From ControlTheoryPro.com

Contents
1 Introduction to Pole Placement (or Polynomial Approach or Polynomial Design) to Controller Design
Pole placement is the most straightforward means of controller design.
 The design starts with an assumption of what form the controller must take in order to control the given plant.
 From that assumption a symbolic characteristic equation is formed.
 At this point the desired closedloop poles must be determined.
 Typically, specifications designate overshoot, rise time, etc. This leads to the formation of a 2^{nd} order equation. Most of the time the final characteristic equation will have more than 2 poles. So additional desired poles must be determined.
 Once the closed loop poles are decided a desired characteristic equation is formed.
 The coefficients for each power of s are equated from the symbolic characteristic equation to the desired.
 Algebra is used to determine the controller coefficients necessary to achieve the desired closedloop poles with the assumed controller form.
Typically, an integrator is used to drive the steadystate error towards 0. This implies that the final characteristic equation will have at least 1 more pole than the uncontrolled system started with.
The following pole placement examples show you how to decide on the desired closedloop poles, determine the "extra" closedloop poles, and create a generic and PID controller to achieve those desired closedloop poles.
2 Generic Control design using Pole Placement
This example is lifted from the hovering helicopter example in which the dynamics of Blackhawk helicopter are defined and controller for pitch attitude is designed. Feedforward commands are generated by integrating gyro measurements.
Let's assume a 2^{nd} order system of with the following form
Generic 2^{nd} Order System 
 where
 is the system gain
 is the system's natural frequency
 is the system's damping ratio.
Also, we assume a compensator of form
1 
is adequate to control the plant.
The resulting characteristic equation is
This can be reduced to
2a 
In matrix form this is
2b 
At this point we must decide what closed loop poles we would like. In order to do this we need to consider system overshoot and settling time (or time to peak). The equations for each are
Overshoot 
Settling Time 
Time to Peak 
I've never worked on a helicopter but I'm going to guess that minimizing overshoot is desired. Using the Overshoot equation we find that a common value, , provides an overshoot of only 4.3%. Examination of the Time to Peak equation lets you know that a value of provides a peak time of \pi seconds. However, a little over 3 seconds is probably too slow. Let's shoot for 0.5 seconds instead. This requires .
Recap
However, this leaves us with only 2 roots (poles) in our desired characteristic equation. Since we want the above parameters to dominate the closed loop system dynamics we choose a 3^{rd} pole that is well above the desired natural frequency.
3 
where
 is our 3^{rd} pole.
This 3^{rd} pole is a high frequency pole that allows the desired poles to the dominate the closedloop system response while allowing the desired characteristic equation to have the correct number of poles.
Our desired characteristic equation, Eqn. 3, can be reduced to
This results in
From here we go back to our characteristic equation (Eqn. 2a or 2b) to determine
3 PID Control design using Pole Placement
Let's assume a 2^{nd} order system of with the following form
Generic 2^{nd} Order System 
 where
 is the system gain
 is the system's natural frequency
 is the system's damping ratio.
Let's assume we have a PID controller of form
PID^{[1]} 
Roll inside the parentheses and add a factor to denominator of the integrator
Do a little algebra...
Finally, rearrange
The characteristic equation of this system is
From the order of we know we need 3 stable closed loop poles so the form of the desired characteristic equation is
Therefore
Result  









Once the desired closed loop poles are determined then and can be determined. Algebra does the rest.
4 Notes on 3^{rd} poles
In the [Helicopter Hover Examplehovering helicopter example] the desired closed loop poles are determined by deciding what a reasonable overshoot and rise time would be. This allowed for the determination of a desired and . From those a standard 2^{nd} order system was formed. Since 3 poles were required, just as in the cases provided here, a 3^{rd} pole was placed at an arbitrarily high frequency. The desired chracteristic equation from that became
where
 is our arbitrarily high 3^{rd} pole.
The addition of the pole at required that a prefilter be included to reduce overshoot. That prefilter was
5 See Also
6 References
 ↑ Franklin et all, pg. 185, Eqn. 4.31