e^{p_1t}-e^{p_2t})$ Comments on $h(t)$
$\zeta<-1$   $M(e^{\vert p_1\vert t}-e^{\vert p_2\vert t})u(t)$ exponential growth
$\zeta=-1$ $\omega_n^2/(s-\omega_n)^2$ $\omega_n^2\;t\;e^{\omega_nt}u(t)$ exponential growth
$ -1<\zeta<0$   $\frac{\omega_n}{\sqrt{1-\zeta^2}}e^{-\zeta\omega_nt}sin(\omega_dt)u(t)$ exponentially growing sinusoid
$\zeta=0$ $\omega_n^2/(s^2+\omega_n^2)$ $\omega_n\; sin(\omega_nt) \;u(t)$ constant sinusoid
$ 0<\zeta<1$   $\frac{\omega_n}{\sqrt{1-\zeta^2}}e^{-\zeta\omega_nt}sin(\omega_dt)u(t)$ exponentially decaying sinusoid
$\zeta=1$ $\omega_n^2/(s+\omega_n)^2$ $\omega_n^2\;t\;e^{-\omega_nt}\;u(t)$ critically damped
$\zeta > 1$   $M(e^{-\vert p_1\vert t}-e^{-\vert p_2\vert t})u(t)$ exponential decay

h2.gif


next up previous
Next: Frequency Response for Small Up: Evaluation of Fourier Transform Previous: Second order system
Ruye Wang 2012-01-28