I want to control an unstable plant system with transfer function $\frac{1}{s-\alpha}$ using controller in the feedforward loop and unity feedback. However, my requirement is my closed loop system has i) No zeros in the complex plane ii) Settling time Ts (2%). No pole zero cancellation allowed.

Using proportional controller with control gain $K_p$, my closed loop transfer becomes $\frac{C(s)}{R(s)}=\frac{K_p}{s+(K_p-\alpha)}$. If $K_p>\alpha$, my system becomes stable. No zeros in the closed loop transfer function and I can choose $K_p$ by equating $T_s=4*T=\frac{4}{K_p-1}$.

Apart from proportional control, is there any way I can stabilize the system and achieve my other constraints?


Well, when we have a transfer function that looks like:


Where $\text{H}\left(\text{s}\right):=1$ because of the unity feedback and $\text{G}\left(\text{s}\right):=\text{K}\cdot\frac{1}{\text{s}-\alpha}$

We get a complete transfer function that looks like:


Now, for the poles and zeros:

  • Poles: $$1+\text{K}\cdot\frac{1}{\text{s}-\alpha}=0\space\Longleftrightarrow\space\text{s}=\alpha-\text{K}\tag3$$
  • Zeros: $$\text{K}\cdot\frac{1}{\text{s}-\alpha}=0\space\Longleftrightarrow\space\text{K}=0\tag4$$

For the settling-time we have that:


We need to use:


Where $\lambda\space\wedge\space\omega\in\mathbb{R}$ and $\text{j}^2=-1$

Now, we need to solve for $\text{K}$ in the pole equation (assuming $\text{K}\ne0$):


Conclusion, we get a transfer function that looks like:

$$\color{red}{\mathcal{T}=\frac{\text{Y}\left(\text{s}\right)}{\text{X}\left(\text{s}\right)}=\frac{\alpha\cdot t\space_{2\text{%}}+\ln\left(50\right)}{\text{s}\cdot t\space_{2\text{%}}+\ln\left(50\right)}}\tag8$$

  • $\begingroup$ Thank you Jan for your thorough answer. However, I would like to emphasize that using a controller with proportional gain $K$ is one way of achieving the stable system which has no zeros in the complex plane and settling time 1 second. If we use PI, PD or PID, we always get zeros, which we don't want. If I use only integrator (I), my system does not become stable for any integrator gain $k_i$. Is there any other way I can achieve my goal? Do we have any second order controller such that $\frac{K}{s^2+bs+K}$ Please correct me if I am wrong. $\endgroup$
    – Kunal
    Mar 8 '17 at 20:39
  • $\begingroup$ @Kunal Yes, we can have a second order controller, but then we will have overshoot, because of the imaginary part of the poles in the complex plane $\endgroup$ Mar 9 '17 at 17:06

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