L 1 Adaptive Control

The objective of this document is to provide a technical response to a widely circulated article [1], co-authored by P. Ioannou, K. Narendra, A. Annaswamy, S. Jafari, L. Rudd, R. Ortega, and J. Boskovic, in which the authors “have come together” to express “with one voice” their “growing concern” about the claims on performance and robustness guarantees of L1 adaptive control. The article has been submitted to IEEE Transactions on Automatic Control, and was also sent to funding agencies, government, industry, as well as academic communities. I was invited to provide a technical review for the article by IEEE Transactions on Automatic Control, which I formally declined because of two reasons. First, the document violates the objective standards for scholarly publications, and it seems to be written in an ill-spirited tone. Second, it is so wrong that it is clear that the authors did not put any effort to follow up on our work and rushed to create a document that shows complete lack of understanding of the main claims of the theory. Instead of providing a formal review, I composed this document to state –once more– the main properties of L1 adaptive control and present the technical mistakes of [1] for the community at large. The first paper on L1 adaptive control was submitted to IEEE CDC in 2005, which was declined. It later appeared in ACC 2006 as a two-part paper, [2], [3]. It was later followed by a series of articles, culminating in the book, [4]. The development of L1 adaptive control was motivated by some of the apparent issues that model reference adaptive control (MRAC) has been facing for a number of years, [5]. Although every attempt has been made from our end to explain the novelty and the contribution of L1 adaptive control over the years in various papers and presentations, including [4], [6], [7], it seems that some of its benefits have not been well understood by several members of our community. The key difference between MRAC and L1 adaptive control is in the problem formulation, which consequently leads to significant architectural differences. The wrong conception in [1] is that L1 adaptive control is perceived like an input filtered MRAC, which is not the case. While the L1 adaptive control architectures do have a filtering structure, there are two key points to be kept in mind: i) this filtering structure stands in a very particular point of the control architecture, ensuring that the estimation loop is decoupled from the control loop; and ii) the synthesis of this filtering structure depends on the class of systems considered and, in general, does not correspond to the mere insertion of a low-pass filter. These two points have to be carefully addressed with every particular system, dependent upon the presence of unmodeled dynamics, disturbances, noise, etc. To make this exposition clear, this document includes three main parts: i) a description of the problem formulation, key ideas, and main properties of L1 adaptive control; ii) a brief overview of the current challenges of the theory; and iii) a discussion of the claims in [1], which –as it will become evident– are unsubstantiated and, for the most part, plainly wrong. In particular, we note that Section II-C analyzes in detail the simulation example from [1] with a correct implementation of an L1 adaptive controller.