**Poles and Stability**

If p* _{i}* is a pole of G(s), then the natural, or zero-input, the response of G(s) will consist of the mode functions e

^{pit}if p

*is distinct, and t*

_{i}^{q }e

^{pit}, q = 0, 1,. . ., r – 1, if p

*has multiplicity r. Thus the natural response will decay to zero if Re[p*

_{i}*] < 0 for i = 1, . . . , n-that is, if all the poles are in the open left-half of the s-plane, i.e., the left half-plane excluding the imaginary axis. Such a system is said to be*

_{i}**asymptotically stable.**

If all the poles are in the open left-half plane except for distinct poles on the imaginary axis and at the origin, the natural response will consist of undamped sinusoids or a nonzero constant, and the system is said to be **marginally stable**.

If some of the poles are in the right-half of the s-plane or on the imaginary axis with multiplicity greater than one, then the natural response will be unbounded and the system is said to be **unstable**.

** Determine Transfer Function Poles**

For G(s) given in TF form, the MATLAB function **tf2zp** can be used to determine the poles of G(s) and its stability. Alternatively, since we are interested only in the poles, we can apply the polynomial roots function **root** to the denominator of G(s) to ﬁnd its poles. The following example illustrates the use of roots to determine the system’s stability.

**System Stability using Matlab Example**

Find the poles of the transfer function

$G(s)=\frac{1.5s+1}{{{s}^{3}}+2{{s}^{2}}+2.5s+0.5}$

and determine whether the system is stable. Plot the poles and zeros of G(s) in the s-plane. Finally, demonstrate the system stability by simulating the impulse response.

**Solution**

Deﬁne the row vectors **numG = [1.5 1]** and **denG = [1 2 2.5 0.5]** to represent the numerator and denominator of G(s), respectively. Using the command **roots (denG)**, we ﬁnd the poles of G(s) to be at s = -0.2408 and -0.8796 ± j1.1414. Since all the poles have negative real parts, that is, they are all in the open left-half plane, the system is **asymptotically stable**. The impulse response can be found from the command** impulse (numG, denG)**, and will decay to zero as shown in figure 1. The MATLAB commands in Script 7 will compute the poles and zeros and make the plot as shown in figure 2.

From roots command, we obtained the following result:

Poles |

-0.8796 + 1.1414i |

-0.8796 – 1.1414i |

-0.2408 + 0.0000i |

**Matlab Code to find poles and impulse response**

% Script 7: Matlab Code to find poles and impulse response clear all;close all;clc numG = [1.5 1]; % Create G(s) as a ratio of numerator &amp;amp;amp;amp; denumerator denG = [1 2 2.5 0.5]; roots(denG) % find poles of G(s) pzmap(numG,denG) % plot the poles and zeros impulse(numG,denG) % simulate impulse response

**Result**

Fig.1: Impulse Response of a Transfer Function

Fig.2: Pole Zero Plot for System Stability