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## Regulating Pressure in a Drum Boiler

This example shows how to use Simulink Control Design, using a drum boiler as an example application. Using the operating point search function, we illustrate model linearization as well as subsequent state observer and LQR design. In this drum-boiler model, the control problem is to regulate boiler pressure in the face of random heat fluctuations from the furnace by adjusting the feed water flow rate and the nominal heat applied. For this example, 95% of the random heat fluctuations are less than 50% of the nominal heating value. This is not unusual for a furnace-fired boiler.

Step 1: Initialize and Open the Model

To begin, let's open the Simulink model.

```Boiler_Demo
```

The boiler control model's pre-load function initializes the controller sizes. This is necessary because to compute the operating point and linear model, the Simulink model must be executable. Note that u0, y0 are set after the operating point computation and are thus initially set to zero. The observer and regulator are computed during the controller design step and are also initially set to zero.

Step 2: Find a Nominal Operating Point and Linearize the Model

The model's initial state values are defined in the Simulink model. Using these state values find the steady state operating point using the findop function.

First, we'll create an operating point specification where the state values are known.

```opspec = operspec('Boiler_Demo');
opspec.States(1).Known = 1;
opspec.States(2).Known = 1;
opspec.States(3).Known = [1;1];
```

Now, let's adjust the operating point specification to indicate that the inputs must be computed and that they are lower bounded.

```opspec.Inputs(1).Known = [0;0];     %Inputs unknown
opspec.Inputs(1).Min = [0;0];   %Input minimum value
```

Finally, we'll add an output specification to the operating point specification; this is necessary to ensure that the output operating point is computed during the solution process.

```opspec = addoutputspec(opspec,'Boiler_Demo/Boiler',1);
opspec.Outputs(1).Known = 0;    %Outputs unknown
opspec.Outputs(1).Min = 0;      %Output minimum value
```

Next, we compute the operating point and generate a report.

```[opSS,opReport] = findop('Boiler_Demo',opspec);
```
``` Operating Point Search Report:
---------------------------------

Operating Report for the Model Boiler_Demo.
(Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States:
----------
(1.) Boiler_Demo/Boiler/Steam volume
x:           5.6      dx:      7.85e-13 (0)
(2.) Boiler_Demo/Boiler/Temperature
x:           180      dx:     -5.93e-14 (0)
(3.) Boiler_Demo/Observer/Internal
x:             0      dx:             0 (0)
x:             0      dx:             0 (0)

Inputs:
----------
(1.) Boiler_Demo/Input
u:      2.41e+05    [0 Inf]
u:           100    [0 Inf]

Outputs:
----------
(1.) Boiler_Demo/Boiler
y:         1e+03    [0 Inf]

```

Before linearizing the model around this point, we'll specify the input and output signals for the linear model.

First we specify the input points for linearization.

```Boiler_io(1)=linio('Boiler_Demo/Sum',1,'input');
Boiler_io(2)=linio('Boiler_Demo/Demux',2,'input');
```

Now we specify the open loop output points for linearization.

```Boiler_io(3)=linio('Boiler_Demo/Boiler',1,'openoutput');
setlinio('Boiler_Demo',Boiler_io);
```

In this code, we find a linear model around the chosen operating point.

```Lin_Boiler = linearize('Boiler_Demo',opSS,Boiler_io);
```

Finally, using the minreal function, make sure that the model is a minimum realization, (e.g., there are no pole zero cancellations).

```Lin_Boiler = minreal(Lin_Boiler);
```
```1 state removed.
```

Step 3: Designing a Regulator and State Observer

Using this linear model, we will design an LQR regulator and Kalman filter state observer. First find the controller offsets to make sure that the controller is operating around the chosen linearization point by retrieving the computed operating point.

```u0 = opReport.Inputs.u;
y0 = opReport.Outputs.y;
```

Now design the regulator using the lqry function. Note that tight regulation of the output is required while input variation should be limited.

```Q = diag(1e8);                  %Output regulation
R = diag([1e2,1e6]);            %Input limitation
[K,S,E] = lqry(Lin_Boiler,Q,R);
```

Design the Kalman state observer using the kalman function. Note that for this example the main noise source is process noise. It enters the system only through one input, hence the form of G and H.

```[A,B,C,D] = ssdata(Lin_Boiler);
G = [B(:,1)];
H = [0];
QN = 1e4;
RN = 1e-1;
NN = 0;
[Kobsv,L,P] = kalman(ss(A,[B G],C,[D H]),QN,RN);
```

Step 4: Simulate and Test

For the designed controller the process inputs and outputs are shown below.

```sim('Boiler_Demo')
```

Here is the feed water actuation signal in kg/s

```figSize = [0 0 360 240];
h = figure(1); plot(FeedWater.time/60,FeedWater.signals.values)
set(h,'color',[1 1 1])
set(h,'Position',figSize)
title('Feedwater flow rate [kg/s]');
ylabel('Flow [kg/s]')
xlabel('time [min]')
grid on
```

This illustrates the heat actuation signal in kJ:

```h = figure(2); plot(Heat.time/60,Heat.signals.values/1000)
set(h,'color',[1 1 1])
set(h,'Position',figSize)
title('Applied heat [kJ]');
ylabel('Heat [kJ]')
xlabel('time [min]')
grid on
```

The next figure shows the heat disturbance in kJ. Note that the disturbance varies by as much as 50% of the nominal heat value.

```h = figure(3); plot(HeatDist.time/60,HeatDist.signals.values/1000)
set(h,'color',[1 1 1])
set(h,'Position',figSize)
title('Heat disturbance [kJ]');
ylabel('Heat [kJ]')
xlabel('time [min]')
grid on
```

The figure below shows the corresponding drum pressure in kPa. Notice how the pressure varies by about 1% of the nominal value even though the disturbance is relatively large.

```h =figure(4); plot(DrumPressure.time/60,DrumPressure.signals.values)
set(h,'color',[1 1 1])
set(h,'Position',figSize)
title('Drum pressure [kPa]');
ylabel('Pressure [kPa]')
xlabel('time [min]')
grid on

bdclose('Boiler_Demo')
```