Showing papers by "Kumpati S. Narendra published in 2014"

${\rm L}_{1}$ -Adaptive Control: Stability, Robustness, and Interpretations

Abstract: An adaptive control design approach that involves the insertion of a strictly proper stable filter at the input of standard Model Reference Adaptive Control (MRAC) schemes has been proposed in the recent years. This approach was given the name ${\rm L}_{1}$ -Adaptive Control ( ${\rm L}_{1}$ -AC) due to the ${\rm L}_{1}$ bounds obtained for various signals. As part of the approach it is recommended to use very high adaptive gains for fast and robust adaptation. The purpose of this note is to analyze whether ${\rm L}_{1}$ -AC provides any improvements to existing MRAC schemes by focusing on a simple plant whose states are available for measurement presented in [1] . Our analysis shows that the insertion of the proposed filter deteriorates the performance and robust stability margin bounds compared to standard MRAC, i.e., when the filter is removed. The use of high adaptive gains recommended in the ${\rm L}_{1}$ -AC approach may cause two major problems. First, it makes the differential equation of the adaptive law very stiff leading to possible numerical instabilities. Second, it makes the adaptive scheme less robust with respect to unmodeled dynamics.

Stability, robustness, and performance issues in second level adaptation

Abstract: A new approach, described as second level adaptation, was introduced in [1] for the control of unknown linear time-invariant plants using multiple identification models. If θ p ∈ R n , the unknown parameter vector of an LTI system lies in the convex hull P(t 0 ) of (n+1) vectors θ i (t 0 ) (initial values of adaptive vectors) in parameter space, it was shown that it lies also in the convex hull of θ i (t)(i = 1, 2, ..., n+1), of the adaptive parameters of the identification models. If the representations of the plants and models are in companion form, and all the state variables are accessible, simulation results were presented to demonstrate that the new method would result in much better performance than conventional adaptive control. In this paper, all aspects of second level adaptation are critically reviewed. Following this, an analysis of the stability and robustness of the approach is undertaken, and detailed reasons are provided for the observed improvement in performance. Due to space limitations, most of the results described in the paper pertain to plants in companion form, with all state variables accessible. Towards the end of the paper, an effort is made to indicate how the same concepts can be extended to more general cases. These include plants whose matrices are not in companion form, and systems in which only the input and the output of the plant are accessible. The details of the latter problems will be included in a forthcoming paper [7]. The authors believe that the two papers, together, will make a convincing case for the use of multiple models in adaptive control.

Unstable systems stabilizing each other through adaptation

Abstract: An asymptotically stable reference model plays a crucial role in the entire literature on adaptive systems. Given a plant with unknown parameters, the objective is to adapt the parameters of a controller so that the behavior of the controlled plant emulates that of the reference model in some sense. This paper addresses the following question, which is markedly different from that encountered in conventional adaptive control: “Can two or more unstable plants adaptively stabilize each other? ”. This is because neither system has a stable model to emulate, and each depends upon the other to stabilize itself. It is not surprising that this seemingly innocuous question has far reaching implications in widely different fields such as biology, psychology, economics and robotics, where it is found to arise frequently. If simple rules of adaptation were adequate to answer the above question, it would not merit much attention. However, preliminary investigations have revealed that such adaptation often does lead to instability. In fact the answer to the question is far from simple, and depends upon the assumptions made regarding the adaptive subsystems, and the manner in which they interact with each other. This has given rise to a vast spectrum of interesting questions. The objective of this paper is consequently not to provide an exhaustive set of answers, but merely to pose several problems in what the authors hope will be a new area of research, and to present preliminary results concerning some of them.

Classical Results on the Stability of Linear Time-Invariant Systems, and the Schwarz Form

Abstract: The paper deals with the classical results of Routh and Hurwitz, Biehler and Kharitonov, concerning the stability of a linear time invariant differential equation. It is shown that these results follow directly from a Schwarz matrix representation of stable systems.