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L04: FOPTD Transfer Functions

Prerequisites: L00-L03 | Effort: 45 min | Seborg: Chapter 4

Learning Objectives

By the end of this lesson you will:

  1. βœ… Understand FOPTD (First-Order Plus Time Delay) transfer function parameters
  2. βœ… Implement time delay using manual shift with multiple prev() calls
  3. βœ… Parameterize YAML configurations for reusability
  4. βœ… Validate step response characteristics (gain, time constant, delay)
  5. βœ… Use scenario flags for comparing different process configurations

Theory Recap: FOPTD Transfer Functions (Seborg Ch. 4)

The FOPTD model is the most important transfer function in process control:

G(s) = K Γ— e^(-ΞΈs) / (Ο„s + 1)

Parameters: - K = Process gain (steady-state change in output per unit input) - Example: K = 2.0 means 1% heater change β†’ 2Β°F temperature change - Ο„ = Time constant (how fast the process responds, in minutes) - 63.2% of final value reached at t = Ο„ - 98% reached at t β‰ˆ 4Ο„ (settling time) - ΞΈ = Dead time / time delay (lag before ANY response starts) - Example: Analyzer 50 ft from reactor β†’ 2-minute sample transport delay

Why FOPTD? - 80% of process units can be approximated as FOPTD - Simple to identify from step tests - Easy to tune controllers with (IMC, Direct Synthesis methods)

Discrete-time implementation: 

# At each timestep:
u_delayed = prev(prev(...prev(input)))  # Apply delay with nested prev()
y[k] = y[k-1] + (Ξ”t/Ο„) Γ— (K Γ— u_delayed - y[k-1])  # First-order lag

CRITICAL: Time Delay Implementation

⚠️ IMPORTANT: Odibi does NOT support prev(..., lag=N) syntax.

WRONG (does not work): 

expression: "prev('input', 0.0, lag=5)"  # ❌ This syntax doesn't exist!

CORRECT (manual shift with nested prev()): 

# For 2-timestep delay:
- name: input_delayed_1
  generator:
    type: derived
    expression: "prev('input', 0.0)"

- name: input_delayed
  generator:
    type: derived
    expression: "prev('input_delayed_1', 0.0)"

# Now use input_delayed (2 timesteps behind input)

For larger delays: - ΞΈ = 1 timestep: Use prev('input', default) directly - ΞΈ = 2 timesteps: 2 columns (as shown above) - ΞΈ = 3 timesteps: 3 columns (input β†’ delayed_1 β†’ delayed_2 β†’ delayed) - ΞΈ = 5+ timesteps: Consider reducing timestep or accepting limitation

Odibi Hands-On

Example 1: Tank Temperature with FOPTD

Process: Tank with heater, first-order thermal response plus transport delay

Parameters: - K = 0.5 Β°F/% (heater power to temperature gain) - Ο„ = 10 minutes (thermal time constant) - ΞΈ = 2 minutes (temperature sensor delay)

# tank_foptd.yaml
name: tank_temperature_foptd
engine: pandas

connections:
  output:
    type: local
    path: ./output/tank_foptd.parquet
    format: parquet

pipelines:
  - name: foptd_pipeline
    nodes:
      - name: generate_foptd_data
        read:
          connection: null
          format: simulation
          options:
            simulation:
              scope:
                start_time: "2024-01-01T00:00:00Z"
                timestep: "1min"
                row_count: 240  # 4 hours
                seed: 42

              entities:
                count: 1
                id_prefix: "TK-"

              columns:
                - name: entity_id
                  data_type: string
                  generator:
                    type: constant
                    value: "{entity_id}"

                - name: timestamp
                  data_type: timestamp
                  generator:
                    type: timestamp

                - name: minutes_elapsed
                  data_type: float
                  generator:
                    type: sequential
                    start: 0
                    step: 1

                # ─────────────────────────────────
                # INPUT: Heater power (step change)
                # ─────────────────────────────────
                - name: heater_pct
                  data_type: float
                  generator:
                    type: derived
                    expression: "60.0 if minutes_elapsed >= 30.0 else 50.0"

                # ─────────────────────────────────
                # TIME DELAY (ΞΈ = 2 minutes)
                # Manual implementation using nested prev()
                # ─────────────────────────────────
                - name: heater_delayed_1
                  data_type: float
                  generator:
                    type: derived
                    expression: "prev('heater_pct', 50.0)"

                - name: heater_delayed
                  data_type: float
                  generator:
                    type: derived
                    expression: "prev('heater_delayed_1', 50.0)"

                # ─────────────────────────────────
                # FIRST-ORDER LAG (K=0.5, Ο„=10)
                # ─────────────────────────────────
                - name: tank_temp_true_f
                  data_type: float
                  generator:
                    type: derived
                    expression: >
                      prev('tank_temp_true_f', 100.0) +
                      (1.0 / 10.0) *
                      (0.5 * (heater_delayed - 50.0) -
                       (prev('tank_temp_true_f', 100.0) - 100.0))

                # ─────────────────────────────────
                # SENSOR (with noise)
                # ─────────────────────────────────
                - name: sensor_noise
                  data_type: float
                  generator:
                    type: range
                    min: -0.3
                    max: 0.3
                    distribution: normal

                - name: tank_temp_measured_f
                  data_type: float
                  generator:
                    type: derived
                    expression: "tank_temp_true_f + sensor_noise"

        write:
          connection: output

Working example: /examples/cheme_course/L04_foptd/tank_foptd.yaml

Run:

cd examples/cheme_course/L04_foptd
odibi run tank_foptd.yaml

Example 2: Parameterized FOPTD

Working example: /examples/cheme_course/L04_foptd/foptd_parameterized.yaml

Note: Due to manual delay implementation, theta is limited to small values (0-3 timesteps typical).

Example 3: Heat Exchanger FOPTD

Process: Heat exchanger with negative gain (more coolant β†’ lower temperature)

Parameters: - K = -0.4 Β°F/gpm (negative gain!) - Ο„ = 8 minutes - ΞΈ = 3 minutes (sensor location)

Working example: /examples/cheme_course/L04_foptd/heat_exchanger_foptd.yaml

Exercises

See ../solutions/L04.md for full solutions.

All YAML files: /examples/cheme_course/L04_foptd/

Reflection: How This Relates to Real Plants

In a real plant: - Process engineers identify FOPTD models from step tests (bump tests) - K tells you sensitivity (high K = touchy process, hard to control) - Ο„ tells you speed (low Ο„ = fast, high Ο„ = slow/sluggish) - ΞΈ is often from measurement delays (analyzers, sample transport) - Controllers are tuned based on K, Ο„, ΞΈ (IMC, Lambda tuning)

Common pitfalls: - Ignoring dead time (ΞΈ) β†’ controller oscillates - Wrong gain sign (K positive vs negative) β†’ runaway - Time constant too large (Ο„ >> actual) β†’ sluggish control

What you just learned: - How to implement FOPTD in discrete time (manual delay with prev()) - Parameterize configs for reusability - Validate process characteristics from data - Dead time is CRITICAL for control (we'll see this in PID lesson)

Next Steps

You now know: - βœ… FOPTD transfer function (K, Ο„, ΞΈ) - βœ… Time delay implementation with manual shift (nested prev()) - βœ… Parameterized YAML configurations - βœ… Step response validation

Next lesson: πŸ‘‰ L05: Second-Order Systems + Overshoot

We'll learn damping ratio (ΞΆ), natural frequency (Ο‰n), and how to model underdamped oscillations.

Lesson L04 complete!