Digital control: what a sample costs

The loop runs on a microcontroller

• The controller runs in discrete time. Sampling, word length and computation time each cost phase margin, affordable only if they are counted.
• An ADC samples every $T$ seconds; a zero-order hold keeps the output constant until the next sample. On average that is a delay of $T/2$, costing $\omega T/2$ of phase.

Tustin and aliasing

• Tustin replaces $s$ by $\tfrac{2}{T}\tfrac{z - 1}{z + 1}$; for the integral this is the trapezium rule, $I_k = I_{k-1} + \tfrac T2(e_k + e_{k-1})$.
• Anything above the Nyquist frequency, half the sample rate, folds back into the band, so filter it out before the ADC.

Choosing the sample rate

Quantisation and latency

• The encoder gives 100 000 counts per valve turn: $\mathbf{0.0036°}$ per count, far below the 0.5° spec.
• Speed from counting at 1 kHz resolves only $\mathbf{3.14}$ rad/s at the motor, one more reason to avoid derivative action. A 0.2 ms computation delay adds 0.6° of lag.

The integral in code

• Tustin's integral at $T = 1$ ms is one line per sample: $I_k = I_{k-1} + 0.4965(e_k + e_{k-1})$, since $K_iT/2 = 993 \times 0.0005$. Then $u_k = 134e_k + I_k$, clamped to ±12 V.
• The clamp sits inside the loop: if $u_k$ saturates, do not store the new $I_k$. That is anti-windup in two lines of code.

Worked example: mains hum in the temperature

• Sample the tank temperature at 10 Hz. 50 Hz pickup on the sensor wires is exactly five times the sample rate, so every sample catches the hum at the same phase: it aliases to 0 Hz and looks like a constant offset.
• No later filter can remove it. An RC filter at 1 Hz before the ADC passes the slow temperature and cuts 50 Hz to $1/\sqrt{1 + 50^2} = \mathbf{0.020}$, 2% of its size.

Three digital slips

Recap