International Conference on Sensors, Systems, Signals and Advanced Technologies (SSS’18) Robust integral sliding mode control of time-varying input and state delays systems with mismatched uncertainties Dhouha Ben Salem 1 , Wajdi Saad 1 , Anis Sellami 1 and Germain Garcia 2 1 Research Laboratory of Industrial Systems and Renewable Energies (LISIER) ENSIT, TUNIS university, 5 Av. Taha Hussein, BP 56, Tunis 1008, Tunisia 2 LAAS, CNRS, Toulouse university, 7 Av. Colonel Roche, F-31400 Toulouse, France dhouha.bensalem9@gmail.com, (wajdi.saad, anis.sellami)@esstt.rnu.tn, garcia@laas.fr Abstract—This paper proposes a robust integral sliding mode control (ISMC) of uncertain systems with state and input time- varying delays in the presence of unmatched uncertainties. An integral sliding surface is firstly constructed. Then, by using the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI), a sufficient condition is obtained to ensure the asymptotic stability in the sliding mode. Furthermore, a robust control law is synthesized to guarantee that the system trajectories can be driven onto the specified sliding surface in a finite time and maintained there for all subsequent time. Finally, a truck-trailer model is used to illustrate the advantages and effectiveness of the design method. keywords- integral sliding mode control, state and input delays, linear matrix inequality (LMI), robust control. I. I NTRODUCTION Time-delay is frequently encountered in various engineering systems such as chemical processes systems, nuclear reactor, hydraulic systems, biological systems and electrical networks systems [1]. Its existence is often a source of instability and poor performance. Furthermore, the stabilization of control systems which have delays is more difficult that systems with- out delay. Hence, the control problem of time-delay systems is a true challenge and has received considerable attention in recent years [2], [3]. Among the various methods developed to control time-delay systems, the well-known sliding mode control (SMC, [4]) has been extensively used. It has attractive features, such as fast response, good transient response and insensitive to external disturbances and parameter variations [5]. Generally, the typi- cal SMC design consists of two steps: sliding step and reaching step. Firstly, design of a sliding surface to provide the desired behavior for the closed-loop system during the sliding mode. Secondly, synthesize a control law which induces a sliding motion on the sliding surface in finite time [6], [7]. In the literature, much effort has been made on SMC of time delay systems. The study of state delay systems has obtained some attention, for example, the SMC design method for a class of time-delay systems with state delay has been investigated by Li and Decarlo [8] . The work in [9] is concerned with the optimal guaranteed cost sliding mode control problem for interval type-2 Takagi-Sugeno fuzzy systems with time- varying state delays and exogenous disturbances. However, all these results mentioned above did not obtain the input delays which are often encountered in real control systems. The problem of SMC systems with input delays is very significant. SMC for systems with matched bounded disturbances in the presence of input time-varying delay with using a singular perturbation approach is used by Han et al. [10]. In [11], the discrete time sliding mode controller for the robust tracking of time-delay systems is presented. It is important to note that most of the literature focuses on systems with either state delay or with input delay. There is a little work undertaken on the SMC problem of systems featuring both state and input delays. In [12], a SMC scheme has been designed to ensure the asymptotic stability of a linear system with constant time-delay in both the input and state delays. Later, in [13], a proportional-integral sliding mode control methodology for the robust control of vibration in a linear system with state and input delays is proposed. Note that these works have considered only the constant time-delay. In fact, there have been few researches that investigate the time-varying delay. The problem of SMC for systems with simultaneous input and state time-varying delays is addressed by Xia et al. [14]. The proposed methods have the advantage that they can be implemented numerically very efficiently using standard LMI algorithms. On the other hand, uncertainties can be found in real systems. Their presence with the time-delays often result in poor performance or even systems instability. The robust control of uncertain time-delay systems have attracted many researchers attention in order to overcome difficulties caused by the uncertainties and the time-delays. The existence and reachability problems for time-delay systems in the presence of matched uncertainty are considered by Han et al. [15]. In [16], the theory of sliding mode control is employed to develop a fuzzy controller for uncertain dynamic time-delayed systems. It is worth noting that in the aforementioned works, it is usually assumed that there only exist matched uncertainties in the system. SMC technique has been generalized to more gen-