Question

For what range of k is the following transfer function stable? (Use the Routh stability test to estimate values of k)

Answer 1

The transfer function is stable for 0 < k < 3. This range ensures the system's stability based on the Routh stability test.

Step-by-step explanation:

The stability of a system with a given transfer function can be determined using the Routh-Hurwitz stability criterion. For the given transfer function, let's denote the characteristic equation as
`\(1 + k\textsuperscript{2}s - 3ks - 2k = 0\)`. The coefficients of the characteristic equation are [1, -3k, -2k]. Now, we construct the Routh array:

`\[ \begin{array}{cccc} s\textsuperscript{2} & 1 & -2k \\ s\textsuperscript{1} & -3k & 0 \\ s\textsuperscript{0} & -2k & 0 \end{array} \]`

For the system to be stable, all the elements in the first column of the Routh array must be positive. In this case, for stability, we require -3k > 0 and -2k > 0, leading to k < 0. However, since k is squared in the second row of the characteristic equation, we need
`\( k\textsuperscript{2} > 0 \)`, which implies
`\( k \\eq 0 \)`. Combining these conditions, we get
`\( 0 < k < 3 \)` for the system to be stable.

In conclusion, the Routh-Hurwitz stability test determines that the system is stable for values of k within the range 0 < k < 3. This means that the stability of the system is guaranteed when the proportional gain k falls within this specified interval. Beyond this range, the system may become unstable, leading to undesirable behaviors such as oscillations or divergence in response to input signals.

Answer 2

To determine the range of k for which a transfer function is stable, you can use the Routh stability test. The first column of the Routh array should have all positive coefficients for the system to be stable.

Step-by-step explanation:

The question asks for the range of k for which a given transfer function is stable.

To determine this, we can use the Routh stability test. The Routh stability criterion states that for a system to be stable, all the coefficients in the first column of the Routh array must be positive.

In our case, the transfer function can be expressed as a polynomial in k, and we can construct the Routh array using the coefficients of this polynomial.

We can then analyze the first column of the Routh array to determine the range of k for stability.