# CH-5120: Modern Control Theory Assignment 5

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## Problem 1: SISO System: Setting up DMC Algorithm (2+1+1+2 points)

This is a hand-written problem. Although, you can do it in MATLAB, I will suggest doing by
hand (except for matrix multiplication) to get yourself used to solving by hand.

In the previous assignments, you developed step-response model for the first-order system:
�(�) = 5
�� + 1 �!”.\$%&, � = 1 + �
2 a = Last digit of roll number, Δ� = 0.5

I will upload a snipped of code that will generate S matrix for you. Please use this to construct the
following that will be required for SISO DMC Algorithm

1. With p = 5 and m = 2 as the prediction and control horizons, respectively, please compute the
matrix �’ used in DMC algorithm for future predictions: �((�) = ℳ�H(�) + �’Δ�

2. Let the output weights be Q = 1 and input weights be R = 0.5. Please compute the Hessian ℋ
3. If the constraints are −0.1 ≤ Δ�(�) ≤ 0.1 and 0 ≤ �(�) ≤ 1, please express the left-hand side
of the constraint equation �Δ�(�) ≤ �)*+.

4. If the previous inputs were �(� − �) = 0, compute the RHS of the constraint equation, i.e., �,-.
This problem is worth double points if submitted before 5 pm on Monday 19th October.

## Problem 2: Step-Response Model of Reactor (3 + 3 points)

In Problem-3 of the previous Assignment, we considered a step response model for a reactor,
with Δ� = 0.2 and with �/ = 1, �0 = 2, and � = 25 steps in the step-response model. As in the
previous assignment, the model parameters will be provided as a R�0. �S × 1 matrix Smodel.

5. With p = 5 and m = 2 as the prediction and control horizons, respectively, please compute the
matrix �’ used in the DMC algorithm and report it in 10 × 2 matrix bigSu.
6. Let the output weights be � = V
0.25 0
0 1
W and input weights be R = 0.1. With these values, please
compute the Hessian ℋ and report it in a 2 × 2 matrix Hess.

## Problem 3: Step-Response Model for (4 + 4 points)

In Problem-3 of the Assignment-2, we considered a step response model for the following
two-input two-output system (with sampling interval Δ� = 2 and n = 25)
�(�) =

⎡ 2
40�1 + 16� + 1
0.5
20�1 + 7� + 1
1.2
10�1 + 5� + 1
1
36�1 + 12� + 1⎦

As in the previous problem, the R�0. �S × �/ matrix Smodel will be provided to you.

Like the previous two problems, you will compute the matrices �’, Γ0, Γ/, ℋ. However,
unlike the previous problem, you will not know the values of p and m. You will write a MATLAB
function [bigSu,Hess]=mimo_dmc_fcn(p,m), where p,m are accepted as inputs and and the
matrices bigSu (�’) and Hess (ℋ) are returned as outputs.

7. For input values of p and m, please compute the matrix �’ used in the DMC algorithm. This must
be returned by your function as a (2� × 2�) matrix bigSu.

8. Let the output weights be � = V
1 0
0 1
W and input weights be R = � = V
0.25 0
0 0.25W. With these
values, please compute the Hessian ℋ and report it in (2� × 2�) matrix Hess.