How to solve system's general stability from transfer function?

Question

I have a homework which should solve by me. My problem is questions are really simple or should I think outside of the box? Like, bode diagram, nyquist or etc.? And, are my answers correct?

Thanks.

Question-1

$$G(s) = K\dfrac{As+1}{Bs+1}$$

For which values $K, A$ and $B$ is the system always stable? Should I look directly to the pole of the system?

• $Bs+1=0$
• $s=-1/B \implies \text{So, must } B>0$

Is it enough? Or, anything else? What about K, A?

Question-2

$$G(s) = K\dfrac{As+1}{(Bs+1)(Cs+1)}$$

For which values $K, A, B$ and $C$ is the system always stable? Should I look directly to the pole of the system or anything else?

$$Bs+1=0 \wedge Cs+1=0$$ $$s=-1/B \wedge s=-1/C$$ $$\implies B>0 \wedge C>0$$

Is it enough? Or, anything else? What about $K$ and $A$?

Answer 1

I would like to extent the already given answer by MrYouMath.

So question 1 is pretty straight forward and you already got it right. If there's no right half plane (RHP) pole then it doesn't matter what gain you chose. Even for $A = B$, $G(s) = K$ yields a finite response.

For Question 2 have a look at the Routh Hurwitz Array

\begin{array} {|r|r|} \hline s^2 & B \cdot C & 1 \\ \hline s^1 & B+C & 0\\ \hline s^0 & 1 & \\ \hline \end{array}

In order for the system to be stable there must not be any sign changes in the first column, hence

$$BC > 0 \quad \land \quad B+C > 0$$

From $BC > 0$ we derive that B and C must have the same sign. $B+C > 0$ yields that the sign has to be positive.

As you see neither $A$ nor $K$ are involved in that.

If you want to explore other methods like root locus, bode, ... Keep in mind that you have variables ($A$,$B$,$C$) in there. I know that you can see the gain margins for root loci in Python, Matlab, etc. but I think that's it. I don't think (but I stand to be corrected) that you can derive the values for $A$,$B$,$C$ that way. I think with Bode plots this may work, however as you've seen it's much easier to solve with Hurwitz or by just looking at the poles.

Answer 2

You did answer both questions correctly. You could also consider solving the both questions by using the Hurwitz criterion. The first one is directly solvable with the Hurwitz criterion the second one is a little bit involved :).