Z-Domain Analysis

Just as the Laplace transform is the fundamental tool for analyzing continuous-time control systems, the Z-transform serves the same role for discrete-time and digital control systems. The z-domain provides a way to analyze sampled signals and digital controllers using algebraic methods rather than difference equations, and the mapping from the s-domain to the z-domain connects discrete behavior to continuous-time poles and stability. This is a heavily tested topic in GATE.

Core Concept of Z-Domain Analysis

The Z-transform of a discrete-time sequence x[n] = x(nT) is defined as X(z) = Σ x(nT)·z⁻ⁿ for n = 0 to infinity, where z is a complex variable. This transforms a sequence described by a difference equation into an algebraic expression in z. The transfer function G(z) = Y(z)/U(z) of a digital system is the ratio of the Z-transform of the output to that of the input, with zero initial conditions, exactly analogous to the Laplace-domain transfer function G(s) for continuous systems.

The complex variable z is related to the Laplace variable s through the mapping z = e^(sT), where T is the sampling period. This is the fundamental relation connecting the s-domain (continuous) and z-domain (discrete) representations. Since s = σ + jω, we have z = e^(σT)·e^(jωT), meaning the magnitude of z is |z| = e^(σT) and the angle is ωT. This mapping is the key to understanding stability in the z-domain.

An important property is time shifting: Z{x[n-k]} = z⁻ᵏ·X(z). This means z⁻¹ represents a unit delay of one sampling period. This property converts difference equations (which involve delayed terms) into polynomial equations in z, making them easy to handle algebraically.

Mathematical Expression: s to z Mapping

The mapping z = e^(sT) transforms regions of the s-plane into corresponding regions of the z-plane. The left half of the s-plane (where Re(s) < 0, corresponding to stable poles in continuous systems) maps to the interior of the unit circle |z| < 1 in the z-plane. The imaginary axis jω of the s-plane maps to the unit circle |z| = 1. The right half s-plane maps to the exterior |z| > 1.

Because of the periodic nature of e^(jωT), the entire imaginary axis maps onto the unit circle repeatedly with period ωₛ = 2π/T. This means the z-plane mapping is many-to-one. One primary strip of the s-plane (a horizontal band of height ωₛ) maps uniquely to the entire z-plane, and all other strips are aliases, reinforcing the Nyquist condition.

Practical Understanding

In z-domain analysis, the poles and zeros of G(z) determine the system behavior. Poles inside the unit circle contribute decaying modes (stable). Poles outside cause growing oscillations (unstable). A pole exactly on the unit circle gives a marginally stable sustained oscillation. The step response, impulse response, and frequency response can all be computed from G(z) using the inverse Z-transform and final value theorem.

The final value theorem in the z-domain states that the steady-state value of x[n] as n approaches infinity is lim(n→∞) x[n] = lim(z→1) (z-1)·X(z), provided all poles of (z-1)·X(z) lie inside the unit circle. This is directly used in GATE problems to find steady-state error of digital control systems.

Solved Numerical Example

A digital system has the transfer function G(z) = (0.5z) / (z² - 1.5z + 0.5). To find the poles and determine stability, and to find the steady-state output for a unit step input, apply z-domain analysis as follows.

Given:
G(z) = 0.5z / (z² - 1.5z + 0.5)
Input: unit step U(z) = z/(z-1)

Why this formula applies:
Pole locations determine stability (inside unit circle = stable)
Final value theorem gives steady-state output

Formula:
Poles: roots of z² - 1.5z + 0.5 = 0
Final value: lim(z→1) (z-1)·G(z)·U(z)

Substitution:
z² - 1.5z + 0.5 = 0
z = [1.5 ± √(2.25 - 2)] / 2 = [1.5 ± 0.5] / 2

Calculation:
z₁ = 1.0,   z₂ = 0.5
z₁ = 1 → on unit circle (marginally stable mode)
z₂ = 0.5 → inside unit circle (stable mode)

Steady-state: lim(z→1) (z-1) · [0.5z/(z²-1.5z+0.5)] · [z/(z-1)]
= lim(z→1) 0.5z²/(z-0.5)(z-1) · (z-1)/(1)
Note: (z²-1.5z+0.5) = (z-1)(z-0.5)
= lim(z→1) 0.5z² / (z-0.5)
= 0.5(1) / (1-0.5) = 0.5/0.5 = 1

Final Answer:
Poles at z = 1 and z = 0.5. System is marginally stable.
Steady-state output = 1 (unit gain at DC)

Exam Tip: For GATE z-domain problems, always first factor the denominator to find poles. Check |z| vs 1 for stability. For steady-state error, use the z-domain final value theorem: lim(z→1)(z-1)X(z). Remember z = e^(sT): stable s-plane poles (LHP) map to inside the unit circle in z-plane.

• Z-transform: X(z) = Σ x(nT)·z⁻ⁿ converts discrete sequences to algebraic expressions, enabling transfer function analysis.
• The mapping z = e^(sT) connects s-domain and z-domain. Stable LHP poles (σ < 0) map to inside the unit circle (|z| < 1).
• Time-shift property: Z{x[n-k]} = z⁻ᵏ·X(z), so z⁻¹ acts as a unit delay operator in digital systems.
• Poles inside unit circle: stable. On unit circle: marginally stable. Outside unit circle: unstable.
• Final value theorem: lim(n→∞) x[n] = lim(z→1) (z-1)·X(z), valid when poles of (z-1)X(z) are inside unit circle.

Quick Revision

• Z-transform definition: X(z) = Σ x(nT)·z⁻ⁿ, n from 0 to ∞.
• Key mapping: z = e^(sT). LHP → inside unit circle. jω-axis → unit circle. RHP → outside unit circle.
• Time delay property: Z{x[n-k]} = z⁻ᵏ·X(z); the z⁻¹ operator represents one sample delay.
• Common pairs: Z{unit step} = z/(z-1), Z{e^(-anT)} = z/(z-e^(-aT)), Z{impulse} = 1.
• Stability criterion: all poles of G(z) must be strictly inside the unit circle (|z| < 1).
• Final value theorem: lim(z→1)(z-1)X(z); useful for computing steady-state error.
• Exam trap: Do not simply substitute z = e^(sT) to find G(z) from G(s); use ZOH formulation for proper discretization.