Lab 04: Gyro Modeling and Control

Experiment Overview

Gyroscopes are at the heart of inertial navigation systems, spacecraft attitude control, and platform stabilization for cameras, antennas, and weapons systems. A spinning gyroscope resists angular displacement through gyroscopic precession — but without active feedback control, external disturbances cause the deflection angle to wander unpredictably. This lab modeled the open-loop gyroscope dynamics, designed a PI controller to regulate deflection angle, and experimentally demonstrated the dramatic difference between controlled and uncontrolled system behavior when the base is disturbed.

Derive the closed-loop transfer function for a PI-controlled rotary gyroscope from the open-loop plant model
Design PI gains to track a 2° deflection reference angle α with specified second-order transient behavior
Compare the open-loop and closed-loop responses to base rotation disturbances
Verify system stability for both configurations using the Routh-Hurwitz criterion

Open-loop a response with 63.2% cursor annotation – t=0.87 s identified experimentally — Figure 1: Open-loop α response — 63.2% crossing at τ=0.87 s

Open-loop a (alpha) – transfer function model vs. system output, showing model–hardware discrepancy — Figure 2: Open-loop α — transfer function model vs. hardware output

Open-loop theta – servo angle continuously increasing under constant voltage input — Figure 3: Open-loop θ — servo angle continuously increasing under constant voltage

Equipment & Tools

Quanser SRV02 with rotary gyroscope attachment (spinning disk on gimbal)
Gyroscope angle sensor for deflection angle α measurement
Servo encoder for base rotation angle θ measurement
Q2-USB data acquisition board
VoltPAQ power amplifier
MATLAB/Simulink with QUARC real-time toolbox

Approach & Controller Design

The open-loop gyroscope plant is a first-order system: α(s)/V_m(s) = (K/G_g) / (τs + 1), where K = 1.53 rad/s/V, τ = 0.9 s, and G_g = 5.20 1/(rad·s). Closing the loop with a PI controller V_m(s) = (k_p + k_i/s) · e(s) produced a second-order closed-loop transfer function. Matching its denominator coefficients to the standard form gave the PI gain expressions, solved with ζ = 1.05 and ω_n = 3.61 rad/s:

k_p = G_g(2ζω_nτ − 1) / K = 19.79

k_i = G_gτω_n² / K = 39.86

The experimental time constant was identified from the open-loop response: τ_exp = 0.87 s at the 63.2% crossing, compared to the theoretical 0.9 s — close agreement despite the gyroscope’s more complex dynamics.

Open-loop voltage input – constant 0.5 V applied to identify gyroscope time constant — Figure 4: Open-loop voltage input — constant 0.5 V for time constant identification

Closed-loop a – PI controller tracking 2° reference, ess=0.003° — Figure 5: Closed-loop α — PI controller tracking 2° reference, e_ss=0.003°

Closed-loop theta – servo angle ramping to generate gyroscopic torque — Figure 6: Closed-loop θ — servo angle ramping to generate gyroscopic torque

Closed-loop voltage — sinusoidal PI control effort reflecting gyroscopic coupling — Figure 7: Closed-loop voltage — sinusoidal PI control effort

Key Results

Open-loop: Theoretical prediction was α_ss = 8.43°, but measured peak was only 2.8° (66.8% error) — unmodeled gyroscopic coupling, nonlinear friction, and sensor noise caused the real response to oscillate rather than smoothly converge.
Closed-loop tracking: With PI control active, α converged to 1.997° (e_ss = 0.003° ≈ 0), demonstrating that the integral term successfully eliminated steady-state tracking error despite the model inaccuracies.
PI control ON — disturbance: Motor voltage responded dynamically to oppose base rotation; gyroscope deflection held at ~0° (e_ss = 0°). The base rotated continuously to generate the gyroscopic torque needed to maintain heading.
PI control OFF — disturbance: Deflection reached ±7.4° and took ~20 seconds to passively damp out — the gyroscope had no ability to maintain its heading without active feedback.
Routh-Hurwitz: Both open-loop (single pole at −1.11) and closed-loop (all Routh table entries positive for k_p = 19.79, k_i = 39.86) are stable.

Closed-loop a with data cursor – steady-state value 1.997°, ess=0.003° — Figure 8: Closed-loop α — steady-state 1.997°, e_ss=0.003°

Closed-loop with disturbance – a response: PI control holds heading near 0° during base rotation — Figure 9: Closed-loop disturbance — PI holds α near 0° during base rotation

Closed-loop with disturbance – theta transient during base disturbance, servo actively counteracting — Figure 10: Closed-loop disturbance — θ transient, servo actively counteracting

Closed-loop with disturbance – voltage control effort responding dynamically to reject disturbance — Figure 11: Closed-loop disturbance — voltage control effort rejecting disturbance

Open-loop with disturbance – a deflection spike to ±7.4° with no active control — Figure 12: Open-loop disturbance — α spike to ±7.4° with no active control

Open-loop with disturbance – theta oscillates and settles at new position without feedback correction — Figure 13: Open-loop disturbance — θ oscillates without feedback correction

Open-loop with disturbance – voltage remains constant at 0 V, no corrective action — Figure 14: Open-loop disturbance — voltage at 0 V, no corrective action

Open-loop disturbance – a annotated with peak 7.36° and trough -7.18°, 20 s passive damping — Figure 15: Open-loop disturbance — α peak 7.36°, trough −7.18°, 20 s passive damping

Valuable Takeaways

Passive gyroscopic stability is insufficient for precision heading control. The open-loop disturbance test showed 7.4° of deflection; the closed-loop held near 0°. This is why every real inertial navigation system and spacecraft attitude controller uses active feedback, not just a spinning mass.
The PI controller converts a passive gyroscope into an actively stabilized platform. This is the operating principle of gimbal stabilizers in camera drones, ship-mounted antennas, and the Hubble Space Telescope’s pointing system.
Open-loop model accuracy matters less when you have good feedback. The 66.8% discrepancy between the open-loop model and measured response did not prevent the PI controller from achieving near-zero tracking error — demonstrating the robustness that feedback control provides over model-dependent open-loop approaches.
Gained experience designing PI controllers for a multi-physics system involving gyroscopic precession, applying the Routh-Hurwitz stability criterion analytically, and validating controller performance experimentally under both tracking and disturbance conditions.