Novel Modeling and Control Strategies for a HVAC System Including Carbon Dioxide Control

Conventional heating, ventilating, and air conditioning (HVAC) systems have traditionally used the temperature and the humidity ratio as the quantitative indices of comfort in a room. Recently, the carbon dioxide (CO2) concentration has also been recognized as having an important contribution to room comfort. This paper presents the modeling of an augmented HVAC system including CO2 concentration, and its control strategies. Because the proposed augmented HVAC system is multi-input multi-output (MIMO) and has no relative degree problem, the dynamic extension algorithm can be employed; then, a feedback linearization technique is applied. A linear-quadratic regulator (LQR) is designed to optimize control performance and to stabilize the proposed HVAC system. Simulation results are provided to validate the proposed system model, as well as its linearized control system.


Introduction
HVAC systems are automatic systems that control temperature and humidity in buildings, providing people with a comfortable environment.The use of HVAC systems represents more than 50% of the world energy consumption [1][2][3][4].Thus, balancing occupant comfort and energy efficiency is a main goal of HVAC control strategies.
In most previous studies, HVAC systems have been modeled considering only the temperature and the humidity ratio [5][6][7][8].A nonlinear HVAC model that includes dynamics of temperature and humidity ratio is proposed in [5], which includes the design of an observer to estimate the thermal and moisture loads.In [6], an adaptive fuzzy output feedback controller is proposed, based on an observer for the HVAC system.In [7,8], a back-stepping controller and a decentralized nonlinear adaptive controller are respectively applied to the same model.
Recently, the CO 2 concentration has been recognized as having an important contribution to room comfort [9,10].Some researchers have proposed hybrid HVAC systems that represent the temperature and humidity ratio as continuous states and CO 2 concentration as a discrete state [11,12].However, because these states are strongly interrelated, it is more appropriate to integrate these continuous and discrete dynamics into a single model that includes temperature, humidity ratio, and CO 2 concentration as states.
This paper presents a modeling and control strategy for a novel HVAC system that considers temperature, humidity ratio, and CO 2 concentration.In the process of modeling, the dynamic extension algorithm of [13] is employed to deal with non-interacting control problem and no relative degree problem.After the dynamic extension process, a feedback linearization method can be applied to the proposed HVAC system to convert a bilinear system into a linear system.Linear controllers, pole placement and LQR can be designed for the linearized novel HVAC system to stabilize it and improve its control performance.
This paper is organized as follows: in Section 2, we present the bilinear model for the conventional HVAC system, including valve dynamics.Section 3 presents a novel HVAC system including CO 2 concentration and its applicability in the feedback linearization method.Also, dynamic extension algorithm is applied for solving the no relative degree and interacting control problems in the MIMO system.In Section 4, we describe the design of linear controllers for the linearized HVAC system, such as pole placement and LQR controllers, to improve the system's control performance and to verify the effectiveness of the proposed model.

Conventional HVAC System with Temperature and Humidity Ratio
As mentioned, conventional HVAC systems control only temperature and humidity.In this paper, we consider the single-zone system shown in Figure 1 as a representative conventional HVAC system.It consists of the following components: a heat exchanger; a chiller, which provides chilled water to the heat exchanger; a circulating air fan; the thermal space; connecting ductwork; dampers; and mixing air components [5].The conventional HVAC system controls the temperature and humidity ratio as follows [5]:  Fresh air is introduced into the system and is mixed in a 25:75 ratio with recirculated air (position 5) at the flow mixer. Second, air mixed at the flow mixer (position 1) enters the heat exchanger, where it is conditioned.
 Third, the conditioned air is moved out of the heat exchanger; this air is ready to enter the thermal space, and is called supply air (position 2). Fourth, the supply air enters the thermal space (position 3), where it offsets the sensible (actual heat) and latent (humidity) heat loads acting upon the system. Finally, the air in the thermal space is drawn through a fan (position 4); 75% of this air is recirculated and the rest is exhausted from the system.The control inputs for a conventional HVAC system are the flow rate of air, which is varied using a variable-speed fan (position 2), and the flow rate of water from the chiller to the heat exchanger.However, in our proposed HVAC system, the air recirculation rate (position 4) is added as a new control input, and some modifications are made to the basic operation rules listed above.That is to say, the 75% recirculation rate listed above in the first and last steps becomes a variable quantity, and is used as the third control input.

Mathematical Modeling of Conventional HVAC System
The conventional HVAC system is a model considering the temperature and the humidity ratio as states.The differential equations describing the dynamic behavior of the HVAC system in Figure 1 can be derived from energy conservation principles and are given by [5]: (1) The dynamic system given by Equation ( 1) can be converted into a state variable form for the purposes of control.Let , , , , and define the following the dynamic equations given in Equation ( 1) can be written in the following state variable form.(2) The conventional HVAC system of Equation ( 2) is a 2-input, 2-output MIMO system: its inputs are the volumetric air flow rate and the chilled water flow rate, and its outputs are the temperature and the humidity ratio of the thermal space.

Adding Valve Dynamics to the Conventional HVAC Model
The Control input signals in the system described in Equation ( 2) are implemented using liquid valves.The valve dynamics can be modeled as follows in which s is the valve inherent characteristic and is the flow rate of the liquid which enters the valve [14,15]: By considering the characteristic of a linear valve as s , the valve transfer function can be written as: where , , , and are the constant gains and the time constants, respectively; is the control signal applied to the actuator; and is the signal that is input to the HVAC system.An augmented state space model with the new state vector, can be derived as: where: Thus, the system in Equation ( 5) represents a conventional HVAC system to which valve dynamics have been applied to implement the control signals.This conventional system can regulate only the temperature and the humidity ratio; other factors such as CO 2 concentration, which affects the health of occupants or workers indoors, cannot be considered.

Novel Modeling of HVAC System including CO 2 Concentration
If the recirculated air contains too much CO 2 , it can affect the health and work efficiency of the building's occupants.Therefore, CO 2 concentration should be one of the quantitative indices of room comfort, along with temperature and humidity ratio.In this paper, we propose an HVAC system that continuously controls all three of these indices.
From the mass balance equation, the average CO 2 concentration in the room can be represented as [14]: where , is the amount of CO 2 generated in the room; is the CO 2 concentration in the inlet air; is the CO 2 concentration of air leaving the room, and 1 , 0 1, is the air exchange rate.

Proposed HVAC System Model
The proposed HVAC system model includes CO 2 concentration as a state.The differential Equation ( 6) can be integrated into the dynamic equations in (1).The valve dynamics of are added to the control input vector , and the control signal applied to the actuator of also can be added to the actuator input vector .
Let 1 , x , x , x , x , and let an augmented state vector x x x x .Then, the whole dynamics can be written in the state variable form as: where: The novel HVAC system of ( 7) is a 3-input, 3-output MIMO system: its inputs are the volumetric air flow rate, the chilled water flow rate, and the outdoor air flow rate, and its outputs are the temperature, humidity ratio, and CO 2 concentration of the thermal space.This proposed HVAC system can be linearized using a feedback linearization control method, as shown in Figure 2. Thus, we can finally obtain a linearized HVAC system that can be controlled using linear controllers.

Conditions for Input-Output Feedback Linearization
An input-output feedback linearization method can be applied to the state space model given in (7) to track the desired temperature, humidity ratio, and CO 2 concentration only when the decoupling matrix is non-singular.However, the decoupling matrix (refer to Appendix A2): is singular; here, • represents the Lie derivative and the total relative degree is 6 n, where n is the system order [16].When the invertibility condition is violated, some method is needed to carry out an input-output linearization; the dynamic extension algorithm used herein is such a method.

Dynamic Extension Algorithm
Because input-output linearization can be achieved only when the decoupling matrix is non-singular, employing the dynamic extension algorithm involves choosing some new inputs that are the derivatives of some of the original system inputs, in such a way that the decoupling matrix becomes non-singular, as shown in Figure 3 [16].

Let state vector
x x x x .Then, the state space can be written as: where: The decoupling matrix as changed by the dynamic extension (refer to Appendix A3): is non-singular.The vector relative degree is 3 3 3 and the total relative degree is equal to the system order n, which means that there are no internal dynamics [16].Therefore, we can achieve relative degree and non-interacting control.
This equivalent system can apply the feedback linearization law to linearize the HVAC system.By putting proper control gain, the linearized HVAC system can be regulated to maintain the set points of temperature, humidity ratio, and CO 2 concentration.

Control Strategies Using Feedback Linearization Control
The proposed HVAC system is linearized by a feedback linearization method.The linearized HVAC system shown can be controlled by linear controllers such as pole placement and LQR controllers.
Table 1 shows the numerical values of system parameters used in the simulations.The initial state and reference values are given in Table 2.The feedback linearization control law for the proposed HVAC system (9) is designed as: (10) choosing diag , , so that the polynomial 0, 1, 2, 3 has all its roots strictly in the left-half complex plane, thereby meeting the desired performance specifications such as those for the transient response of the steady-state error.By substituting Equation (10) into Equation ( 9), the linearized HVAC system can finally be obtained as follows: (11) where .
The gain K 1 corresponds to the case in which the pole is located at −2, −2, and −4, whereas the gain K 2 corresponds to the case in which the pole is located at −2, −3, and −5, and the gain K 3 corresponds to the case in which the pole is located at −5, −6, and −7.According to the pole placement, the control performance is varied.Figure 4 show the system responses in terms of temperature, humidity ratio, and CO 2 concentration, respectively.Table 3 shows the control performance metrics of settling time, rising time, settling max value, and settling min value for each value of gain.

Design of Linear Quadratic Regulator for Linearized HVAC System
The linearized HVAC system of ( 11) can be controlled by a linear controller.A linear-quadratic regulator (LQR) aims at designing stable controller which can minimize the cost function represents the performance characteristic requirement as well as the controller input limitation [17].The cost function is: (12) where is a positive semi-definite weight matrix and is a positive definite weight matrix.The weighting matrices and are chosen by the Bryson's rule (refer to Appendix A5) [18].
The feedback control law that minimizes the values of cost is: (13) where ; is given by ; and is found by solving the continuous time algebraic Riccati equation: Table 4 shows the gain values resulting from the LQR process solving algebraic Riccati equation.The LQR controller was applied to the proposed HVAC system model (11) and the simulation results are shown in Figure 5.The temperature response of the proposed HVAC system is shown in Figure 5(a), and its humidity ratio response and CO 2 concentration response are shown respectively in Figure 5(b,c).
From these simulation results, we can see that the proposed HVAC system is effective, and that linear controllers are suitable for application to the proposed HVAC system model.Table 5 shows the control performance metrics of settling time, rising time, settling max value, and settling min value for each value of gain.

Conclusions
Herein we have presented a novel HVAC system model that considers not only temperature and humidity ratio, but also CO 2 concentration as the quantitative indices of comfort in a room.In applying an input-output feedback linearization method to linearize the HVAC system, problems of singularity, no relative degree, and interacting controls were encountered and a dynamic extension algorithm was used to solve these problems.The key contribution of this report is the addition of a continuous CO 2 concentration state and corresponding valve dynamics to a conventional HVAC system to allow continuous control of CO 2 concentration.Two types of linear controllers, pole placement and LQR controllers, were able to regulate the linearized HVAC system at the desired set point.Simulation results validated the proposed HVAC model, demonstrating its effectiveness in maintaining comfortable conditions.In future work, we will conduct further study on developing disturbance observer based controllers or intelligent controllers using fuzzy logic or artificial neural networks for a HVAC system considering parameter uncertainty and disturbance effect.
The above procedure is similar to that applied for output : Thus, the relative degree 2 with respect to output .By Equation (A2): In the case of output : x x x ∵ .
Therefore, the relative degree 2 with respect to output .By Equation (A2): From the above results, the vector relative degree is 2 2 2 , the total relative degree is 6, and the decoupling matrix is:

Appendix A3
From the system given in (7): x x x and the output : Therefore, the relative degree ̅ 3 with respect to output .According to Equation (A2): The above procedure is similar to that applied for output : Thus, the relative degree ̅ 3 with respect to output .By Equation (A2): In the case of output : Therefore, the relative degree 3 with respect to output .By Equation (A.2):

Figure 1 .
Figure 1.Model of the representative conventional HVAC system.

Figure 2 .
Figure 2. Overall block diagram for controlling the proposed HVAC system.

Figure 4 .
Figure 4. (a) Temperature response for each pole placement; (b) Humidity ratio response for each pole placement; (c) CO 2 concentration response for each pole placement.

Table 1 .
Numerical values for system parameters.

Table 2 .
Initial state and reference values.

Table 3 .
Control performance metrics of the pole placement controller.

Table 5 .
Control performance metrics of the LQR controller.