In this study, we propose new adaptive laws for adjusting the controller parameters of the model reference adaptive control (MRAC) and MRAC with integral feedback schemes. The innovation presented in this study is considering a new form for the Lyapunov function candidate to prove the stability of the closed-loop system. In general, a Lyapunov function candidate contains two sets of quadratic expressions. The first set contains the state tracking error variable, while the second one consists of the controller parameter estimation errors. We prove that by choosing the tracking error quadratic expressions in the form of the exponential function, new adaptive laws that contain the tracking error quadratic expressions are obtained. The difference between the standard MRAC adaptive laws and the proposed new adaptive laws is the state tracking error exponential quadratic expression appears in the adaptive laws. It is shown that the adaptive laws obtained by the exponential quadratic Lyapunov function are similar to those obtained by the quadratic Lyapunov function except that the adaptive gains are variable with time. The advantage of using these new adaptive laws is improving the tracking performance of the closed-loop system, which has been proven analytically and verified by numerical simulations. Also, the robustness analysis of the proposed MRAC controller in the presence of the exogenous disturbance is studied. We consider the single degree of freedom of the wing rock example to evaluate the performance of the designed controllers.
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