Thermal Control

In this chapter, we’ll consider another system that includes a plant, controller, sensor and actuator. The application will be thermal control of a three zone house. The plant will be the house itself, sensor will be a temperature sensor and the actuator will be the furnace in the house. Using these models, we will explore a few different control strategies.

We’ll also follow the architecture driven approach to building the system that we followed in the previous section. However, we’ll start using one set of interfaces and then, after discussing their limitations, restart taking a different approach that will provide us with greater flexibility.

Initial Approach

Architecture

Let’s start with the following architecture:

Initial architecture

Here we see the same basic pieces we saw in the previous section, a plant model, a sensor, a controller and an actuator. In fact, this is a pretty typical architecture. In some cases, people may break down the plant model into a few subsystems and/or include multiple controllers and control loops. But many closed loop system control problems have a similar structure.

What tends to change from application are the specific signals exchanged between these parts. In this case, we can see from the architecture schematic that our interface definitions are such:

  • The actuator receives a commanded temperature and then injects heat through a thermal connection to the plant
  • The sensor model also has a thermal connector (to the plant) and an output signal containing the measured temperature.
  • The plant has two thermal connections. One represents where the furnace heat is added to the system and the other is where the sensor is located.
  • The controller takes the measured temperature (from the sensor) as an input and outputs a commanded heat output (to the actuator)

The Modelica code for this base system looks like this:

within ModelicaByExample.Architectures.ThermalControl.Architectures;
partial model BaseArchitecture "A basic thermal architecture"
  replaceable Interfaces.PlantModel plant
    annotation (Placement(transformation(extent={{-10,-10},{10,10}})));
  replaceable Interfaces.ControlSystem controller
    annotation (Placement(transformation(extent={{-10,30},{10,50}})));
  replaceable Interfaces.Sensor sensor
    annotation (Placement(transformation(extent={{32,-10},{52,10}})));
  replaceable Interfaces.Actuator actuator
    annotation (Placement(transformation(extent={{-50,-10},{-30,10}})));
equation
  connect(plant.room, sensor.room) annotation (Line(
      points={{10,0},{32,0}},
      color={191,0,0}, smooth=Smooth.None));
  connect(sensor.temperature, controller.temperature) annotation (Line(
      points={{53,0},{70,0},{70,40},{12,40}},
      color={0,0,127}, smooth=Smooth.None));
  connect(actuator.furnace, plant.furnace) annotation (Line(
      points={{-30,0},{-10,0}},
      color={191,0,0}, smooth=Smooth.None));
  connect(controller.heat, actuator.heat) annotation (Line(
      points={{-11,40},{-70,40},{-70,0},{-52,0}},
      color={0,0,127}, smooth=Smooth.None));
end BaseArchitecture;

Initial Implementations

Plant

Our plant model looks like this:

Three zone plant model

Here we can see that the zone where furnace heat is added is separated from the zone where the temperature is measured by a third zone. Our furnace model is a simple heat source:

A conventional actuator

This actuator takes a commanded heat level as an input and then injects that amount of heat into the system.

The sensor is similarly simple:

A conventional sensor

This sensor doesn’t introduce any artifact. Instead, it provides the exact temperature as a continuous signal.

We will use the following PI controller to control the temperature:

The control system

Initial Results

Populating our architecture with these implementations, our model now looks like this:

Initial system configuration

Note how the icons for the various subsystems have changed. This is because when we perform a redeclare, the icon for the new type associated with that subsystem is used. The Modelica code for this system is:

within ModelicaByExample.Architectures.ThermalControl.Examples;
model BaseModel "Base model using a conventional architecture"
  extends Architectures.BaseArchitecture(
    redeclare Implementations.ThreeZonePlantModel plant(
      C=2, G=1, h=2, T_ambient=278.15),
    redeclare
      ModelicaByExample.Architectures.ThermalControl.Implementations.ConventionalPIControl
      controller(setpoint=300, T=1, k=20),
    redeclare Implementations.ConventionalActuator actuator,
    redeclare Implementations.ConventionalSensor sensor);
end BaseModel;

If we simulate this system, we get the following results:

../../../_images/TCB.png

As we can see, this approach works very well. The furnace heat required to achieve this degree of control looks like this:

../../../_images/TCBh.png

Bang Bang Control

So far, this approach seems like it has been quite successful. We have a nice architecture that we can use to consider different actuators, sensors, controllers or even plant models. The control system we’ve developed seems to do a fairly good job of controlling our plant.

But one thing worth noting is that the furnace heat required in this case is continuous. However, home heating systems do not typically use this type of control strategy. Instead, they tend to use something called “bang-bang” control where the furnace is either “off” or “on”.

We have this flexible architecture, so perhaps to address this situation, we should create an implementation of our controller and actuator models where the controller command is a boolean indicating whether the furnace should be on or off. However, if we start this process, we quickly run into the following problem:

A control strategy with on/off control

Note that the output from our controller is Boolean value but the commanded heat signal from our ControlSystem interface requires a Real value. We have the same problem on the actuator side:

The interface supplies an actuator that is a Real value but again we see that if our furnace expects an “on” or “off” command, we have a mismatch.

So the question then becomes, how can we handle situations where choosing different subsystems requires us to have different interfaces?

Expandable Approach

The solution to this problem is expandable connector definitions. With this approach, our subsystem interface would be the same regardless of whether the control strategy generates a Boolean or Real. What changes is the contents of the connector instances.

To understand how these expandable connectors work, we’ll reformulate our architecture to include expandable connectors and then see how these can be used for both continuous and “bang-bang” approaches.

Expandable Connectors

The key feature that allows us to make more flexible architectures is the expandable connector. For example, previously we defined the Actuator interface as:

within ModelicaByExample.Architectures.ThermalControl.Interfaces;
partial model Actuator "Actuator subsystem interface"

  Modelica.Blocks.Interfaces.RealInput heat "Heating command" annotation (
      Placement(transformation(
        extent={{-20,-20},{20,20}},
        origin={-120,0})));

  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_b furnace
    "Connection point for the furnace"
    annotation (Placement(transformation(extent={{90,-10},{110,10}})));
end Actuator;

This interface contains two connectors, the heat connector and the furnace connector. The furnace connector is a thermal connector that allows the furnace to interact thermally with the plant. The heat connector is a Real valued input signal that comes from the controller and specifies the desired heat output level. The fact that this is a Real valued signal was the problem when we wanted to switch to a type of control that required a Boolean signal. To address this, we’ll use the following interface definition for our actuators:

within ModelicaByExample.Architectures.ThermalControl.Interfaces;
partial model Actuator_WithExpandableBus
  "Actuator subsystem interface with an expandable bus"

  ExpandableBus bus
    annotation (Placement(transformation(extent={{-110,-10},{-90,10}})));

  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_b furnace
    "Connection point for the furnace"
    annotation (Placement(transformation(extent={{90,-10},{110,10}})));
end Actuator_WithExpandableBus;

Here we see the furnace connector is still present. But the heat connector is gone. Instead, it has been replaced by a new connector instance, bus, of type ExpandableBus. The connector definition for ExpandableBus is:

within ModelicaByExample.Architectures.ThermalControl.Interfaces;
expandable connector ExpandableBus "An example of an expandable bus connector"
end ExpandableBus;

In other words, it is empty. But what is significant is the presence of the expandable qualifier. If a given bus is required to always have certain signals, they should be listed within the connector definition. But the fact that there are no variables or sub-connectors listed in the ExpandableBus class means there is no minimum requirement for information to be carried on the bus. But the bus can be expanded to include additional information.

Of course, we could use inheritance to add new signals. But that introduces a new type. The types of connectors are fixed by the type used in the interface definition. So creating richer connectors via inheritance doesn’t really help.

Note that there is no formal definition of a “bus” in Modelica. The term is often used in such contexts to connote a connector that is carrying multiple pieces of information.

Expandable connectors work in a special way. The signals on an expandable bus are determined by the connections themselves. By connecting something to the expandable bus, a signal is implicitly added to that connector. Then the Modelica compiler looks at all the connectors in a connection set and expands each one so that they match. We’ll go into more details about this process once we get to the point where we have some implementation models to discuss.

The interface for the plant model is unaffected by the use of expandable connectors, but the interfaces for the sensor and controller are as follows:

within ModelicaByExample.Architectures.ThermalControl.Interfaces;
partial model Sensor_WithExpandableBus
  "Sensor subsystem interface using an expandable bus"

  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a room
    "Thermal connection to room"
    annotation (Placement(transformation(extent={{-110,-10},{-90,10}})));

  ModelicaByExample.Architectures.ThermalControl.Interfaces.ExpandableBus bus
             annotation (Placement(transformation(extent={{90,-10},{110,10}})));
end Sensor_WithExpandableBus;
within ModelicaByExample.Architectures.ThermalControl.Interfaces;
partial model ControlSystem_WithExpandableBus
  "Control system interface using an expandable bus connector"
  ExpandableBus bus annotation (Placement(transformation(extent={{-10,-110},{10,
            -90}}), iconTransformation(extent={{-10,-110},{10,-90}})));
end ControlSystem_WithExpandableBus;

Note how simple the controller interface has become. This is because with an expandable connector, we can put the temperature measurement received from the sensor and the heat command sent to the actuator on the same bus. As such, we only need one connector. A developer may still choose to use multiple buses simply to organize signals, to make them more representative of the physical partitioning or to avoid confusion. Here we will use a single connector simply to demonstrate that this is now possible.

Using expandable connectors, we can create the following revised architecture:

Expandable architecture

Expandable Implementations

With this more flexible architecture, let’s first recreate our original configuration with the continuous control system:

Continuous control using the expandable architecture

If we plot the results from this system, we get the following response:

../../../_images/TCE.png

Note that the measured temperature corresponds to the signal controller.bus.temp where bus is an instance of the expandable connector. Further recall that the ExpandableBus definition didn’t contain a signal called temperature. So the question is, how did it get on the connector. The answer lies in the implementation of the sensor model. The diagram for the sensor model looks like this:

Temperature sensor model using expandable bus

The corresponding Modelica code is:

within ModelicaByExample.Architectures.ThermalControl.Implementations;
model TemperatureSensor "Temperature sensor using an expandable bus"
  extends Interfaces.Sensor_WithExpandableBus;
protected
  Modelica.Thermal.HeatTransfer.Sensors.TemperatureSensor sensor
    annotation (Placement(transformation(extent={{-10,-10},{10,10}})));
equation
  connect(sensor.T, bus.temperature) annotation (Line(
      points={{10,0},{100,0}},
      color={0,0,127},
      smooth=Smooth.None));
  connect(room, sensor.port) annotation (Line(
      points={{-100,0},{-10,0}},
      color={191,0,0},
      smooth=Smooth.None));
end TemperatureSensor;

Of particular importance is the highlighted line.

In the diagram, we can see that the output signal from the temperature sensor component is connected to the bus. But when we look at the connect statement, it is more than just connected to the bus. It is connected to something named temperature inside the bus. This temperature connector doesn’t exist in the definition of ExpandableBus. Instead, it is created by the connect statement itself! This is precisely what the expandable qualifier allows.

In general, we don’t want all connectors to be expandable. In cases where we know a priori the names and types of all signals, it is better to list them explicitly. This allows the Modelica compiler to make several important checks to ensure the correctness of the model. It is worth noting that by adding the expandable qualifier on a connector, the risk of accidentally creating an unintended signal (e.g., as a result of a typing error) becomes a possibility that would otherwise be caught by the compiler.

Reconfiguration

Now that we’ve shown that we can use the expandable approach to model the continuous control version of our system, let’s return our attention to the “bang-bang” version.

We’ve already seen the temperature sensor subsystem configured to work with the expandable connector. What remains is the controller and actuator models. The actuator model diagram looks like this:

Actuator attached via expandable connector

Again, looking at the Modelica code is important to see how the signals on the bus connector are referenced:

within ModelicaByExample.Architectures.ThermalControl.Implementations;
model OnOffActuator "On-off actuator implemented with an expandable bus"
  extends Interfaces.Actuator_WithExpandableBus;
  parameter Real heating_capacity "Heating capacity of actuator";
protected
  Modelica.Thermal.HeatTransfer.Sources.PrescribedHeatFlow heater
    annotation (Placement(transformation(extent={{-10,-10},{10,10}})));
  Modelica.Blocks.Math.BooleanToReal command(realTrue=heating_capacity,
      realFalse=0)
    annotation (Placement(transformation(extent={{-60,-10},{-40,10}})));
equation
  connect(heater.port, furnace) annotation (Line(
      points={{10,0},{100,0}}, color={191,0,0},
      smooth=Smooth.None));
  connect(command.y, heater.Q_flow) annotation (Line(
      points={{-39,0},{-10,0}}, color={0,0,127},
      smooth=Smooth.None));
  connect(command.u, bus.heat_command) annotation (Line(
      points={{-62,0},{-100,0}}, color={255,0,255},
      smooth=Smooth.None));
end OnOffActuator;

Again, note the emphasized line. It references something called heat_command on the bus connector. Again, that signal doesn’t exist in the definition of ExpandableBus, but it is implicitly created simply because it is referenced in the highlighted connect statement.

From the sensor model, we see that the measured temperature is added to the bus connector as a Real signal named temperature. From the actuator model, we see that the command expected by the actuator from the controller is a Boolean signal named heat_command. As such, we should expect to see both of these signals used by the controller model. The diagram for the controller looks like this:

PI controller connected to expandable bus

But the diagram doesn’t include sufficient detail to know the precise names of the signals being referenced on the bus connector. For that, we need to look at the actual source code:

within ModelicaByExample.Architectures.ThermalControl.Implementations;
model ExpandablePIControl "PI controller implemented with an expandable bus"
  extends Interfaces.ControlSystem_WithExpandableBus;
  parameter Real setpoint "Desired temperature";
  parameter Real k=1 "Gain";
  parameter Modelica.SIunits.Time T "Time Constant (T>0 required)";
protected
  Modelica.Blocks.Sources.Trapezoid setpoint_signal(
    amplitude=5, final offset=setpoint, rising=1,
    width=10, falling=1, period=20)
    annotation (Placement(transformation(extent={{-20,-40},{0,-20}})));
  Modelica.Blocks.Math.Feedback feedback
    annotation (Placement(transformation(extent={{30,-10},{10,10}})));
  Modelica.Blocks.Continuous.PI PI(final T=T, final k=-k)
    annotation (Placement(transformation(extent={{-10,-10},{-30,10}})));
equation
  connect(setpoint_signal.y, feedback.u2)
    annotation (Line(
      points={{1,-30},{20,-30},{20,-8}},
      color={0,0,127}, smooth=Smooth.None));
  connect(PI.u,feedback. y) annotation (Line(
      points={{-8,0},{11,0}},
      color={0,0,127}, smooth=Smooth.None));
  connect(bus.temperature, feedback.u1) annotation (Line(
      points={{0,-100},{60,-100},{60,0},{28,0}},
      color={0,0,0}, smooth=Smooth.None));
  connect(PI.y, bus.heat) annotation (Line(
      points={{-31,0},{-60,0},{-60,-100},{0,-100}},
      color={0,0,127}, smooth=Smooth.None));
end ExpandablePIControl;

Again, note the highlighted lines. Not only do these connect statements implicitly add the heat_command and temperature signals to the bus connector, those names match the names that the sensor and actuator models expect.

Pulling all of these subsystems together, we get the following diagram for our system:

System using bang-bang control

The source code for our system model is quite simple:

within ModelicaByExample.Architectures.ThermalControl.Examples;
model OnOffVariant "Variation with on-off control"
  extends ExpandableModel(
    redeclare replaceable
      Implementations.OnOffActuator actuator(heating_capacity=500),
    redeclare replaceable
      Implementations.OnOffControl controller(setpoint=300));
end OnOffVariant;
../../../_images/TCE_BB.png

However, there is still one remaining issue with these models and it can be seen more clearly if we look at the duty cycle of the furnace:

../../../_images/TCE_BBh.png

This is exactly the same issue we demonstrated in the previous section on Hysteresis. It is precisely the fact that our control strategy lacks any hysteresis that we see the furnace constantly turning on and off. If we add hysteresis, our controller model becomes:

Bang-bang controller with hysteresis

Nothing else has changed. We will use the same sensor and actuator models and we still use the same bus signals since this is still a bang-bang controller. So the only change to our system level model (compared to the OnOffVariant model) is the use of a different controller model. As we can see, these configuration management features in Modelica do a nice job of conveying that in our system level model:

within ModelicaByExample.Architectures.ThermalControl.Examples;
model HysteresisVariant "Using on-off controller with hysteresis"
  extends OnOffVariant(redeclare Implementations.OnOffControl_WithHysteresis
      controller(setpoint=300, bandwidth=1));
end HysteresisVariant;

Using hysteresis control, our simulation results look like this:

../../../_images/TCE_Hy.png

But the most important difference is the fact that the hysteresis doesn’t lead to the kind of chattering we saw in our previous bang-bang controller:

../../../_images/TCE_Hyh.png

Conclusion

This is the second example of how we can use the configuration management features in Modelica to take an architecturally based approach to building system models. This architectural approach is very useful when there are many variations of the same architecture that require analysis. Using the redeclare feature, it is possible to easily substitute alternative designs for each subsystem or to consider more or less detail in any given subsystem as necessary for any given engineering analysis.

In this particular example, we saw how an expandable connector can provide greater flexibility than a standard connector. However, it also comes with some risk because the type checking normally done by the Modelica compiler is less rigorous.