A proportional feedback controller is added to the heating system. This controller measures the room temperature error and provides the control signal to the LPHW flow rate final control device. Draw the complete closed?loop block diagram for the room heating system using the data from IX3.1 and PF1.4.
Derive the closed?loop transfer function relating the outside air temperature to the inside air temperature, and the characteristic equation for this closed?loop system.
If oscillation is to be avoided, what is the maximum value of the proportional constant?
If the value of proportional constant calculated in 2.3 is used, what is the steady?state error in the room temperature when the outside
Comment on the acceptability of the value calculated in 2.4, and suggest means of improving the control system.
A PID controller is used to replace the proportional feedback controller, derive the closed? loop transfer function relating the outside air temperature to the inside air temperature, and the characteristic equation for this closed?loop system..
What would be the effect of the transport delay between the radiator and room air temperature sensor?
Interaction has been implicitly ignored by assuming that the heat output from the radiator is not affected by the room temperature. Comment on whether this assumption is reasonable, and its effect on the controller design.
Infiltration heat loss has not been considered. Suggest how this might be added to the system block diagram and comment on its effect on the room temperature control.
Closed-Loop Transfer Function and Characteristic Equation
Radiators and convectors are the key means of heat emission in most building services. There are alternatives which may be used to transfer up to 40 percent of the heat by radiation. These radiators provide a local spot heating and are efficiently implemented in buildings with high air change rates. The low pressure water panel radiator has lower energy consumption and lower air temperatures with reduced stratification at high levels of implementation. It has a rapid warm-up, responsive control and no air movement. The radiator heat output is implemented to warm the building but heating system is susceptible to effects from the environment.
At this point, the open loop system is described using the first order system. There is no feedback channel hence the system output may either be as intended or not. The system is susceptible to disturbances in the surrounding and as a result the system output is not bounded. This is an undesirable characteristic in the system design and implementation as it may result in errors transmitted to other systems in cascaded system design. Open loop systems do not observe the system output and as a result, they do not find application in machine learning. These systems depend solely on the model of the plant and its current state. Such a system counteracts against the environmental noises and is designed such that the disturbances are not eliminated. The system is not unstable.
The room heater dynamic model represented in the first order follows the function,
K = DC gain. It is given as the ratio of the output voltage and the input voltage.
The DC gain of the open loop system is given as 3
- The noise is a constant value; zero-mean additive white Gaussian noise is introduced in the system as the source of the system disturbance.
- The room temperature reader is well calibrated depending on the ambient temperature of the room.
- The room temperature is not disrupted by air flow from the ventilation sections of the room such as the windows, louvers, and other ventilation spaces.
Adding a proportional feedback controller to the heating system as a controller improves the control and system response. The proportional gain controller is adjusted to provide different response. The gain parameter reduces the steady state error in the system response though when it exceeds a given value, the output oscillates and may cause system instability. Based on the information provided, the response of the heater in the LPHW heater, the time constant is small enough to render the response instantaneous. The proportional controller decreases the rise time, increases the overshoot, makes small changes to the settling time and decreases the steady state error considerably.
In the closed loop, there is a feedback loop with a sensor that reads the output value and compares it with the intended value. The error obtained is corrected by the controller and passed through the radiator panel to provide the intended heat amount to the room. The closed loop block diagram is as illustrated below,
Effect of Transport Delay
In the block diagram the valve is used to set the desired temperature for the room. At the feedback loop, there is a temperature sensor that measures the actual room temperature and sends it to the summing point to determine the difference from the desired input or the reference input. Let the sensor in the feedback loop be a first order transfer function, such that,
Implementing in MATLAB,
- Assuming a unity feedback in the closed loop system.
It is observed that there is an increase in the overshoot and a slight decrease in the rise time. The proportional gain parameter used is .
1. Rise time
3. Settling time
3 seconds (DC Gain)
4. Peak response
To confirm our results on the choice of proportional controller used, an automatic PID tuning is performed. The proportional gain parameter used here is ,
To compare the automatic tuner with the implemented controller for the system,
The automatic tuner slightly reduces the rise time but has a higher overshoot as compared to the implemented controller. To improve the system response or output, the designer ought to include more controllers such as the integral and the derivative components.
Effect of transport delay between the radiator and the room air temperature sensor
In this case study, the time constant used in the design is 2 minutes. It is considerably low and it is treated as virtually instantaneous. The transfer delays and the dead time are as a result of resistance and capacitances in the plant system elements. Poor positioning of the sensor on the feedback loop may result in dead time and it is quite difficult to compensate for it once the system has been configured. Dead time causes a sensor to detect nothing at a given period where it should be detecting the room temperature instead. The lag may cause wrong values to be fed back to the summing point and as a result, the radiator panel may give the wrong output. In building and construction services, the designers and construction managers must take keen interest in the LPHW system implementation.
Assumption: the heat output from the radiator is not affected by the room temperature
The assumption is justified to the extent that the time constant used in the system allows for the transport delays and the dead time. As a result, it is possible to state that the sensor reads the room temperature on a given periodic basis so that the system is constant monitoring the room temperature and making adjustments at the valve accordingly.
Assumption: Heat Output from Radiator Is Not Affected by Room Temperature
An addition of the infiltration heat loss in the system block diagram & its effect on the room temperature control.
In the previous sections, the analysis has been carried out with the assumption that the room being heated is ideal and no heat loss is encountered. Every designer must account for heat loss during design based on the system implementation environment. During winter, for instance, the temperature, wind, humidity, and radiation; affect the room temperature in a building in a significant manner. Heat losses from the room are attributed to conductive heat losses such as through the building walls, room ceilings, the floor and interiors. It is also attributed to convective infiltration through wall openings or cracks as well as ventilation sections of the room.
Heat loss through conductive movement of heat,
The total evaluation in the entire building or room under design,
Through the slab on the grade foundation,
Heat loss through the basement walls and the floor of a room under design,
The heat loss through infiltration and ventilation is quite common in most rooms. It is crucial for the designer to know the volume of the space and the amount of air that leaks in a given duration of time. Typical values show that the infiltration of a standard room size ranges from 0.15 to 0.5 changes per hour, ach, in winter. The value of heat loss by infiltration can be computed on the basis of:
- Air change method
- Infiltration through the cracks
- Based on occupancy, number of people in the space
%% room heating using the LPHW Panel Radiator
tau=2; %Time constant
K=3; % DC Gain
s=tf('s'); %symbolic transfer function
R1=K*exp(-L*s)/(tau*s+1) % G(s)
g=dcgain(R1) % DC gain of the transfer function
%% performing proportional control
Kp=0.75; %proportional gain parameter
C=pid(Kp) %Defines the proportional controller
T=feedback((C*R1),1) %Assuming unity feedback
%% For an automatic pid tuner of the system
In a nutshell, the open loop system allows adjustment of the radiator panel but it does not get to know what value the system output obtains. The system output is considered unbounded and any other environmental factors that may affect the output are not captured. The closed loop system eliminates this problem by including a feedback loop with a sensor that reads the room temperature and compares it to the set values. Any differences are corrected by the actuator and the radiator panel provides the accurate output.
Further, the designer considers other environment factors that may cause heat loss affecting the room temperature read by the sensor. Some of these heat loss modes are infiltration, ventilation, and conductive or convectional heat losses. The heat loss meter is inserted in the feedback loop to compensate for the heat loss.
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