BIBO stability criteria for certain class of discrete nonlinear systems. (English) Zbl 0815.93074
This paper deals with the bounded input bounded output (BIBO) stability criteria of discrete-time nonlinear systems in the form of
\[
\begin{aligned} y(t) &+ \sum a_ i y(t-i)+ \sum a_{ij} y(t-i) y(y-j)+ \cdots+ \sum a_{i_ 1\dots i_ n} y(i-i_ 1) y(t-i_ n)= \\=u(t) &+ \sum b_ i u(t-i)+ \sum b_{ij} u(t-i) u(t-j) +\cdots+ \sum b_{i_ 1 \dots i_ n} u(t- i_ 1) u(t- i_ n)+\tag{1}\\ &+ \sum c_{i_ 1\dots i_ n; j_ 1\dots j_ m} y(t-i_ 1) \dots y(t-i_ n) u(t-j_ 1) u(t-j_ m)\end{aligned}
\]
which may be viewed as a finite Taylor series expansion of the following sufficiently smooth function
\[
y(t)= f(y(t-1), \dots, y(t- k), u(t-1), \dots, u(t-l)). \tag{2}
\]
The paper gives several theorems on the sufficient conditions derived by using an equivalent \(\delta \varepsilon\)-series representation, whose coefficients are exponentially bounded. The result obtained may be viewed as an extension of the stability criteria for discrete linear time-invariant systems (i.e. a system is BIBO stable, if and only if its impulse response is absolutely summable). The theorems presented can be applied to systems identification and adaptive control.
Reviewer: Xu Ningshou (Beijing)
MSC:
93D25 | Input-output approaches in control theory |
93C10 | Nonlinear systems in control theory |
93C55 | Discrete-time control/observation systems |
Keywords:
bounded input bounded output stability; series representation; discrete- time nonlinear systemsReferences:
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