How to Derive the Mathematical Equation of a Non-Linear ARX Model in MATLAB

When working with time-series data, it is essential to understand the underlying mathematical models that can capture the dynamics of the data. Among these, Non-Linear Auto-Regressive models with Exogenous inputs (Non-Linear ARX models) are particularly useful for capturing complex behaviors that linear models cannot handle. This article will guide you through the process of deriving the mathematical equation of a Non-Linear ARX model in MATLAB, which can be a bit intricate due to its complex nature. We will cover the necessary steps and commands to achieve this.

Introduction to Non-Linear ARX Models

To begin with, it is important to understand that Non-Linear ARX models represent a class of time-series models that can capture non-linear relationships between inputs and outputs. Unlike classical linear models, Non-Linear ARX models are composed of a network of time-delayed subsystems, making them more complex to represent mathematically. There is no general classical representation of a NL ARX model as it involves a complex interplay of linear and non-linear components.

Deriving the Mathematical Equation in MATLAB

Fortunately, MATLAB provides powerful tools that assist in deriving the mathematical equation of a Non-Linear ARX model. Below, we will walk through the steps involved in obtaining the equation using the getreg and getFunction commands.

Steps to Obtain the Equation

First, you need to ensure that you have a Non-Linear ARX model defined in your MATLAB environment. If you do not already have one, you can create it using the idnlarx function. For example:

model  idnlarx([2 2 1],'wavenet','reg');

To obtain the regressors, you can use the getreg command. This command returns the regressors of the model as a cell array of strings. This helps in identifying the input variables that are included in the model. For example:

reg  getreg(model);

To get the output function that defines the relationship between the regressors and the output, you can use the getFunction command. This command returns a structure containing the linear and nonlinear terms and the offset. For example:

fun  getFunction(model);

Once you have the regressors and the output function, you can display the equation in the MATLAB command window using the following code snippet:

disp(['y(t)  ' num2str(1) ' ' reg{1} '   ' num2str(2) ' ' reg{2} '   ' reg{3} ' ' mat2str() '   ' mat2str() '   ' num2str() '   ' error']);

This code will display the equation in the format:

y(t) a_1 * x_1(t) a_2 * x_2(t) f(x_3(t)) b error

where x_1(t), x_2(t), and x_3(t) are the regressors, a_1 and a_2 are the linear coefficients, f is the nonlinear function, b is the offset, and error is the random noise term.

Example and Further Applications

Let's go through a concrete example to demonstrate how to use these commands. Suppose we have a Non-Linear ARX model named model in our MATLAB workspace. We can derive its equation as follows:

Get the regressors:

reg  getreg(model);

Get the output function:

fun  getFunction(model);

Display the equation:

disp(['y(t)  ' num2str(1) ' ' reg{1} '   ' num2str(2) ' ' reg{2} '   ' reg{3} ' ' mat2str() '   ' mat2str() '   ' num2str() '   ' error']);

The exact values of the coefficients, the nonlinear function, and the offset will depend on the specific Non-Linear ARX model you are working with and the estimation settings used.

Conclusion

Deriving the mathematical equation of a Non-Linear ARX model in MATLAB is a powerful technique that can provide valuable insights into the model's behavior. By following the steps outlined in this article, you can easily obtain the equation and further analyze the dynamics of your time-series data. For more details and advanced applications, refer to the MATLAB documentation on Non-Linear ARX models.