To meet the everincreasing demand for modeling accuracy, the models of largescale systems are highorder, computationally complex, and with complex nonlinearities. However, these characteristics greatly limit the development of effective control strategies. In order to make a tradeoff between the modeling accuracy and the convenience in controller design, model structure design plays an important role and draw intensive research interests over the last two decades^{[1, 2]}.
Vapor compression cycle (VCC) is a core element in the heating, ventilating, and airconditioning (HVAC) systems. The VCC system is a high dimensional system that has obviously nonlinear thermodynamic coupling characteristics and timevarying dynamics characteristics. As a result, direct numerical simulation for such a large scale system becomes an intractable task. Through model reduction to choose appropriate model structure is an approach to overcome this problem^{[3]}. It aims to approximate a large scale system by low dimensional models that have similar response characteristics as the original system so that the feasible controllers can be designed.
There are a number of systematic strategies for model reduction proposed in recent year. One of the most common model reduction schemes is balanced truncation which was applied in the complex systems by Moore (1981)^{[4]}. Another popular model reduction method is the balanced residualization introduced by Fernando and Nicholson^{[5]}. Furthermore, this approach was extended by Rasmussen^{[6]} to the modeling and controller design of VCC system. The reduction results of these researches were to some extent simplified the model but still presented great challenges for the controller design and implementation. Based on Laguerre polynomials, Wang and Jiang^{[7]} proposed a model reduction method for coupled systems in timedomain. According to Laguerre coefficients, projection matrices were defined, and low order coupled systems were generated to match a desired number of these coefficients. This method retained the stability of coupled systems, however, it was not applied to an actual HVAC system. A novel model reduction strategy is based on the proper orthogonal decomposition (POD) technique, with which a reduced order model was obtained by Guha and Mishra^{[8]} verified by a nonlinear induction heating system.
For controller design, PI (proportional integration) or PID (proportion integration differentiation) feedback control algorithm is widely used in HVAC (heating ventilation and air conditionig) fields due to its simplicity^{[913]}. However, it is difficult for the multivariable systems to overcome the coupling effects among each degree of freedom resulting in poor performance. Neural network control and robust control have been introduced to deal with nonlinearities or uncertainties in HVAC processes^{[1416]}. In [17], a gainscheduling approach was utilized to handle the nonlinearity. The development of model predictive control (MPC) is considered as the recent major achievement in control literature, which has been widely accepted as the next generation of a practical control technology^{[1819]}. The application of MPC in HVAC systems can be found in the research of Elliott and Rasmussen^{[20]}. He designed a twoinputtwooutput MPC controller and a singleinputsingleoutput PI controller to control a multiple evaporator system. Xu et al.^{[21]} presented a linear matrix inequalitybased robust MPC strategy for the temperature control of an airconditioning system. An offsetfree MPC controller, comprising of a Luenberger observer, was implemented on a VCC system by Wallace to resolve the problem of plantmodel mismatch^{[18]}. Many studies presented different multivariable control strategies to improve the system energy efficiency; however, based on the premise of simplifying the controller complexity, most of them are based on the twoinputtwooutput model structure.
In this paper, a structure selection criterion, which can be used to evaluate the performance of different model structures and consequently to choose the optimal model structure for MPC controller design, is proposed. In order to select the impressionable variables which have main influence on the system performance, the relationships among variables of VCC system are analyzed first. According to the computational results under different loworder models, the optimized simplified model is determined. Then MPC based controller is designed for the optimized loworder model. Experiment results indicate that the proposed method has high modeling accuracy and great control performance.
The remainder of the paper is organized as follows. Section 2 details the dynamic model of the VCC system. Section 3 proposes a structure selection criterion which is used to evaluate the performance of different reduced order models and choose the optimal simplified model. Section 4 proposes the MPC based controller design. Section 5 presents the experimental results which justify some of the conclusions discussed in the previous sections. Section 6 summarizes the main conclusions.
2 Dynamic model of VCC systemA typical VCC system is composed of a compressor, a condenser, an expansion valve, and an evaporator. Fig. 1 illustrates the iterative cycle of VCC system. The refrigerant with high pressure and temperature enters the condenser and is cooled into subcooled liquid phase by heat exchange with cold fluid flowing across the coil or tubes. Then the condensed refrigerant enters the expansion valve where the pressure is reduced abruptly. The twophase refrigerant at low pressure and temperature exits the expansion valve and routes through the evaporator, in which the refrigerant evaporates and absorbs the heat from ambient air. At the outlet of evaporator, the refrigerant is a vapor at superheat state, and then enters the compressor where it is further compressed into a superheated vapor which has high pressure and temperature. After the compression process, the circulating refrigerant is routed back into the condenser to complete the refrigeration cycle. From the point of view of energy consumption, the system can be described as Fig. 2. It displays the relationship between pressure and enthalpy which is characterized as the energy parameter of VCC system. The dotted line in Fig. 2 is the saturation curve of refrigerant. The condensation process and evaporation process are considered as isobaric processes, while the expansion process is considered as isenthalpic process.
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Fig. 1. The schematic diagram of VCC system 
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Fig. 2. Pressure and enthalpy diagram of VCC system 
Using the lumpedparameter and movingboundary method, the dynamic model of each component of VCC was derived by Rasmussen and Alleyan^{[6]}, and is briefed as below.
Compressor. The dynamics of the compressor are considered to be much faster than those of heat exchangers, therefore, its mass flow rate can be modeled as a static component
$ \begin{align} \label{eq1} \dot {m}_k =F_k V_k \rho _k \left( {1+C_k +D_k \left( {\frac{P_{ko} }{P_{ki} }} \right)^{\frac{1}{n}}} \right) \end{align} $  (1) 
where
$ \begin{align*} f_c \left( {x_c, u_c } \right)= \\ \left[{{\begin{array}{*{20}l} {\dot {m}_{ci} \left( {h_{ci}h_{cg} } \right)+A_{ci} \alpha _{ci1} \displaystyle\frac{L_{c1} }{L_{cT} }\left( {T_{cw1}T_{cr1} } \right)\left( {\displaystyle\frac{1}{2}\left( {\displaystyle\frac{\partial \rho _{c1} }{\partial h_{c1} }} \right)(h_{c1}h_{cg} )+\displaystyle\frac{1}{2}\rho_{c_1}} \right)A_c L_{c1} \dot {h}_{ci} } \\ {\dot {m}_{ci} h_{cg} \dot {m}_{co} h_{cf} +\alpha _{ci2} A_{ci} \left( {\displaystyle\frac{L_{c2} }{L_{cT} }} \right)\left( {T_{cw2} T_{cr2} } \right)\left( {\displaystyle\frac{1}{2}\displaystyle\frac{\partial \rho _{c1} }{\partial h_{c1} }} \right)A_c h_{cg} L_{c1} \dot {h}_{ci} } \\ \!\!\!{\begin{array}{l} \dot {m}_{co} (h_{cf} h_{co} )+A_{ci} \alpha _{ci3} \displaystyle\frac{L_3 }{L_{cT} }\left( {T_{cw3} T_{cr3} } \right)\quad \\ \dot {m}_{ci} \dot {m}_{co} \left( {\displaystyle\frac{1}{2}\displaystyle\frac{\partial \rho _{c1} }{\partial h_{c1} }} \right)A_c L_{c1} \dot {h}_{ci} \\ \end{array}} \\ {\alpha _{co} A_{co} \left( {T_{ca} T_{cw1} } \right)\displaystyle\frac{\alpha _{ci1} A_{ci} \left( {T_{cw1} T_{cr1} } \right)}{L_1 }} \\ \!\!\! {\begin{array}{l} \alpha _{co} A_{co} \left( {T_{ca} T_{cw2} } \right)\displaystyle\frac{\alpha _{ci2} A_{ci} \left( {T_{cw2} T_{cr2} } \right)}{L_2 } \\ \! \alpha _{co} A_{co} \left( {T_{ca} T_{cw3} } \right)\displaystyle\frac{\alpha _{ci3} A_{ci} \left( {T_{cw3} T_{cr3} } \right)}{L_{c3} } \\ \end{array}} \\ \end{array} }} \right]. \end{align*} $ 
Condenser. According to the state of refrigerant, the condenser can be divided into three regions: a subcooled liquid region, a twophase region and a superheated vapor region. Based on the conservation of the refrigerant mass and energy, the dynamic equations of condenser can be established in a nonlinear state space form with 7 states and 5 inputs, shown as (2)(4):
$ \begin{align} &Z_{c} \left( {x_c, u_c } \right)\cdot \dot {x}_c =f_c \left( {x_c, u_c } \right) \end{align} $  (2) 
$ \begin{align} &\label{eq3} x_c =\left[{{\begin{array}{*{20}l} {\dot {L}_{c1} } \hfill&{\dot {L}_{c2} } \hfill&{\dot {P}_c } \hfill & {\dot {h}_{cro} } \hfill&{\dot {T}_{cw1} } \hfill&{\dot {T}_{cw2} } \hfill&{\dot {T}_{cw3} } \hfill \\ \end{array} }} \right]^{\rm T} \end{align} $  (3) 
$ \begin{align} &\label{eq4} u_c =\left[{{\begin{array}{*{20}l} {\dot {m}_{ci} } \hfill&{\dot {m}_{co} } \hfill&{h_{cri} } \hfill & {T_{ca} } \hfill&{\dot {m}_{ca} } \hfill \\ \end{array} }} \right]^{\rm T} \end{align} $  (4) 
where the state variables are: length of the two condensation regions
Expansion valve. The expansion valve is also modeled as a static component; its mass flow rate can be calculated from the orifice equation
$ \begin{align} \dot {m}_v =C_v A_v \left[{\rho _v \left( {P_{vi}P_{vo} } \right)} \right]^n \end{align} $  (5) 
where
Evaporator. Similar to the condenser model, the evaporator can be divided into two regions, i.e., a twophase region with a mean void fraction, and a superheated region. According to the refrigerant mass conservation and energy balance, it can be formulated by a fifth order nonlinear model:
$ \begin{align} &Z_e \left( {x_e, u_e } \right)\times \dot {x}_e =\notag\\ &\quad \left[{{\begin{array}{*{20}c} {\dot {m}_{ei} (h_{ei}h_{eg} )+\alpha _{ei1} A_{ei} \left( {\frac{L_{e1} }{L_{eT} }} \right)\left( {T_{ew1}T_{er1} } \right)} \hfill \\ {\dot {m}_{eo} (h_{eg}h_{eo} )+\alpha _{ei2} A_{ei} \left( {\frac{L_{e2} }{L_{eT} }} \right)\left( {T_{ew2} T_{er2} } \right)} \hfill \\ {\dot {m}_{ei} \dot {m}_{eo} } \hfill \\ {\alpha _{eo} A_{eo} (T_{ea} T_{ew1} )\alpha _{ei1} A_{ei} (T_{ew1} T_{er1} )} \hfill \\ {\alpha _{eo} A_{eo} (T_{ea} T_{ew2} )\alpha _{ei2} A_{ei} (T_{ew2} T_{er2} )} \hfill \\ \end{array} }} \right] \end{align} $  (6) 
$ \begin{align} &\label{eq7} x_e =\left[{{\begin{array}{*{20}c} {L_{e1} } \hfill&{P_e } \hfill&{h_{eo} } \hfill&{T_{ew1} } \hfill & {T_{ew2} } \hfill \\ \end{array} }} \right]^{\rm T} \end{align} $  (7) 
$ \begin{align} &\label{eq8} u_e =\left[{{\begin{array}{*{20}c} {\dot {m}_{ei} } \hfill&{\dot {m}_{eo} } \hfill&{h_{eri} } \hfill & {T_{ea} } \hfill&{\dot {m}_{ea} } \hfill \\ \end{array} }} \right]^{\rm T} \end{align} $  (8) 
where the state variables are: length of two phase flow
The combined cyclic model of integrated VCC system can be obtained by appropriately combining the component models according to the relations between the variables. The manipulated variables and output variables of the complete model are shown in (9) and (10). Different model structures can be obtained by different compositions of the manipulated variables and output variables. For the purposes of high accuracy and simple calculation, the optimized simplified model needs to be chosen for advancing towards further studies.
$ \begin{align} &\quad u=\left[{{\begin{array}{*{20}c} {F_k } \hfill&{u_v } \hfill&{\dot {m}_{ca} } \hfill&{T_{cai} } \hfill {\dot {m}_{ea} } \hfill&{T_{eai} } \hfill \\ \end{array} }} \right]^{\rm T} \end{align} $  (9) 
$ \begin{align} &\begin{array}{l} y=[{\begin{array}{*{20}c} {L_{c1} } \hfill&{L_{c2} } \hfill&{P_c } \hfill&{h_{cro} } \hfill & {T_{wc1} } \hfill&{T_{wc2} } \hfill \\ \end{array} }\\ {\begin{array}{*{20}c} \quad {T_{wc3} } \hfill&{T_{cao} } \hfill&{T_{cro} } \hfill&{T_{crsc} } \hfill&{\dot {m}_{ca} } \hfill&{L_{e1} } \hfill&{P_e } &\hfill {h_{ero} }\hfill \\ \end{array} }\\ {\begin{array}{*{20}c} \quad \hfill {T_{crsh} } \hfill&{T_{we1} } \hfill&{T_{we2} } \hfill&{T_{eao} } \hfill&{T_{ero} } \hfill&{T_{esh} } \hfill & {\dot {m}_{ea}]^{\rm T}}. \end{array} } \end{array} \end{align} $  (10) 
A structure selection criterion, which is used to evaluate the performance of different reduced order models and choose the optimal simplified model, is proposed in the following section. The nonlinear VCC system can be represented by the discretized timevarying statespace model which can be formulated as
$ \begin{align} \begin{array}{l} x\left( {k+1} \right)=A\left( k \right)x\left( k \right)+B\left( k \right)u\left( k \right) \\ y\left( k \right)=C\left( k \right)x\left( k \right) \\ \end{array} \end{align} $  (11) 
where
To maintain the model accuracy while reducing the difficulty of controller design, appropriate model structure selection is an important part of process control. A popular model reduction method is the proper orthogonal decomposition (POD) method by which the order of model is reduced. The reduced order model can be obtained using snapshots. Since low order model is simple and easy to be controlled, the POD algorithm can to some extent simplify the complex models, but it does not show whether it is the optimized model for controller design.
According to the POD theory, the states and outputs of model can be represented by the reduced states and outputs:
$ \begin{align} \begin{array}{l} x\left( k \right)=\Omega _1 x_n \left( k \right) \\ y\left( k \right)=\Omega _2 ^{1}y_n \left( k \right) \\ \end{array} \end{align} $  (12) 
where the
Substituting (12) into (11), the reduced order statespace model is formulated as
$ \begin{align} \begin{array}{l} x_n \left( {k+1} \right)=A_n \left( k \right)x_n \left( k \right)+B_n \left( k \right)u\left( k \right) \\ y_n \left( k \right)=C_n \left( k \right)x_n \left( k \right) \\ \end{array} \end{align} $  (13) 
where
In this section, an analysis on the system variables of VCC system is carried out first to find which variables have a significant impact on the refrigeration cycle system. Then the important variables are selected for the following structure selection criterion.
Because of the differing units and disparate scaling of the variables, the inputs and outputs of the system are normalized first. The system is excited with random inputs, and an experimentally derived linear model is created under selected operating condition. Fig. 3 shows the frequency responses of four outputs (evaporator pressure, superheat, condenser pressure, and subcool) derived by step changes of the compressor speed. Clearly, compressor speed has a strong effect on all of the four outputs, while it has a stronger effect on evaporator pressure and superheat than condenser pressure and subcool especially at lower speeds.
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Fig. 3. Normalized frequency responses to step changes in compressor speed 
Fig. 4 details the responses to changes in the expansion valve. Compared to the subcool, the effect is stronger on the superheat, while the effect is stronger on evaporator pressure than condenser pressure.
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Fig. 4. Normalized frequency responses to step changes in expansion valve 
According to the analysis results, the evaporator pressure and the superheat of evaporator are chosen in the proposed criterion as the crucial influencing variables for controller design of this system. The overall system model is firstly reduced from 12thorder to reduced order models using POD method, and then giving random varying inputs to the fullorder model and the same varying inputs to the reduced order models, the structure selection criterion is defined as
$ \begin{align} J(i)\mbox{=}(\bar {P}_e (i)P_e (i))^2+(\bar {T}_{esh} (i)T_{esh} (i))^2, ~~i=1, \cdots, Z \end{align} $  (14) 
where
Denote the cost value
$ \begin{align} \label{eq12} \Delta J_{n} =\sum\limits_{i=1}^{Z1} \left[\frac{( {J_n (i+1)}J_n (i))^2}{J_n^2 (i)} \right]. \end{align} $  (15) 
This criterion is to ensure a smoothly varying profile of the output variables. The structure selection criterion aims to find the optimal structure which minimizes the sum of quadratic partial variances of the evaporator pressure and the superheat of evaporator. With different dimension
Thus, the structure selection criterion algorithm performs the following steps:
Step 1. Reduce the overall system model first from 12thorder model to a particular low order model using POD method, and then further reduce from that reduced order model to 2ndorder model.
Step 2. Give same various random inputs to the fullorder model and the reduced order models, respectively.
Step 3. Compute the criterion
Step 4. Calculate the differences
Step 5. Compare the values of
MPC is a control algorithm which computes a sequence of control inputs based on an explicit prediction of outputs within some future horizon^{[22]}. One of the most important advantages of MPC is that it accounts for the constraints of input and output variables that can be inherent to the real industrial systems, e.g., a valve cannot open past 100% open or close past 0% open. Another advantage of MPC is that additional constraints can be defined by the user to keep the system operating in a safe range, e.g., keeping evaporator superheat above a desired minimum.
The performance of MPC depends on a number of design parameters, such as length of the control time interval, the number of future moves for the manipulated variable, and the number of time intervals in the output prediction. In order to define how well the predicted process tracks the set points, an objective function
$ \begin{align} \label{eq13} \begin{array}{l} J_{MPC} (k)= \\ \sum\limits_{i=1}^P {\left\{ {\left( {w(k+i)y(k+i)} \right)^{\rm T}Q\left( {w(k+i)y(k+i)} \right)} \right\}}+ \\ \, \sum\limits_{i=1}^M {\left\{ {\left( {\Delta u(k+i1)} \right)^{\rm T}{\pmb R}\left( {\Delta u(k+i1)} \right)} \right\}} \\ \end{array} \end{align} $  (16) 
where
Equation (16) computes the weighted sum of squared deviations of the outputs from the set points and the weighted sum of squared incremental manipulated variables.
The optimization algorithm searches for values of
$ \Delta u_{\min } \le \Delta u(k)\le \Delta u_{\max } $  (17) 
$ u_{\min } \le u(k)\le u_{\max } $  (18) 
$ y_{\min } \le y(k)\le y_{\max }. $  (19) 
The objective function
$ \begin{align} &J_{MPC, n} (k)= \notag\\ &\quad \sum\limits_{i=1}^P \left\{ \left( {w(k+i)y_n (k+i)}\right)^{\rm T}\Omega _2^{\rm T}\right.\notag\\ &\quad \left.Q\Omega _2^{1} \left( {w(k+i)y_n (k+i)} \right) \right\} +\notag\\ &\quad \sum\limits_{i=1}^M {\left\{ {\left( {\Delta u_n (k+i1)} \right)^{\rm T}{\pmb R}\left( {\Delta u_n (k+i1)} \right)} \right\}} \end{align} $  (20) 
$ \begin{align} &\Delta u_{\min, n} \le \Delta u_n (k)\le \Delta u_{\max, n} \notag\\ &u_{\min, n} \le u_n (k)\le u_{\max, n} \notag\\ &y_{\min, n} \le y_n (k)\le y_{\max, n} \end{align} $  (21) 
where
The MPC control algorithm for VCC system is as follows:
Step 1. Compute the objective function
Step 2. Calculate the best control signal
Step 3. Apply
Step 4. At the next sampling time, go back to calculate
The experimental platform used in this research is developed at the process instrumentation laboratory of Nanyang Technological University of Singapore. The photograph and schematic diagram of the experimental platform are shown in Figs. 5 and 6.
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Fig. 5. Photograph of the experimental platform in this research 
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Fig. 6. Schematic of the experimental platform in this research 
The system includes a variable speed compressor, an electronic expansion valve, an aircooled condenser and evaporator, a liquid receiver after the condenser, an accumulator after the evaporator, and the fans of condenser and evaporator with variable frequency. In addition, the pressure measurement devices are installed on the system, and the measurement range and measuring error of these devices are 0
To verify that the structure selection criterion can choose optimized model structure effectively, a comparison study has been carried out. The model has been firstly linearized around a steady state operating point indicated in Table 1. Then with the POD method, the system model is reduced to 4thorder model with the selection of input variables as
The
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Fig. 7. Pressure of evaporator for random variations in compressor speed 
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Fig. 8. Superheat of the evaporator for random variations in compressor speed 
Through the simulation, the structure selection criteria of the three simplified structures are calculated according to (14)(15) respectively, and the corresponding structure selection criteria are shown in (22)(24).
$ \begin{align} &J_2 (i)\mbox{=}(\bar {P}_e (i)P_{e2} (i))^2+(\bar {T}_{esh} (i)T_{esh2} (i))^2 \notag\\ &\Delta J_2 =\sum\limits_{i=1}^{Z1} {\left[{\frac{\left( {J_2 (i+1)J_2 (i)} \right)^2}{J_2^2 (i)}} \right]}~~i=1, \cdots, Z \end{align} $  (22) 
$ \begin{align} &J_3 (i)\mbox{=}(\bar {P}_e (i)P_{e3} (i))^2+(\bar {T}_{esh} (i)T_{esh3} (i))^2\notag \\&\Delta J_3 =\sum\limits_{i=1}^{Z1} {\left[{\frac{\left( {J_3 (i+1)J_3 (i)} \right)^2}{J_3^2 (i)}} \right]} ~~ i=1, \cdots, Z \end{align} $  (23) 
$ \begin{align} &J_4 (i)\mbox{=}(\bar {P}_e (i)P_{e4} (i))^2+(\bar {T}_{esh} (i)T_{esh4} (i))^2 \notag\\ &\Delta J_4 =\sum\limits_{i=1}^{Z1} {\left[{\frac{\left( {J_4 (i+1)J_4 (i)} \right)^2}{J_4^2 (i)}} \right]}~~i=1, \cdots, Z \end{align} $  (24) 
where
The results of the proposed structure selection criterions are shown in Table 2 which indicate that the 3rdorder model structure has the minimum structure selection criterion. Thus the 3rdorder model structure is finally chosen as optimal model structure, which is convenient to the controller design of the VCC system. The coefficient matrices of the 3rdorder model are expressed as (25).
$ \begin{align} &A_3 \mbox{=}\left[{{\begin{array}{*{20}c} {\mbox{$$0.152 5}} \hfill&{\mbox{$$0.010 6}} \hfill&{\mbox{0.046 6}} \hfill \\ {\mbox{$$0.000 0}} \hfill&{\mbox{$$0.171 8}} \hfill&{\mbox{0.244 1}} \hfill \\ {\mbox{$$0.000 0}} \hfill&{\mbox{$$0.128 1}} \hfill&{\mbox{$$0.010 3}} \hfill \\ \end{array} }} \right]\notag\\ &B_3 \mbox{=}\left[{{\begin{array}{*{20}c} {\mbox{$$0.007 2}} \hfill&{\mbox{0.131 5}} \hfill&{\mbox{2.233 9}} \hfill \\ {\mbox{$$0.011 5}} \hfill&{\mbox{3.810 7}} \hfill&{\mbox{$$4.729 4}} \hfill \\ {\mbox{0.018 9}} \hfill&{\mbox{$$0.507 7}} \hfill&{\mbox{$$1.412 1}} \hfill \\ \end{array} }} \right]\notag\\ &C_3 \mbox{=}\left[{{\begin{array}{*{20}c} {\mbox{3.416 6}} \hfill&{\mbox{$$1.365 6}} \hfill&{\mbox{$$1.998 3}} \hfill \\ {\mbox{0.937 6}} \hfill&{\mbox{0.173 5}} \hfill&{\mbox{$$1.145 9}} \hfill \\ {\mbox{$$0.069 0}} \hfill&{\mbox{0.273 5}} \hfill&{\mbox{0.522 1}} \hfill \\ \end{array} }} \right]\notag\\ &D_3 \mbox{=}\left[{{\begin{array}{*{20}c} \mbox{0} \hfill&\mbox{0} \hfill&\mbox{0} \hfill \\ \mbox{0} \hfill&\mbox{0} \hfill&\mbox{0} \hfill \\ \mbox{0} \hfill&\mbox{0} \hfill&\mbox{0} \hfill \\ \end{array} }} \right]. \end{align} $  (25) 
In this section, the MPC controller for the proposed 3rdorder model is designed based on objective function
To demonstrate the effectiveness of the designed control system, an MPC controller for the 2ndorder model is designed as follows: the evaporator pressure and superheat are the system outputs, meanwhile the compressor speed, expansion valve opening are inputs. The weight of 1 and 100 are placed on the evaporator pressure and superheat, and rate weights of 0.1 and 0.01 on the compressor speed, expansion valve in MPC controller, respectively. Other tuning parameters are the same with controller for 3rdorder model. The performance comparison between the two controllers are carried out, and the output and input profiles are shown in Figs. 9 and 10.
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Fig. 9. Outputs comparison of VCC system under the two MPC controllers 
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Fig. 10. Inputs comparison of VCC system under the two MPC controllers 
Fig. 9 shows the output performance in response to the disturbances due to the changes in the air mass flow rate of evaporator at 1000s. The 3rdorder controller tracks the setting value faster than the 2ndorder controller when the disturbance occurs, which demonstrate that the proposed control system has satisfactory tracking performance and robustness against disturbance.
The corresponding input variables are given as Fig. 10. When the disturbance is changed, the compressor speed and the expansion valve opening respond to change quickly in order to reject the disturbance and keep the superheat and pressure as before.
6 ConclusionsThis paper presented a model predictive control system based on a structure selection criterion which is used to simplify model and select the optimal reduced order model for MPC controller. The overall system model is reduced from 12thorder to reduced order models using POD method, and the model which minimizes the cost value is chosen as the optimized simplified model. Based on the proposed model, the MPC based controller is designed, and the objective function for the simplified model is optimized. To validate the effectiveness of the proposed control system, comparison experiments have been carried out, and the experimental results indicate that the proposed controller has excellent advantages of tracking performance and disturbance rejection.
AcknowledgementsThis work was supported by National Natural Science Foundation of China (Nos. 61233004, 61221003, 61374109 and 61473184), National Basic Research Program of China (973 Program)(No. 2013CB035500), and partly sponsored by the Higher Education Research Fund for the Doctoral Program of China (No. 20120073130006), and National Research Foundation of Singapore (No. NRF2011 NRFCRP001090).
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