Digital Control Systems: Lecture Notes (KPI course, 2014)

This 87-page course is the supplied source for the digital half of TAU. The file has no title page, institution line, author name, or bibliography; PDF metadata contains only the informal author string Sany. The registry therefore does not invent authorship. The lecture sequence and terminology identify it as course material, but every technical claim used in GTK is checked against named, current sources.

Lecture map #

The lecture sequence is unusually valuable because it does not stop at “replace $s$ with $z$.” It follows the full chain from a continuously evolving plant and intersample hold to a causal difference equation that can actually run. It also exposes why a rational $D(z)$ is not yet an implementation: the transfer function must be realized as state updates with a defined evaluation order.

Mechanisms worth preserving #

The course's central model is the hybrid loop. Sampling creates a sequence, but the plant continues between sample instants. A zero-order hold turns each command into a piecewise-constant waveform; its dynamics and computation delay are part of the loop. The pulse transfer function describes the sample-to-sample map for a specified sampler/hold convention, not the entire intersample trajectory.

The stability map is another durable idea. Under $z=e^{sT}$, the continuous left half-plane maps inside the unit circle, the imaginary axis maps to the unit circle, and the right half-plane maps outside. This makes unit-circle pole location the direct discrete-time analogue of left-half-plane poles, while also showing how continuous frequencies wrap around the unit circle and why Nyquist frequency is a hard boundary.

Direct synthesis is retained with explicit realizability conditions. If a desired closed-loop map implies an improper, noncausal, internally unstable, or fragile pole-zero-cancelling controller, the algebraic solution is not a valid design. A deadbeat response is finite in sample count only under ideal model conditions; it can demand large control moves and be highly sensitive to unmodeled zeros, delay, saturation, and quantization.

What must be replaced or extended #

• Sampling terminology in the notes sometimes groups time sampling, amplitude quantization, relay behavior, and digital coding too loosely. GTK keeps them distinct because they have different dynamics and failure modes.
• Statements about microprocessor speed and real-time capability are historical. The current design variable is measured worst-case sample-to-actuation latency and jitter on the target runtime.
• The notes do not provide a complete anti-alias filtering, reconstruction, intersample, multirate, or network-delay treatment.
• Numerical realizations are presented algebraically but not fully audited for coefficient quantization, state scaling, accumulator width, overflow policy, rounding mode, limit cycles, and hot state transfer.
• Exact cancellation of unstable or nonminimum-phase plant dynamics is not treated as acceptable. GTK fails such a design closed and preserves the forbidden dynamics in the sensitivity analysis.

How it feeds the bank #

The course seeds four control-specific topics beyond the general z-transform: sampled-data loop modeling, discrete stability, direct digital synthesis, and executable controller realizations. Current MathWorks documentation supplies exact ZOH semantics, conversion-method limits, and finite-precision behavior; MIT and Stanford course materials supply independent derivations and design examples; ARM CMSIS-DSP documentation supplies a current concrete realization boundary.

Visual audit #

All 87 pages were rendered and inspected. Page 24 has no extractable text but contains visual content, so it was included in the audit. The important visuals are the sampler–ADC–controller–DAC/hold–plant path, sequence and quantization plots, $s$-plane to $z$-plane mappings, unit-circle frequency loci, sample-period comparisons, and direct/cascade/parallel realization diagrams.

No source visual is copied. The original diagrams mix notation from several conventions and often become unreadable at article width. GTK independently redraws the hybrid signal path, exponential $s\mapsto z$ map, finite-settling response, and cascaded second-order realization. Each diagram identifies continuous versus discrete signals, includes sample-time units, and distinguishes a mathematical delay $z^{-1}$ from wall-clock computation latency.