Linearization of higly non-linear model

Does someone know how to linearize this highly non linear functions

dTz = ((alfa_dispersione*U_dispersione)*(Tw-Tz) + (1/Ts)*IntGain*People + alpha*(Tr-Tz) )/C_z; 
dTw = (alfa_dispersione*U_dispersione*(Tz-Tw)+alfa_dispersione*U_dispersione*(Text-Tw))/C_disp;
dTr = 1/(md_load*cp_water)*(Qhp + (1/0.75)*alpha*(Tz-Tr) );

where [Tw Tz Tr] are our state variables and [Text People Tz_ref Cf Tz_ref_fc T5]

Text = u{1}; 
People = u{2}; 
Tz_ref = u{3};
Cf = u{4}; 
Tz_ref_fc = u{5};
T5 = u{8};

and other parameters are described

COPQ3 = a+b*Text+c*T5+d*Cf+e*Text^2+n*T5^2+g*Cf^2+h*Text*T5+m*T5*Cf+l*Text*Cf;
Php = Pmax*Cf;
Qhp = Php*COPQ3;

a =  -1.806e+06 ;
b =   1.988e+06 ;
alpha = a*(0.11*atan(10*((Tz_ref_fc-Tz)-100))) + b*(0.1*atan(20*((Tz_ref_fc-Tz)-0.32)+5.52));

Answer 1

[Avni Agrawal]

I understand that you want to linearize a highly non-linear model.You can use the concept of small-signal linearization. This involves approximating the non-linear functions as linear functions around an operating point.

This can be achieved in 2 ways.

1. Using plain MATLAB code
2. Using llinear analysis tool and linearize (inbuilt function) by MATLAB

In-order to achieve that, you can refer to the following steps:

Step 1: calculate linearized equations around operating point.

Step 2: Define Operation point value based on your model

Step 3: Substitute the operating point values into the linearized equations using the subs function in-order to calculate co-efficients

Step 4: Calculate linearized co-efficients, which are then used to construct the state-space matrices (A, B, C, D) for the linearized model.

Step 5: Create the linear state-space model using in-built 'ss' function from MATLAB

Here is the code snippet for the reference:

% Define the symbolic variables
syms Tw Tz Tr Text People Tz_ref Cf Tz_ref_fc T5
% Define the symbolic parameters
syms alfa_dispersione U_dispersione Ts IntGain alpha C_z C_disp md_load cp_water Qhp
% Define the symbolic equations
dTz = ((alfa_dispersione*U_dispersione)*(Tw-Tz) + (1/Ts)*IntGain*People + alpha*(Tr-Tz) )/C_z;
dTw = (alfa_dispersione*U_dispersione*(Tz-Tw)+alfa_dispersione*U_dispersione*(Text-Tw))/C_disp;
dTr = 1/(md_load*cp_water)*(Qhp + (1/0.75)*alpha*(Tz-Tr));
% Linearize the equations around an operating point
operating_point = [Tw Tz Tr Text People Tz_ref Cf Tz_ref_fc T5];
linearized_eqns = jacobian([dTz; dTw; dTr], operating_point); % Compute the Jacobian matrix
% Evaluate the linearized equations at the operating point
linearized_eqns = subs(linearized_eqns, operating_point, operating_point_values);  % Define the operating point values from your model
% Extract the coefficients of the linearized equations
linearized_coeffs = coeffs(linearized_eqns, [Tw Tz Tr]);
% Extract the state-space matrices
A = double(linearized_coeffs(1:3, 1:3));
B = double(linearized_coeffs(1:3, 4:end));
C = eye(3);
D = zeros(3, size(B, 2));
% Create the linearized state-space model
linear_sys = ss(A, B, C, D);

Resources which may help for further assistance:

https://in.mathworks.com/help/simulink/ug/linearizing-models.html

https://in.mathworks.com/help/slcontrol/ug/linearizing-nonlinear-models.html

https://in.mathworks.com/help/control/linear-analysis.html

https://www.youtube.com/watch?v=3H_hglTg8fo