Download On the Use of a General Quadratic Lyapunov Function for Studying

c Allerton Press, Inc., 2017.
ISSN 1066-369X, Russian Mathematics, 2017, Vol. 61, No. 5, pp. 66–72. c N.O. Sedova, Zh.E. Egrashkina, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 5, pp. 77–85.
Original Russian Text On the Use of a General Quadratic Lyapunov Function for Studying
the Stability of Takagi–Sugeno Systems
N. O. Sedova* and Zh. E. Egrashkina**
Ulyanovsk State University
ul. L. Tolstogo 42, Ulyanovsk, 432017 Russia
Received November 15, 2015
Abstract—We study the stability of the zero solution to a nonlinear system of ordinary differential
equations on the base of its Takagi–Sugeno (TS) representation. As is known, the most constructive stability and stabilization conditions for TS systems stated as linear matrix inequalities
are established with the help of a general quadratic Lyapunov function (GQLF). However, such
conditions are often too rigid. Using a modification of the Lyapunov direct method, we propose
asymptotic stability conditions with weaker requirements to GQLF. They allow an application to a
wider class of systems. We also give some illustrative examples.
DOI: 10.3103/S1066369X17050097
Keywords: Takagi–Sugeno system, stability, general quadratic Lyapunov function.
INTRODUCTION
In recent decades the study of nonlinear systems by methods based on their Takagi–Sugeno linear
representation (TS-systems) is being actively developed. An attractive feature of this approach is the
applicability of linear methods for studying nonlinear systems ensured by the TS-representation of the
system under consideration.
The idea is to represent (either exactly or approximately) a nonlinear function f (x) : Rn → Rn as
p
µi (x)(Ai x + bi ), where Ai and bi are constant matrices of dimensions 0 ≤ µi (x) ≤ 1 for
the sum
i=1
i = 1, 2, . . . , p, respectively,
p
µi (x) ≡ 1 [1–3]. Properties of functions µi (x) allow us, on one hand,
i=1
to consider the function f (x) as a convex combination of linear functions, and do the system, whose
dynamics obeys the function f (x) (more precisely, the differential equation ẋ = f (x) in a continuous
case or the difference equation xk+1 = f (xk ) in a discrete one), as a convex combination of linear
“subsystems” described by functions Ai x + bi . On the other hand, these properties allow us to treat
functions µi as membership functions for certain fuzzy sets, and to do the obtained representation as
a fuzzy TS-system defined by p fuzzy inference rules [3]. The “standard” control for such systems is
r
µi (x)(Ki x + ci ). This control synthesis method is
a function with an analogous structure: u(x) =
i=1
said to be a parallel distributed compensation (a PDC-control); it allows an analogous interpretation in
terms of fuzzy logic.
There are many papers dedicated to studying the stability and to solving stabilization problems for
TS-system; see, for example, [1], [3–5] for their survey. A specific feature of the problem consists
in the fact that in spite of the linearity of “subsystems” of a TS-system, in a general case, frequency
methods do not give the desired result, and the main research tool is the direct Lyapunov method. As
is well-known, the efficiency of this method in each concrete case depends on the proper choice of the
*
**
E-mail: [email protected].
E-mail: [email protected].
66