2. Consider the objective function for an MPC (single controlled variable, single manipulated variable) that may be developed for an artificial pancreas. M min 'elk + ik)Qie(k +i/k) + (Au(k+i[k)"R; (Au(k+ik) i=1 where elk) is the predicted error at time k: e(k) = YReference(k) - Ypredicted(k). The terminology x(k+ik) represents the value of x computed at time k+i based on information available at time k. Au(k)=u(k)- u(k-1). The objective function is not complete. All of its parameters must be specified. (See part b). a) What are the independent variables whose values are changed during the optimization in order to minimize this objective function? b) Expand and complete the objective function and additional expressions needed for setting up the MPC optimization problem. Assign numerical values to the parameters of the MPC based on the additional information listed below: The prediction horizon is 4 The control horizon is 2 The penalty on the controlled variable error is assigned different values for various future times. The weight for the first time in the future is set to 1, the weight for the second time in the future is set to 2 and the weight for the higher times in the future is set to 5. The penalty on the change in the manipulated variable error is assigned different values for various future times. It is 0.1 for the first future time step and twice as high for each successive value in time. The maximum value of manipulated variable values is limited to 5. The change in the manipulated variable from a time step to the next time step (i.e., from k+i to k+i+1) cannot be greater than 0.8. You need to expand the two summations in the objective function and show the final form which is the sum of several terms and enter the values for the weights. For example, the first term of the expanded sum is (you need to figure out the value of the weight 91 from the information given in the text): min[91(elk + 1|k)2 + ...]