Stability of Systems: Routh-Hurwitz Criterion Quiz Quiz

Challenge your understanding of system stability using the Routh-Hurwitz criterion. This quiz covers key concepts, procedures, and application scenarios essential for analyzing the stability of linear time-invariant systems.

1. Identifying Stability from Characteristic Equations

   Given the characteristic equation s^3 + 2s^2 + s + 3 = 0, how many sign changes are present in the first column of the Routh array if constructed correctly?

   1. One
   2. Two
   3. Three
   4. Zero

   Explanation: The Routh array for the equation s^3 + 2s^2 + s + 3 = 0 will have one sign change in the first column, indicating one right-half-plane root. Zero sign changes would indicate stability, while two or three sign changes would suggest two or three unstable roots, respectively. This particular case makes 'One' the correct answer since the system has precisely one sign change after building the array.

2. Purpose of the Routh-Hurwitz Criterion

   What is the primary objective of applying the Routh-Hurwitz criterion to a system’s characteristic equation?

   1. Calculate the time response
   2. Find the transfer function
   3. Check system stability
   4. Determine the natural frequency

   Explanation: The Routh-Hurwitz criterion is specifically designed to check the stability of a system by analyzing its characteristic equation. Determining natural frequency or calculating time response are different analyses, and finding the transfer function is not the objective of the Routh-Hurwitz method. Only the system's stability is directly determined using this approach.

3. Conditions for All Roots in Left-Half Plane

   Under which condition does the Routh-Hurwitz criterion guarantee that all roots of the characteristic equation have negative real parts?

   1. All coefficients are positive and no sign changes in the first column
   2. Presence of at least one zero in the first row
   3. Highest order coefficient is negative
   4. Complex coefficients in the equation

   Explanation: The Routh-Hurwitz criterion ensures all roots are in the left-half plane when all equation coefficients are positive and there are no sign changes in the array’s first column. If the leading coefficient is negative, stability cannot be assured; zeros in the first row require additional steps but do not guarantee stability. Complex coefficients invalidate the standard Routh-Hurwitz approach.

4. Handling a Zero in the First Column of the Routh Array

   If a zero appears as the first element of a row in the Routh array during the criterion process, which is the correct procedure?

   1. Multiply the row by -1
   2. Replace the zero with one
   3. Use a small positive number (epsilon) in place of zero
   4. End the Routh array and conclude stability

   Explanation: When a zero appears as the leading element, a small positive number, epsilon, is substituted to continue calculations and analyze stability trends. Simply replacing the zero with one changes the mathematical meaning, and stopping the process gives incomplete information. Multiplying rows by -1 does not address the issue or preserve the Routh array’s logic.

5. Special Case: Entire Row of Zeros

   What does an entire row of zeros indicate in the Routh array when analyzing a fourth-order system?

   1. System is always stable
   2. Characteristic equation has all real roots
   3. System gain is too high
   4. System is marginally stable and repeated roots exist on the imaginary axis

   Explanation: An entire row of zeros indicates the presence of repeated roots on the imaginary axis, signifying marginal stability. It does not guarantee overall stability or only real roots, nor does it necessarily relate to system gain. This is a special case requiring further analysis with an auxiliary equation to determine root positions.