Find the allowable regions in the s-plane for the poles of a transfer function.?

Find the allowable regions in the s-plane for the poles of a transfer function of a


standard second-order system if the system response requirements are tr less than or equal to 0.6


seconds, M p less than or equal to 10%, and ts less than or equal to 3 seconds. Use the definitions given


tr is about 1.8/蠅n


ts=4.6/蟽


Mp=e^(-pi*味/sqrt(1-味))





I have no idea how to answer... please tell me how to start this??





thanks to whoever answers this|||The characteristic eqn of a 2nd order system is:





s虏 + 2味蠅ns + 蠅n虏 = 0





蠅n = 1.8/tr = 1.8/0.6 = 3 rad/s





MPO = e^[-蟺味 /鈭?1 - 味)] = 0.1





蟺味 /鈭?1 - 味) = ln10 = 2.3





蟺虏味虏 = 2.3虏 (1 - 味 ), =%26gt; 味虏 = 0.537(1 - 味), =%26gt; 味虏 + 0.537味 - 0.537 = 0,





味 = [ -0.537 +- 鈭歿0.28858 + 4(0.537)}]/2 = -0.2686 + 0.78 = 0.5118





s虏 + 2(0.5118)3s + 3虏 = 0, s = [-3.07126 +- 鈭歿9.4326 - 4*9}]/2





s = -1.535 +- j2.577, =%26gt; complex pole pair in the left-half plane.





but since, settling time, ts = 4.6/蟽





蟽 = 4.6/ts = 4.6/3 = 1.53





s = -蟽 + j蠅d





蠅d = 蠅n鈭?1 - 味) = 3鈭?1 - 0.5118) = 3鈭?.488 = 2.095





Therefore, the allowable regions of the left-half s-plane


is the rectangle defined by:





s = -1.53 +- j2.095, and s = -1.535 +- j2.577





Note:


rise time: tr


maximum % overshoot: MPO


settling time: ts


damping factor: 味


natural angular frequency: 蠅n


damped frequency: 蠅d


2nd order transfer function poles: s


dissipation parameter: 蟽





I hope this helped.