Mathematical proof for non linear system

In summary: The input to the system is not just "x1" and "x2", it is also "g" which is not being accounted for in your equations. The system is nonlinear because the output of the system is not just a linear combination of the inputs, but also a product with the impulse response. Therefore, the system does not satisfy the condition for linear stability.
  • #1

Homework Statement


Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.

Block diagram is the product of x1(t) and x2(t) going through a block step of h(t) (exp-2t)

Give a mathematical proof that this is a nonlinear system.

Question attached in picture.

Homework Equations


α1y1(t) + α2y2(t) = H {α1x1(t) + α2x2(t)} , condition for linear stability

where x1(t) and x2(t) are two input signals and y1(t) and y2(t) are the corresponding output signals for system H

The Attempt at a Solution



Would my equations become

y1(t) = x1(t)e(-2t)

y2(t) = x2(t)e(-2t)

then calculating one way

y(t) = α1x1(t)e(-2t) + α2x2(t)e(-2t)

then the other

y(t) = e(-2t)[α1x1(t) + α2x2(t)] = α1x1(t)e(-2t) + α2x2(t)e(-2t)

But that would suggest linearity... Is there something I'm missing?
 

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  • #2
Jimmy Johnson said:

Homework Statement


Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.

Block diagram is the product of x1(t) and x2(t) going through a block step of h(t) (exp-2t)

Give a mathematical proof that this is a nonlinear system.

Question attached in picture.

Homework Equations


α1y1(t) + α2y2(t) = H {α1x1(t) + α2x2(t)} , condition for linear stability

where x1(t) and x2(t) are two input signals and y1(t) and y2(t) are the corresponding output signals for system H

The Attempt at a Solution



Would my equations become

y1(t) = x1(t)e(-2t)

y2(t) = x2(t)e(-2t)

then calculating one way

y(t) = α1x1(t)e(-2t) + α2x2(t)e(-2t)

then the other

y(t) = e(-2t)[α1x1(t) + α2x2(t)] = α1x1(t)e(-2t) + α2x2(t)e(-2t)

But that would suggest linearity... Is there something I'm missing?

You seem to be missing totally that the input to H is g(t) and inventing two outputs y when there is only one.

Seem. Thisnis not my field but it sure seems.
 
  • #3
Jimmy Johnson said:

Homework Statement


Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.

Block diagram is the product of x1(t) and x2(t) going through a block step of h(t) (exp-2t)

Give a mathematical proof that this is a nonlinear system.

Question attached in picture.

Homework Equations


α1y1(t) + α2y2(t) = H {α1x1(t) + α2x2(t)} , condition for linear stability

where x1(t) and x2(t) are two input signals and y1(t) and y2(t) are the corresponding output signals for system H

The Attempt at a Solution



Would my equations become

y1(t) = x1(t)e(-2t)

y2(t) = x2(t)e(-2t)

then calculating one way

y(t) = α1x1(t)e(-2t) + α2x2(t)e(-2t)

then the other

y(t) = e(-2t)[α1x1(t) + α2x2(t)] = α1x1(t)e(-2t) + α2x2(t)e(-2t)

But that would suggest linearity... Is there something I'm missing?

Yes, you are missing the effects of "g".
 

1. What is a mathematical proof for a non-linear system?

A mathematical proof for a non-linear system is a rigorous demonstration that a non-linear system of equations has a unique solution or set of solutions based on the given constraints and parameters. This proof relies on logical reasoning and employs mathematical concepts and techniques to show that the solutions to the non-linear system are valid and consistent.

2. Why is it important to have a mathematical proof for a non-linear system?

A mathematical proof for a non-linear system is important because it provides a solid foundation and justification for the solutions to the system. It allows us to confidently use and rely on the solutions for further analysis and applications. Additionally, a proof helps to identify any potential errors or inconsistencies in the solutions, ensuring their accuracy and reliability.

3. What are some common techniques used in mathematical proofs for non-linear systems?

Some common techniques used in mathematical proofs for non-linear systems include substitution, elimination, and linearization. These techniques allow us to manipulate the equations and constraints of the system to simplify and solve for the solutions. Other techniques may include induction, contradiction, and direct or indirect proofs.

4. Can a non-linear system have multiple solutions?

Yes, a non-linear system can have multiple solutions. In fact, it is common for non-linear systems to have multiple solutions or even an infinite number of solutions. This is because non-linear systems can exhibit complex behavior and have multiple intersecting curves or surfaces, leading to multiple solutions that satisfy the given constraints.

5. How can we verify the validity of a mathematical proof for a non-linear system?

To verify the validity of a mathematical proof for a non-linear system, we can check the logical reasoning and mathematical techniques used in the proof, as well as the assumptions and constraints of the system. Additionally, we can test the solutions found using the proof by plugging them back into the original equations and confirming that they satisfy all of the constraints and accurately predict the behavior of the system.

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