Linear and Non-Linear Systems

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Linear System

A system is said to be linear if it obeys the principle of homogeneity and principle of superposition.

Principle of Homogeneity

The principle of homogeneity says that a system which generates an output y(t) for an input x(t) must produce an output ay(t) for an input ax(t).

Superposition Principle

According to the principle of superposition, a system which gives an output y₁(t) for an input x₁(t) and an output y₂(t) for an input x₂(t) must produce an output [y₁(t) + y₂(t)] for an input [x₁(t) + x₂(t)].

Therefore, for a continuous-time linear system,

$$\mathrm{[ay^{1}(t) \:+\: by^{2}(t)] \:=\: T[ax^{1}(t) \:+\: bx^{2}(t)] \:=\: aT[x^{1}(t)] \:+\: bT[x^{2}(t)]}$$

Also, for a discrete-time linear system,

$$\mathrm{[ay^{1}(n) \:+\: by^{2}(n)] \:= \:T[ax^{1}(n) \:+\: bx^{2}(n)] \:=\: aT[x^{1}(n)] \:+\:bT[x^{2}(n)]}$$

Hence, we can say that a system is linear if the output of the system due to weighted sum of inputs is equal to the weighted sum of outputs.

Filter circuits, communications channels, etc. are few examples of linear systems.

Non-Linear System

A system is said to be a non-linear system if it does not obey the principle of homogeneity and principle of superposition.

Generally, if the equation describing the system contains square or higher order terms of input/output or product of input/output and its derivatives or a constant, the system will be a non-linear system. Triangulation of GPS signals is an example of non-linear system.

Numerical Example

Check whether the given systems are linear or non-linear systems −

$$\mathrm{\frac{d^{2}y(t)}{d t^{2}}\:+\:3ty(t)\:=\:t^{2}x(t)}$$

$$\mathrm{3\frac{dy(t)}{d t}\:+\:4y(t)\:=\:x^{2}(t)}$$

Solution

The given system is,

$$\mathrm{\frac{d^{2}y(t)}{dt^{2}}\:+\:3ty(t)\:=\:t^{2}x(t)} $$

Consider an input x₁(t) which produces an output y₁(t), then,

$$\mathrm{\mathrm{\frac{d^{2}y_{1}(t)}{dt^{2}}\:+\:3ty_{1}(t)\:=\:t^{2}x_{1}(t)}\: \: \cdot \cdot \cdot (1)} $$

And an input x₂(t) which produces an output y₂(t), then,

$$\mathrm{\mathrm{\frac{d^{2}y_{2}(t) }{dt^{2}}\:+\:3ty_{2}(t)\:=\:t^{2}x_{2}(t)}\: \: \cdot \cdot \cdot (2)} $$

Now, the linear combination of the equations (1) and (2) gives,

$$\mathrm{ \begin{Bmatrix} a\mathrm{\frac{\mathrm{d^{2}}y_{1}(t) }{\mathrm{d} t^{2}}\:+\:a3ty_{1}(t)}\ \end{Bmatrix}+\begin{Bmatrix} b\mathrm{\frac{\mathrm{d^{2}}y_{2}(t) }{\mathrm{d} t^{2}}\:+\:b3ty_{2}(t)}\ \end{Bmatrix}\:=\:at^{2}x_{1}(t)\:+\:bt^{2}x_{2}(t) } $$

$$\mathrm{\Rightarrow \frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\left [ ay_{1}(t)+by_{2}(t) \right ]\:+\:3t\left [ ay_{1}(t)\:+\:by_{2}(t) \right ]}$$

$$\mathrm{=\: t^{2}\left [ ax_{1}(t)\:+\:bx_{2}(t) \right ]\: \: \cdot \cdot \cdot (3)}$$

Where, [ay₁(t) + by₂(t)] is the weighted sum of outputs and [ax₁(t) + bx₂(t)] is the weighted sum of inputs.

Hence, the equation (3) shows that the weighted sum of inputs to the given system generates an output that is equal to the weighted sum of outputs to each of the individual inputs. Therefore, the given system is a linear system.

The given system is,

$$\mathrm{3\frac{\mathrm{d} y(t)}{\mathrm{d} t}\:+\:4y(t)\:=\:x^{2}(t)}$$

Let the output y₁(t) is corresponding to the input x₁(t), then,

$$\mathrm{3\frac{\mathrm{d} y_{1}(t)}{\mathrm{d} t}\:+\:4y_{1}(t)\:=\:x_{1}^{2}(t)\: \: \cdot \cdot \cdot (1)}$$

Similarly, the output y₂(t) is corresponding to the input x₂(t), then,

$$\mathrm{3\frac{\mathrm{d} y_{2}(t)}{\mathrm{d} t}\:+\:4y_{2}(t)\:=\:x_{2}^{2}(t)\: \: \cdot \cdot \cdot (2)}$$

Then, the linear combination (i.e., homogeneity and superposition) of the equations (1) and (2) can be written as,

$$\mathrm{\begin{Bmatrix} a3\frac{\mathrm{d} y_{1}(t)}{\mathrm{d} t}\:+\:a4y_{1}(t)\ \end{Bmatrix}\:+\:\begin{Bmatrix} b3\frac{\mathrm{d} y_{2}(t)}{\mathrm{d} t}\:+\:b4y_{2}(t)\ \end{Bmatrix}\:=\:ax_{1}^{2}(t)\:+\:bx_{2}^{2}(t)}$$

$$\mathrm{\Rightarrow\: 3\frac{\mathrm{d} }{\mathrm{d} t}\left [ ay_{1}(t)\:+\:by_{2}(t) \right ]\:+\:4\left [ ay_{1}(t)\:+\:by_{2}(t) \right ]\:=\:ax_{1}^{2}(t)\:+\:bx_{2}^{2}(t)}$$

Where, $\mathrm{\left [ ay_{1}(t)\:+\:by_{2}(t) \right ]}$ is the weighted sum of outputs but $\mathrm{\left [ ax_{1}^{2}(t)\:+\:bx_{2}^{2}(t) \right ]} $ is not the weighted sum of inputs. Here, the principle of superposition is not satisfied. Therefore, the given system is a non-linear system.