L 1 Adaptive Controller in the Presence of Input Saturation * Dapeng Li, Naira Hovakimyan † and Chengyu Cao ‡ This paper presents a proof of stability and performance bounds of the L 1 adap- tive control architecture in the presence of input constraints. We prove that by appropriate modification of the state predictor, which is used for the definition of the error signal in the adaptive laws, the stability and the performance bounds of the L 1 adaptive controller can be quantified within an appropriately defined do- main of attraction. While for open-loop stable system the results are global, for open-loop unstable systems the domain of attraction is shown to depend upon the conservative knowledge of the uncertain parameters, the choice of the filter and the amplitude constraint of the actuator. The performance bounds can be sys- tematically improved by increasing the rate of adaptation. Simulations verify the theoretical findings. I. Introduction Recent papers 1, 2 introduced the L 1 adaptive control architecture, which has guaranteed tran- sient and steady-state performance bounds for system’s input and output signals in the presence of fast adaptation. This paper addresses the performance of L 1 adaptive controller in the presence of input saturation. Control design in the presence of input saturation has been addressed in various papers, a good historical overview of which can be found in Ref. 3 In adaptive systems this issue is especially challenging, due to the lack of appropriate performance metrics that would quantify the time-history of the control signal. In Ref., 4 a modification of the reference input is introduced to prevent control saturation, but without a formal stability proof. In Ref., 5 a rigorous proof of asymptotic stability is pursued by considering an adaptive modification of the reference input in conventional model reference adaptive control (MRAC) scheme. Reference 6 considers a fixed gain adjustment (PCH component) to the reference model. In Ref., 7 the results from Ref. 5 have been further elaborated to ensure that the control signal never incurs saturation. This paper considers the L 1 adaptive control architecture from Refs. 1, 2 in the presence of input amplitude saturation. Similar to Ref., 5 we show that for open-loop stable (unstable) scalar systems, global (local) stability results can be guaranteed. For open-loop unstable systems, the local domain of attraction is explicitly quantified and is shown to depend upon the conservative bounds of uncertain parameters, the filter choice and the actuator saturation bound. The transient performance of L 1 adaptive controller is quantified similar to the results in Refs. 1, 2 Modifications are made to both the state predictor and the reference system that depend upon the control * This material is based upon work supported by the United States Air Force under Contract No. FA9550-08-1-0135 and NASA under Contracts NNX08BA64A, NNX08BA65A. † Dapeng Li and Naira Hovakimyan are with Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign (UIUC), Urbana, IL 61801, USA, {li63,nhovakim}@illinois.edu ‡ Chengyu Cao is with Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA, ccao@engr.uconn.edu 1 of 11 American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control Conference 10 - 13 August 2009, Chicago, Illinois AIAA 2009-6064 Copyright © 2009 by Dapeng Li. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.