Article - Control Transactions of the Institute of Measurement and Control 1–9 Ó The Author(s) 2020 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0142331220952201 journals.sagepub.com/home/tim Robust dynamic sliding mode observer design for a class of nonlinear systems Mahnoosh Shajiee 1 , Seyed Kamal Hosseini Sani 1 , Mohammad Bagher Naghibi-Sistani 1 and Saeed Shamaghdari 2 Abstract In this paper, a novel method for the design of robust nonlinear observer in the H ‘ framework for Lipschitz nonlinear systems is proposed. For this purpose, a new dynamical structure is introduced that ensures the stability of observer error dynamics. Design innovation is the use of dynamic gain in the sliding mode observer. The additional degree of freedom provided by this dynamic formulation is exploited to deal with the nonlinear term. The performance of this stable H ‘ observer is better than conventional static gain observers and the dynamic Luenberger observer. The compensator is designed in such a way that, while ensuring the stability of the closed-loop system, it prevents performance loss in the presence of the nonlinearities. By the proposed approach, the observer is robust to nonlinear uncertainties because of increasing the Lipschitz constant. Also, the design procedure in the presence of system and measurement noises is addressed. Finally, the simulation of our methodology is conducted on a nonlinear system to illustrate the advantage of this work in comparison with other observers. Keywords Dynamic sliding mode observer, robust nonlinear observer, Lipschitz nonlinear system, H N synthesis, H N observer design Introduction This article considers the design of the dynamic sliding mode observer (DSMO) for a class of nonlinear systems. Nonlinear observer design has been considered by researchers to esti- mate system states, outputs and fault detection in recent years (Yang et al., 2020). However, due to the existence of the non- linear term in the error dynamics between the system and the observer, it has a complexity in design compared with the lin- ear observer. The idea of using an observer was first devel- oped by Luenberger (1972). He showed that the state of a linear system can be obtained from the input and measured output. By developing this idea, observer design for the Lipschitz nonlinear systems was first suggested by Thau (1973). Thau’s technique pursued by Raghavan based on an iterative algorithm by solving a Riccati equation for the observer stability condition (Raghavan and Hedrick, 1994). Rajamani obtained a condition to guarantee observer asymp- totic stability. He also concerned the equivalence between this condition and the minimization of the H ‘ norm of a system in the standard form (Rajamani, 1998). This condition was developed by Pertew et al. (2006). The conventional method of nonlinear observer design is applying the linear observer to a nonlinear system, including the generalization of the linear Luenberger and Kalman filter to the nonlinear system (Ekramian 2013), unknown input observer (Zarei and Poshtan, 2010), high gain observer (Farza et al., 2010) and adaptive observer (Azmi and Khosrowjerdi, 2016). Another nonlinear observer is the sliding mode obser- ver (SMO). [It has a linear and nonlinear switching term that acts as a feedback on the output estimation error (Rongchang and Qingling, 2018). For nonlinear systems, the synthesis and computation of these gains in the SMOs are more challenging and complex. Due to its success in design techniques, this paper is focused on dynamic SMO design. To investigate pre- vious works on nonlinear SMO design, Koshkouei and Zinober (2004) proposed a SMO for Lipschitz nonlinear sys- tems based on the Lyapunov equation. Conservatism and complexity is the shortcoming of this article. One of the classi- cal methods of nonlinear SMO design include finding a trans- formation that linearizes the system and then applying linear observer design techniques (He and Zhang, 2012). Our article does not address the complexity of computing the coordinate transformation. In our work, the discontinuous term in the SMO is not replaced with a continuous one because it degrades the accuracy (Mu et al., 2015). High order SMOs like twisting algorithms need knowledge of the output deriva- tive and include differentiators (Hammouda et al., 2015). The design idea of the dynamic gain observer was pre- sented by Pertew et al. (2006) for the Luenberger observer. 1 The Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 2 The Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran Corresponding author: Seyed Kamal Hosseini Sani, Department of Electrical Engineering, Ferdowsi University of Mashhad, P.O. Box 917751111 Mashhad, Iran. Email: k.hosseini@um.ac.ir