Stable and convergent iterative feedback tuning of fuzzy controllers for discrete-time SISO systems Radu-Emil Precup a,⇑ , Mircea-Bogdan Ra ˘dac a , Marius L. Tomescu b , Emil M. Petriu c , Stefan Preitl a a Department of Automation and Applied Informatics, ‘‘Politehnica’’ University of Timisoara, Bd. V. Parvan 2, RO-300223 Timisoara, Romania b Faculty of Computer Science, ‘‘Aurel Vlaicu’’ University of Arad, Complex Universitar M, Str. Elena Dragoi 2, RO-310330 Arad, Romania c School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward, Ottawa, ON, Canada K1N 6N5 article info Keywords: Convergence Discrete-time input affine SISO systems Iterative feedback tuning PI-fuzzy controllers Real-time experimental results Stability abstract This paper proposes new stability analysis and convergence results applied to the Iterative Feedback Tun- ing (IFT) of a class of Takagi–Sugeno–Kang proportional-integral-fuzzy controllers (PI-FCs). The stability analysis is based on a convenient original formulation of Lyapunov’s direct method for discrete-time sys- tems dedicated to discrete-time input affine Single Input-Single Output (SISO) systems. An IFT algorithm which sets the step size to guarantee the convergence is suggested. An inequality-type convergence con- dition is derived from Popov’s hyperstability theory considering the parameter update law as a nonlinear dynamical feedback system in the parameter space and iteration domain. The IFT-based design of a low- cost PI-FC is applied to a case study which deals with the angular position control of a direct current servo system laboratory equipment viewed as a particular case of input affine SISO system. A comparison of the performance of the IFT-based tuned PI-FC and the performance of the PI-FC tuned by an evolutionary- based optimization algorithm shows the performance improvement and advantages of our IFT approach to fuzzy control. Real-time experimental results are included. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The stability analysis of fuzzy control systems has been investi- gated extensively in the context of nonlinear autonomous/nonau- tonomous systems in close connection with their stabilization. The current approaches reported in the literature concerning the stable design of fuzzy control systems with Takagi–Sugeno–Kang fuzzy controllers are based mainly on linear matrix inequalities (LMIs) (Blaz ˇic ˇ et al., 2009; Chiang & Liu, 2012; Feng, 2006; Sala, Guerra, & Babuška, 2005; Tanaka, Yoshida, Ohtake, & Wang, 2009; Tsai, 2011; Zhang, Shi, & Xia, 2010) making use of quadratic, piecewise quadratic, non-quadratic, parameter-dependent or poly- nomial Lyapunov functions (Boulkroune & M’Saad, 2011; Kruszew- ski, Wang, & Guerra, 2009; Lee, Park, & Joo, 2011; Li & Ge, 2011; Precup & Hellendoorn, 2011). The LMIs are computationally solv- able even in relaxed versions, and they require numerical algo- rithms embedded in well acknowledged software tools or implementations in other programming languages. The design of optimal control systems is of permanent interest because it ensures very good control system performance indices by the minimization of objective functions expressed as integral quadratic performance indices (Bayam, Liebowitz, & Agresti, 2005; Cazarez-Castro, Aguilar, & Castillo, 2010; Linda & Manic, 2011; Ruano, Fleming, Teixeira, Rodríguez-Vázquez, & Fonseca, 2003; Tikk, Johanyák, Kovács, & Wong, 2011; Vašc ˇák & Pal ˇ a, 2012). The Iterative Feedback Tuning (IFT) carries out the gradi- ent-based minimization of the objective functions making use of the input-output data from the closed-loop system in several experiments conducted per iteration (Hjalmarsson, Gevers, Gun- narsson, & Lequin, 1998; Hjalmarsson, Gunnarsson, & Gevers, 1994). A good overview on IFT is given in Hjalmarsson (2002) while ensuring the unbiased estimates of the objective function with re- spect to controller parameters. Various extensions of IFT to Multi Input-Multi Output (MIMO) systems are investigated in Huusom, Poulsen, and Jørgensen (2009a). The extension of IFT according to Huusom, Poulsen, and Jørgensen (2009a, 2009b) provides addi- tional ways to disturbance rejection, and it improves the conver- gence. Recent IFT applications to industrial control problems in servo systems and drives are discussed in Kissling, Blanc, Myszko- rowski, and Vaclavik (2009), McDaid, Aw, Xie, and Haemmerle (2010), and Ra ˘dac, Precup, Petriu, and Preitl (2011). The combination of IFT and fuzzy control aims the fuzzy control system performance enhancement. A combination of IFT and fuzzy control is analyzed in Nafaa, Hadjadj-Aoul, and Mehaoua (2005), and the fuzzy control system enables the run-time adaptation based on IFT and knowledge acquired from past experience. A fuz- zy-based supervisor that modifies the parameters of an iteratively 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.07.023 ⇑ Corresponding author. Tel.: +40 2564032 29/30/40 (lab), 26 (office); fax: +40 256403214. E-mail address: radu.precup@aut.upt.ro (R.-E. Precup). Expert Systems with Applications 40 (2013) 188–199 Contents lists available at SciVerse ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa