Digital Control and Implementation
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Digital Control and Implementation.
§7.1 Sampling rates and zero-order hold
**Mechanical Engineering · Control Systems Engineering · Unit 7: Digital Control and Implementation.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§7.2 Discrete-time models and z-transforms
**Mechanical Engineering · Control Systems Engineering · Unit 7: Digital Control and Implementation.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§7.3 Discrete stability and pole mapping
**Mechanical Engineering · Control Systems Engineering · Unit 7: Digital Control and Implementation.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§7.4 Digital PI/PID, saturation and computation delay
**Mechanical Engineering · Control Systems Engineering · Unit 7: Digital Control and Implementation.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§7.5 Quantization, filtering, latency and deployment
**Mechanical Engineering · Control Systems Engineering · Unit 7: Digital Control and Implementation.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
Step-by-Step Control Systems Engineering Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement:
Worked Engineering Derivation:
For a 5 Hz oscillation, a 12 Hz sample rate barely exceeds the Nyquist threshold; 100 Hz yields twenty samples per cycle and better waveform resolution. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Discrete-time models and z-transforms** under steady operational loading.
Worked Engineering Derivation:
For τ=2 s and h=0.1 s, a=e⁻⁰·⁰⁵≈0.951229 and the input coefficient for K=3 is ≈0.146312. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Discrete stability and pole mapping** under steady operational loading.
Worked Engineering Derivation:
A pole at s=−4 s⁻¹ with h=0.05 s maps to z=e⁻⁰·²≈0.81873, strictly inside the unit circle. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Digital PI/PID, saturation and computation delay** under steady operational loading.
Worked Engineering Derivation:
With K_i=0.4 s⁻¹, h=0.2 s and e[k]=5 units, one rectangular integrator update adds 0.4 actuator units. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Quantization, filtering, latency and deployment** under steady operational loading.
Worked Engineering Derivation:
A 12-bit ADC over 0–10 V has step size 10/4096≈2.44 mV. A plant with 0.05°C/mV sensor gain therefore has ≈0.122°C per code in this synthetic illustration. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.