|Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models
Luís F. P. Silva, Valter J. S. Leite*, Eugênio B. Castelan, and Gang Feng
International Journal of Control, Automation, and Systems, vol. 16, no. 3, pp.1435-1447, 2018
Abstract : "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, xk, and the maximum delayed
state vector, xkd¯. 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."
Keyword : "Fuzzy Lyapunov-Krasovskii function, LMIs, nonlinear discrete-time systems, Takagi-Sugeno fuzzy models, time-varying delay in states."