Solving Non-Linear Systems with Higher Order Differential Equations

In summary, the conversation discusses the use of mathematical models in control engineering, specifically in regards to input, output, and state variables related by differential equations. It also mentions the use of transfer functions and state space models for solving linear and non-linear systems. The question is how to solve non-linear systems with higher order differential equations using state space representation and what mathematical concepts to refer to.
  • #1
pairofstrings
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Homework Statement


In control engineering, I want to have a mathematical model of a physical system as a set of input, output and state variables related by higher order differential equations.

2. Relevant concepts
As we all know that, in control engineering, we can solve linear-system using transfer functions. The transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s). And we use state space models for Multiple input Multiple output systems and/or for non-linear systems. Right?

Y(s) = H(s) X(s)

The Attempt at a Solution



I am able to solve non-linear system using state space representation where mathematical model of a physical system is a set of input, output and state variables are related by first-order differential equations.

But, my question is, how do I solve non-linear system using state space representation where physical system is a set of input, output and state variables are related by higher order differential equations.

Which concepts in mathematics should I refer?

I hope my question is clear. Thanks for the help.
 
Last edited:
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  • #2


An nth order differential equation can allways be represented as a set of n first order differential equations.
 

Related to Solving Non-Linear Systems with Higher Order Differential Equations

1. What are non-linear systems with higher order differential equations?

Non-linear systems with higher order differential equations are mathematical models that describe the relationship between a variable and its rate of change, where the rate of change is not proportional to the variable itself. These systems often involve multiple variables and equations of at least second order.

2. Why are non-linear systems with higher order differential equations important?

Non-linear systems with higher order differential equations are important because they allow us to model and understand complex phenomena in nature and science. Many real-world problems cannot be accurately described with simple linear equations, and higher order differential equations provide a more accurate representation.

3. What are some methods for solving non-linear systems with higher order differential equations?

Some common methods for solving non-linear systems with higher order differential equations include substitution, elimination, and numerical methods such as Euler's method or Runge-Kutta methods. These methods require a combination of algebraic manipulation and calculus techniques.

4. Can non-linear systems with higher order differential equations have multiple solutions?

Yes, non-linear systems with higher order differential equations can have multiple solutions. This is because they are more complex and can have multiple sets of initial conditions that satisfy the equations. It is important to carefully analyze the solutions to ensure they are physically meaningful.

5. Are there any applications of non-linear systems with higher order differential equations?

Yes, non-linear systems with higher order differential equations have many applications in various fields such as physics, engineering, biology, and economics. They are commonly used to model systems with non-linear behavior, such as chaotic systems, population dynamics, and chemical reactions.

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