Automatica 49 (2013) 1012–1018 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Tube model reference adaptive control ✩ Boris Mirkin 1 , Per-Olof Gutman Faculty of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel article info Article history: Received 13 February 2012 Received in revised form 21 October 2012 Accepted 31 October 2012 Available online 22 February 2013 Keywords: Robust model reference adaptive control Optimization abstract By using the concept of on-line goal adaptation, we develop a new paradigm of performance shaping in MRAC. The general idea is to replace the single reference model generated trajectory in classical adaptive design with a tube reference model. Two alternative adaptive control schemes that lead to tractable design formulations are developed in which the performance is adapted on-line to satisfy a new specification in addition to maintaining the usual stability and robustness properties. For this purpose an additional optimization problem is formulated within the MRAC framework to find a correction control term at each instant of time. The proposed approach provides a convenient intuitive interpretation of the design problem, while retaining the fundamental ideas on which model reference adaptive control is based. The system performance is found to be as desired by simulation. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Model reference adaptive control (MRAC) is one of the main approaches of adaptive control, see e.g. the popular text- books (Åström & Wittenmark, 1995; Ioannou & Sun, 1996; Krstić, Kanellakopoulos, & Kokotović, 1995; Narendra & Annaswamy, 1989; Sastry & Bodson, 1989; Tao, 2003). Most model reference adaptive control design techniques have paid attention only to control problem solutions for one particular performance index — the tracking error which is the difference between the plant out- put and the reference model output. The reference model is chosen to generate a single desired trajectory that the plant output has to follow. Choosing a single reference trajectory is an idealization in order to obtain a solution based on the relevant mathematics. In many applications, such as industrial process control or flight control, it is not necessary to exactly follow a single reference trajectory; usually some specified deviation is allowed. Several approaches can be taken when a control specification is given in the form of an admissible set, and interesting problems can be posed. Here, we mention only some works in the context of adaptive control: so called ‘‘funnel control’’ in Ilchman, Ryan, and Townsend (2007) where approximate tracking and prescribed transient behaviour of the tracking error are both captured by ✩ This work was supported by the Israel Science Foundation under Grant 588/07. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic. E-mail addresses: bmirkin@technion.ac.il (B. Mirkin), peo@technion.ac.il (P.-O. Gutman). 1 Tel.: +972 48320974; fax: +972 48295696. the concept of a performance funnel; a reference model chosen to be piecewise linear e.g. Sang and Tao (2010); and an on-line adaptation of the reference model as is demonstrated in Joshi, Tao, and Patre (2011) when the plant-model matching conditions are violated whereby the adaptive controller includes tuning of the controller gain and simultaneous estimation of the plant-model mismatch. In the framework of the concept of goal adaptation, and with the aim to provide additional desirable properties of the closed- loop system, see Mirkin (2001) and Mirkin and Mirkin (1999), a new paradigm of performance shaping in MRAC, called Tube based Model Reference Adaptive Control (TMRAC) was recently developed in Mirkin, Gutman, and Shtessel (2012); Mirkin, Gutman, and Sjöberg (2011). It is advised to design a controller which not only guarantees closed loop stability, asymptotical tracking, and robustness to various uncertainties in the plant model and to external disturbances, but also diminishes the control cost with respect to some criterion. For this the control signal is split into two parts – an adaptive component and a component that corrects the control objective – and an additional optimization problem is formulated in order to find the value of the newly defined correction control component in each time instant. In the context of TMRAC we develop here two alternative adap- tive control schemes that lead to tractable design formulations where the performance is adapted on-line to satisfy additional re- quirements. The problem is treated from two different viewpoints: when the correcting component is included in the so-called ‘‘re- gressor’’ vector, and when it is treated as an external bounded per- turbation. 2. Statement of the MRAC problem with performance tube Before proceeding to the main results, we describe the principal idea of tube reference model based adaptive control 0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2013.01.022