Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

• Published in: International journal of control, automation, and systems Vol. 16; no. 3; pp. 1435 - 1447
• Main Authors: Silva, Luís F. P., Leite, Valter J. S., Castelan, Eugênio B., Feng, Gang
• Format: Journal Article
• Language: English
• Published: Bucheon / Seoul Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers 01-06-2018; Springer Nature B.V; 제어·로봇·시스템학회
• Subjects: Attraction; Automation; Computational geometry; Control; Controllers; Convex analysis; Convexity; Delay; Discrete time systems; Engineering; Fuzzy control; Fuzzy systems; Initial conditions; Linear matrix inequalities; Mathematical analysis; Matrix methods; Mechatronics; Nonlinear systems; Optimization; Regular Papers; Robotics; Stabilization; Validity; 제어계측공학; LMIs; time-varying delay in states; Fuzzy Lyapunov-Krasovskii function; nonlinear discrete-time systems; Takagi-Sugeno fuzzy models
• ISSN: 1598-6446; 2005-4092
• DOI: 10.1007/s12555-017-0526-z

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state,
x
k
, and the maximum delayed state vector,
x
k
−
d̅
. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.