High-order sliding-mode control for blood glucose: Practical relative degree approach Ana Gabriela Gallardo Herna ´ ndez a,n , Leonid Fridman b , Arie Levant c , Yuri Shtessel d , Ron Leder b , Cristina Revilla Monsalve a , Sergio Islas Andrade a a Centro Me´dico Nacional Siglo XXI Av, Cuauhte´moc 330 Col, Doctores C.P., 06725 Me´xico D.F., Mexico b Universidad Nacional Auto ´noma de Me´xico (UNAM), Department of Control, Engineering Faculty. C.P. 04510 Me ´xico D.F., Mexico c Applied Mathematics Department in Tel-Aviv University, Tel-Aviv 69978, Israel d Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, USA article info Article history: Received 30 March 2011 Accepted 27 November 2012 Available online 24 January 2013 Keywords: Sliding mode control Biomedical control Uncertain systems Nonlinear control abstract High order sliding mode controller (HOSMC) is proposed for blood glucose regulation. With this aim a novel concept of Practical Relative Degree (PRD) and a method of its identification are suggested. First, for PRD identification method is applied for the most simple (Bergman Minimal Model) and the most complicated (Sorensen Model) models concluding that the common PRD for both models is three. Then, a third order quasi-continuous control law was designed. The proposed control law has been tested on simulations for a third model (Hovorka Model), and both above mentioned models. Finally, the experiments are performed with rats just to show that control design based on PRD three is efficient. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Automatic insulin infusion for diabetic patients has been the subject of extensive research, some examples can be seen in Weinzimer et al. (2008), Salzsieder, Albrecht, Fischer, and Freyse (1985), Hovorka et al. (2010), Shtessel and Kaveh (2008), Zambrano, Garcia-Gabin, Bondia Company, and Vehi (2009), Abu-Rmileh, Garcia-Gabin, and Zambrano (2010) and Wang, Zisser, Dassau, Jovanovi, and Doyle (2010). To date there is one pump that works in closed loop mode; it stops insulin infusion when hypoglycemia is detected via a subcutaneous sensor. All the other commercially available insulin pumps work in open loop mode. The glucose–insulin regulation system is nonlinear and time variable. Identification of the patient’s parameters is expensive, invasive and uncertainties are always present due to the fact that most important parameters are time variable. For example, insulin resistance, can change with aerobic exercise routine, Cuff et al. (2003); and during exercise, muscles can uptake glucose without insulin mediation. The operating range of blood glucose in a diabetic patient is wide, it can vary between 40 and 500 mg/dl, Islas-Andrade and Revilla Monsalve (2000). These characteristics make difficult to use linear control. High Order Sliding Mode Control (HOSMC) was presented in Fridman and Levant (2002), Pisano and Usai (2008), Bartolini, Punta, and Zolezzi (2004), Orlov (2009) and Levant (2011). It is a black-box oriented control, i.e. one only needs the knowledge of the relative degree of the system and appropriate bounds for a few expressions. Thus HOSMC presents an attractive alternate approach to blood glucose control due to its nonlinearity, it can work in the whole operating range of the system. Its design does not depend on the model’s parameters, which guarantees the required robustness with respect to parameter uncertainties. There are several known mathematical models describing the glucose–insulin regulatory system; the order and the relative degree of each model depend on the number of dynamics considered and the assumptions made. Bergman Minimal Model (BeM) has relative degree three, and it contains the smallest number of parameters that describe the glucose–insulin regula- tory system with sufficient accuracy (Bergman, Ider, Bowden, & Cobelli, 1979). There are some other models, such as the Candas and Radziuk Model (Candas & Radziuk, 1994), and the Cobelli Model (Cobelli & Mari, 1985), which concur with BeM and also have relative degree three. More detailed models are the Hovorka Model (HoM), (Hovorka et al., 2004), and the adaptation of the Dalla Man model (Dalla Man, Rizza, & Cobelli, 2007) for Type 1 Diabetes presented in Magni et al. (2008), and they have relative degree five. One of the most complete models is the Sorensen Model (SoM). It describes the action of each group of organs, having some influence on glucose regulation. SoM has also relative degree five. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/conengprac Control Engineering Practice 0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.11.015 n Corresponding author. E-mail address: anagabygh@gmail.com (A.G. Gallardo Herna ´ ndez). Control Engineering Practice 21 (2013) 747–758