Energies 2014, 7(6), 3599-3617; doi:10.3390/en7063599

Article
Novel Modeling and Control Strategies for a HVAC System Including Carbon Dioxide Control
Chang-Soon Kang 1, Jong-Il Park 1, Mignon Park 1 and Jaeho Baek 2,*
1
Department of Electrical & Electronic Engineering, Yonsei University, Seoul 120-749, Korea; E-Mails: clozer@yonsei.ac.kr (C.-S.K.); pjjang84@naver.com (J.-I.P.); mignpark@yonsei.ac.kr (M.P.)
2
Advanced R&D Team, Samsung Electronics Co. Ltd., Suwon 443-742, Korea
*
Author to whom correspondence should be addressed; E-Mail: jhoya97@gmail.com; Tel.: +82-2-2123-2868; Fax: +82-2-312-2333.
Received: 17 February 2014; in revised form: 27 April 2014 / Accepted: 20 May 2014 /
Published: 2 June 2014

Abstract

: 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.
Keywords:
heating ventilating and air conditioning (HVAC) system; multi-input multi-output (MIMO) system; dynamic extension algorithm; feedback linearization

1. 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 CO2 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 CO2 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 CO2 concentration as states.

This paper presents a modeling and control strategy for a novel HVAC system that considers temperature, humidity ratio, and CO2 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 CO2 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.

2. 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.

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Figure 1. Model of the representative conventional HVAC system.

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

2.1. 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]:

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The dynamic system given by Equation (1) can be converted into a state variable form for the purposes of control. Let u 1 = F , u 2 = g p m , z 1 = T 3 , z 2 = W 3 , z 3 = T 2 and define the following parameters: α 1 = 60 V s , α 2 = 60 h f g C p V s , α 3 = 1 ρ a C p V s , α 4 = 1 ρ a V s , β 1 = 60 V h e , β 2 = 1 ρ a C p V h e , and β 3 = 60 h w C p V h e . Then, the dynamic equations given in Equation (1) can be written in the following state variable form.

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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.

2.2. Adding Valve Dynamics to the Conventional HVAC Model

The Control input signals u = [ u 1 u 2 ] T in the system described in Equation (2) are implemented using liquid valves. The valve dynamics can be modeled as follows in which ψ ( s ) is the valve inherent characteristic and u ( s ) is the flow rate of the liquid which enters the valve [14,15]:

u ( s ) = 1 ( 1 + τ s ) ψ ( s )

By considering the characteristic of a linear valve as ψ ( s ) = k v ( s ) , the valve transfer function can be written as:

u 1 = k 1 1 + τ 1 s ν 1 ,  u 2 = k 2 1 + τ 2 s ν 2
where   k 1 k 2 τ 1 and τ 2 are the constant gains and the time constants, respectively; ν = [ ν 1 ν 2 ] T  is the control signal applied to the actuator; and u = [ u 1 u 2 ] T is the signal that is input to the HVAC system.

An augmented state space model with the new state vector, z = [ z 1 z 2 z 3 u 1 u 2 ] T = [ z 1 z 2 z 3 z 4 z 5 ] T can be derived as:

{ z ˙ = f ( z ) + g ( z ) ν =   [ a 1 ( z ) a 2 ( z ) a 3 ( z ) a 4 ( z ) a 5 ( z ) ] + [ 0 0 0 0 k 1 τ 1 0 0 0 0 k 2 τ 2 ] ν y = [ z 1     z 2 ]   T
where:
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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 CO2 concentration, which affects the health of occupants or workers indoors, cannot be considered.

3. Novel Modeling of HVAC System including CO2 Concentration

If the recirculated air contains too much CO2, it can affect the health and work efficiency of the building’s occupants. Therefore, CO2 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 CO2 concentration C s in the room can be represented as [14]:

V s C ˙ s = C g + ( 1 μ ) ( C i C o )
where C s = C i C o , C g is the amount of CO2 generated in the room; C i is the CO2 concentration in the inlet air; C o is the CO2 concentration of air leaving the room, and ( 1 μ ) ,   0 μ 1 , is the air exchange rate.

3.1. Proposed HVAC System Model

The proposed HVAC system model includes CO2 concentration as a state. The differential Equation (6) can be integrated into the dynamic equations in (1). The valve dynamics of u 3 are added to the control input vector u = [ u 1 u 2 ] T , and the control signal applied to the actuator of ν 3 also can be added to the actuator input vector ν .

Let u 3 = 1 μ , x 1 = z 1 , x 2 = z 2 , x 3 = z 3 , x 4 = C s , and let an augmented state vector x = [ x 1 x 2 x 3 x 4 u 1 u 2 u 3 ] T . Then, the whole dynamics can be written in the state variable form as:

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where:
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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 CO2 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.

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

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Figure 2. Overall block diagram for controlling the proposed HVAC system.
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3.2. 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 CO2 concentration only when the decoupling matrix is non-singular. However, the decoupling matrix (refer to Appendix A2):

D ( x ) = [ L g 1 L f 1 x 1 L g 2 L f 1 x 1 L g 3 L f 1 x 1 L g 1 L f 1 x 2 L g 2 L f 1 x 2 L g 3 L f 1 x 2 L g 1 L f 1 x 4 L g 2 L f 1 x 4 L g 3 L f 1 x 4 ] = [ γ 1 k 1 τ 1 0 0 γ 2 k 1 τ 1 0 0 0 0 γ 5 k 3 τ 3 ]
is singular; here, L ( ) 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.

3.2.1. Dynamic Extension Algorithm

Because input-output linearization can be achieved only when the decoupling matrix D ( x ) 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 D ( x ) becomes non-singular, as shown in Figure 3 [16].

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Figure 3. Concept of the dynamic extension algorithm [13].

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Figure 3. Concept of the dynamic extension algorithm [13].
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Let state vector x - = [ x v 1 v 3 ] T = [ x 1 x 2 x 3 x 4 u 1 u 2 u 3 v 1 v 3 ] T . Then, the state space can be written as:

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where:
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The decoupling matrix as changed by the dynamic extension (refer to Appendix A3):

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is non-singular. The vector relative degree is { 3 3 3 } and 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.

3.3. Equivalent Linearization HVAC System

To be able to apply the feedback linearization method conveniently, we can change system (8) into the equivalent system (9) by using Equation (A3):

{ x ˙ = A x + B [ F ( x ) + G ( x ) ζ ] y = [ y 1 y 2 y 3 ] T = [ x 1 x 2 x 4 ] T
where:
x = [ x 1 x 1 ( 1 ) x 1 ( 2 ) x 2 x 2 ( 1 ) x 2 ( 2 ) x 4 x 4 ( 1 ) x 4 ( 2 ) ] T ;
F ( x ) = [ f 1 f 2 f 3 ] T ;
G ( x ) = [ g 11 g 12 g 13 g 21 g 22 g 23 g 31 g 32 g 33 ] ;
A = d i a g [ [ 0 1 0 0 0 1 0 0 0 ] [ 0 1 0 0 0 1 0 0 0 ] [ 0 1 0 0 0 1 0 0 0 ] ] ;
B = d i a g [ [ 0 0 1 ] [ 0 0 1 ] [ 0 0 1 ] ] ;
and C = diag [ [ 1 0 0 ] [ 1 0 0 ] [ 1 0 0 ] ] .

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 CO2 concentration.

4. 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.

Table Table 1. Numerical values for system parameters.

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Table 1. Numerical values for system parameters.
ParameterValueUnit
ρ 0.0741b/ft3
Vhe60.75ft3
Vs58464ft3
Cp0.24Btu/lb °F
Ws0.0070lb/lb
Mo166.06lb/hour
Qo289,897.52Btu/hour
To85°F
Wo0.0181b/1b
Co400ppm
τ1230.008hour
k1,k2,k35-
Table Table 2. Initial state and reference values.

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Table 2. Initial state and reference values.
Tinitial76 °FTinitial0.021 lb/lbTinitial1300ppm
Tinitial71 °FTinitial0.0092 lb/lbTinitial1200ppm

4.1. Design of Pole Placement Control for Linearized HVAC System

Let the reference signal y d = [ y 1 d y 2 d y 3 d ] T , y d ( 3 ) = [ y 1 d ( 3 ) y 2 d ( 3 ) y 3 d ( 3 ) ] T , and matrix Y d = [ y 1 d y 1 d ( 1 ) y 1 d ( 2 ) y 2 d y 2 d ( 1 ) y 2 d ( 2 ) y 3 d y 3 d ( 1 ) y 3 d ( 2 ) ] T . Define the tracking errors as e 1 = y 1 y 1 d , e 2 = y 2 y 2 d and e 3 = y 3 y 3 d , and let the error matrix e = x Y d = [ e 1 e 1 ( 1 ) e 1 ( 2 ) e 2 e 2 ( 1 ) e 2 ( 2 ) e 3 e 3 ( 1 ) e 3 ( 2 ) ] T .

The feedback linearization control law for the proposed HVAC system (9) is designed as:

ζ = G 1 ( F + y d ( 3 ) K e )
choosing K = diag ( [ k 11 k 12 k 13 ] , [ k 21 k 22 k 23 ] , [ k 31 k 32 k 33 ] ) so that the polynomial s 3 + k i 1 s 2 + k i 2 s 1 + k i 3 = 0 , i = 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:
x ˙ = A x + B η
where η = y d ( 3 ) K e .

The gain K1 corresponds to the case in which the pole is located at −2, −2, and −4, whereas the gain K2 corresponds to the case in which the pole is located at −2, −3, and −5, and the gain K3 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 CO2 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.

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

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Table 3. Control performance metrics of the pole placement controller.
TemperatureHumidity ratioCO2 concentration
K1K2K3K1K2K3K1K2K3
Ts3.26952.84311.16183.38892.73121.15953.38572.72701.1581
Tr1.89191.51840.68041.89711.51240.67951.89701.51190.6794
Settling min70.996271.017270.99270.00920.00920.00921200.112001199.9
Settling max71.492971.474171.42970.01040.01040.01021209.912101208.5
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Figure 4. (a) Temperature response for each pole placement; (b) Humidity ratio response for each pole placement; (c) CO2 concentration response for each pole placement.

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Figure 4. (a) Temperature response for each pole placement; (b) Humidity ratio response for each pole placement; (c) CO2 concentration response for each pole placement.
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4.2. 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 J represents the performance characteristic requirement as well as the controller input limitation [17]. The cost function is:

J = 1 2 0 [ x T Q x + η T R η ] d t
where Q is a positive semi-definite weight matrix and R is a positive definite weight matrix. The weighting matrices Q and R are chosen by the Bryson’s rule (refer to Appendix A5) [18].

The feedback control law that minimizes the values of cost is:

η = k e + y d ( 3 )
where e = x Y d = [ e 1 e 1 ( 1 ) e 1 ( 2 ) e 2 e 2 ( 1 ) e 2 ( 2 ) e 3 e 3 ( 1 ) e 3 ( 2 ) ] T ; K is given by K = R 1 B T P ; and P is found by solving the continuous time algebraic Riccati equation:
A T P + P A P B R 1 B T P + Q = 0

Table 4 shows the gain values resulting from the LQR process solving algebraic Riccati equation.

Table Table 4. LQR gain values.

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Table 4. LQR gain values.
9 × 9 Gain matrix
k 1 diag([100 44.7850 9.9785],[100 44.7850 9.9785],[100 44.7850 9.9785])
k 2 diag([(100 54.7214 14.4721)],[(100 54.7214 14.4721)],[(100 54.7214 14.4721)])
k 3 diag([(100 88.6706 34.3124)],[(100 88.6706 34.3124)],[(100 88.6706 34.3124)])

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 CO2 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.

Table Table 5. Control performance metrics of the LQR controller.

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Table 5. Control performance metrics of the LQR controller.
TemperatureHumidity ratioCO2 concentration
K1K2K3K1K2K3K1K2K3
Ts1.48862.01073.23151.49821.98803.20281.50491.98733.1964
Tr0.52790.71871.30930.52770.71891.30980.52770.71891.3098
Settling min70.660170.792270.87020.00840.00870.00891193.211961197.5
Settling max71.424971.358671.47770.01020.01000.01031208.51207.21209.6
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Figure 5. (a) Temperature response by LQR; (b) Humidity ratio response by LQR; (c) CO2 concentration response by LQR.

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Figure 5. (a) Temperature response by LQR; (b) Humidity ratio response by LQR; (c) CO2 concentration response by LQR.
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5. Conclusions

Herein we have presented a novel HVAC system model that considers not only temperature and humidity ratio, but also CO2 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 CO2 concentration state and corresponding valve dynamics to a conventional HVAC system to allow continuous control of CO2 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.

Author Contributions

A co-author, Joil-Il Park, cooperates on dynamic extension algorithm for achieving relative degree and applying linearization method to the HVAC system in Section 3.2.1. A co-author, Mignon Park, cooperates on deriving modeling of HVAC system including CO2 concentration in Section 3.1. And Jaeho Baek cooperates on the equivalent linearization HVAC system in Section 3.3.

Nomeclature

h w Enthalpy of liquid water
W o Humidity ratio of outdoor air
h f g Enthalpy of water vapor
V h e Volume of heat exchanger
W s Humidity ratio of supply air
W 3 Humidity ratio of thermal space
C p Specific heat of air
T o Temperature of outdoor air
M o Moisture load
Q o Sensible heat load
T 2 Temperature of supply air
T 3 Temperature of thermal space
V s Volume of thermal space
ρ Air mass density
F Volumetric air flow rate ( ft 3 / min )
g p m Chilled water flow rate ( gal / min )
μ Air recirculation rate

Appendix A1

Consider the MIMO system of the form:

x ˙ = f ( x ) + G ( x ) u
where X is the n × 1 state vector; u is the m × 1 control input vector (of components u i ) ; y is the m × 1 vector of system outputs (of components y i ) ; f and h are smooth vector fields; and G is a n × m matrix whose columns are smooth vector fields g i .

Assume that r i is the smallest integer for which at least one of the inputs appears in y i ( r i ) ; then:

y i ( r i ) = L f r i h i + j = 1 m L g j L f r i 1 h i u j
with L g j L f r i 1 h i ( x ) 0 for at least one j . Applying Equation (A2) for each output y i yields:
[ y 1 ( r 1 ) y m ( r m ) ] = [ L f r 1 h 1 ( x ) L f r m h m ( x ) ] + D ( x ) u

Therefore, the decoupling matrix D ( x ) is defined as:

D ( x ) = [ L g 1 L f r i 1 h 1 L g m L f r i 1 h 1 L g 1 L f r i 1 h 2 L g m L f r i 1 h 2       L g 1 L f r i 1 h m   L g m L f r i 1 h m ]
where 1 i ,  j m .

For example, in our proposed HVAC system case, m = 3 and r = 3 :

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Appendix A2

From the system given in (6):

y = h ( x ) = [ h 1 ( x ) h 2 ( x ) h 3 ( x ) ] T = [ x 1 x 2 x 4 ] T
and the output y 1 :
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Therefore, the relative degree r 1 = 2 with respect to output y 1 . According to Equation (A2):

y 1 ( 2 ) = L f 2 h 1 + L g 1 L f 1 h 1 v 1 + L g 2 L f 1 h 1 v 2 + L g 3 L f 1 h 1 v 3

  • L g 1 L f 1 h 1   :   γ 1 k 1 τ 1 ;

  • L g 2 L f 1 h 1   :   0 ;

  • L g 3 L f 1 h 1   : 0.

The above procedure is similar to that applied for output y 2 :

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Thus, the relative degree r 2 = 2 with respect to output y 2 . By Equation (A2):

y 2 ( 2 ) = L f 2 h 2 + L g 1 L f 1 h 2 v 1 + L g 2 L f 1 h 2 v 2 + L g 3 L f 1 h 2 v 3

  • L g 1 L f 1 h 2   :   γ 2 k 1 τ 1 ;

  • L g 2 L f 1 h 2   :   0 ;

  • L g 3 L f 1 h 2   : 0.

In the case of output y 3 :

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Therefore, the relative degree r 3 = 2 with respect to output y 3 . By Equation (A2):

y 3 ( 2 ) = L f 2 h 3 + L g 1 L f 1 h 3 v 1 + L g 2 L f 1 h 3 v 2 + L g 3 L f 1 h 3 v 3

  • L g 1 L f 1 h 3   :   0 ;

  • L g 2 L f 1 h 3   :   0 ;

  • L g 3 L f 1 h 3   :   γ 5 k 3 τ 3 ;

From the above results, the vector relative degree is { 2 2 2 } , the total relative degree is 6, and the decoupling matrix is:

D ( x ) = [ L g 1 L f 1 h 1 L g 2 L f 1 h 1 L g 3 L f 1 h 1 L g 1 L f 1 h 2 L g 2 L f 1 h 2 L g 3 L f 1 h 2 L g 1 L f 1 h 3 L g 2 L f 1 h 3 L g 3 L f 1 h 3 ] = [ γ 1 k 1 τ 1 0 0 γ 2 k 1 τ 1 0 0 0 0 γ 5 k 3 τ 3 ] .

Appendix A3

From the system given in (7):

y = h ( x ) = [ h 1 ( x ) h 2 ( x ) h 3 ( x ) ] T = [ x 1 x 2 x 4 ] T
and the output y 1 :
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.

Therefore, the relative degree r 1 = 3 with respect to output y 1 . According to Equation (A2):

y 1 ( 3 ) = L f 3 h 1 + L g 1 L f 2 h 1 ζ 1 + L g 2 L f 2 h 1 ζ 2 + L g 3 L f 2 h 1 ζ 3

  • L g 1 L f 2 h 1   :   γ 1 k 1 τ 1

  • L g 2 L f 2 h 1   :   α 1 γ 4 k 2 u 1 τ 2

  • L g 3 L f 2 h 1   :   0.

The above procedure is similar to that applied for output y 2 :

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Thus, the relative degree r 2 = 3 with respect to output y 2 . By Equation (A2):

y 2 ( 3 ) = L f 3 h 2 + L g 1 L f 2 h 2 ζ 1 + L g 2 L f 2 h 2 ζ 2 + L g 3 L f 2 h 2 ζ 3

  • L g 1 L f 2 h 2   :   γ 2 k 1 τ 1

  • L g 2 L f 2 h 2   :   0

  • L g 3 L f 2 h 2   :   0.

In the case of output y 3 :

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Therefore, the relative degree r 3 = 3 with respect to output y 3 . By Equation (A.2):

y 3 ( 3 ) = L f 3 h 3 + L g 1 L f 2 h 3 ζ 1 + L g 2 L f 2 h 3 ζ 2 + L g 3 L f 2 h 3 ζ 3

  • L g 1 L f 2 h 3   :   0 ;

  • L g 2 L f 2 h 3   :   0 ;

  • L g 3 L f 2 h 3   :   γ 5 k 3 τ 3 .

From the above results, the vector relative degree is { 3 3 3 } , the total relative degree is 9, and the decoupling matrix is:

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Appendix A4

From Equation (A2) and our proposed 3-input ( ζ i ) and 3-output ( y i ) MIMO HVAC system given in (7):

y = h ( x ) = [ h 1 ( x ) h 2 ( x ) h 3 ( x ) ] T = [ x 1 x 2 x 4 ] T
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Appendix A5

Bryson’s rule specifies the weighting of the Q and R matrices where they are selected to be diagonal matrices:

Q i i = 1 maximum acceptance values of   x i 2 ,   i { 1 , 2 , , l }
R j j = 1 maximum acceptance values of   η j 2 ,   j { 1 , 2 , , m }

This corresponds to the following criteria:

J = 1 2 0 [ i = 1 l Q i i x 2 + ρ j = 1 m R j j η 2 ] d t
where x is state vector; η is control input, and ρ = ( max state error max control input   ) 2 .

Conflicts of Interest

The authors declare no conflict of interest.

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