Reset Observers for Linear Time-Delay Systems. A Delay-Independent Approach D. Paesa, A. Ba˜ nos, and C. Sagues Abstract— A Reset observer (ReO) is a novel sort of observer consisting of an integrator, and a reset law that resets the output of the integrator depending on a predefined condition over its input and/or output. The introduction of the reset element in the adaptive laws can decrease the overshooting and settling time of the estimation process without sacrificing the rising time. Motivated by the interest in the design of state observers for systems with time-delay, which is an issue that often appears in process control, this paper contributes with the extension of the ReO to the time-delay system framework. The time-independent stability analysis of our proposal is addressed by means of linear matrix inequalities (LMIs). Simulation results show the potential benefit of the proposed reset observer compared with traditional linear observers. I. I NTRODUCTION State observers are recursive algorithms that play a key role in many applications such as failure detection and re- covery, monitoring and maintenance, or fault tolerant control. Initially, the research on state observers was focused on linear time invariant (LTI) systems [1], and afterwards on nonlinear systems [2], and on time-delay systems as well [3]. All those works are characterized by having only a proportional feedback term of the output observation error, and are known as proportional observers (POs). This proportional approach ensures a bounded estimation of the state and the unknown parameter, assuming a persistent excitation condition as well as the lack of disturbances. The performance of proportional observers can be improved by adding an integral term to the adaptive laws, and the resultant observers are known as proportional integral observers (PIOs) [4]. This additional integral term can increase the steady state accuracy and improve the robustness against modeling errors and dis- turbances [5]. Although PIOs were initially introduced in LTI systems for robustness improvement and loop transfer recovery, their effectiveness have been also checked with nonlinear system [6] and time-delay systems [7]. However, since the adaptive laws are still linear, they have the inherent limitations of linear feedback control. Namely, they cannot decrease the settling time and the overshoot of the estimation process simultaneously. Therefore, a trade- off between both requirements is needed. Nevertheless, this fundamental limitation can be overcome by adding a reset element. A simple reset element consists of an integrator and a reset law that resets the output of the integrator as long as This work was supported in part by DGA project PI065/09 and BSH Home Appliances Group. D. Paesa, and C. Sagues are with DIIS, University of Zaragoza, Spain. e-mail: dpaesa@unizar.es, csagues@unizar.es A. Ba˜ nos, is with Department of Informatica y Sistemas, University of Murcia, Spain. e-mail: abanos@um.es the reset condition holds. This reset element is commonly referred to as the Clegg integrator after the work of Clegg in 1958 [8], who proposed an integrator which was reset to zero when its input is zero (zero crossing reset law). In 1974, Horowitz generalized that initial work substituting the Clegg integrator by a more general structure called the first order reset element (FORE) [9]. During the last years, the research on the stability analysis and stabilization for reset systems is attracting the attention of many academics and engineers. A main difference between the state-of-art reset control works is the definition of the reset law: in [10] reset instants are fixed and thus the stability analysis is much simpler; the work [11] develops stability conditions for the zero crossing reset law; in [12] stability conditions are obtained when the reset is performed at those instants in which the input and output of the reset element have different signs (sector reset condition); finally in [13] a dwell-time stability condition over reset instants is given that is applicable to any reset law. It is important to note that all these definitions of reset laws results in different reset systems dynamics, and all of them have advantages and disadvantages in relation to obtain stability and performance specifications in control practice. It should be noted that although there are a relevant number of stability results for the different types of reset law, synthesising reset elements for optimal performance is still an open issue. Recently, stability analysis of reset control systems has been also extended to time-delay systems for the case of zero crossing reset law. There are two main approaches to study the stability of time-delay systems, which depend on whether the time-delay is included in the stability analysis or not [14]. Regarding the stability of reset control systems, the delay-dependent approach was addressed in [15], whereas the delay-independent stability analysis was given in [16]. In both cases, stability conditions were given using a set of LMI. Although the research on reset elements is still an open and challenging topic, this research has been mainly focused on control issues. The first application of the reset elements to the state observer framework is [17]. There, the authors proposed a new sort of observer called reset observer (ReO). A ReO is an state observer whose integral term has been substituted for a reset element. The introduction of the reset element in the adaptive laws can improve the performance of the observer, as it is possible to decrease the overshoot and settling time of the estimation process simultaneously. This paper extends our previous work about ReO [17], [18] to the time-delay system framework, using the ideas 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 4152