Invoking Halanay inequality to conclude on closed-loop stability of a process with input-varying delay D. Bresch-Pietri ∗ J. Chauvin ∗∗ N. Petit ∗ ∗ MINES ParisTech, Centre Automatique et Syst` emes, Unit´ e Math´ ematiques et Syst` emes, 60 Bd St-Michel, 75272 Paris, Cedex 06, France (e-mail : delphine.bresch-pietri@mines-paristech.fr) ∗∗ IFP Energies nouvelles, D´ epartement Contrˆ ole, Signal et Syst` eme, 1 et 4 avenue du Bois-Pr´ eau, 92852 Rueil-Malmaison, France Abstract: At the light of a simple tutorial example, this paper discusses the merits of a recently introduced technique to control a class of systems with a delay depending on the past values of the control variables. The convergence proof of the proposed control strategy is obtained by a boundedness analysis of a class of delay differential equation, inspired by the Halanay inequality. The relations of the proposed technique with previous works from the literature on predictor-based controllers are discussed. The treated example is representative of a wide class of systems often observed in process control and distributed parameter systems. 1. INTRODUCTION Numerous industrial applications involve a physical dead-time which reveals troublesome in the design and tuning of output feedback control laws, needed to achieve the still increasing expectations of dynamic performances. The source of this dead- time is that the sensors and the actuators are often not co- located. Such processes may involve transportation of material, such as gas in automotive engine (Bresch-Pietri et al. [2010]), liquid in mixing processes for chemical reactors (Harmand and Dochain [2005]) and heater collector plant (Sbarciog et al. [2008]), or blending (Ch` ebre et al. [2010]) and batch processes (Petit et al. [1998]). Very importantly, the lag directly depends on the manipulated variable. Therefore, the considered delay is inherently input-dependent and, then, non-linear. A prime and very simple example of this type of non-linear dynamical systems is the shower system, or bath depicted in Fig. 1, for which a temperature regulation is considered 1 . In this example, the involved liquid holdups contained in the pipes are directly correlated to the history of the input flow rates. The holdup has a “group” velocity (due to the incompressible nature of the flow) which depends on the current value of the control. The temperature of the holdup is not spatially uniform, as it depends on the past values of the control which may not have been constant over a sufficiently long past. Due to its relative simplicity and its occurrence in everyday life, this example is often used to introduce time-delay systems in lectures and textbooks, like in Zhong [2006], and to illustrate some corresponding control challenges. Surprisingly, as similar problems involving input-varying delay, such as the crushing- mill problem presented by Richard [2003], it seems that this problem has never been studied theoretically. More generally, it seems that the stabilization problem of a non- linear process, with input-dependent time-delay in the input, of 1 Of course, for this application, an open-loop result of asymptotic conver- gence can be straightforwardly obtained. Yet, closed-loop controllers lead to substantial transient improvements, which may be crucial in real-life applica- tions. type ˙ x = f (x, u(t − D(u))) has never been theoretically considered. A classically proposed approach in such situations is to treat the delay dependence on the control by robustness, i.e. by neglecting this dependence as D(u) ≈ D(t ) or even D(u) ≈ D and by requiring the controller to be able to deal with a certain level of model error. Robust stability of systems with time-varying delay in the input has been widely studied lately : in Han and Gu [2001], He et al. [2007], Jiang and Han [2008] or Liu and Hu [2011], “memoryless” controllers are employed at the expense of sub- stantial Linear Matrix Inequalities (LMIs) to check. On the other hand, predictor-based strategies have been proposed to improve closed-loop dynamic performance,(see Smith [1959], Artstein [1982], Manitius and Olbrot [2002]). Such techniques, which are widely used for a constant input time-delay (see for instance Jankovic [2008], Krstic [2008b], Mondi´ e and Michiels [2003], Moon et al. [2001] or Gu and Niculescu [2003] and Richard [2003] and the reference therein) are less popular for time-varying ones. Yet, in Nihtila [1991] or, more recently, in Krstic [2009], a time-varying delay version of predictor-based control has been proposed. To compensate the input delay, this prediction is calculated over a time window of which length matches the value of the future delay 2 . In other words, it is required to be able to predict the future variations of the de- lay, which is not always practically achievable when the delay depends on the input. For this reason, in this paper, although we use a prediction technique, we do not aim at exactly compensating the delay. We blend the previously described techniques and propose a robust compensation approach as has been done recently in Yue and Han [2005] for example. We follow the overture proposed in Krstic [2008a] and Krstic [2009] to analyze the stability of linear input time-delay systems (and which have been devel- 2 In details, defining the delay operator φ (t )= t − D(t ) and assuming that its inverse is well-defined, exact delay-compensation is obtained with the feedback law U (t )= KX pred (φ −1 (t )).