What is L1 adaptive control

Abstract

Recently, a class of adaptive control schemes called L1 Adaptive Control (L1-AC) has been proposed and widely advertised in aerospace control for achieving fast and robust adaptation and better performance than the existing Model Reference Adaptive Control (MRAC) Schemes. The L1-AC scheme is designed mainly for plants with full state measurement even though the name L1-AC has been used as an umbrella name for more general classes of plants. In this paper, we show that the L1-AC for plants with measured states is simply a standard MRAC with a low pass filter inserted in front of the control input. The analysis of the scheme is almost identical to that of MRAC as the same Lyapunov function is used to establish stability. The motivation for using the filter is the fact that for this class of adaptive schemes i.e. MRAC for plants with full state measurement the tracking error can be made arbitrarily small during transient by increasing the adaptive gain. A high adaptive gain however makes the differential equation of the adaptive law or estimator very stiff and leads to numerical problems that cause high oscillations in the estimated parameters leading to loss of adaptivity and deviations from what the theoretical properties dictate. The L1-AC approach mistook these numerical oscillations as properties of the adaptive scheme and inserted an input low pass filter in order to filter them out. While the filter helps reduce the frequency of these oscillations in the control law the price paid is high. First the numerical instability does not go away and the estimated parameters continue to oscillate without converging to the true parameters even in the presence of suffciently rich signals. Second, due to the filter the tracking error is no longer guaranteed to converge to zero and the transient bounds for the tracking error also depend on the filter. As a result, the tracking properties of the L1-AC scheme are worse than what a simple MRAC scheme can generate with adaptive gains that could be high but away from the region of numerical instability. In addition, the presence of the filter reduces the robust stability margins in the presence of unmodeled dynamics and provides literally no advantage as simple robust MRAC techniques can solve the same problem achieving much better properties. The authors of L1-AC often compare the properties of a numerically unstable MRAC due to extremely high adaptive gains something that is prohibited by robust adaptive control with those of the filtered MRAC aka L1-AC to show that L1-AC performs better. Such comparisons are not only misleading but do not reveal what causes what giving the reader a false impression of a new theory that results to new performance and robustness improvements. The use of filters in adaptive control is not new and they are used to improve the performance and robustness of certain adaptive control schemes with-out destroying their ideal tracking properties. In this paper we present such a scheme that guarantees good transient performance and robustness and reveals the trade off between transient performance and robust stability.