linearize a second-order system without symbolic math toolbox, partial differential equation toolbox or simulink control design in matlab

To linearize a second-order system in MATLAB without using the Symbolic Math Toolbox or other advanced toolboxes, you can use the state-space representation of the system. The state-space representation allows you to convert a system of differential equations into a set of linear algebraic equations that can be solved using matrices.

To do this, you will need to define the state-space equations for your second-order system. These equations describe the relationship between the inputs, states, and outputs of the system:

main.m
x' = Ax + Bu
y = Cx + Du
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where x is a vector of the system's states, u is a vector of inputs to the system, y is a vector of outputs from the system, and A, B, C, and D are matrices that describe the system's behavior.

For a second-order system, you can use the following state-space equations:

main.m
x1' = x2
x2' = -a*x2 - b*x1 + c*u
y = x1
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where x1 and x2 are the states of the system, u is the input, y is the output, and a, b, and c are constants that describe the system's behavior.

To linearize the system, you will need to find the matrices A, B, C, and D that describe the linear behavior of the system at a particular operating point. You can do this using the jacobian function in MATLAB, which calculates the Jacobian matrix of a set of equations at a specified point. Here's the MATLAB code to linearize the system:

main.m
% Define the system's parameters
a = 1;
b = 0.5;
c = 2;

% Define the state-space equations
A = [0 1; -b -a];
B = [0; c];
C = [1 0];
D = 0;

% Define the operating point
x1_0 = 0;
x2_0 = 0;
u_0 = 0;

% Calculate the Jacobian matrix
J = jacobian([A*x + B*u; C*x + D*y], [x; u]);
J_op = subs(J, [x; u; y], [x1_0; x2_0; u_0; x1_0]);

% Extract the A, B, C, and D matrices
A_lin = double(J_op(1:2, 1:2));
B_lin = double(J_op(1:2, 3));
C_lin = double(J_op(3, 1:2));
D_lin = double(J_op(3, 3));
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This code defines the state-space equations for the second-order system and calculates the Jacobian matrix of those equations. It then substitutes in the operating point values and extracts the A, B, C, and D matrices for the linearized system.

Once you have the linearized system, you can use MATLAB's control systems toolbox to analyze and design controllers for the system. You can also use MATLAB's simulation tools to simulate the behavior of the system over time.