Stability of Control Systems
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Questions and Answers

What condition is required for the Routh-Hurwitz stability?

  • The total number of coefficients must be odd.
  • At least one coefficient of the characteristic polynomial must be negative.
  • All coefficients of the characteristic polynomial must be positive. (correct)
  • Only the first coefficient of the characteristic polynomial must be positive.

    • What does a sign change in the first column of the Routh array indicate?

  • The control system is stable.
  • The control system is unstable. (correct)
  • The characteristic equation has a repeated root.
  • There are roots in the left half of the 's' plane.

    • What is done if the first element of a row in the Routh array is zero?

  • Use a small positive integer in its place. (correct)
  • Leave the row unchanged.
  • Eliminate that row from the Routh array.
  • Replace the row with all ones.

    • How do you fill the elements of the Routh array after the first two rows?

    <p>Based on the auxiliary equation derived from the zero row.</p> Signup and view all the answers

    What does it mean if all the elements in the first column of the Routh array have the same sign?

    <p>The control system is stable.</p> Signup and view all the answers

    In which case is it necessary to differentiate the auxiliary equation?

    <p>When all elements of a row are zero.</p> Signup and view all the answers

    What is the required outcome for the number of sign changes in the Routh array to deem a system unstable?

    <p>At least one sign change.</p> Signup and view all the answers

    What is an example of a special case in the Routh array formation?

    <p>The first element of a row is zero.</p> Signup and view all the answers

    What is the first step in using the Routh-Hurwitz method to analyze stability?

    <p>Check if all coefficients of the characteristic polynomial are positive.</p> Signup and view all the answers

    What is the role of the Routh array method in control systems?

    <p>To determine the stability of control systems without finding roots.</p> Signup and view all the answers

    What characterizes a stable system?

    <p>It produces a bounded output for a bounded input.</p> Signup and view all the answers

    Which type of system is stable for all ranges of system component values?

    <p>Absolutely stable system</p> Signup and view all the answers

    What defines a conditionally stable system?

    <p>Stability over a certain range of system component values</p> Signup and view all the answers

    What indicates that an open loop control system is absolutely stable?

    <p>All poles of the transfer function are in the left half of the ‘s’ plane.</p> Signup and view all the answers

    What is true regarding marginally stable systems?

    <p>They can be marginally stable if any two poles are on the imaginary axis.</p> Signup and view all the answers

    What is a necessary condition for the Routh-Hurwitz stability criterion?

    <p>Coefficients of the characteristic polynomial must be positive.</p> Signup and view all the answers

    Which condition indicates that a system is unstable?

    <p>The system does not satisfy the necessary condition of the Routh-Hurwitz criterion.</p> Signup and view all the answers

    What is indicated by poles located in the right half of the ‘s’ plane?

    <p>The system is unstable.</p> Signup and view all the answers

    What is a distinguishing feature of a marginally stable system's output?

    <p>Constant amplitude with frequency oscillations.</p> Signup and view all the answers

    For which type of system does stability depend on a specific range of component values?

    <p>Conditionally stable system</p> Signup and view all the answers

    What is the first non-zero row of the Routh array obtained from the given characteristic polynomial?

    <p>s5</p> Signup and view all the answers

    What must be done when a row in the Routh array consists entirely of zeros?

    <p>Replace it with the derivatives of the previous non-zero row</p> Signup and view all the answers

    How many sign changes were observed in the first column of the Routh table?

    <p>Two</p> Signup and view all the answers

    What becomes of the coefficients when the elements of a row share a common factor in the Routh array?

    <p>They are divided by the common factor</p> Signup and view all the answers

    What is the resulting auxiliary equation from the differentiation step?

    <p>A(s) = s^4 + s^2 + 1</p> Signup and view all the answers

    What is indicated if the closed-loop poles are not found in the left half of the 's' plane?

    <p>The system is unstable</p> Signup and view all the answers

    What mathematical process is used to derive the coefficients for the Routh array from the auxiliary equation?

    <p>Differentiation</p> Signup and view all the answers

    Which of the following statements describes the Routh-Hurwitz stability criterion?

    <p>It assesses stability based on the signs of the first column in the Routh table</p> Signup and view all the answers

    What should be done with the coefficients when performing the operation (2×1) - (1×1)?

    <p>Subtract directly to get the result</p> Signup and view all the answers

    What technique can be utilized to analyze systems where Routh-Hurwitz gives inconclusive results?

    <p>Root locus technique</p> Signup and view all the answers

    Study Notes

    Stability of Control Systems

    • A control system is considered stable if its output remains controlled for any given bounded input.
    • Stable systems produce a bounded output response for a bounded input signal.

    Types of Stability

    • Absolutely Stable System: Stable for all values of system components. Both open and closed loop systems are absolutely stable if all poles of their transfer function are located in the left half of the 's' plane.
    • Conditionally Stable System: Stable only for a specific range of system component values.
    • Marginally Stable System: Stable but oscillates with constant amplitude and frequency for a bounded input. The open and closed loop systems are marginally stable if two poles of their respective transfer function are located on the imaginary axis.

    Routh-Hurwitz Stability Criterion

    • Necessary Condition: All coefficients of the characteristic polynomial must be positive. This implies that all roots of the characteristic equation have negative real parts.
    • Sufficient Condition: All elements in the first column of the Routh array must have the same sign (all positive or all negative).

    Routh Array Method

    • The Routh array is constructed using the coefficients of the characteristic polynomial.
    • The number of sign changes in the first column of the Routh array indicates the number of roots of the characteristic equation that are present in the right half of the 's' plane.
    • A system is unstable if at least one root of the characteristic equation is located in the right half of the 's' plane.

    Special Cases of Routh Array

    • First Element of any row is zero: Replace the zero element with a small positive value (epsilon) and proceed with constructing the Routh array. Analyze the sign changes as epsilon approaches zero.
    • All Elements of any row are zero: Determine the auxiliary equation of the row above the row of zeros. Differentiate this auxiliary equation with respect to 's' and place these coefficients in the row of zeros.

    Limitations of Routh-Hurwitz

    • The Routh-Hurwitz stability criterion does not allow for the identification of the system's behavior if poles are located on the imaginary axis.
    • The Root Locus technique offers a more comprehensive analysis of system stability by visualizing the locations of closed loop poles for varying gain values.

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    Description

    This quiz covers the concepts of stability in control systems, including definitions and types of stability such as absolutely stable, conditionally stable, and marginally stable systems. It also introduces the Routh-Hurwitz Stability Criterion and its importance in determining system stability. Challenge your understanding of these key concepts in control theory!

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