Higher order DE to State space

In summary, the conversation discusses how to convert a second order differential equation to state space format for use in ODE45 in Matlab. It suggests using methods such as Phase Canonical Forms or Nested Integrals to generate the state space equation, and advises using two state variables for equations with second order derivatives.
  • #1
anirudh_s
1
0
Hey,

I've been trying to run a few simulations in Matlab using ODE45. This algorithm requires a function which gives the first order differential as an output i.e a state space format (Correct me if I'm wrong here).

If its a normal N order differential such as d2x/dt2 + dx/dt -1 =0 . dx/dt can be substituted as y and hence the differential equation becomes:

y(dot) = 1-y i.e. the required state space format.

Now comes the question , what do I do if the DE is something like
d2z/dt2 + d2x/dt2 +dx/dt -1=0 ? How Do I convert this to state space format ?

And more specifically , how do I code this as a function to use in ODE45 in Matlab ?

Thanks in advance!
 
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  • #2
There are second order derivatives so you you will have two state variables. You then use one of following methods:

- Phase Canonical Forms of State Space Equation
- State Space Equation Generation via 'Nested Integrals'

There are other methods as well.
 

1. What is the difference between a higher order differential equation and a state space representation?

A higher order differential equation is an equation that involves derivatives of a dependent variable with respect to an independent variable. It can be written in the form of a single equation with multiple derivatives, such as y'' + y' + y = 0. A state space representation, on the other hand, is a set of first-order differential equations that describe the evolution of a system over time. It is often used in control theory and can be written in the form of x' = Ax + Bu, where x is the state vector, A is the system matrix, and B is the input matrix.

2. Why would someone want to convert a higher order differential equation to a state space representation?

Converting a higher order differential equation to a state space representation can make it easier to analyze and solve the system. State space representations are also useful for designing control systems and can provide insight into the behavior of the system over time. Additionally, state space representations can be easily simulated and implemented in computer programs.

3. How do you convert a higher order differential equation to a state space representation?

The process of converting a higher order differential equation to a state space representation involves several steps. First, the equation must be rewritten as a set of first-order differential equations by introducing new variables for each derivative. Then, the state vector x and input vector u can be defined. Finally, the system matrix A, input matrix B, and output matrix C can be determined using coefficients from the original equation.

4. Are there any limitations or drawbacks to using a state space representation?

One potential limitation of using a state space representation is that it may not be as intuitive or easy to understand as a higher order differential equation. Additionally, some systems may be difficult to model using state space representations, especially if they have many inputs and outputs. Furthermore, state space representations may not be as accurate as higher order differential equations for certain types of systems.

5. Can state space representations be used for nonlinear systems?

Yes, state space representations can be used for nonlinear systems. However, in order to accurately model a nonlinear system using state space, the equations must be linearized around a specific operating point. This involves finding the Jacobian matrix of the system and evaluating it at the desired operating point. Then, the linearized state space equations can be used to approximate the behavior of the nonlinear system.

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