System Stability

Why it matters

System stability is crucial in engineering because it determines whether a system will behave predictably over time. In practical terms, stable systems ensure that signals do not grow unbounded, which is essential for the reliability and safety of electrical and electronic systems.

Key ideas

• Stability Definition: A system is stable if, for every bounded input, the output is also bounded. This is known as BIBO (Bounded Input, Bounded Output) stability.
• Types of Stability:
  • Asymptotic Stability: The system returns to equilibrium after a disturbance.
  • Marginal Stability: In the internal zero-input sense, responses remain bounded but need not decay; oscillation is one possibility. This does not imply BIBO stability.
  • Unstable: The system output grows without bound.
• Poles and Zeros: The location of poles in the s-plane (for continuous systems) or z-plane (for discrete systems) determines stability.
  • For causal proper rational transfer functions, BIBO stability requires all uncancelled poles strictly in the left half-plane. Internal asymptotic stability requires every state-matrix eigenvalue there, including hidden modes.
  • The corresponding discrete-time criterion places poles/eigenvalues strictly inside the unit circle under the same distinctions.
• Routh-Hurwitz Criterion: A method to determine the stability of a linear time-invariant system by examining the characteristic equation.
• Nyquist Criterion: A graphical method used to determine the stability of a control system.

Formulas

• Characteristic Equation: a_n·s^n + a_(n-1)·s^(n-1) + ... + a_1·s + a_0 = 0
  • s: Complex frequency variable
  • a_n, a_(n-1), ..., a_0: Coefficients of the polynomial
• Routh-Hurwitz Criterion: Requires constructing the Routh array and checking the first column for sign changes.

For a continuous-time LTI convolution system, BIBO stability requires an absolutely integrable impulse response. In a regular Routh array, first-column sign changes count right-half-plane roots; zero leading entries or all-zero rows require special treatment.

Worked example

Given: A system with the characteristic equation s^3 + 2s^2 + 3s + 4 = 0.

1. Construct the Routh Array:
   • First row: 1, 3
   • Second row: 2, 4
   • Third row: (2*3 - 1*4)/2 = 1, 0
   • Fourth row (s⁰): 4
2. Check for sign changes in the first column:
   • The first column is 1, 2, 1, 4, with no sign changes.
3. Conclusion: The system is stable.

Final Answer: Stable

Common mistakes

• Confusing the conditions for stability in continuous and discrete systems.
• Incorrectly constructing the Routh array, leading to wrong conclusions about stability.
• Ignoring the effects of zeros on system behavior, although they do not affect stability directly.

For GATE EE

Questions often involve determining the stability of a given system using the Routh-Hurwitz criterion or analyzing pole-zero plots. Practice constructing Routh arrays and interpreting Nyquist plots.

Quick check

1. What is BIBO stability?
2. How do you determine stability using the Routh-Hurwitz criterion?
3. Where should the poles be located for a discrete system to be stable?

Answers: 1. A system is BIBO stable if every bounded input leads to a bounded output. 2. By constructing the Routh array and checking for sign changes in the first column. 3. Inside the unit circle.