Design Lead/Lag Controller for 3rd Order Transfer Function

[dEEP6ix]

I need to design a lead/lag controller for the following 3rd order close loop transfer function using the root locus method:

(11.35)(1.62)
-------------------------------
(s^2 + 1.84 s + 11.35)(s + 1.62)

The design requirement is just to reduce the settling time from the orginal 2.85s to 1.2s for a unit step input. I have tried adding a lead controller hoping to improve the transient response of the system. However, I soon discovered that I was just varying the gain, zero and pole in a trial and error method and it is getting me nowhere.

I have tried looking for information in control textbook but most of the examples are on 2nd order system. Those examples on higher order systems are always approximated to 2nd order systems which could not be done in the above transfer function.

Is there any systematic approach to zero/pole placement for higher order systems? Should I use a lag compensator instead? Any help is greatly appreciated.

http://www.freeimagehosting.net/image.php?72420b1ba8.jpg

http://www.freeimagehosting.net/image.php?6b83e7173f.jpg

 

[CEL]

I need to design a lead/lag controller for the following 3rd order close loop transfer function using the root locus method:

(11.35)(1.62)
-------------------------------
(s^2 + 1.84 s + 11.35)(s + 1.62)

The design requirement is just to reduce the settling time from the orginal 2.85s to 1.2s for a unit step input. I have tried adding a lead controller hoping to improve the transient response of the system. However, I soon discovered that I was just varying the gain, zero and pole in a trial and error method and it is getting me nowhere.

I have tried looking for information in control textbook but most of the examples are on 2nd order system. Those examples on higher order systems are always approximated to 2nd order systems which could not be done in the above transfer function.

Is there any systematic approach to zero/pole placement for higher order systems? Should I use a lag compensator instead? Any help is greatly appreciated.

http://www.freeimagehosting.net/image.php?72420b1ba8.jpg

http://www.freeimagehosting.net/image.php?6b83e7173f.jpg

A lag compensator will only improve your steady-state response (reduction of ss error). For improving transient response you need a lead compensator.
Since you want a 1.2s settling time, you can calculate the real part of your dominant poles.
Normally yoou would want the damping coefficient to remain constant. Now you can calculate the desired position of your dominant poles. Since you want those poles to belong to the root-locus, you want that the angles from them to the open loop poles and zeros (including plant and compensator) to be an odd multiple of 180.
Place the zero of the compensator over the real pole of the plant and calculate the position of the pole of the compensator in order to achieve the angle condition.

 

[Mene]

I would first suggest approaching the design of a lead/lag controller for a 3rd order transfer function using a systematic approach rather than trial and error. This will ensure a more efficient and effective design process.

One approach could be to use the root locus method, as mentioned in the question. This method involves plotting the roots of the characteristic equation of the closed-loop system as a function of a parameter (such as the gain or controller parameters) and analyzing the resulting locus to determine the stability and performance of the system.

To design a lead/lag controller using this method, the first step would be to determine the desired closed-loop pole locations for the specified settling time of 1.2s. These pole locations can be determined using the desired damping ratio and natural frequency for the system. Once the desired pole locations are determined, the lead/lag controller can be designed to shift the closed-loop poles to these locations.

Another approach could be to use frequency response techniques, such as Bode plots, to design the controller. This method involves analyzing the frequency response of the system and designing the controller to achieve the desired performance specifications, such as reducing the settling time.

In terms of whether to use a lag or lead compensator, it depends on the specific system and the desired performance specifications. A lag compensator is typically used to improve steady-state error, while a lead compensator is used to improve transient response. In this case, since the design requirement is to reduce the settling time, a lead compensator may be more suitable.

In conclusion, a systematic approach, such as the root locus method or frequency response techniques, should be used to design a lead/lag controller for a 3rd order transfer function. This will ensure a more efficient and effective design process and help achieve the desired performance specifications.