Adaptive Control | Problem Bank

Problem Setup

Single-Axis Spacecraft with Unknown Inertia

This problem uses a simplified spacecraft attitude model. The true spacecraft inertia may be different from the inertia assumed by the controller.

$$ \dot{\theta} = \omega $$ $$ \dot{\omega} = \frac{\tau}{J_{true}} + d $$

The controller is designed using an assumed inertia:

$$ J_{assumed} \neq J_{true} $$

The main question: what happens when the controller is designed for the wrong spacecraft?

Why Fixed Gains Can Fail

Fixed Gains Assume the Model Is Correct

A fixed-gain controller can work well when the assumed model is close to the real spacecraft. But if inertia changes because of fuel usage, payload deployment, configuration changes, or modelling error, the same gains may become too weak or too aggressive.

correct model → expected response
wrong model → degraded response

Model Uncertainty

The Spacecraft May Not Match the Model

In real systems, mass properties, disturbance torques, actuator behaviour, and environmental effects are never known perfectly.

uncertainty sources:
inertia mismatch + disturbance torque + unmodelled dynamics

Adaptive control tries to compensate by changing the controller behaviour based on observed error.

Parameter Mismatch

True Inertia vs Assumed Inertia

The baseline controller uses $J_{assumed}$, but the plant evolves using $J_{true}$. This mismatch is the central teaching point of the simulation.

$$ \text{Mismatch} = \frac{|J_{true} - J_{assumed}|}{J_{true}} \times 100\% $$

Large inertia mismatch means the controller is not acting on the system it thinks it is controlling.

Adaptive Gain Logic

Changing the Controller During Operation

Adaptive control adds a correction term that evolves during the simulation. When tracking error and angular velocity indicate poor response, the adaptive term changes the effective control action.

fixed controller: gains stay constant
adaptive controller: correction evolves with tracking behaviour

Reference Model

What Response Should the Spacecraft Follow?

A reference model describes the desired behaviour. In this simplified page, the desired response is:

fast error reduction + controlled damping + no excessive torque

The adaptive correction attempts to reduce the difference between actual behaviour and the desired response pattern.

Adaptation Law

How the Adaptive Term Evolves

The baseline controller is:

$$ \tau = J_{assumed}(-K_p e - K_d \omega) $$

The adaptive correction evolves using:

$$ \dot{K}_{adapt} = \gamma e\omega $$

The adaptive controller becomes:

$$ \tau = J_{assumed}(-K_p e - K_d\omega) - K_{adapt}e $$

Adaptation helps with uncertainty, but too much adaptation can introduce oscillation or over-correction.

Interactive Demo

Compare Baseline and Adaptive Control

Adjust the true inertia, assumed inertia, disturbance torque, baseline gain, and adaptation rate. The simulation compares fixed-gain control against adaptive correction.

Initial Attitude Error: 1.0 rad

True Inertia Jtrue: 2.0

Assumed Inertia Jassumed: 1.0

Disturbance Torque: 0.12

Baseline Gain Kp: 3.0

Adaptation Rate γ: 1.2

Adjust sliders to test model mismatch, disturbance torque, and adaptation rate.

Engineering insights will appear after the simulation runs.

Engineering Interpretation

Why Adaptive Control Matters

Adaptive control is important when a spacecraft cannot be represented by one perfectly known model. This may happen during deployment, docking, fuel usage, payload release, actuator degradation, or environmental disturbance.

Model Mismatch

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Baseline Final Error

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Adaptive Final Error

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Max Adaptive Gain

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Max Adaptive Torque

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Takeaway

What This Problem Shows

Fixed-gain control assumes the model is known.
Adaptive control responds when the model is uncertain.

Model uncertainty → parameter mismatch → tracking error → adaptive correction

Adaptive control is powerful, but adaptation rate must be tuned carefully.