Model Reference Adaptive Control (MRAC)

🚧 Under Development This tutorial is actively being updated. Check back for more content and refinements.

Introduction

Model Reference Adaptive Control (MRAC) is a powerful technique for controlling systems with unknown or slowly varying parameters. Instead of designing a controller for a fixed plant model, MRAC designs a controller that adapts its own parameters in real time so the closed-loop plant behaves like a desired reference model.

This tutorial covers two approaches:

Lyapunov-based Gradient — Guaranteed stability, simpler to implement
MIT Rule — Gradient-based, historically significant

Try the interactive MRAC Demo while reading this tutorial — adjust plant parameters mid-simulation and watch the controller adapt.

The Problem Setup

Plant (Unknown Parameters)

We have a first-order SISO plant with unknown parameters $a$ and $b$ :

\dot{y} = -a \cdot y + b \cdot u

Symbol	Meaning
$y$	Plant output
$u$	Control input
$a$	Unknown pole location (stable if $a > 0$ )
$b$	Unknown input gain

Reference Model (Desired Behavior)

We want the plant to behave like this reference model with known parameters $a_m$ and $b_m$ :

\dot{y}_m = -a_m \cdot y_m + b_m \cdot r

Symbol	Meaning
$y_m$	Reference model output
$r$	Reference command input
$a_m$	Desired pole location
$b_m$	Desired DC gain

Control Law Structure

The adaptive controller has the structure:

u = \theta_1 \cdot r + \theta_2 \cdot y

where $\theta_1$ and $\theta_2$ are adaptive parameters that will be updated in real time.

Derivation: Lyapunov-Based Approach

Step 1: Define the Error

The tracking error is the difference between the plant output and the reference model output:

e = y_m - y

We want $e \to 0$ as $t \to \infty$ .

Step 2: Find the Ideal Parameters

If the plant parameters $a$ and $b$ were known, we could choose $\theta_1$ and $\theta_2$ so that the closed-loop dynamics match the reference model exactly.

Substitute the control law $u = \theta_1 \cdot r + \theta_2 \cdot y$ into the plant:

\dot{y} = -a \cdot y + b \cdot (\theta_1 \cdot r + \theta_2 \cdot y) = -(a - b\theta_2) \cdot y + b\theta_1 \cdot r

To match $\dot{y}_m = -a_m \cdot y_m + b_m \cdot r$ , we need:

a - b\theta_2^* = a_m \quad \Rightarrow \quad \theta_2^* = \frac{a - a_m}{b}

b\theta_1^* = b_m \quad \Rightarrow \quad \theta_1^* = \frac{b_m}{b}

These $\theta_1^*$ and $\theta_2^*$ are the ideal parameters that would give perfect tracking if the plant were known.

Step 3: Error Dynamics with Adaptive Parameters

Since $a$ and $b$ are unknown, the ideal parameters $\theta_1^*$ and $\theta_2^*$ cannot be computed. Instead, we use the adaptive parameters $\theta_1$ and $\theta_2$ from the control law and define the parameter errors as the gap between the current adaptive values and their ideal counterparts:

\tilde{\theta}_1 = \theta_1 - \theta_1^* \quad \text{and} \quad \tilde{\theta}_2 = \theta_2 - \theta_2^*

Differentiate the tracking error:

\dot{e} = \dot{y}_m - \dot{y} = (-a_m y_m + b_m r) - (-a y + b u)

Substitute the control law $u = \theta_1 r + \theta_2 y$ :

\dot{e} = -a_m y_m + b_m r + a y - b(\theta_1 r + \theta_2 y)

Rearrange:

\dot{e} = -a_m(y_m - y) + (b_m - b\theta_1)r + (a - b\theta_2)y

Now substitute $\theta_1 = \tilde{\theta}_1 + \theta_1^*$ and $\theta_2 = \tilde{\theta}_2 + \theta_2^*$ , using the ideal parameter relationships:

b_m - b\theta_1 = b_m - b(\tilde{\theta}_1 + \theta_1^*) = -b\tilde{\theta}_1

a - b\theta_2 = a - b(\tilde{\theta}_2 + \theta_2^*) = a_m - b\tilde{\theta}_2

So:

\dot{e} = -a_m e - b\tilde{\theta}_1 r + (a_m - b\tilde{\theta}_2)y - a_m y

Simplify the $y$ terms:

\dot{e} = -a_m e - b\tilde{\theta}_1 r - b\tilde{\theta}_2 y

This is the error dynamics in terms of the parameter errors:

\boxed{\dot{e} = -a_m e - b(\tilde{\theta}_1 r + \tilde{\theta}_2 y)}

Step 4: Choose a Lyapunov Function

To guarantee stability, we choose a Lyapunov function that includes both the tracking error and the parameter errors:

V = \frac{1}{2}e^2 + \frac{b}{2\gamma_1}\tilde{\theta}_1^2 + \frac{b}{2\gamma_2}\tilde{\theta}_2^2

where $\gamma_1, \gamma_2 > 0$ are adaptation gains that control how fast the parameters adapt.

Step 5: Compute $\dot{V}$ and Choose Adaptation Laws

Differentiate $V$ :

\dot{V} = e \cdot \dot{e} + \frac{b}{\gamma_1}\tilde{\theta}_1 \cdot \dot{\tilde{\theta}}_1 + \frac{b}{\gamma_2}\tilde{\theta}_2 \cdot \dot{\tilde{\theta}}_2

Since $\theta_1^*$ and $\theta_2^*$ are constant, $\dot{\tilde{\theta}}_1 = \dot{\theta}_1$ and $\dot{\tilde{\theta}}_2 = \dot{\theta}_2$ .

Substitute $\dot{e} = -a_m e - b(\tilde{\theta}_1 r + \tilde{\theta}_2 y)$ :

\dot{V} = e \cdot [-a_m e - b(\tilde{\theta}_1 r + \tilde{\theta}_2 y)] + \frac{b}{\gamma_1}\tilde{\theta}_1 \cdot \dot{\theta}_1 + \frac{b}{\gamma_2}\tilde{\theta}_2 \cdot \dot{\theta}_2

Expand:

\dot{V} = -a_m e^2 - b\tilde{\theta}_1 e r - b\tilde{\theta}_2 e y + \frac{b}{\gamma_1}\tilde{\theta}_1 \dot{\theta}_1 + \frac{b}{\gamma_2}\tilde{\theta}_2 \dot{\theta}_2

Group the $\tilde{\theta}_1$ and $\tilde{\theta}_2$ terms:

\dot{V} = -a_m e^2 + \tilde{\theta}_1 \left(\frac{b}{\gamma_1}\dot{\theta}_1 - b e r\right) + \tilde{\theta}_2 \left(\frac{b}{\gamma_2}\dot{\theta}_2 - b e y\right)

Step 6: Choose Adaptation Laws to Make $\dot{V} \leq 0$

To ensure stability, we want $\dot{V} \leq 0$ . The first term $-a_m e^2$ is already negative semi-definite (since $a_m > 0$ ). We can make the remaining terms zero by choosing:

\frac{b}{\gamma_1}\dot{\theta}_1 - b e r = 0 \quad \Rightarrow \quad \boxed{\dot{\theta}_1 = \gamma_1 e r}

\frac{b}{\gamma_2}\dot{\theta}_2 - b e y = 0 \quad \Rightarrow \quad \boxed{\dot{\theta}_2 = \gamma_2 e y}

With these adaptation laws:

\dot{V} = -a_m e^2 \leq 0

This proves that:

The tracking error $e$ converges to zero
The parameter errors $\tilde{\theta}_1$ , $\tilde{\theta}_2$ remain bounded
The adaptive system is globally stable

Complete Lyapunov-Based MRAC Algorithm

Initialize: $y(0) = 0$ , $y_m(0) = 0$ , $\theta_1(0) = 0$ , $\theta_2(0) = 0$

At each time step $t$ :

Read reference: $r(t)$
Compute control:
$u = \theta_1 \cdot r + \theta_2 \cdot y$
Integrate plant and reference model:
$\dot{y} = -a \cdot y + b \cdot u \quad \text{(unknown true plant)}$ $\dot{y}_m = -a_m \cdot y_m + b_m \cdot r$
Compute error:
$e = y_m - y$
Update adaptive parameters:
$\dot{\theta}_1 = \gamma_1 \cdot e \cdot r$ $\dot{\theta}_2 = \gamma_2 \cdot e \cdot y$

Tuning the Adaptation Gains $\gamma$

Gain	Effect	Typical Value
$\gamma_1$	Speed of $\theta_1$ adaptation	0.5–2.0
$\gamma_2$	Speed of $\theta_2$ adaptation	0.5–2.0

Too low ( $\gamma \ll 0.1$ ): Adaptation is sluggish; error decays slowly.

Too high ( $\gamma \gg 5$ ): Parameters oscillate; may cause instability with large $r$ or $y$ .

Best practice: Start with $\gamma_1 = \gamma_2 = 1.0$ and adjust based on response.

MIT Rule: Derivation and Comparison

The MIT rule is the original approach to model-reference adaptive control, developed at the Instrumentation Laboratory at MIT. It is based on gradient descent on the squared error.

The MIT Rule

Let $e$ be the error between the output of the closed-loop system and the output of the reference model. One possibility is to adjust parameters in such a way that the loss function

J(\theta) = \frac{1}{2} e^2

is minimized. The gradient method gives

\boxed{\dot{\theta} = -\gamma \frac{\partial J}{\partial \theta} = -\gamma e \frac{\partial e}{\partial \theta}}

The partial derivative $\frac{\partial e}{\partial \theta}$ , which is called the sensitivity derivative of the system, tells how the error is influenced by the adjustable parameter. If it is assumed that the parameter changes are slower than the other variables in the system, then the derivative $\frac{\partial e}{\partial \theta}$ can be evaluated under the assumption that $\theta$ is constant.

MIT Rule for First-Order Systems (Example 5.2)

Consider the first-order plant:

\dot{y} = -a y + b u

where $u$ is the control variable and $y$ is the measured output. Assume that we want to obtain a closed-loop system described by

\dot{y}_m = -a_m y_m + b_m r

Let the controller be given by

u = \theta_1 r - \theta_2 y

The controller has two parameters. If they are chosen to be

\theta_1^* = \frac{b_m}{b}, \quad \theta_2^* = \frac{a - a_m}{b}

the input-output relations of the system and the model are the same. This is called perfect model-following.

To apply the MIT rule, introduce the error

e = y - y_m

where $y$ denotes the output of the closed-loop system. It follows from the plant and controller equations that

y = \frac{b}{p + a + b\theta_2} \cdot u_c

where $p = d/dt$ is the differential operator. The sensitivity derivatives are obtained by taking partial derivatives with respect to the controller parameters $\theta_1$ and $\theta_2$ :

\frac{\partial e}{\partial \theta_1} = \frac{b}{p + a + b\theta_2} \cdot r

\frac{\partial e}{\partial \theta_2} = -\frac{b}{p + a + b\theta_2} \cdot y

These formulas cannot be used directly because the process parameters $a$ and $b$ are not known. Approximations are therefore required. One possible approximation is based on the observation that $p + a + b\theta_2 \approx p + a_m$ when the parameters give perfect model-following. We will therefore use the approximation

p + a + b\theta_2 \approx p + a_m

which will be reasonable when parameters are close to their correct values. With this approximation we get the following equations for updating the controller parameters:

\boxed{\dot{\theta}_1 = \gamma \cdot e \cdot \frac{a_m}{p + a_m} \cdot r}

\boxed{\dot{\theta}_2 = -\gamma \cdot e \cdot \frac{a_m}{p + a_m} \cdot y}

In these equations we have combined the parameters $\gamma'$ (the raw adaptation gain from the MIT rule) and $a_m$ with the adaptation gain $\gamma$ , since they appear as the product $\gamma' b/a_m$ . The sign of parameter $b$ must be known to have the correct sign of $\gamma$ . Notice that the filter has also been normalized so that its steady-state gain is unity.

(This follows Example 5.2 in Åström & Wittenmark, 2008, p. 190–191.)

Filter Implementation

The operator $\frac{a_m}{p + a_m}$ represents a first-order filter with unity steady-state gain. In state-space form:

\dot{x}_1 = -a_m x_1 + a_m r

\dot{x}_2 = -a_m x_2 + a_m y

The adaptation laws become:

\dot{\theta}_1 = -\gamma \cdot e \cdot x_1

\dot{\theta}_2 = \gamma \cdot e \cdot x_2

Complete MIT Rule Algorithm

Initialize: $y(0) = 0$ , $y_m(0) = 0$ , $\theta_1(0) = 0$ , $\theta_2(0) = 0$ , $x_1(0) = 0$ , $x_2(0) = 0$

At each time step $t$ :

Read reference: $r(t)$
Compute control:
$u = \theta_1 \cdot r - \theta_2 \cdot y$
Integrate plant and reference model:
$\dot{y} = -a \cdot y + b \cdot u$ $\dot{y}_m = -a_m \cdot y_m + b_m \cdot r$
Update filters:
$\dot{x}_1 = -a_m x_1 + a_m r$ $\dot{x}_2 = -a_m x_2 + a_m y$
Compute error:
$e = y - y_m$
Update adaptive parameters:
$\dot{\theta}_1 = -\gamma \cdot e \cdot x_1$ $\dot{\theta}_2 = \gamma \cdot e \cdot x_2$

Why the Lyapunov Approach is Preferred

The MIT rule is less robust than the Lyapunov approach because:

Stability not guaranteed: For arbitrary adaptation gains $\gamma$ , the MIT rule can become unstable
Approximation dependency: The filter approximation $p + a + b\theta_2 \approx p + a_m$ is only valid near convergence
Parameter drift: Can occur with persistent excitation (mitigated by $\sigma$ -modification or leakage)
Sign requirement: The sign of $b$ must be known to choose the correct sign of $\gamma$

The Lyapunov approach replaces the sensitivity derivative approximation with a stability proof, giving guaranteed convergence without requiring knowledge of the sign of $b$ .

Comparison: MIT Rule vs. Lyapunov

Aspect	MIT Rule	Lyapunov-based
Stability guarantee	❌ No	✅ Yes
Extra dynamics	2 first-order filters	None
Tuning	$\gamma$ must be small	$\gamma$ can be larger
Derivation	Gradient descent	Lyapunov stability theory
Sign of $b$	Must be known	Not required
Approximation	$p + a + b\theta_2 \approx p + a_m$	Exact
Historical note	MIT Instrumentation Lab (1950s)	Russian school (1960s)

Try It Yourself

Open the interactive MRAC Demo and experiment with:

Changing plant parameters mid-simulation — drag the $a$ and $b$ sliders while the simulation is running. Watch $\theta_1$ and $\theta_2$ adapt to the new plant dynamics.
Different reference signals — switch between square wave, sine wave, and step inputs.
Adaptation gain tuning — increase $\gamma$ for faster adaptation (but watch for oscillations).

References

Primary Source

Åström, K. J., & Wittenmark, B. (2008). Adaptive Control, 2nd ed. Dover. ISBN: 978-0-486-46278-3.

The standard graduate-level text on adaptive control. Example 5.2 (pp. 190–191) in Chapter 5 (Model Reference Adaptive Control) derives the MIT rule for first-order systems using filtered sensitivity signals — the formulation used in this tutorial. The Lyapunov-based approach presented here draws on the stability framework developed in Chapter 4 (Lyapunov Stability).