Define linear and non linear systems

In control theory, systems theory, and signal processing, systems can be classified based on the linearity of their behavior. Here’s a detailed explanation of linear and non-linear systems:

Linear System

A linear system is one that satisfies the principles of superposition and homogeneity (or scaling). In other words, the output of a linear system is directly proportional to its input, and the response to a combination of inputs is the sum of the responses to each input separately.

Characteristics of Linear Systems:

1. Superposition Principle:
   • If x1(t)x_1(t)x1?(t) and x2(t)x_2(t)x2?(t) are inputs to a linear system, and y1(t)y_1(t)y1?(t) and y2(t)y_2(t)y2?(t) are the corresponding outputs, then for any inputs x1(t)x_1(t)x1?(t) and x2(t)x_2(t)x2?(t): If x(t)=x1(t)+x2(t), then y(t)=y1(t)+y2(t)\text{If } x(t) = x_1(t) + x_2(t), \text{ then } y(t) = y_1(t) + y_2(t)If x(t)=x1?(t)+x2?(t), then y(t)=y1?(t)+y2?(t)
2. Homogeneity (Scaling) Principle:
   • If x(t)x(t)x(t) is an input to a linear system and y(t)y(t)y(t) is the corresponding output, then for any scalar α\alphaα: If x(t) results in y(t), then αx(t) results in αy(t)\text{If } x(t) \text{ results in } y(t), \text{ then } \alpha x(t) \text{ results in } \alpha y(t)If x(t) results in y(t), then αx(t) results in αy(t)

Mathematical Representation:

• Linear systems are often described by linear differential equations in continuous time or linear difference equations in discrete time.

For continuous-time systems: andny(t)dtn+…+a1dy(t)dt+a0y(t)=bmdmx(t)dtm+…+b1dx(t)dt+b0x(t)a_n \frac{d^n y(t)}{dt^n} + \ldots + a_1 \frac{dy(t)}{dt} + a_0 y(t) = b_m \frac{d^m x(t)}{dt^m} + \ldots + b_1 \frac{dx(t)}{dt} + b_0 x(t)an?dtndny(t)?+…+a1?dtdy(t)?+a0?y(t)=bm?dtmdmx(t)?+…+b1?dtdx(t)?+b0?x(t)

For discrete-time systems: any[n]+an−1y[n−1]+…+a0y[0]=bmx[n]+bm−1x[n−1]+…+b0x[0]a_n y[n] + a_{n-1} y[n-1] + \ldots + a_0 y[0] = b_m x[n] + b_{m-1} x[n-1] + \ldots + b_0 x[0]an?y[n]+an−1?y[n−1]+…+a0?y[0]=bm?x[n]+bm−1?x[n−1]+…+b0?x[0]

Examples of Linear Systems:

1. Resistor: Follows Ohm's law, V=IRV = IRV=IR, where the voltage drop across a resistor is directly proportional to the current through it.
2. Mass-Spring System: The force is proportional to displacement (Hooke's Law: F=−kxF = -kxF=−kx).
3. Linear Filters: Electrical or digital filters that obey the superposition principle.

Non-Linear System

A non-linear system is one that does not satisfy the principles of superposition and homogeneity. The output of a non-linear system is not directly proportional to its input, and the response to a combination of inputs is not necessarily the sum of the responses to each input separately.

Characteristics of Non-Linear Systems:

1. Violation of Superposition Principle:

   • For inputs x1(t)x_1(t)x1?(t) and x2(t)x_2(t)x2?(t) with corresponding outputs y1(t)y_1(t)y1?(t) and y2(t)y_2(t)y2?(t), a non-linear system may not satisfy: If x(t)=x1(t)+x2(t), then y(t)≠y1(t)+y2(t)\text{If } x(t) = x_1(t) + x_2(t), \text{ then } y(t) \neq y_1(t) + y_2(t)If x(t)=x1?(t)+x2?(t), then y(t)?=y1?(t)+y2?(t)

2. Violation of Homogeneity (Scaling) Principle:

   • For an input x(t)x(t)x(t) with corresponding output y(t)y(t)y(t), a non-linear system may not satisfy: If x(t) results in y(t), then αx(t) results in αy(t)\text{If } x(t) \text{ results in } y(t), \text{ then } \alpha x(t) \text{ results in } \alpha y(t)If x(t) results in y(t), then αx(t) results in αy(t)

Mathematical Representation:

• Non-linear systems are described by non-linear differential equations in continuous time or non-linear difference equations in discrete time.

For continuous-time systems: an(dny(t)dtn)2+…+a1sin?(dy(t)dt)+a0y(t)=bmlog?(dmx(t)dtm)+…+b1(dx(t)dt)3+b0x(t)a_n \left(\frac{d^n y(t)}{dt^n}\right)^2 + \ldots + a_1 \sin\left(\frac{dy(t)}{dt}\right) + a_0 y(t) = b_m \log\left(\frac{d^m x(t)}{dt^m}\right) + \ldots + b_1 \left(\frac{dx(t)}{dt}\right)^3 + b_0 x(t)an?(dtndny(t)?)2+…+a1?sin(dtdy(t)?)+a0?y(t)=bm?log(dtmdmx(t)?)+…+b1?(dtdx(t)?)3+b0?x(t)

For discrete-time systems: any[n]2+an−1sin?(y[n−1])+…+a0y[0]=bmlog?(x[n])+bm−1x[n−1]3+…+b0x[0]a_n y[n]^2 + a_{n-1} \sin(y[n-1]) + \ldots + a_0 y[0] = b_m \log(x[n]) + b_{m-1} x[n-1]^3 + \ldots + b_0 x[0]an?y[n]2+an−1?sin(y[n−1])+…+a0?y[0]=bm?log(x[n])+bm−1?x[n−1]3+…+b0?x[0]

Examples of Non-Linear Systems:

1. Diode: The current-voltage relationship of a diode is exponential and does not obey superposition.
2. Pendulum: For large angles, the restoring force is not directly proportional to the displacement (non-linear differential equation).
3. Population Models: Logistic growth models where the rate of growth is not proportional to the population size.

Comparison of Linear and Non-Linear Systems:

Summary

• Linear Systems: Satisfy superposition and homogeneity, described by linear equations, and exhibit proportional responses to inputs.
• Non-Linear Systems: Do not satisfy superposition and homogeneity, described by non-linear equations, and exhibit complex, non-proportional responses to inputs.

Understanding the distinction between linear and non-linear systems is fundamental in various fields of science and engineering, as it affects the analysis, design, and control of systems.