Properties of Nonlinear System (Related to Definition)
November 25, 2008 1 Comment
Definition 1.1 (Linearity of The Function)
The function f(x) is linear with respect to independent variable x if and only if it satisfies two conditions:
1. Additivity : ,
V in domain of the function f,
2. Homogenity: ,
Vx in domain of function f and all scalar
Definition 1.2 (Locally Lipschitz Function)
The function f is said to be locally Lipschitz for the variable x(t) if near the point it satisfies the Lipschitz criterion:
for all x and y in the vicinity of where k is a positive constant and the norm is Euclidian.
Definition 1.3 (Forced and Unforced System)
A continuous system is said to be forced if an input signal is present:
A continuous system is said to be unforced if there is no input (excitation) signal:
Definition 1.4 (Time-Varying and Time-Invariant System)
A system is time-invariant if the function f does not explicitly depend on time, i.e. the system can be described with:
A system is time-varying if the function f explicitly depend on the time, i.e. the system can be described with:
Definition 1.5 (Equilibrium State)
An equilibrium state can be defined as a state which the system retain if no external signal is present. Mathematically, the equilibrium state is expressed by the vector . The system remains in equilibrium state if the initial moment the state is the equilibrium state, i.e. at . In the other words, if the system begins from the equilibrium state, it remains in this state ,V under presumption that no external signal act, i.e.
Definition 1.6 (System’s Trajectory-Solution)
An unforced system described by the vector differential equation:
where , and continuous function
Definition 1.7 (Attractive Equilibrium State-Attractor)
Equilibrium state is attractive if for every there exist a number such that
Definition 1.8 (Uniformly Attractive Equilibrium State)
Equilibrium state is uniformly attractive if for every , there exist a number so that the following condition is true:
, as uniformly in and.
Vukic, Zoran, Kuljaca, Ljubomir, Donlagic, Dali, Tesnjak, Sejid, Nonlinear Control Systems, Control Engineering Series, Marcel Dekker, New York USA, 2003.