Let be a process described by the following ODE: =3^2(1+32)+12, (0)=0 Consider that 1 can be used as a control input and the input 2 as a disturbance. Consider that the nominal values of the inputs are: 1_nominal = 6 and 2_nominal= 1

6. For each of the linear models, consider a servo-control type control problem. In terms of deviation variables, the nominal value of the inputs and status is equal to zero. (i) Determine an effective value in the Proportional Gain and in the Integral Gain, in such a way that the control system (in terms of deviation variables) reaches a new stationary state equal to to 0.1, in a time that is less than the settling time of the linear model. Justify how I arrive at the establishment of the value of the gain of the controller (without this justification, the answer of this test item will be considered null). Illustrate the behavior of the control system for each linear model. Make a Matlab program that describes this part as well as calculations to support it.

Complete the Matlab program that describes this part using this first part of matlab code as a start.

function controlSystem()

% Nominal values

x_nominal = 0;

u1_nominal = 0;

u2_nominal = 0;

% Linear model coefficients

A = 1 - (u1_nominal + 3*u2_nominal);

B = 1;

% We can choose any non-zero value for simplicity

C = 3*x_nominal^2 - (3*x_nominal^2 - (u1_nominal + 3*u2_nominal)*x_nominal);

% Control system parameters