In this topic, you study the Linear and Nonlinear Systems theory, definition & solved examples.
Linear System
A system is called linear if it satisfies two properties
(i) Additivity
(ii) Homogeneity
Additivity
If an input $x_1(t)$ produces output $y_1(t)$ and another input $x_2(t)$ also acting along produces output $y_2(t)$, then, when both inputs acting on the system simultaneously, produces output $y_1(t) + y_2(t)$. Mathematically,
If
\[{x_1}(t)\xrightarrow{{system}}{y_1}(t)\]
and
\[{x_2}(t)\xrightarrow{{system}}{y_2}(t)\]
then
\[{x_1}(t) + {x_1}(t)\xrightarrow{{system}}{y_1}(t) + {y_2}(t)\]
Homogeneity
It states that if input is scaled by any scalar $k$, then output also scaled by the same amount.
If
\[x(t)\xrightarrow{{system}}y(t)\]
then
\[k \cdot x(t)\xrightarrow{{system}}k \cdot y(t)\]
Nonlinear System
Any system is called nonlinear that does not satisfy two properties
(i) Additivity
(ii) Homogeneity
Example : Determine whether or not each of the following systems are linear with input $x(t)$ and output $y(t)$.
(i) \[y(t) = ax(t) + b\]
(ii) \[y(t) = xsin(t)\]
Solution : (i) \[y(t) = ax(t) + b\]
Additivity
\[{y_1}(t) + {y_2}(t) = a{x_1}(t) + b + a{x_2}(t) + b\]
\[y(t) = a{x_1}(t) + a{x_2}(t) + b\]
\[y(t) \ne {y_1}(t) + {y_2}(t)\]
hence the system is Nonlinear.
Solution : (ii) \[y(t) = xsin(t)\]
Additibity
\[{y_1}(t) + {y_2}(t) = {x_1}sin(t) + {x_2}sin(t)\]
\[y(t) = {x_1}sin(t) + {x_2}sin(t)\]
\[y(t) = {y_1}(t) + {y_2}(t)\]
Homogeneity
\[k \cdot y(t) = k \cdot xsin(t)\]
\[k \cdot x(t) = k \cdot xsin(t)\]
so
\[k \cdot y(t) = k \cdot x(t)\]
hence the system is Linear.
