Stability of Control Systems
Determine the stability of a system using Routh-Hurwitz criterion and understand the concept of poles and zeros.
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Stability definitions
For a continuous-time finite-dimensional LTI state model, asymptotic internal stability requires every eigenvalue of A to have strictly negative real part. For a causal proper rational transfer function, BIBO stability requires every uncancelled pole to lie strictly in the left half-plane. Hidden unstable modes can make a realization internally unstable despite a stable input-output transfer. Simple imaginary-axis poles can produce bounded free oscillations, but do not give BIBO stability: a resonant bounded input can produce unbounded output. Repeated imaginary-axis modes require special care.
Routh-Hurwitz method
Form the Routh array from the real characteristic-polynomial coefficients. In a nonsingular array, the number of sign changes in the first column equals the number of right-half-plane roots. A zero first-column entry requires the limiting epsilon procedure; an entire zero row requires the auxiliary-polynomial procedure. Positive coefficients are necessary for a strictly Hurwitz polynomial with positive leading coefficient, but not sufficient at higher order.
Worked example
For p(s) = s³ + 2s² + 3s + K, the rows are:
| Row | First entry | Second entry |
|---|---|---|
| s³ | 1 | 3 |
| s² | 2 | K |
| s¹ | (6 − K)/2 | 0 |
| s⁰ | K | 0 |
| All first-column entries are positive exactly when 0 < K < 6. At K = 6, p(s) = (s + 2)(s² + 3), with poles −2 and ±j√3, so strict stability is lost. At K = 0 there is a pole at zero. Neither endpoint belongs to the asymptotically stable interval. |
Common mistakes
Do not apply Routh to an open-loop denominator when asked for closed-loop stability; first obtain the characteristic equation. Do not declare stability from coefficient signs alone or cancel unstable modes without considering internal behavior.
Quick check
- Is s² + 2s + 2 Hurwitz? Yes, its poles are −1 ± j.
- Does a right-half-plane zero by itself make a transfer function unstable? No; it affects response and design limitations, while poles determine causal rational BIBO stability.
- What does a first-column sign change count? A right-half-plane root.
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