5

This is a basic control theory question, since Control Theory is a part of applied mathematics but also of engineering I was unsure whether to ask this here.

The question says:

Given the transfer function of a system is $G(s)=1/(s^2+3s+2)$, consider the design of a PI closed loop control system with unit feedback using proportional gain $k_p$ and integral gain $k_i$, both of which are positive. Determine the range of gain for which the closed loop system is stable. What I did was this, I went the Routh-Hurwitz way:

My solution

After completing the Routh table, I went ahead and reasoned a little about what conditions need to be met in order to avoid sign changes on the main column, however it seems that these conditions are never met! I get that $k_i$ should be less than zero when the problem clearly specifies it will always be positive. Have I done something wrong? Is my reasoning right? Is the answer "The controller is never stable"?

Fred
  • 9,782
  • 13
  • 36
  • 48
HCalderon
  • 53
  • 4

1 Answers1

5

There is a mistake in your expressions. The coefficient of $s$ is $2+k_p$.

The conditions are: $$\frac{1}{3} \left(3 \left(k_p+2\right)-k_i\right)>0$$ $$k_i>0$$ $$k_p>0$$

This simplifies to: $$k_p>0$$ $$0<k_i<3 k_p+6$$

Update:

If the transfer function is $$ \frac{1}{s^2+3 s + c} $$ where c is some positive constant, then the conditions simplify to: $$k_p>0$$ $$0<k_i<3 ( k_p+c)$$

Suba Thomas
  • 2,036
  • 11
  • 15