Region of root locus not inside the feasible region for a given performance specification I wanted to plot the root locus for a system, whose plant transfer function is
$$
P(s)= \frac{k}{Ts+1}e^{-ts}.
$$
A PI controller was suggested for this system, and the task is to find $K_p$ and  $K_i$ values, such that it satisfies the following performance specs:

Maximum peak overshoot &lt; 5%
Settling time (2% band) &lt; 0.8 seconds

This is how I modified the open loop transfer function to suit the format of root locus finding.
$$
G(s)H(s) = C(s)P(s)= K_p\frac{k(s+A)(2-ts)}{s(Ts+1)(2+ts)},
$$
where $A = K_i/K_p$ and Pade approximation was used. For starters, $A$ was assumed to be 1.
But, when I plot the root locus of the above, this is what I get.

Here, the white region is the feasible region which satisfies the performance specs.
As we can see, the root locus of the pole introduced by the integrator (at Im axis) ends up being completely inside the unfeasible region. This is the case for any $A$ value I take.
What should I do? Is my approach correct? And more importantly, how will this pole affect my performance?
I'm new to this field, and I'm a bit confused by this. Please help me out here. Thanks in advance.
The root locus will start at the open loop poles and move toward the open loop zeros as the gain is varied. Your pole cannot get past the zero that is located in the yellow region. This pole that is close to the imaginary axis will be dominant, meaning it will dominate your response times, slowing down the entire system. One approach is to cancel that zero with an additional open loop pole at it's location, unless that zero was introduced by your controller in which you have the ability to move it to be out in the white area, or eliminate it entirely.
It appears that the zero that terminates the closed loop pole as shown is due to $(s+A)$ in the numerator, in which case choosing a value for A that is within the white region should allow the pole to extend further to that point as the gain is increased. Also, the pole and zero due to the Pade approximation does not appear to be visible in the root locus.
To see how poles close to the imaginary axis are dominant, consider the simple case of $x(t) = e^{-at}$ which has a Laplace Transform $X(s)=\frac{1}{s+a}$  (which is a pole at $s= -a$. The time domain equation is a decaying exponential with time constant $a$, if a is very small this will take a very long time to decay. If $a$ is very large, the decay rate will be very fast.