Stability in Z-Domain

Stability analysis is central to any control system design, and in digital control systems, stability must be evaluated in the z-domain rather than the s-domain. The unit circle criterion is the z-domain counterpart of the Routh-Hurwitz criterion, and the Jury stability test is the primary algebraic method for checking discrete system stability without explicitly computing all roots of the characteristic polynomial.

Core Concept: Unit Circle Stability Criterion

A discrete-time system described by the transfer function G(z) = N(z)/D(z) is asymptotically stable if and only if all roots of the characteristic polynomial D(z) = 0 lie strictly inside the unit circle in the z-plane, i.e., |zᵢ| < 1 for all poles zᵢ. This is the direct z-domain analog of the Routh-Hurwitz criterion, which requires all characteristic roots to have negative real parts (lie in the left half s-plane).

The physical interpretation is straightforward: a pole at z = r·e^(jθ) inside the unit circle (r < 1) corresponds to a mode in the time domain that decays exponentially as rⁿ. If r > 1 (pole outside), the mode grows without bound. A pole on the unit circle (r = 1) represents a sustained oscillation. A pair of complex poles inside the unit circle at z = r·e^(±jθ) corresponds to a damped oscillation at digital frequency θ rad/sample.

For simple second-order systems, stability can be checked directly from the characteristic polynomial z² + a₁z + a₀ = 0 using three necessary and sufficient conditions: (1) 1 + a₁ + a₀ > 0, (2) 1 - a₁ + a₀ > 0, and (3) a₀ < 1. These are special cases of the Jury conditions.

Jury Stability Test

The Jury stability test is an algebraic procedure that determines whether all roots of a polynomial lie inside the unit circle, without computing the roots. It is the discrete-time equivalent of the Routh array for continuous systems. Given the characteristic polynomial D(z) = a₀ + a₁z + a₂z² + ... + aₙzⁿ (with aₙ > 0), the Jury array is constructed row by row and conditions on the first-column elements are checked.

The Jury array is built by writing the polynomial coefficients in ascending powers (row 1) and reverse order (row 2). The next pair of rows is generated by a determinant-based recursion. The necessary and sufficient conditions for stability are: D(1) > 0, (-1)ⁿ·D(-1) > 0, and all (n-1) Jury conditions on the array elements. For GATE purposes, the second-order conditions are most tested, and the procedure for up to third-order systems is important.

Practical Understanding

In digital control design, stability margins in the z-domain are assessed by how far the poles are from the unit circle. Poles close to the unit circle (|z| just under 1) imply slow decay and poor damping. The w-domain transformation (also called bilinear or w-transform) w = (z-1)/(z+1) maps the interior of the unit circle to the left half w-plane, allowing the application of Nyquist and Bode techniques from continuous control theory. This is a useful bridge between z-domain analysis and frequency-domain design.

Gain margin and phase margin concepts extend to digital systems through the w-domain. In practice, as the sampling period T increases, poles of the digital closed-loop system move toward the unit circle, degrading stability margins. This is why choosing an appropriate T is critical, not just for performance but for maintaining adequate stability margins.

Solved Numerical Example

A digital control system has the closed-loop characteristic polynomial D(z) = z² - 1.5z + 0.7. Determine stability using both the direct root method and the second-order Jury conditions.

Given:
D(z) = z² - 1.5z + 0.7  (second-order characteristic polynomial)

Why this formula applies:
For stability, all roots must satisfy |z| < 1.
Second-order Jury conditions: D(1)>0, D(-1)>0, |a₀|<1

Formula:
Roots from quadratic formula, and Jury conditions:
D(1) = 1 - 1.5 + 0.7 > 0 ?
D(-1) = 1 + 1.5 + 0.7 > 0 ?
|a₀| < 1 ?

Substitution:
D(z) = z² + a₁z + a₀  →  a₁ = -1.5,  a₀ = 0.7

Calculation:
Jury condition 1: D(1) = 1 - 1.5 + 0.7 = 0.2 > 0  (satisfied)
Jury condition 2: D(-1) = 1 + 1.5 + 0.7 = 3.2 > 0  (satisfied)
Jury condition 3: |a₀| = |0.7| = 0.7 < 1  (satisfied)

Direct roots check:
z = [1.5 ± √(2.25 - 2.8)] / 2 = [1.5 ± √(-0.55)] / 2
z = 0.75 ± j0.37
|z| = √(0.75² + 0.37²) = √(0.5625 + 0.1369) = √0.6994 = 0.836 < 1

Final Answer:
All three Jury conditions satisfied. Both poles at |z| = 0.836 < 1.
System is ASYMPTOTICALLY STABLE.

Exam Tip: For second-order z-domain systems, memorize the three Jury conditions: D(1) > 0, D(-1) > 0, and |a₀| < a₂ (constant term < leading coefficient). For a monic polynomial z²+a₁z+a₀, the conditions simplify to D(1)>0, D(-1)>0, and |a₀|<1. These three conditions are necessary and sufficient for second-order systems.

• Unit circle criterion: a digital system is stable iff all poles of the closed-loop transfer function are strictly inside the unit circle (|zᵢ| < 1).
• Poles inside unit circle: stable decaying modes. On unit circle: sustained oscillation. Outside: unstable growing modes.
• Jury test is the discrete-time counterpart of Routh-Hurwitz. It determines stability without computing roots by checking array conditions.
• For second-order D(z) = z² + a₁z + a₀, three conditions apply: D(1) > 0, D(-1) > 0, and |a₀| < 1.
• The w-transform w = (z-1)/(z+1) maps the unit circle interior to the left half w-plane, enabling Bode/Nyquist analysis.

Quick Revision

• Stability condition: all closed-loop poles must satisfy |z| < 1 (inside the unit circle).
• Pole at |z| > 1: unstable. |z| = 1: marginally stable. |z| < 1: asymptotically stable.
• Jury test: algebraic test for all roots inside unit circle; no root computation needed.
• Second-order Jury conditions: D(1) > 0, D(-1) > 0, |a₀| < a₂ (or |a₀| < 1 for monic polynomial).
• Jury test complexity increases with order; for GATE, second and third-order cases are key.
• w-transform: w = (z-1)/(z+1) maps z-plane unit circle to w-plane LHP; enables standard frequency domain analysis.
• Exam trap: Routh-Hurwitz applied directly to D(z) does NOT give correct stability results for discrete systems. Always use Jury test or direct pole computation for z-domain stability.