Suppose we want to know how does the controller affect pole locations? Routh table just tells us if the system is stable or not and how many poles are in the RHS, but it does not tell us how the poles behave as the gain \(K\) changes. Suppose our controller \(H\left ( s\right ) \) is a function of \(\left ( K_{1},K_{2},\cdots \right ) \) where we the designer set the values of these \(K_{i}\) for example, selecting \(K_{1}\) for overshoot specification as we did before. Now we will talk about how to select \(K\) for other purpose, which is pole locations of the closed loop system. Closed loop pole location gives us many information about the system, and pole locations are informative about behavior of closed loop response. For example
The above diagram tell us about the speed of the response or speed of decay. \(e^{-\sigma t}\) is indicator of speed to decay of response. If we have large \(\sigma \) then the system will go to its final value (steady state much faster). There is also what is called the damping cone. This is the cone where the poles lie inside.
The angle \(\theta \) above is informative about damping. Large damping implies small angle \(\theta \). Suppose we have combination of needs: speed of decay of response and low damping, hence we have region where we want the poles be located.
So many performance specs (but not all) can be found by location of poles. Suppose that the open loop pole locations are not desired, and we want to move them to desired region. We use feedback with controller \(K\) such that these locations are moved to the desired location in the closed loop poles. Root locus is such method to allow us to do this in systematic way (computer aided design) rather by trial and error. The classical setup for root locus design is this
Suppose we designed controller \(H(s)\) and are not happy with the pole locations. i.e. \(G\left ( s\right ) H\left ( s\right ) \) are given. We need to select \(K\geq 0\) to move the poles to desired location. We view \(G\left ( s\right ) H\left ( s\right ) \) above as the open loop system.
Root locus is the locus of the closed loop in the complex plane obtained by changing \(K\) from \(0\) to \(\infty \).
A common sense approach to use root locus. Example:
The open loop poles are at \(s=0\) and \(s=-2\). The closed loop is \(T\left ( s\right ) =\frac{K}{s\left ( s+2\right ) +k}=\frac{K}{s^{2}+2s+K}\) and the pole location are \(s=-1\pm \sqrt{1-k}\). Now we increase \(K\) from \(0\) and see how the locus of the poles change. We get this
So for large \(k\) the poles will move outside the cone of interest.
Reader: How large can \(K\) be to satisfy damping constraint of \(\alpha =45^{0}\)?
Formal root locus. Will develop 9 lemmas. Each lemma gives more information about the locus. We begin with \(G\left ( s\right ) H\left ( s\right ) \) which is the open loop. Write it as \(\frac{N\left ( s\right ) }{D\left ( s\right ) }\) where it is proper (degree of numerator (\(m\)) \(\leq \) degree of denominator \(\left ( n\right ) \)). Closed loop is \(T\left ( s\right ) =\frac{KGH}{1+KGH}\). To find closed loop poles, we write
\begin{align*} 1+KGH & =0\\ 1+K\frac{N\left ( s\right ) }{D\left ( s\right ) } & =0\\ D\left ( s\right ) +KN\left ( s\right ) & =0 \end{align*}
Observer that, for \(K\neq 0\), the above have \(n\) poles.
Lemma 1 root locus has \(n\) branches for \(K>0\).
Root locus (R.L.) geometry: Central idea. A point \(s\) is on R.L. if \(1+KGH=0\) for some \(K\) value. \(GH=-\frac{1}{K}\). Hence the phase of \(GH\) is \(\pi \). And the corresponding magnitude of \(K\) is \(\left \vert \frac{1}{GH}\right \vert \). So to decide of point \(s\) is on the R.L. quickly, look at the angle. For example, for open loop poles \(s=-2\) and \(s=0\), and suppose to want to know if some point \(s^{\ast }\) is on the R.L., then we draw this
Then \[ \sphericalangle GH=-\theta _{1}-\theta _{2}\neq \pi \]
Then \(s^{\ast }\) is not on the root locus path.