System stability tells engineers whether a circuit, aircraft controller, robot, or chemical process will return to a desired condition after a disturbance. For a linear time-invariant system, stability is determined by the locations of the poles of its transfer function in the complex s-plane. Poles in the left half-plane produce decaying responses, while poles in the right half-plane produce growing responses.
This makes pole location one of the most important links between algebra, graphs, and physical behavior.
Key Facts
- A continuous-time LTI system is asymptotically stable if all poles have negative real parts.
- For characteristic equation a_n s^n + a_(n-1) s^(n-1) + ... + a_0 = 0, all coefficients must have the same sign for stability, but this is not sufficient.
- Routh-Hurwitz criterion: the number of sign changes in the first column of the Routh array equals the number of right half-plane poles.
- For a second-order polynomial a s^2 + b s + c, stability requires a > 0, b > 0, and c > 0.
- For s^3 + a s^2 + b s + c, stability requires a > 0, b > 0, c > 0, and ab > c.
- Pure imaginary poles indicate marginal stability only when they are simple and no poles lie in the right half-plane.
Vocabulary
- Pole
- A pole is a value of s that makes the transfer function denominator equal to zero and strongly shapes the system response.
- s-plane
- The s-plane is the complex plane used in control engineering, with horizontal real axis sigma and vertical imaginary axis j omega.
- Asymptotic stability
- Asymptotic stability means the natural response decays to zero as time goes to infinity.
- Routh array
- A Routh array is a tabular arrangement of characteristic equation coefficients used to count right half-plane poles without solving for the roots.
- Marginal stability
- Marginal stability means the response remains bounded but does not decay to zero, often due to simple poles on the imaginary axis.
Common Mistakes to Avoid
- Assuming positive coefficients guarantee stability is wrong because higher-order polynomials can have all positive coefficients and still have right half-plane poles.
- Counting any negative entry in the Routh table instead of sign changes is wrong because the stability test uses changes of sign only in the first column.
- Ignoring a zero in the first column is wrong because it can make the next row undefined and requires the epsilon method or a special case procedure.
- Calling every imaginary-axis pole stable is wrong because repeated imaginary-axis poles or any right half-plane pole make the system unstable.
Practice Questions
- 1 Use the Routh-Hurwitz criterion to determine the number of right half-plane poles for s^3 + 2s^2 + 3s + 10 = 0.
- 2 Construct the Routh array for s^4 + 3s^3 + 5s^2 + 4s + 2 = 0 and decide whether the system is stable.
- 3 A system has poles at -2, -1 + 3j, -1 - 3j, and 0 + 4j. Explain whether the system is asymptotically stable, marginally stable, or unstable.