L03: First-Order Dynamics

Prerequisites: L02 | Effort: 45 min | Seborg: Chapter 5

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Learning Objectives

By the end of this lesson you will:

1. ✅ Understand first-order system behavior (exponential response)
2. ✅ Implement first-order dynamics using prev() and ema()
3. ✅ Calculate time constants from data
4. ✅ Use exponential moving averages (EMA) for filtering
5. ✅ Generate KPIs from time-series data (settling time, rise time)

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Theory Recap: First-Order Systems (Seborg Ch. 5)

What is a First-Order System?

Definition: A system whose behavior is described by a single time constant τ (tau)

Transfer function:

G(s) = K / (τs + 1)

Where: - K = Process gain (steady-state output/input ratio) - τ = Time constant (time to reach 63.2% of final value)

Step response:

y(t) = K × (1 - e^(-t/τ))

Key properties: - At t = τ: reaches 63.2% of final value - At t = 3τ: reaches 95% of final value - At t = 5τ: reaches 99.3% of final value (considered "settled")

Discrete-Time Implementation:

Continuous ODE:

τ dy/dt + y = K×u

Discrete form (Euler):

y[k] = y[k-1] + (Δt/τ) × (K×u[k] - y[k-1])

This is what we implement with prev() in Odibi!

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Odibi Hands-On

Example 1: Tank Temperature (First-Order Response)

Process: Steam-heated tank - Input: Steam flow rate (step change) - Output: Tank temperature (first-order response) - Time constant τ = 10 minutes

# tank_temperature_fo.yaml
name: first_order_tank
engine: pandas

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

pipelines:
  - name: fo_tank_pipeline
    nodes:
      - name: generate_temperature_response
        read:
          connection: null
          format: simulation
          options:
            simulation:
              scope:
                start_time: "2024-01-01T00:00:00Z"
                timestep: "1min"
                row_count: 600  # 10 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

                # ──────────────────────────────────
                # PROCESS PARAMETERS
                # ──────────────────────────────────
                - name: time_constant_min
                  data_type: float
                  generator:
                    type: constant
                    value: 10.0

                - name: process_gain
                  data_type: float
                  generator:
                    type: constant
                    value: 2.0

                # ──────────────────────────────────
                # INPUT (step change at t=60 min)
                # ──────────────────────────────────
                - name: minutes_elapsed
                  data_type: float
                  generator:
                    type: sequential
                    start: 0
                    step: 1

                - name: steam_flow_lb_hr
                  data_type: float
                  generator:
                    type: derived
                    expression: "150.0 if minutes_elapsed >= 60.0 else 100.0"

                # ──────────────────────────────────
                # OUTPUT (first-order response)
                # ──────────────────────────────────
                - name: tank_temp_f
                  data_type: float
                  generator:
                    type: derived
                    expression: >
                      prev('tank_temp_f', 200.0) +
                      (1.0 / time_constant_min) *
                      (process_gain * steam_flow_lb_hr - prev('tank_temp_f', 200.0))

                # ──────────────────────────────────
                # SENSOR (with measurement noise)
                # ──────────────────────────────────
                - name: noise
                  data_type: float
                  generator:
                    type: range
                    min: -0.5
                    max: 0.5
                    distribution: normal

                - name: temp_sensor_f
                  data_type: float
                  generator:
                    type: derived
                    expression: "tank_temp_f + noise"

                # ──────────────────────────────────
                # KPIs (calculated)
                # ──────────────────────────────────
                - name: percent_of_final_value
                  data_type: float
                  generator:
                    type: derived
                    expression: >
                      0.0 if minutes_elapsed < 60.0 else
                      ((tank_temp_f - 200.0) / 100.0) * 100.0

        write:
          connection: output

Working example: /examples/cheme_course/L03_first_order/tank_temperature_fo.yaml

Expected output: - At t=60: step input occurs - At t=70 (τ later): reached 63.2% of new value - At t=90 (3τ later): reached 95% - At t=110 (5τ later): essentially settled

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Example 2: Using ema() for Filtering

Use case: Noisy sensor data → smooth with exponential moving average

EMA formula:

y_filtered[k] = α × y_raw[k] + (1-α) × y_filtered[k-1]

Where α = smoothing factor (0 to 1): - α = 1.0: No filtering (raw data) - α = 0.1: Heavy filtering (slow response) - α relates to time constant: α ≈ Δt/τ

Working example: /examples/cheme_course/L03_first_order/ema_filtering.yaml

Trade-off: - More filtering (lower α) → Less noise BUT more lag - Less filtering (higher α) → More responsive BUT noisier

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Exercises

See ../solutions/L03.md for full solutions.

All YAML files: /examples/cheme_course/L03_first_order/

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Next Steps

You now know: - ✅ First-order system theory and discrete implementation - ✅ prev() and ema() stateful functions - ✅ Exponential response characteristics (63.2%, 95%, settling) - ✅ KPI calculations from time-series

Next lesson: 👉 L04: FOPTD Transfer Functions

We'll add process gain (K) and dead time (θ) to create realistic FOPTD models with parameterized YAML.

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Lesson L03 complete!