---
title: Extremum-Seeking MRAC in Adaptive Control
url: https://www.emergentmind.com/topics/extremum-seeking-mrac-es-mrac
type: topic
---

# Extremum-Seeking MRAC in Adaptive Control

Extremum-Seeking Model Reference Adaptive Control (ES-MRAC) is a methodology for the adaptive control of linear time-invariant (LTI) systems, combining classical model reference adaptive control (MRAC) with extremum seeking (ES) techniques. ES-MRAC was developed to enable global asymptotic tracking via adaptation mechanisms that do not require perturbation of the reference trajectory or prior knowledge of parameter signs. The approach leverages high-frequency dither signals in parameter estimation and employs averaging theory for frequency-domain separation of adaptation and control, ensuring robust performance and convergence without restrictive plant assumptions [1207.5642].

## 1. Problem Formulation

The ES-MRAC framework addresses adaptive control for a single-input, multi-output (SIMO) LTI plant of order $n$ in companion canonical form:
\[
a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)
\]
with unknown constant coefficients $a_0, \dots, a_n$. The objective is to ensure that the output $y(t)$ tracks the states of a reference model of the same order $n$:
\[
a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)
\]
where $a_{mj}$ are known and the characteristic polynomial is Hurwitz. The performance goal is precise asymptotic tracking:
\[
\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)
\]
The state vector $x(t)$ is composed of the first $n$ derivatives of the tracking error.

## 2. Control Law and Adaptation Loop

The ES-MRAC architecture is built from several key components:

- **Auxiliary Signal:** 
  \[
  z(t) = y_m^{(n)}(t) - \sum_{j=0}^{n-1} \beta_j\,e^{(j)}(t)
  \]
  where the designer selects $\beta_j$ so that $p^n + \beta_{n-1}p^{n-1} + \cdots + \beta_0$ is Hurwitz.

- **Regressor Vector:**
  \[
  v(t) = [ y(t),\,\dot y(t),\,\cdots,\,y^{(n-1)}(t),\,z(t) ]^T \;\in\mathbb{R}^{n+1}
  \]

- **Parameter Estimation via "Dither":** 
  For each unknown $a_i$ ($i=0,\dots,n$), the estimate is modulated:
  \[
  \breve a_i(t) = \hat a_i(t) + c_i\sin(\omega_i t)
  \]
  with $c_i>0$ (probe amplitude), $\omega_i > 0$ (frequency).

- **Control Law:**
  \[
  u(t) = \sum_{i=0}^{n-1} \breve a_i(t)\,y^{(i)}(t) + \breve a_n(t)\,z(t) = \breve a(t)^T v(t)
  \]

- **Performance Cost Function:** 
  \[
  J(t) = \frac{1}{2} (q^T x(t))^2
  \]
  with weighting vector $q \in \mathbb{R}^n$.

- **Adaptation Law:** 
  Each estimate $\hat a_i$ is updated by demodulating the cost $J(t)$ at its dither frequency, via a compensator:
  \[
  \dot{\hat a}_i = g_i \Bigl[ \sin(\omega_i t-\phi_i) + d_i\omega_i \cos(\omega_i t-\phi_i) \Bigr] J(t) + g_i d_i \sin(\omega_i t-\phi_i) \dot J(t)
  \]
  where $g_i$ (adaptation gain), $d_i$ (compensator damping), and $\phi_i$ (phase lag) are design parameters.

## 3. Rigorous Averaging-Based Analysis

The ES-MRAC adaptation mechanism induces a time-periodic, non-autonomous closed-loop error-adaptation system of the form
\[
\dot X = \varepsilon f(t, X), \qquad \varepsilon = \frac{1}{\omega} \ll 1
\]
where $\omega$ is the greatest common divisor of $\{\omega_i\}$. Averaging theory (cf. Khalil §10.4) is applied:
\[
f_{\mathrm{av}}(X) = \frac{1}{T} \int_0^T f(\tau, X) d\tau
\]
If $f_{\mathrm{av}}$ is globally asymptotically stable at $X=0$, actual trajectories satisfy $\|X(t) - X_{\mathrm{av}}(t)\| = O(1/\omega)$. The autonomous “averaged” error–adaptation dynamics are
\[
\begin{aligned}
\frac{dx_{\mathrm{av}}}{d\tau} &= \varepsilon \left[ A x_{\mathrm{av}} - \frac{1}{a_n} b\,v_{\mathrm{av}}^T \tilde a_{\mathrm{av}} \right] \\
\frac{d\tilde a_{\mathrm{av}}}{d\tau} &= \varepsilon\, \frac{q_n}{2 a_n} C v_{\mathrm{av}} (q^T x_{\mathrm{av}})
\end{aligned}
\]
with explicit definitions for $A$, $b$, and $C$. This formulation facilitates Lyapunov-based proofs.

## 4. Stability, Convergence, and Design Conditions

The ES-MRAC design guarantees that, under appropriate design (distinct $\omega_i$, sufficiently large $\omega$, proper gains), $x(t) \to 0$ as $t \to \infty$, with $x(t) = O(1/\omega)$. No prior knowledge of the sign of $a_n$ is necessary. The core Lyapunov argument for global asymptotic tracking employs:
\[
V = x_{\mathrm{av}}^T P x_{\mathrm{av}} + 2 \tilde a_{\mathrm{av}}^T \Gamma \tilde a_{\mathrm{av}}
\]
where $P > 0$ solves the Lyapunov equation $P A + A^T P = -Q$ for $Q > 0$ and $\Gamma$ is a positive definite diagonal matrix. The derivative $V'$ simplifies under the key design condition:
\[
q_n\Gamma C v q^T = b^T P q
\]
eliminating cross-terms and yielding $V' \le 0$. System boundedness and Barbalat's lemma yield $x_{\mathrm{av}} \to 0$, thus $x(t) \to O(1/\omega)$. High-frequency conditions required are: $\omega \gg 1$ (so $\varepsilon = 1/\omega$ is small), probe gains $d_i\,\omega_i = O(1)$, distinct $\omega_i = n_i\omega$, and $g_i = O(1)$.

## 5. Design Procedure and Tuning Guidelines

The recommended step-wise tuning procedure is as follows for an $n$th-order LTI plant:
1. Select $\{\beta_j\}$ so that $p^n + \sum_{j=0}^{n-1}\beta_j\,p^j$ is Hurwitz ($A$ fixed).
2. Choose cost weighting vector $q$ (commonly $q_i=1$).
3. Choose $Q>0$ and solve $P A + A^T P = -Q$ for $P>0$.
4. Choose incommensurate dithering frequencies $\omega_i=n_i\omega$, set $d_i=O(1/\omega_i)$.
5. Select amplitudes $c_i$ for measurable output effect but small enough to avoid excitation of unmodeled dynamics.
6. Choose $(g_i,\gamma_i,\phi_i)$ to satisfy the eigenvalue condition:
   \[
   q_n g_i d_i c_i \cos\phi_i \gamma_i = \frac{b^T P q}{\|q\|^2}
   \]
where $\gamma_i$ are the diagonal entries of $\Gamma$. Larger $g_i$ accelerate adaptation; larger $c_i$ improve gradient sensitivity; $\gamma_i$ manages parameter adaptation step-size versus stability.

## 6. Application Example: Second-Order System

The ES-MRAC methodology is exemplified with a scalar second-order plant:
\[
\ddot y + 3\,\dot y + 6.25\,y = u
\]
with all coefficients unknown. The reference model is:
\[
\ddot y_m + 4.2\,\dot y_m + 9\,y_m = r(t)
\]
with initial states $y(0) = -0.1$, $\dot y(0) = 0.2$, $y_m(0) = \dot y_m(0) = 0$, and $r(t)=1$ for $t \ge 0$. Parameters chosen include $\beta_0 = 9, \beta_1 = 3$ for $z = \ddot y_m - 3\,\dot e - 9 e$, cost weights $q_1 = q_2 = 1$, $Q=I_2$, probe amplitudes $c_0 = 0.3, c_1 = c_2 = 0.2$, frequencies $\omega_0 = 5, \omega_1 = 8, \omega_2 = 14$ rad/s, damping gains $d_i = 0.1$, adaptation gains $g_0 = 9000, g_1 = 3200, g_2 = 2000$, phase lags $\phi_i = 0$, and $\Gamma = 0.01 I_3$.

Simulation results confirm rapid $e(t) \to 0$ and parameter convergence $\hat a_0, \hat a_2 \to$ ground truth, with all convergence rates compatible with the theoretical $O(1/\omega)$ scaling, thus validating global asymptotic tracking for arbitrary-order LTI plants via ES-MRAC [1207.5642].

Source: https://www.emergentmind.com/topics/extremum-seeking-mrac-es-mrac