Abstract
This paper studies the exponential stability of switched positive nonlinear systems defined by cooperative and homogeneous vector fields. In order to capture the decay rate of such systems, we first consider the subsystems. A sufficient condition for exponential stability of subsystems with time-varying delays is derived. In particular, for the corresponding delay-free systems, we prove that this sufficient condition is also necessary. Then, we present a sufficient condition of exponential stability under minimum dwell time switching for the switched positive nonlinear systems. Some results in the previous literature are extended. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.
Highlights
Positive systems are those systems whose state variables and output signals are always contained in the first quadrant whenever both the initial conditions and input signals are nonnegative
This paper studies the exponential stability of switched positive nonlinear systems defined by cooperative and homogeneous vector fields
In [4], it was shown that the delayed positive linear system ẋ(t) = Ax(t) + Bx(t − τ) was globally asymptotically stable (GAS) for all τ ≥ 0 if the corresponding delay-free system ẋ(t) = (A + B)x(t) was GAS
Summary
Positive systems are those systems whose state variables and output signals are always contained in the first quadrant whenever both the initial conditions and input signals are nonnegative. [6] derived a necessary and sufficient condition for exponential stability of such positive systems with the homogeneity of a degree of one. A switched system is a type of hybrid dynamical system consisting of family (either discrete-time or continuous-time) subsystems and a rule that regulates the switching among them It has been widely applied in many areas, such as chemical processing and traffic control. In [19], the author showed a necessary and Complexity sufficient condition for exponential stability of a class of switched positive nonlinear systems under average dwell time switching. The author of [19] only studied the switched positive systems with the homogeneity of a degree of one, in which the common exponential decay rate for all subsystems independent of initial conditions can be found for all subsystems.
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