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

1. Determine the two steady states that can arise under the nominal values of the inputs.

2. Illustrate in a single Figure the behavior of the process operated with the nominal values of the inputs, with trajectories that start in each of the stationary states and at different points to stationary states. That is, solve the ODE for various initial conditions with the nominal values of the inputs and put all the trajectories in a single Figure. In the Figure it should be possible Observe that one of the stationary states is stable and the other is unstable. Write a Matlab program that describes this part.

3. Insert the process into a control system with a PI controller. (i) Draw the schematic of the control system. (ii) Obtain the ODE that describes the control system.

4. Obtain the linear model of the process. For this case, there are two steady state options around which the linearization can be performed; then, obtain the two linear models. Hint: put them in terms of deviance variables.

5. Insert each of the linear models in a control system, with a PI controller. (i) Draw the schematic of the control system. (ii) Obtain the ODE, for each linear model, that describes the control system.

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.

7. Regulatory control problem in process (non-linear model): Consider the process control system (with non-linear model) (Item 3(i)). Consider that the value of 2 has changed from 1 to 1.2. Illustrate the control system performance that is obtained with the same gains as Item 6 of this exam, considering that the process is required to be kept at its nominal steady state. Take into account that there are two stationary states. Illustrative for both stationary states. Write a Matlab program that describes this part.