Stability Criteria for Polynomials
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Questions and Answers

What is the difference between absolute and relative stability?

  • Absolute stability refers to whether a system is stable or unstable, while relative stability refers to the degree of stability. (correct)
  • Absolute stability is used for BIBO stability, while relative stability is used for impulse response.
  • Absolute stability refers to the degree of stability, while relative stability refers to whether a system is stable or unstable.
  • Absolute stability is used for impulse response, while relative stability is used for BIBO stability.

    • What is the necessary and sufficient condition for a feedback system to be stable?

  • All poles of the system transfer function have imaginary real parts.
  • All poles of the system transfer function have negative real parts. (correct)
  • All poles of the system transfer function have zero real parts.
  • All poles of the system transfer function have positive real parts.

    • What is the effect of pole locations on the unit step response of a 2nd-order system?

  • The system is unstable if all poles have zero real parts.
  • The system is stable if all poles have negative real parts. (correct)
  • The system is stable if all poles have positive real parts.
  • The system is unstable if all poles have negative real parts.

    • What is the condition for a system to be BIBO stable?

    <p>The system has a bounded response to a bounded input.</p> Signup and view all the answers

    What is the form of the poles of the standard transfer function of a 2nd-order system?

    <p>s1, 2 = -ζωn ± j1 - ζ2</p> Signup and view all the answers

    Is the system G(s) = 10(s+2)/(s+1)(s-3)(s+4) stable?

    <p>No, the system is unstable.</p> Signup and view all the answers

    What is the Routh array used for?

    <p>To determine the stability of a system</p> Signup and view all the answers

    What is the form of the response transform X1(s)?

    <p>X1(s) = bn s^n + bn-1 s^n-1 +... + b1s + b0</p> Signup and view all the answers

    What is the characteristic equation of the system?

    <p>Q(s) = bn s^n + bn-1 s^n-1 +... + b1s + b0 = 0</p> Signup and view all the answers

    How are the constants c1, c2, c3, etc. evaluated in the Routh array?

    <p>By using the coefficients of the characteristic equation</p> Signup and view all the answers

    What is the condition for stability of a system according to Routh's criterion?

    <p>All terms in the first column have the same sign</p> Signup and view all the answers

    What is the number of roots of the characteristic equation with positive real parts equal to?

    <p>The number of changes of sign of the coefficients in the first column</p> Signup and view all the answers

    What does the presence of imaginary roots in the characteristic equation indicate about the output of the system?

    <p>The output includes a sinusoidally oscillating component</p> Signup and view all the answers

    What is the next step after finding the characteristic equation of a system?

    <p>Form the auxiliary equation from the preceding row</p> Signup and view all the answers

    What is the condition for a system to be stable according to the Routhian array method?

    <p>There are no changes of sign in the first column</p> Signup and view all the answers

    What is the purpose of differentiating the auxiliary equation in the Routhian array method?

    <p>To complete the Routhian array</p> Signup and view all the answers

    What is the condition for the numerator of the first term in the s1 row to be positive for a stable system?

    <p>–K2 – 43K + 2000 &gt; 0</p> Signup and view all the answers

    What is the range of K for which the system is stable in the given example?

    <p>K &lt; 80 and K &gt; 0</p> Signup and view all the answers

    What is the necessary condition for a stable system?

    <p>All the coefficients of the characteristic equation have the same sign</p> Signup and view all the answers

    What is the Routh-Hurwitz stability criterion used for?

    <p>To determine the number of roots of q(s) with positive real parts</p> Signup and view all the answers

    What is the first step in the Routh-Hurwitz stability test?

    <p>To construct the Routh array from the system characteristic equation</p> Signup and view all the answers

    What is the characteristic equation of a system?

    <p>The equation q(s) = 0</p> Signup and view all the answers

    What is the purpose of the Routh-Hurwitz stability test?

    <p>To determine the stability of a SISO-LTI system</p> Signup and view all the answers

    What is the necessary and sufficient condition for a stable system?

    <p>The Routh-Hurwitz test is satisfied</p> Signup and view all the answers

    What is the condition for all roots of the 1st Order polynomial Q(s) = a1s + a0 to be in the Left Half Plane (LHP)?

    <p>a1 and a0 have the same sign</p> Signup and view all the answers

    What is the condition for all roots of the 2nd Order polynomial Q(s) = a2s^2 + a1s + a0 to be in the LHP?

    <p>a2, a1, and a0 have the same sign</p> Signup and view all the answers

    What is the condition for all roots of the Higher Order polynomial Q(s) = an s^n + an-1s^n-1 +... + a1s + a0 to be in the LHP?

    <p>All coefficients ak have the same sign</p> Signup and view all the answers

    What is the limitation of the Routh-Hurwitz criterion?

    <p>It is only applicable to polynomials</p> Signup and view all the answers

    What is the value of K for which the system is stable in the given example?

    <p>0 &lt; K &lt; 28.1</p> Signup and view all the answers

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