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L05: Second-Order Systems + Overshoot

Prerequisites: L00-L04 | Effort: 60 min | Seborg: Chapter 5

Learning Objectives

By the end of this lesson you will:

  1. βœ… Understand second-order system parameters (ΞΆ, Ο‰n)
  2. βœ… Implement underdamped responses using state-space form
  3. βœ… Calculate overshoot, peak time, and settling time from data
  4. βœ… Validate transient response metrics
  5. βœ… Recognize overdamped vs critically damped vs underdamped behavior

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

Standard second-order transfer function:

G(s) = Ο‰nΒ² / (sΒ² + 2ΞΆΟ‰nΒ·s + Ο‰nΒ²)

Parameters: - Ο‰n = Natural frequency (rad/time) - how fast the system oscillates - ΞΆ = Damping ratio (dimensionless) - determines overshoot

Damping ratio categories: - ΞΆ > 1: Overdamped - slow, no overshoot (like opening a heavy door) - ΞΆ = 1: Critically damped - fastest response without overshoot - 0 < ΞΆ < 1: Underdamped - fast with overshoot (like a car suspension) - ΞΆ = 0: Undamped - pure oscillation (spring with no friction)

Key metrics for underdamped (ΞΆ < 1):

Overshoot: 

Mp = exp(-π΢ / √(1-ΞΆΒ²))
Example: ΞΆ = 0.3 β†’ Mp β‰ˆ 37%

Peak time: 

tp = Ο€ / (Ο‰n√(1-ΞΆΒ²))

Settling time (2% criterion): 

ts β‰ˆ 4 / (ΞΆΟ‰n)

Why second-order matters: - Pressure gauges with damping - U-tube manometers - Mechanical systems (valves, actuators) - Nested control loops (cascade)

Odibi Hands-On

Example 1: Underdamped Pressure Response (ΞΆ = 0.3)

Process: Pressure transmitter with mechanical damping

State-space implementation: 

dx₁/dt = xβ‚‚                           (velocity)
dxβ‚‚/dt = -Ο‰nΒ²x₁ - 2ΞΆΟ‰nΒ·xβ‚‚ + Ο‰nΒ²u     (acceleration)
y = x₁                                 (output)


# pressure_underdamped.yaml
name: pressure_second_order
engine: pandas

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

pipelines:
  - name: pressure_pipeline
    nodes:
      - name: generate_pressure_data
        read:
          connection: null
          format: simulation
          options:
            simulation:
              scope:
                start_time: "2024-01-01T00:00:00Z"
                timestep: "10sec"
                row_count: 600  # 100 minutes
                seed: 42

              entities:
                count: 1
                id_prefix: "PT-"

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

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

                - name: seconds_elapsed
                  data_type: float
                  generator:
                    type: sequential
                    start: 0
                    step: 10

                # ─────────────────────────────────
                # PARAMETERS
                # ─────────────────────────────────
                - name: zeta
                  data_type: float
                  generator:
                    type: constant
                    value: 0.3  # Damping ratio

                - name: omega_n
                  data_type: float
                  generator:
                    type: constant
                    value: 0.0167  # rad/sec (1.0 rad/min)

                # ─────────────────────────────────
                # INPUT: Pressure step change
                # ─────────────────────────────────
                - name: true_pressure_psi
                  data_type: float
                  generator:
                    type: derived
                    expression: "10.0 if seconds_elapsed >= 600.0 else 0.0"

                # ─────────────────────────────────
                # STATE VARIABLES (state-space)
                # ─────────────────────────────────

                # Velocity (dxβ‚‚/dt)
                - name: velocity
                  data_type: float
                  generator:
                    type: derived
                    expression: >
                      prev('velocity', 0.0) +
                      10.0 *
                      (-omega_n * omega_n * prev('pressure_reading_psi', 0.0) -
                       2.0 * zeta * omega_n * prev('velocity', 0.0) +
                       omega_n * omega_n * true_pressure_psi)

                # Position (output)
                - name: pressure_reading_psi
                  data_type: float
                  generator:
                    type: derived
                    expression: "prev('pressure_reading_psi', 0.0) + 10.0 * velocity"

        write:
          connection: output

Working example: /examples/cheme_course/L05_second_order/pressure_underdamped.yaml

Expected behavior: - Step input at t=600 sec (10 min) - Response overshoots by ~37% (for ΞΆ=0.3) - Oscillates before settling

Example 2: Damping Ratio Comparison

Compare underdamped, critically damped, and overdamped:

Working example: /examples/cheme_course/L05_second_order/damping_comparison.yaml

Observations: - Underdamped (ΞΆ=0.3): Fastest but overshoots ~37% - Critical (ΞΆ=1.0): No overshoot, reasonably fast - Overdamped (ΞΆ=2.0): Slow, no overshoot

Example 3: Valve Actuator Model

Process: Control valve with second-order dynamics (ΞΆ=0.7, fast response)

Working example: /examples/cheme_course/L05_second_order/valve_actuator.yaml

Typical for industrial valves: - ΞΆ β‰ˆ 0.7 (slightly underdamped) - Ο‰n β‰ˆ 3 rad/s (fast response) - Small overshoot (~5%) - Settling time ~1-2 seconds

Exercises

See ../solutions/L05.md for full solutions.

All YAML files: /examples/cheme_course/L05_second_order/

Reflection: How This Relates to Real Plants

In a real plant: - Pressure gauges often underdamped (oscillate after disturbances) - Control valves tuned for critical damping (fast, no overshoot) - Level controllers can be overdamped (slow but stable) - Cascade loops create second-order behavior (inner + outer loop)

Design trade-offs: - Fast response β†’ small ΞΆ β†’ overshoot (bad for safety-critical) - No overshoot β†’ large ΞΆ β†’ slow response (bad for throughput) - Optimal: ΞΆ β‰ˆ 0.7 (compromise: fast with ~5% overshoot)

What you just learned: - State-space implementation (position + velocity) - Damping ratio determines overshoot - KPI calculation from time-series data - Metrics tables for reporting

When you see plant data: - Oscillation after setpoint change β†’ likely underdamped (ΞΆ < 1) - Slow, sluggish response β†’ overdamped (ΞΆ > 1) or large Ο„ - Measure overshoot to estimate ΞΆ: ΞΆ β‰ˆ -ln(Mp) / √(π² + lnΒ²(Mp))

Next Steps

You now know: - βœ… Second-order systems (ΞΆ, Ο‰n) - βœ… Underdamped vs critically damped vs overdamped - βœ… State-space implementation in discrete time - βœ… Computing transient metrics (overshoot, peak time, settling time)

You've completed Part I: Foundations! πŸŽ‰

Next lesson: πŸ‘‰ L06: PID Basics + Constraints

Now we start Part II: Feedback Control. You'll implement PID controllers using Odibi's pid() transformer.

Lesson L05 complete! Ready for Part II: Control!