Properties of Nonlinear System (Related to Definition)

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 : f\left({x}_{1},{x}_{2} \right)=f\left({x}_{1} \right)+f\left({x}_{2} \right), V{x}_{1},{x}_{2} in domain of the function f,

2. Homogenity: f\left(\alpha x \right)=\alpha f\left(x \right), Vx in domain of function f and all scalar \alpha

Definition 1.2 (Locally Lipschitz Function)

The function f is said to be locally Lipschitz for the variable x(t) if near the point x\left(0 \right)={x}_{0} it satisfies the Lipschitz criterion:

\left|\left|f\left(x \right)-f\left(y \right) \right| \right|\leq \left|\left|x-y \right| \right|

for all x and y in the vicinity of {x}_{0} 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:

\dot{x}\left(t \right)=f\left[t,x\left(t \right),u\left(t \right) \right]; Vt\geq 0, u\left(t \right)\neq 0

A continuous system is said to be unforced if there is no input (excitation) signal:

\dot{x}\left(t \right)=f\left[t,x\left(t \right)\right]; Vt\geq 0, u\left(t \right)= 0

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:

\dot{x}\left(t \right)=f\left[x\left(t \right),u\left(t \right) \right]; Vt\geq 0

A system is time-varying if the function f explicitly depend on the time, i.e. the system can be described with:

\dot{x}\left(t \right)=f\left[t,x\left(t \right),u\left(t \right) \right]; Vt\geq 0

Definition 1.5 (Equilibrium State)

An equilibrium state can be defined as a state which the system retain if no external signal u\left(t \right) is present. Mathematically, the equilibrium state is expressed by the vector {x}_{e}\in {R}^{n}. The system remains in equilibrium state if the initial moment the state is the equilibrium state, i.e. at t={t}_{0}-x\left({t}_{0} \right)={x}_{e}. In the other words, if the system begins from the equilibrium state, it remains in this state x\left(t \right)={x}_{e},Vt\geq {t}_{0} under presumption that no external signal act, i.e. u\left(t \right)=0

Definition 1.6 (System’s Trajectory-Solution)

An unforced system described by the vector differential equation:

\dot{x}\left(t \right)=f\left[t,x\left(t \right) \right];Vt\geq 0

where x\in {R}^{n}, t\in {R}_{+}and continuous function f:{R}_{+}\times {R}^{n}\rightarrow {R}^{n}

Definition 1.7 (Attractive Equilibrium State-Attractor)

Equilibrium state {x}_{e}is attractive if for every  {t}_{0}\in {R}_{+}there exist a number \eta \left({t}_{0} \right)>0such that

\left|\left|{x}_{0} \right| \right|<\eta \left({t}_{0} \right)\Rightarrow s\left({t}_{0}+t,{t}_{0},{x}_{0} \right)\rightarrow {x}_{e}as t\rightarrow \sim

Definition 1.8 (Uniformly Attractive Equilibrium State)

Equilibrium state {x}_{e}is uniformly attractive if for every {t}_{0}\in {R}_{+}, there exist a number \eta \left({t}_{0} \right)>0so that the following condition is true:

\left|\left|{x}_{0} \right| \right|<\eta ,{t}_{0}\geq 0\Rightarrow s\left({t}_{0}+t,{t}_{0},{x}_{0} \right)\rightarrow {x}_{e} as t\rightarrow \sim uniformly in {x}_{0}and{t}_{0}.

Source:

Vukic, Zoran, Kuljaca, Ljubomir, Donlagic, Dali, Tesnjak, Sejid, Nonlinear Control Systems, Control Engineering Series, Marcel Dekker, New York USA, 2003.

One Response to Properties of Nonlinear System (Related to Definition)

  1. Pingback: Properties of Nonlinear System (Related to Theorem) « kang Irwan Purnama’s Blog

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