Stability Analysis of Continuous PI-like Fuzzy Control Systems based on Vector Norms Approach
Anis Sakly, Basma Zahra, Mohamed Benrejeb · European Society for Fuzzy Logic and Technology Conference · 2009
Abstract —This paper proposes a practical stability analysis of aparticular class of continuous PI-like fuzzy control systems based onthe convergence of regular vector norms. This approach is based onthe comparison, the overvaluing principle and the application ofBorne and Gentina criterion. The stability conditions issued fromvector norms correspond to a vector Lyapunov function. Acomparison system relative to a regular vector norm will be used inorder to get the simple arrow form of the state matrix that yields to asuitable use of Borne and Gentina criterion for establishing sufficientconditions for global asymptotic stability. Keywords —Continuous PI-like fuzzy control systems, globalasymptotic stability, vector norms, Borne and Gentina criterion,Arrow form state matrix. 1 Introduction Over the last few decades, fuzzy control has received greatattention from engineering and science community and hasmarked rapid growth. However, there isn’t a general methodfor analysis or synthesis of such control strategy. Inparticular, stability analysis of control systems is an essentialstep before synthesizing and elaborating the control law.In this way, many researches have been done on the stabilitystudy of fuzzy control systems since their appearance in themiddle of the 70’s [17].Indeed, the majority of these studies concern TSK fuzzysystems since consequent fuzzy variables of this type (knownas type III) are explicit and many important results have beenobtained [13, 18, 23] based on Lyapunov stability theory.Nevertheless, there have been a limited number of stabilitystudies of Mamdani type fuzzy systems (known as type I)due to their complexity, and for this type, fuzzy variables arelinguistically understandable in both the premises andconsequents. Most of these papers provide a stabilityanalysis of a linear plant controlled by a fuzzy controller andthey regard the latter as a nonlinear controller correspondingto a Lur’e system. So, the stability problem of fuzzy controlsystems comes down to conventional nonlinear stabilitytheory.Among the methods that have been used, Popov’s theoremfor time-invariant nonlinearity [16], circle criterion for time-variant nonlinearity [19, 20] and its extended version, theconicity criterion [1].Other approaches have been used: ones can cite: thehyperstability approach which is equivalent to passivity [9,10, 24], the input-output stability based on the use of smallgain theorem [11, 25] and the Lyapunov function approach[8, 12].We do remark also that the majority of these papers simplifythe study by considering the consequents of the fuzzy systemas singletons [22]. So, this type (known as type II) is aspecial case of both TSK fuzzy systems and those ofMamdani.In this paper, the stability study of particular class of fuzzyPI controllers of type Mamdani is presented. This study isbased on the application of the Borne and Gentina criterion[6] which uses Kotelyanski conditions. In [3], it has beenshown that when system state matrix is in arrow form, thenBorne and Gentina criterion becomes very simple to apply.This approach has been used in many previous works [2, 4,5, 21]. In this way, results proposed in [21] will be used andgeneralized for stability analysis of continuous Mamdanifuzzy systems in the case of nonlinear processes to becontrolled.The paper is organised as follows. The next section isdevoted to the description of the particular class of PI-likefuzzy control system. In section 3, stability conditions of thefuzzy system are established by using Borne and Gentinacriterion and vector norms approach. The stability conditionsproposed are illustrated with an example presented in section4. Finally, concluding remarks are drown in section 5.