Stability from Z-Transform

System stability determines whether a digital output remains bounded or grows out of control. Control engineers analyze this to prevent catastrophic failures in automated machinery.

Core Concept

A stable discrete system produces a limited output for any limited input sequence. We call this bounded input bounded output stability. The transfer function reveals this property through its region of convergence.

For absolute stability, the region of convergence must strictly include the unit circle. The unit circle represents the continuous frequency response contour where absolute z equals exactly one.

When dealing with causal systems, stability depends entirely on the poles. Every single pole of the transfer function must sit strictly inside the unit circle. Even one pole outside guarantees an unstable response.

Key Formula

The stability condition requires the impulse response to be absolutely summable over all time.

sum( |h[n]| ) from -infinity to infinity < infinity. This time domain condition is mathematically equivalent to the ROC including the geometric contour |z| = 1.

Problem: Assess stability of causal system H(z) = z / (z^2 - 1.5z + 0.5).\nGiven: System is causal, meaning ROC extends outward from outermost pole.\nSteps:\nFind roots of denominator: z^2 - 1.5z + 0.5 = 0.\nFactor polynomial: (z - 1)(z - 0.5) = 0.\nIdentify poles: p1 = 1, p2 = 0.5.\nCheck magnitudes: |p1| = 1, |p2| = 0.5.\nEvaluate against unit circle: One pole rests exactly on the unit boundary.\nFinal Answer: System is marginally stable, not strictly stable.

Exam Tip: A pole exactly on the unit circle means the system is only marginally stable. It will oscillate forever without decaying. For strict bounded input bounded output stability, all poles must be strictly less than one in magnitude.

Properties Summary

• Bounded stability requires an absolutely summable impulse response.
• The region of convergence must explicitly contain the unit circle.
• For causal systems, all poles must have a magnitude strictly less than 1.
• For anti-causal systems, all poles must have a magnitude strictly greater than 1.
• Right half plane poles in continuous time map to the outside of the unit circle.
• Left half plane poles in continuous time map to the inside of the unit circle.

Quick Revision

• Stability prevents system outputs from exploding mathematically.
• The unit circle is the ultimate boundary for causal systems.
• Pole magnitude is the absolute geometric distance from the origin.
• Complex poles always come in conjugate pairs and cause oscillations.
• System causality dictates the permissible region for pole locations.
• Exam trap: Stating a system is stable just because the poles are negative, forgetting that magnitude matters, not sign.