# How to get maximum damping factor from this system?

I have a system with a transfer function as $$\frac{(s+3)(s+10)}{s(s^2+4s+5)(s+10)+K_Ts(s+10)(s+3)+600(s+3)}$$ Which has the characteristic equation: $$1+K_T\frac{s(s+3)(s+10)}{s(s^2+4s+5)(s+10)+600(s+3)}$$ And I know that the poles and zeros of this are: $$z_1=0 , z_2=-3,z_3=-10 , p_1=-3.1786 , p_2=-13.2478 , p_{3 , 4}=1.213_-^+6.42j$$ Here is the root locus that comes from said function:

I have been asked to find the maximum damping factor of the complex roots - I am afraid I don't know what this means or how to go about it - I have had a search on the web but am more confused as to how to solve this. Any help / guidance would be greatly appreciated.
Forgive me if this is more of a Mathematics question - it's just part of an Electronic Engineering degree hence asking here.

• Maximum with respect to what? K? May 2, 2016 at 11:19

The damping factor of a complex pair of poles (roots of the characteristic equation) is defined using the the pole position within the complex s-plane. If the angle between the negative-real axis and the pointer to the pole is "alpha" the damping factor d is defined as d=cos(alpha).

It is obvious that cos(alpha) can be expressed by the real part of the pole and the pole frequency (magnitude of the pointer to the pole position).

This definition concerns the complex pole pair only and is, thus, independent on a third pole on the negative-real axis or any zeros of the function.

More than that, the "pole quality factor" Qp is defined as Qp=1/2d.

Hence, for poles in the left half of the s-plane (stable systems) the damping factor varies between "1" and "0" and the pole Qp between "1/2" and "infinite" (oscillation condition).

I am currently studying control theory and one of the things that was emphasised was that terms like damping ratios and natural frequency is only relevant in 2nd order systems ie n = 2 (n is number of poles). However for 4th order systems you can express the system in terms of s^2 and then solve. Let:$$s^2 = s$$ Normally for the system G(s): $$G(s) = \frac{b_0}{s^2+a_1s+a_2}=\frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}$$ where the damping ratio $$\xi = \frac{a_1}{2 \sqrt{a_2}}$$ and coefficients $$a_2=\omega_n^2$$ and $$a_1 = 2\xi\omega_n$$

hope this helps

• Isn't that suggesting / assuming there aren't any zeros? May 1, 2016 at 16:08
• Yes thats right, I think the OPs term maximum damping factor is maybe confused with another term May 1, 2016 at 16:13