How to separate state and unknown parameters in adaptive control using matlab

Some background

I am trying to do adaptive control, specifically for a robot. I have dynamics of the form

I can formulate this for adaptive control as follows

Where a contains the unknown parameters I want to adapt on, and

is a function only of measured state values.

Some details/context you don't need, but which might be helpful for those familiar with adaptive control

I then want to use an adaptation law akin to

Where

The problem

I need to seperate my dynamics into a linear vector a, and the matrix which is a function of measured states, Y.

For a previous robotic arm I did adaptive control on, I did this by hand. As you might see, this was very tedious to do by hand, and my model is going to become infinitely much more complicated, it is unfeasible for me to do this by hand in the future.

My question

Is there a way to do this using matlab's symbolic toolbox or similar? I already have an expression for Y*a as a function of the symbolic variables, q, dq, ddq and m1, m2, I1, I2, etc. I just need to seperate them, possibly using something like equationsToMatrix or solve or something similar. I've tried extensively with solve but been unable to make it work.

I will add the specific symbolic expressions in comments rather than here as I don't expect anyone to familiarize themselves with the entire code, only to answer how they would solve this problem using symbolic or similar.

Answer 1

[akshatsood]

Hi @Johannes Ihle,

I understand that you seek guidance to separate state and unknown parameters in adaptive control using MATLAB. Using Symbolic Toolbox can simplify the process of separating the dynamics into a linear vector and a matrix based on measured states. Here's a general approach to solving this problem using MATLAB's symbolic toolbox:

STEP 1: Define the symbolic variables representing the states and parameters in your system

STEP 2: Define the dynamics of the system using the symbolic variables.

STEP 3: Use the symbolic expressions you have to formulate the product Y*a as a function of the symbolic variables.

STEP 4: To separate Y and a, you can use MATLAB's symbolic manipulation functions such as `solve` or `mldivide`.

STEP 5: Once you have obtained the separate expressions for Y and a, you can use them to implement the adaptation law and proceed with your adaptive control algorithm.

Have a look at the following code snippet to use MATLAB's Symbolic Toolbox to separate 'Y' and 'a':

% Define symbolic variables
syms q dq ddq m1 m2 I1 I2

% Define the dynamics
H = ...; 
C = ...;
D = ...;
    
% Formulate Y*a
Y_times_a = ...;
    
% Separate Y and a
Y = solve(Y_times_a == ..., Y);
a = solve(Y_times_a == ..., a);

I hope this helps.