You are designing a controller for a process with the following transfer function: -1/(s+1) a) Convert the transfer function into a discrete time transfer function (Z-transform) with a sampling time of 1 second. (2 marks)

b) Using the discrete time transfer function obtained write the model of the system in the following form: y[k+1] = a*y[k] + b*u[k], where y is the output, u is the input, a, and b are constants describing the model. (2 marks)

c) You are required to design an MPC controller for the process described by the difference equation. Starting from zero initial conditions for the outputs and the inputs, the setpoints are -1 at time instant 1 and 0 at time instant 2 (r[1] = -1 and r[2] = 0). Write a cost function corresponding to the sum of the squares of the predicted errors in the next two time steps k = 1 and k = 2 (prediction horizon is 2). Develop the cost function to make it a function of u[0] and u[1]. (3 marks)

d) Calculate u[0] and u[1] that will minimize the cost function. (3 marks)