Suppose we have a negative unity feedback system with a plant transfer function $G(s) = \frac{1}{s(s+2)}$. I want to design a PI controller for this system in order to meet some time domain specifications. The specifications are settling time ($\textbf{2%}$ criterion) less than 1 second and damping ratio greater than $\sqrt 3/2$.
Obviously the closed tf will be $G_{cl}(s) = \frac{K_ps + K_I}{s^3+2s^2+K_ps+K_I}$ where $K_p$ and $K_I$ are proportional and integral gains, respectively. Since the closed loop system becomes third order and there is an additional zero, I cannot figure out how to calculate the specifications in here. Can help me to proceed?
Analysis: The plant transfer function is given by $$ G_{p}(s) = \frac{1}{s \left(s + 2\right)} .$$ It features an integrator. Thus, integral action is unnecessary, and a proportional controller alone suffices to achieve zero steady-state error. The closed-loop system transforms into $$ G_{cl}(s) = \frac{k_{p}}{s^{2} + 2s + k_{p}} .$$
By comparing the system's characteristic equation, $ s^{2} + 2\zeta\omega s + \omega^{2} $, and assuming $ \zeta = \frac{\sqrt{3}}{2} $, one can solve the equation $ \zeta \sqrt{k_{p}} = 1 $ to derive $ k_{p} = \frac{4}{3} $. However, the settling time within the 2% tolerance band under the P-only controller is 3.76 seconds.
Update 1: In order for the response to reach and stay within 1 second, two critically-damped closed-loop systems with integer coefficients can achieve that in slightly under a second. They are:
(1). Second-order closed-loop system,
$$ G_{cl}(s) = \frac{36}{s^{2} + 12 s + 36} $$
(2). Third-order closed-loop system,
$$ G_{cl}(s) = \frac{512}{s^{3} + 24 s^{2} + 192 s + 512} .$$
If the compensator is a PI controller, then the closed-loop system transforms into $$ G_{cl}(s) = \frac{k_{p} s + k_{i}}{s^{3} + 2 s^{2} + k_{p} s + k_{i}} .$$
It is clear that no combination of real values for $k_{p}$ and $k_{i}$ can achieve the two target closed-loop systems mentioned above.
It is also possible to place the PI controller $\frac{k_{p} s + k_{i}}{s}$ in the feedback loop and then introduce an integrator $\frac{k_{i}}{s}$ after the reference input to achieve
$$ G_{cl}(s) = \frac{k_{i}}{s^{3} + 2 s^{2} + k_{p} s + k_{i}} .$$
However, there are no combinations of real values for $k_{p}$ and $k_{i}$ that can achieve the target system (2) mentioned above.
Update 2: The critically-damped systems are chosen because the desired damping ratio $\zeta = \sqrt{3}/2$ would result in an insignificantly small percentage overshoot of 0.4333%. A critically-damped system does not exhibit overshoot. Thus, an $n$-order critically-damped system, denoted as $G_{cd}$, can be easily designed using the transfer function $$ G_{cd}(s) = \frac{\omega^n}{\left(s + \omega\right)^n} ,$$ where $\omega$ can be adjusted to achieve the desired settling time. Personally, I prefer integer values for $\omega$, and it doesn't take long to find that $\omega = 6$ for the 2nd-order system and $\omega = 8$ for the 3rd-order system meet the settling time requirement.
It is also possible to design a 3rd-order system by starting with the 2nd-order critically-damped system and introducing a third pole that is positioned far away from the origin. This configuration results in the system behaving approximately like a 2nd-order system. For example, $$ G_{cl}(s) = \frac{6^{2}}{\left(s + 6\right)^{2}} \left(\frac{40}{s + 40}\right) = \frac{1440}{s^{3} + 52 s^{2} + 516 s + 1440} $$ also meets the settling time requirement.
Proposed solution: If a simple lead compensator is allowed, then the controller $$ G_{c}(s) = 36 \frac{s + 2}{s + 12} $$ can achieve the target system (1) and meet the settling time requirement. I'd like to point out that the proposed lead compensator is mathematically equivalent to the PD controller of the form $$ G_{pd}(s) = k_{p} + k_{d} \frac{s}{T_{f} s + 1} ,$$ where $k_{p} = 6$, $k_{d} = 2.5$, and $T_{f} = \frac{1}{12}$.