Solve above by design MAC(model algorthmic control) mathmaticaly and by using matlab

Inlet Outlet W Cold water Hot water Control valve Fig. 3.4. Water heater 3.4 Case Study: A Water Heater This example shows the design of a DMP to control the outlet temperature of a water heater. Notice that a MAC can be designed following the same steps. Consider a water heater where the cold water is heated by means of a gas bumer. The outlet temperature depends on the energy added to the water through the gas bumer (see Figure 3.4). Therefore this temperature can be controlled by the valve which manipulates the gas flow to the heater The step response model of this process must be obtained to design the controller. The step response is obtained by applying a step in the control valve. Coefficients can be obtained directly from the response shown in figure 3.5. It can be seen that the output stabilizes after 30 periods, so the model is given by (0) Auf-i), where the coefficients are shown in the following table: 1999 94 195 196 19 1990 JO1002710.1980.6870.8450.977 1087 1.1791.250 gw 12 gu gu gu gan garg giao 1.320 1.374 1.419 1.4561.4571.513 1.535 1.5531 5651.581 921 1922 1923 1924 1925 1926 1927 925 1926 1920 1.592 1.6001.6081.6141.619 1.623 1.6271.630 1.6331.635 The response shown in Figure 35 corresponds to a system with a transfer function given by: C(=) = 0.2713-3 1 -0.63513--1 Notice that although the coefficients are obtained in practice from real plant tests, for this example, where the response has been generated with a simple model, the step response coefficients can easily be obtained from the transfer function by the expression 910