Sampled Data Systems

Modern control systems increasingly operate in digital environments where continuous physical signals must be processed by digital computers. A sampled data system is one in which signals are sampled at discrete time instants, processed digitally, and then reconstructed to drive the continuous plant. Understanding the ideal sampler, zero order hold, and pulse transfer function is essential for digital control system design and is a core GATE topic.

Core Concept of Sampled Data Systems

In a sampled data system, the continuous error signal e(t) is first converted into a sequence of impulses by an ideal sampler. The ideal sampler operates at a fixed sampling period T (equivalently, sampling frequency fₛ = 1/T Hz or ωₛ = 2π/T rad/s). The sampled signal e*(t) is a train of impulses whose strengths equal the values of e(t) at the sampling instants: e*(t) = Σ e(nT)·δ(t - nT).

The digital controller processes the discrete samples e*(nT) and produces a discrete output sequence. Since the plant is continuous, a zero order hold (ZOH) is used to reconstruct a staircase approximation of the analog signal from the discrete sequence. The ZOH holds each sample value constant for one sampling period T, making it the simplest and most commonly used data hold device in practice.

The sampling theorem (Nyquist criterion) states that to avoid aliasing, the sampling frequency must satisfy fₛ ≥ 2·fmax, where fmax is the highest frequency component in the signal. Aliasing causes high-frequency components to masquerade as low-frequency ones, degrading control performance. In practice, fₛ is chosen 10 to 20 times the system bandwidth.

Mathematical Expression

The transfer function of the ZOH in the Laplace domain is G_ZOH(s) = (1 - e^(-Ts))/s. This represents the action of holding a value for T seconds. To analyze the complete digital loop, the ZOH is combined with the plant Gp(s) to form the pulse transfer function G(z), obtained by taking the Z-transform of the combined ZOH-plant: G(z) = Z{(1 - e^(-Ts))/s · Gp(s)} = (1 - z⁻¹) · Z{Gp(s)/s}.

The closed-loop pulse transfer function is T(z) = G(z)·D(z) / (1 + G(z)·D(z)), where D(z) is the digital controller transfer function. This is entirely analogous to the continuous-time closed-loop formula, but in the z-domain. The z-transform plays the same role for discrete systems as the Laplace transform does for continuous systems.

Practical Understanding

The choice of sampling period T is a critical design decision. Too large a T (slow sampling) results in poor tracking, potential instability, and aliasing. Too small a T (fast sampling) increases computational load and quantization noise. For a system with bandwidth ωb, a rule of thumb is T ≤ π/(10·ωb), ensuring at least 10 samples per cycle of the highest relevant frequency.

In real systems, the ZOH introduces a transport delay of approximately T/2 seconds in the equivalent continuous-time representation. This is because the ZOH's staircase approximation lags the ideal signal by half a sampling interval on average. This effective delay degrades the phase margin of the continuous-time equivalent system and must be accounted for in stability analysis.

Solved Numerical Example

Consider a plant Gp(s) = 1/(s+1) with ZOH and sampling period T = 0.1 s. The pulse transfer function G(z) is found by taking the Z-transform of (1 - e^(-Ts))/s · 1/(s+1). The partial fraction expansion of 1/(s(s+1)) = 1/s - 1/(s+1) leads to Z-transform pairs that are standard table entries.

Given:
Gp(s) = 1/(s+1),  T = 0.1 s,  ZOH used

Why this formula applies:
G(z) = (1 - z⁻¹) · Z{Gp(s)/s}

Formula:
Gp(s)/s = 1/(s(s+1)) = 1/s - 1/(s+1)

Substitution:
Z{1/s} = z/(z-1)
Z{e⁻ᵀ/s} = Z{1/(s+1)} at T=0.1 → z/(z - e⁻⁰·¹) = z/(z - 0.9048)

Calculation:
G(z) = (1 - z⁻¹) · [z/(z-1) - z/(z-0.9048)]
     = (z-1)/z · z·[1/(z-1) - 1/(z-0.9048)]
     = 1 - (z-1)/(z-0.9048)
     = (z - 0.9048 - z + 1) / (z - 0.9048)
     = 0.0952 / (z - 0.9048)

Final Answer:
G(z) = 0.0952 / (z - 0.9048)  — first-order pulse transfer function

Exam Tip: In GATE problems, the pulse transfer function G(z) is computed using G(z) = (1 - z⁻¹)·Z{Gp(s)/s}. Memorize the Z-transform pairs: Z{1/s} = z/(z-1) and Z{1/(s+a)} = z/(z-e^(-aT)). The ZOH transfer function in s-domain is always (1 - e^(-Ts))/s.

• The ideal sampler converts e(t) into e*(t) = Σ e(nT)·δ(t - nT), a train of weighted impulses at multiples of T.
• The ZOH reconstructs a piecewise constant signal by holding each sample for T seconds, with Laplace TF = (1 - e^(-Ts))/s.
• The pulse transfer function G(z) combines the ZOH and plant and is found using G(z) = (1 - z⁻¹)·Z{Gp(s)/s}.
• Nyquist criterion requires fₛ ≥ 2·fmax. In control practice, fₛ is chosen 10-20 times the system bandwidth.
• The ZOH introduces an effective delay of T/2 in the continuous-time equivalent, which reduces phase margin.

Quick Revision

• Ideal sampler: e*(t) = Σ e(nT)·δ(t-nT); samples the continuous error at intervals T.
• ZOH transfer function: G_ZOH(s) = (1 - e^(-Ts))/s; holds each sample constant for T seconds.
• Pulse transfer function: G(z) = (1 - z⁻¹)·Z{Gp(s)/s}; use standard Z-transform pairs for computation.
• Nyquist criterion: fₛ ≥ 2·fmax. Aliasing occurs when this condition is violated.
• ZOH introduces equivalent time delay of T/2, reducing phase margin of the equivalent continuous system.
• Exam trap: The pulse TF G(z) is NOT simply substituting z = e^(sT) into Gp(s); the ZOH must be included.