Extremum-Seeking MRAC in Adaptive Control

• ES-MRAC is a hybrid adaptive control approach that combines model reference adaptive control with extremum seeking to enable global asymptotic tracking without needing prior parameter sign knowledge.
• The method uses high-frequency dither signals and averaging theory to accurately estimate unknown plant parameters while maintaining robust performance without perturbing the reference trajectory.
• A systematic design procedure, including tuning dither frequencies, probe amplitudes, and adaptation gains, is validated through simulations on second-order systems to ensure convergence and stability.

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 (Haghi et al., 2012).

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:

$$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$0

where the designer selects $$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$1 so that $$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$2 is Hurwitz.

• Regressor Vector:

$$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$3

• Parameter Estimation via "Dither":

For each unknown $$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$4 ($$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$5), the estimate is modulated:

$$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$6

with $$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$7 (probe amplitude), $$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$8 (frequency).

• Control Law:

$$a_n y^{(n)}(t) + a_{n-1}\,y^{(n-1)}(t) + \cdots + a_1\,\dot y(t) + a_0\,y(t) = u(t)$$9

• Performance Cost Function:

$a_0, \dots, a_n$0

with weighting vector $a_0, \dots, a_n$1.

• Adaptation Law:

Each estimate $a_0, \dots, a_n$2 is updated by demodulating the cost $a_0, \dots, a_n$3 at its dither frequency, via a compensator:

$a_0, \dots, a_n$4

where $a_0, \dots, a_n$5 (adaptation gain), $a_0, \dots, a_n$6 (compensator damping), and $a_0, \dots, a_n$7 (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

$a_0, \dots, a_n$8

where $a_0, \dots, a_n$9 is the greatest common divisor of $y(t)$0. Averaging theory (cf. Khalil §10.4) is applied: $y(t)$1 If $y(t)$2 is globally asymptotically stable at $y(t)$3, actual trajectories satisfy $y(t)$4. The autonomous “averaged” error–adaptation dynamics are

$y(t)$5

with explicit definitions for $y(t)$6, $y(t)$7, and $y(t)$8. This formulation facilitates Lyapunov-based proofs.

4. Stability, Convergence, and Design Conditions

The ES-MRAC design guarantees that, under appropriate design (distinct $y(t)$9, sufficiently large $n$0, proper gains), $n$1 as $n$2, with $n$3. No prior knowledge of the sign of $n$4 is necessary. The core Lyapunov argument for global asymptotic tracking employs: $n$5 where $n$6 solves the Lyapunov equation $n$7 for $n$8 and $n$9 is a positive definite diagonal matrix. The derivative $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$0 simplifies under the key design condition: $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$1 eliminating cross-terms and yielding $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$2. System boundedness and Barbalat's lemma yield $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$3, thus $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$4. High-frequency conditions required are: $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$5 (so $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$6 is small), probe gains $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$7, distinct $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$8, and $$a_{mn} y_m^{(n)}(t) + a_{m(n-1)}\,y_m^{(n-1)}(t) + \cdots + a_{m0}\,y_m(t) = r(t)$$9.

5. Design Procedure and Tuning Guidelines

The recommended step-wise tuning procedure is as follows for an $a_{mj}$0th-order LTI plant:

1. Select $a_{mj}$1 so that $a_{mj}$2 is Hurwitz ($a_{mj}$3 fixed).
2. Choose cost weighting vector $a_{mj}$4 (commonly $a_{mj}$5).
3. Choose $a_{mj}$6 and solve $a_{mj}$7 for $a_{mj}$8.
4. Choose incommensurate dithering frequencies $a_{mj}$9, set $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$0.
5. Select amplitudes $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$1 for measurable output effect but small enough to avoid excitation of unmodeled dynamics.
6. Choose $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$2 to satisfy the eigenvalue condition:

$$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$3

where $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$4 are the diagonal entries of $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$5. Larger $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$6 accelerate adaptation; larger $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$7 improve gradient sensitivity; $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$8 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: $$\lim_{t\to\infty} e^{(i)}(t) = 0, \quad i=0,\dots,n-1, \qquad \left( e(t) = y(t) - y_m(t) \right)$$9 with all coefficients unknown. The reference model is: $x(t)$0 with initial states $x(t)$1, $x(t)$2, $x(t)$3, and $x(t)$4 for $x(t)$5. Parameters chosen include $x(t)$6 for $x(t)$7, cost weights $x(t)$8, $x(t)$9, probe amplitudes $n$0, frequencies $n$1 rad/s, damping gains $n$2, adaptation gains $n$3, phase lags $n$4, and $n$5.

Simulation results confirm rapid $n$6 and parameter convergence $n$7 ground truth, with all convergence rates compatible with the theoretical $n$8 scaling, thus validating global asymptotic tracking for arbitrary-order LTI plants via ES-MRAC (Haghi et al., 2012).