energies

Article

Robust Sliding Mode Control of Air Handling Unitfor Energy Efficiency Enhancement

Awais Shah, Deqing Huang, Yixing Chen, Xin Kang and Na Qin *

School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China;awaisshah@my.swjtu.edu.cn (A.S.); elehd@home.swjtu.edu.cn (D.H.); bk20110190@my.swjtu.edu.cn (Y.C.);kangxin@my.swjtu.edu.cn (X.K.)* Correspondence: qinna@home.swjtu.edu.cn

Academic Editor: Roberto de Lieto VollaroReceived: 7 October 2017; Accepted: 6 November 2017; Published: 9 November 2017

Abstract: In order to achieve feasible and copacetic low energy consuming building, a robust andefficient air conditioning system is necessary. Since heating ventilation and air conditioning systemsare nonlinear and temperature and humidity are coupled, application of conventional control isinappropriate. A multi-input multi-output nonlinear model is presented. The temperature andhumidity of thermal zone are ascendance by the manipulation of the water and air flow rates.A sliding mode controller (SMC) is designed to ensure robust performance of air handling unit inthe presence of uncertainties. A simple proportional-integral-derivative (PID) controller is usedas a comparison template to highlight the efficiency of the proposed controller. To accomplishtracking targets, a variety of desired temperature and relative humidity commands (includingramp and combination with sequence of steps) are investigated. According to simulation results,SMC transcends the PID controller in terms of settling time, steady state and rise time, which makesSMC more energy efficient.

Keywords: sliding mode control; air handling unit, heating; ventilation and air conditioning (HVAC);multi-input muti-output (MIMO)

1. Introduction

With the progression in lifestyle, Heating, Ventilation and Air Conditioning (HVAC) systems arewidely employed to provide better comfort and indoor air quality (IAQ). The important factor thatneeds to be handled in HVAC systems is energy loss. Around 50% of total energy consumed by amodern building is due to HVAC [1]; therefore, optimal control of HVAC system is crucial. Air handlingunit (AHU), which regulates and conditions (warm or cool) airflow to thermal zones, is the main partof HVAC systems. For AHU design, a mathematical model of components is pressingly needed toanalyze the system performance. Furthermore, it is very difficult to design an exact mathematicalmodel because of its time-varying characteristics and complex nonlinear behavior [2]. The mostcommon parameters of AHU are temperature and humidity. Airflow and water flow control is usedinside AHU to track these parameters. Different investigations, such as variable or constant volume,have been conducted for transient response and dynamic modelling of AHU. Moreover, dynamicequations are achievable for AHU by combining mass balance equations with engineering models [3–5].For the control of AHU, the sliding mode design is perceived as the most effective techniques to layoutrobust controllers for complex higher order nonlinear systems working under uncertainties. Controllerconfiguration gives a deliberate way to deal with the issue of keeping up consistent performance andstability. It also has the probability of balancing out some nonlinear systems that are not stabilizable bycontinuous state input laws. Robustness is the system is on a sliding surface. Then, it creates minimumvariations in parameters.

Energies 2017, 10, 1815; doi:10.3390/en10111815 www.mdpi.com/journal/energies

Energies 2017, 10, 1815 2 of 21

Researchers have carried out several investigations in recent years for efficient energymanagement of air conditioning system [6]. One control strategy of HVAC indoor temperatureand humidity is maintained by proportional-integral-derivative (PID)-fuzzy controller in the presenceof uncertainties [7]. Khan et al. presented multivariable controller based on adaptive fuzzy and geneticalgorithm for AHU [8]. Wang et al. presented a control strategy for indoor environments by usinga number of occupants and applying predictive control technique [9]. Prediction is done by videosurveillance and concentration of room carbon dioxide. A multi-input multi-output (MIMO) controlleris designed to enhance indoor air quality (IAQ) by changing air supply fan and compressor speedin a direct expansion (DX) air conditioning system [10]. Mba et al. presented prediction of indoortemperature and humidity, which is done on an hourly basis by applying artificial neural networks(ANN) [11].

Robust control of AHU with uncertainties and a comparison in details between observers(the minimum and full order) is presented in [12]. On account of rupture in a sensor combinationsystem of air handling units, the full-order observer is a tool that can be utilized for satisfactorycalculation of the state variables. Design of robust optimal control for buildings based on chilledwater systems and considering it with minimum life cycle cost and quantified uncertainties ispresented in [13]. The proposed study upgrades the plan of chilled water pump frameworks with theuncertainities of configuration inputs. This proposed outline strategy limits the yearly aggregate costunder various control techniques. A physical statistical based approach, which considers distributedair conditioning control for commercial building, is discussed in [14]. Control designs are producedfor zone cooling and indicated by inhabitance designs at zone levels, and a novel reaction model iscreated in this paper. The control designs show endeavors to improve cooling system planning onthe premise of inhabitance designs, value motion from utilities and human comfort and profitability.A feedback linearization method for nonlinear dynamics of AHU, which is based on disturbancerejection, is discussed in [15]. In this study, a multivariable control technique in view of feedbacklinearization technique is executed. The air and water stream rates are the inputs to the controller andthermal space temperature and humidity are the outputs. Nevertheless, it is difficult to implementsuch type of approach because optimal realization controller is complex at run time, which is causedby inversion and posses of nonlinear state space model.

Temperature and humidity are decoupled for a hot and humid atmosphere in [16]. Three distincttechniques are planned in a hot and sticky atmosphere area for air handling units and decouplingassessments are considered. The first two techniques utilize a similar feedback control references(temperature and humidity). The air handling unit of the third system is furnished by a wet pre-coolingcurl and a dry fundamental cooling loop to dehumidify natural air that enables the controller to dealwith the coupling issue. H∞ controller is designed for AHU and its comparison is done with poleplacement method in [17]. An observer and a regulator are intended for the estimation of state factorsand unsettling influence dismissal, respectively.

An experimental study is done on ground source heat pump (GSHP), which is assimilated withvariable air volume (VAV) system in [18]. The study demonstrate about online evaluation of the coolingload, the bilinear control could keep up the set-point of the required temperature well. The controlscheme was better than that of a customary PI control. Simulation based dynamic model is presentedfor adaptive control of temperature and humidity with external disturbance in [19]. This strategypresents a control algorithm, in view of a reference adaptive control scheme that empowers the onlineadjustment of the control gains.

Pre cooling coil is added in a VAV system with a residential load factor (RLF) in [20]. The doublecooling coiled system used hybrid modelling of HVAC and RLF for load calculation and energy saving.Temperature and humidity are two key parameters of HVAC, but the proposed method focuses onbetter control of humidity only. Prediction of fan speed control based on wavelet decompositionand neural networks is discussed in [21]. Predictive control of heating and cooling of office buildingwith cooperative fuzzy mode (CFMPC) is discussed in [22]. In [23], predictive control based on

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fuzzy-neural technique of heating buildings is proposed. Cascaded HVAC control of superheattemperature have been applied on AHUs, for which super twisting of sliding mode is presented in [24].In [25], a monetary model-based predictive control (MPC) whose principle quality is the utilizationof the day-ahead price (DAP) and keeping in mind the end goal to foresee the energy utilizationrelated with the HVAC is presented. The study presented in [26] models the HVAC system in TRNSYSsoftware (v.16, Thermal Energy System Specialists, Madison, WI, USA), the principle componentsof which are a reference building, cooling tower, and a water chiller. The cooling tower display isapproved utilizing test information in a pilot plant. In [27], a point by point enhancement of planparameters of warmth wheels is performed with a specific end goal to amplify sensible viability and tolimit pressure drop. The investigation is brought out through a one-dimensional lumped parametersheat wheel model, which settles heat and mass exchange conditions, and through suitable relationshipsto appraise pressure drop.

Contribution of this study, unlike the previous research is, two control strategies, namely,(a) sliding mode control (SMC), and (b) proportional integral derivative (PID) controller have beencompared for the MIMO AHU dynamic model. Advantages and disadvantages of both strategiesare discussed from different point of views such as (a) energy consumption, and (b) meeting controlobjectives. SMC is completely insensitive to external disturbances and parametric uncertainties duringthe sliding mode [28]. Since HVAC system is highly nonlinear and uncertainties based, SMC playsa vital role in HVAC control.

SMC and PID controllers are designed after the dynamic system state space formulation. Indoorrelative humidity and temperature set points (including step ramp and step) are achieved by changingthe air and water flow rates in AHU. Results show that the system controlled by SMC has a morerapid response for required trajectories. The system controlled by PID controller has an unsatisfactoryresponse when compared with SMC on the same set points. In the presence of random disturbance inmodel parameters, the SMC based controller is robust.

The AHU description and its dynamic model are discussed in Section 2. In Section 3, control lawsare developed. Section 4 presents the simulation results achieved from the system. Finally, this articleis concluded in Section 5.

2. Air Handling Unit

A single zone HVAC schematic is presented in Figure 1. It consists of a heat exchanger, a supplyair fan, duct work, filters and air mixing dampers. In our study, we consider the cooling mode of theHVAC system. The inputs to the system are rates of the supply air flow and supply cooling water rates.The operation process of air conditioning system on cooling mode is shown as the following:

• 75% of recirculated air is mixed with 25% of fresh air and the rest is exhausted.• After passing through the filter, this air gets conditioned in the heat exchanger.• The conditioned air (supply air) enters the thermal zone by supply fan.• After offsetting the latent (humidity) and sensible heat (actual heat), the return air is drawn out

by a duct where 25% of recirculated air is exhausted via exhaust dampers.

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Figure 1. Air handling unit schematic drawing view with single thermal zone in the variable airvolume (VAV) system.

Dynamic Modeling

The following assumptions are made for the formulation of system [29].

• There should be no air leakage in the duct work except at exhaust air dampers.• Air flow is homogeneous.• Ideal gasses must be employed.• Zone air pressure is independent of air speed variations.

On the basis of the laws of mass and heat transfer, AHU can be formulated by following differentialequations [29,30]:

Tsa =˙far

Vcu(Tia − Tsa) +

0.25 ˙far

Vcu(Toa − Tia)−

˙farhs

CaVcu(0.25woa + 0.75wia − wsa)− ˙fwr

(ρwdCwδTcu

ρadCaVcu

),

Tia =1

ρadCaVia(Hl − hv Ml) +

˙farhv

CaVia(wia − wsa)−

˙far

Via(Tia − Tsa),

wia =Ml

ρadVia−

˙far

Via(wia − wsa) ,

(1)

where Tsa/wsa, Tia/wia and Toa/woa are the supply conditioned air temperature and humidity ratios,thermal zone air (indoor), and outdoor fresh air, respectively. δTcu is cooling unit temperature gradient,

˙fwr, and ˙far are the cold water and air flow rates, respectively. Hl , Ml are heat and humidity loadstrengths, respectively. ρwd/Ca, ρad/Ca are the cold water, and air mass density/specific heat. hv, hs

are enthalpy of vaporization and saturated water (parameters of thermofluids and operating pointvales are presented in Tables 1 and 2).

Table 1. Air handling unit thermo-fluid parameters.

Parameters of Thermofluids

Via Thermal space volume woa Humidity ratio of outdoor fresh airVcu Cooling unit volume wsa humidity proportion of Supply airρwd Water mass density wia Thermal zone humidity ratioρad Air mass density Toa Outdoor fresh air temperaturehv Enthalpy of vapors Tsa conditioned air supply temperaturehs Enthalpy of saturated water Tia Thermal zone air temperaturecw Specific heat of water δTcu Cooling unit temperature gradientca Specific heat of air Ml Humidity source strengthfwr water flow rate Hl Heat loadfar air flow rate

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Table 2. Thermo-fluid parameters values in air handling unit around an operating point.

Operating Point Values

Via = 400 m3 woa = 0.0082 kg H2O/kg dry airVcu = 1 m3 wsa = 0.0080 kg H2O/kg dry airρwd = 1000 kg/m3 wia = 0.00804 kg H2O/kg dry airρad = 1.18 kg/m3 Toa = 32 ◦Chv = 2500 kJ/kg Tsa = 17 ◦Chs = 80 kJ/kg Tia = 20 ◦Ccw = 4180 J/kg ◦C δTcu = 6 ◦Cca = 1000 J/kg ◦C Ml = 0.000115 kg/sfwr = 0.0009 m3/s Hl = 20 KWfar = 2.6 m3/s

Equation (1) formulates the AHU dynamics, humidity ratio (wia) of indoor air and temperature ofindoor air and supply air (Tia, Tsa), in the form of differential equations. The following definitions aremade to simplify Equation (1):

α1 =1

Via, α2 =

1ρadVia

, α3 =1

Vcu,

β1 =hv

CaVia, β2 =

ρwdCaδTcu

ρadCaVcu,

γ1 =1

ρadCpaVia, γ2 =

hs

CaVcu.

(2)

The nonlinear state space formulation inputs, outputs and state variables of dynamic system aregiven in the following:

u1 = ˙far, u2 = ˙fwr,

y1 = wia, y2 = Tia,

x1 = Tia, x2 = wia, x3 = Tsa.

(3)

The state space formulation can be formed using Equations (1)–(3) for a nonlinear dynamic system [17]:

x1 = γ1(Hl − hs Ml) + β1u1(x2 − wsa)− α1u1(x1 − x3),

x2 = α2Ml − α1u1(x2 − wsa),

x3 = α3u1(x1 − x3) + 0.25α3u1(Toa − x1)− γ2u1[0.25woa + 0.75x2 − wsa]− β2u2,

y1 = wia,

y2 = Tia.

(4)

Simplifying the Equation (4) yields

x1 = f1(x) + g1(x)u1,

x2 = f2(x) + g2(x)u1,

x3 = f3(x) + g3(x)u2,

(5)

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wheref1(x) = γ1[Hl − hs Ml ],

g1(x) = β1(x2 − wsa)− α1(x1 − x3),

f2(x) = α2Ml ,

g2(x) = α1(x2 − wsa),

f3(x) = α3u1(x1 − x3) + 0.25α3u1(Toa − x1)− γ2u1[0.25woa + 0.75x2 − wsa],

g3(x) = −β2.

(6)

3. Controller Design

In this section, controller design for the nonlinear dynamic AHU is discussed. It is observedfrom Equation (1) that temperature and humidity equations are coupled with each other; therefore,the controller must be designed and tuned in order to achieve set points effectively. Figure 2 representsthe block diagram of the general AHU feedback controller:

Figure 2. The block diagram of a general air handling unit (AHU) feedback controller.

3.1. PID Controllers

PID controller is the most popular type of controllers utilized in industrial process. PID controllercontains three sections, which are proportional, integral and derivative modes, respectively.PID controller is used to maintain: (a) pressures, (b) temperatures and (c) flow rates. The simplefeedback control compares a measured variable with a setpoint and generates an error. The output totrack the setpoint is based on the error and input settings. The general form of PID controller can bepresented as [31]:

u(t) = kp · e(t) + ki ·∫

e(t)d(t) + kd ·de(t)

dt(7)

Tuning of PID Controllers

The Ziegler and Nichols method is used to select the values of Kp, Ki and Kd for the proposedstudy. The Ziegler–Nichols technique depends on tests executed on a set up control loop. The followingsteps are involved in selecting the gains [32]:

• Transform the PID controller into a P by adjustment of Ki = ∞ and Kd = 0. At first, set pick up toKp = 0. Close the loop in programmed mode for the controller.

• Increase Kp unless there are maintained motions in the signs in the control system. This Kp esteemis signified a definitive gain, Kpu.

• Measure a specific period Pu of the maintained oscillations.• Figure the controller parameter esteems as per Table 3, and utilize these parameter esteems in

the controller.

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Table 3. Conditions for the controller parameters in the Ziegler–Nichols’ closed loop technique.

Controller Kp Ki Kd

proportional (P) controller 0.5 Kpu ∞ 0proportional-integral (PI) controller 0.45 Kpu

Pu2 0

proportional-integral-derivative (PID) controller 0.6 KpuPu2

Pu8 = Pi

4

3.2. Sliding Mode Control

SMC is based on changing nonlinear system dynamic behavior by discrete control action,which forces the system to operate on its normal mode. SMC is a special class of nonlinear controlsystem that is less sensitive to variations and disturbances in plant parameters. The feedback controllaw for state is a discontinuous time function that can be changed from a structure (in a consistent way)in view of the ground position in space. Accordingly, the SMC is characterized as a variable organizedstrategy. The sliding mode is the specific working method of the framework, as it slides towardsthe predetermined limits of the control scheme. Furthermore, the geometrical locus, fundamentallycomprising of the limits, is referred as the sliding surface of the framework. Figure 3 delineatesan occurrence of the direction of a specific framework related to the SMC procedure. This schemerepresents the sliding surface as s = 0. While the trajectories approaches to predefined sliding surface,then sliding mode begins.

• The direction of trajectories is towards s = 0.• The trajectories have a place with the switching s = 0 and can’t leave it.• Once sliding mode begins, additional movement is administered by the equation s = cx + x = 0.

Figure 3. Working of sliding mode control, a graphical representation [33].

3.3. Selection of SMC over PID

The following reasons are involved in the selection of SMC over PID.

• Typically, the HVAC frameworks are nonlinear with variations in parameters. Henceforth,utilizing PID control strategy may hamper system stability because of the conceivable overlinearization of the system. Then again, a sliding mode control doesn’t overlook nonlinearity.

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• The capability of the framework is dependent on the load. On account of displaying miscalculation,the sliding mode control provides a deliberate approach to maintain the stability and in additionthe coveted reliable performance.

• The sliding mode control implementation on hardware is simple. In the microcontroller, it needsless numerical and computational calculations. The standard protocols are promptly perfect withit, for example, Modbus and the Ethernet/IP, RS-232.

• For the situations, where accuracy and stability are required, SMC requires altogether lessmaintenance and gear costs.

3.4. SMC Design

It is obligatory to choose ‘u’ as a feedback control law for verification of sliding sliding condition.In any case, the control law must be discontinuous over s(t) to represent the presence of the displayingperturbations and additionally imprecision. The chattering phenomena is added as the outcome ofthe blemish of related control switching. Practically speaking, chattering is totally unwanted for theframework because it requires a unique control design. Other than that, this may present higher orderfrequency elements that were disregarded in the modelling scenario. For a nonlinear system, we have

x = f (x) + g(x)u + d(t), (8)

where d(t) is some random unknown disturbance. A sliding variable can be defined as

s(x) = AT(x) =n

∑i=1

aixi =n−1

∑i=1

aixi + xn, (9)

where x denotes a state vector and xi = xn−1i , i = 1, ..., n, A = [a1...., an−1, n]T . If s(x)→ 0, to ensure

xi → 0, a1, a2, ... an are selected in such a way that cn−1 + an−1cn−2 + ...a2c + a1, the polynomial isHurwitz and c is a Laplace operator.

Define the sliding mode function as

s(t) = je(t) + e(t), (10)

where j is satisfying the Hurwitz criteria, j > 0. Define the tracking error and take the derivatives

e = xd − x1,

e = xd − x1,(11)

where xd is an ideal positioning value. We get the following expression

s = je + e

= j(xd − x1) + (xd − x)

= j(xd − x1) + (xd − f − g.u− d).

(12)

Define the exponential reaching law

s = −ζsgns−ms, ζ, m > 0. (13)

By rearranging Equations (12) and (13), we get

j(xd − x1) + (xd − f − g.u− d) = −ζsgns−ms. (14)

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Now, the control law can be defined as

u(t) =1g(ζsgns) + ms + j(xd − x1 + x− f − d). (15)

Clearly, all amounts in the Equation (15) are known, but the aggravation d is obscure. In this way,control law can’t be figured out. Redesign the control law as

u(t) =1g(ζsgns) + ms + j(xd − x1 + x− f − Dsgns), (16)

where d = −Dsgns.After substituting Equation (16) into Equation (12) and simplifying, the following expression

is obtaineds = −ζsgns−ms− Dsgns− d. (17)

With further simplification, we get

ss = s(−ζsgns−ms− Dsgns− d)

= −ms2 − ζ log |s| − D log |s| − ds.(18)

The following Lyapunov candidate function is adopted for stability analysis

V =12

s2.

Therefore, we have

V = ss(t) = −ms2 − ζ log |s| − D log |s| − ds ≤ −2mV. (19)

Ultimately, the expression becomes [34].

V(t) ≤ e−2mtV(0). (20)

It is clear that sliding mode function exponentially goes to zero with m value. The control lawsu1 and u2 are designed in a similar way. The detailed formulations of control laws are given inAppendix A:

u1 =1

α1(x2 − wsa)[− ˙x2r + α2Ml + a1e2 + υ],

u2 =1β2

[a ˙x1r + ˙x3r + ah(x)u1 + aA1 + g(x) + υ],(21)

where υ = V = −β|s|. We use it as negative discontinuous control to ensure stability.

4. Results

Three different setpoints including multilevel steps, steps and ramps, and their combination areused for investigation of tracking temperature and humidity ratios. The three random setpoints forcomfort level are presented in Figure 4. The operation in this study is under summer conditions;however, the controller design based on PID and sliding mode can be used in all weather conditions.

In PID, the kp term diminishes the rise time but builds the overshoot and decreases the steadystate error. The ki term reduces the rise time and overshoot but enlarges the settling time. The kd termwill have the impact of enhancing the stability of the system, decreasing the overshoot and improvingthe transient response. The output temperature response by SMC and PID on multilevel step input ispresented in Figure 5 (case (a) of Figure 4), and the humidity output response is presented in Figure 6.The response of SMC presents almost negligible overshoot and less response time for convergence,

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whereas PID controller lags in tracking the setpoint for both temperature and humidity. The desiredsetpoint (case (b) of Figure 4), multilevel steps and ramp for temperature and humidity trackingresponse are presented in Figures 7 and 8, respectively, PID based controller shows more overshoot inboth temperature and humidity setpoints, but response time for convergence is almost the same asthose of SMC in temperature setpoint cases, whereas tracking of humidity setpoint for PID control isunsatisfactory. Figures 9 and 10 present the temperature and humidity output (for the reference signalpresented in case (c) of Figure 4), respectively. It is clear that SMC outperforms PID in overshoot andtime response for convergence.

Figure 4. Desired input sequence for humidity ratio and temperature including, (a) sequence of steps;(b) steps with ramp; (c) integration of both.

With respect to SMC responsiblity for AHUs in practice, the frequency of control move could be acritical factor for support of actuators and valves. The high frequency of control action can damagethe valves and actuators, and it will also increase the maintenance cost. Figure 11 represents thisperformance analysis for (a) temperature and (b) humidity, with less successive actuator moves.The analysis is done with combination of multilevel steps input (part (a) of Figure 4) for bothtemperature and humidity. It can be seen clearly that SMC shows less convergence time and overshoot.

Performance of controller is also analysed by performance index graph. The performanceindex is a significant measure of system performance and is chosen to emphasize important system

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specifications. The system is the best control system, when settings are adapted to allow the index toreach the minimum value. Two types, (a) integral of the square of the error (ISE), (b) integral of timeinto squared error (ITSE), (c) integral absolute error (IAE) and (d) integral time absolute error (ITAE)are used to analyse the controller performance. The errors are calculated by the following equations:

ISE =∫ T

0e2(t)dt,

ITSE =∫ T

0te2(t)dt,

IAE =∫ T

0|ε|dt,

ITAE =∫ T

0t|ε|dt.

(22)

Table 4. Performance indices, integral of the square of the error (ISE), integral of time into squared error(ITSE), integral absolute error (IAE) and integral time absolute error (ITAE) values for PID and SMC.

Controller

TEMPERATURE HUMIDITY

ISE ITSE ITAE IAE ISE ITSE ITAE IAE

INPUT 1

PID 0.015 5.72 5.5× 1014 614.9 2.55× 1019 6.63× 1020 49.28 0.203SMC 0.00043 0.058 2.79× 107 167 1.208× 1019 2.51× 1020 5.959 0.028

INPUT 2

PID 2.38× 10−5 0.0113 5.44× 103 1.29 7.42× 1016 7.1× 1018 10.15 1.102SMC 8.8× 10−6 0.0036 8.57 0.159 2.73× 1010 4.57× 1015 8.5 0.159

INPUT 3

PID 0.016 5.735 7.6× 108 8.9× 105 1.114× 1018 5.488× 1020 128.8 1.252SMC 0.0018 0.0177 5.5× 108 6.7× 105 4.51× 1017 1.4× 1020 16.07 0.183

Figure 5. Output temperature response of controllers while tracking sequence of steps reference.

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Figure 6. Output humidity response of controllers while tracking sequence of steps reference.

Figure 7. Output temperature response of controllers while tracking sequence of steps and ramp reference.

Figure 8. Output humidity response of controllers while tracking sequence of steps and ramp reference.

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Figure 9. Output temperature response of controllers while tracking combination of sequence of stepsand ramp reference.

Figure 10. Output humidity response of controllers while tracking combination of sequence of stepsand ramp reference.

To improve readability, the performance indices graphs are presented in Appendix B. Performanceindices graph of controller output on multilevel steps reference for temperature and humidity arepresented in Figures A1 and A2, respectively. The numeric values of these indices are presentedin Table 4. SMC values of all four errors are extensively less as compared to PID. Similarly,Figures A3 and A4 represents performance indices of PID and SMC for temperature and humidity onramp and multilevel steps reference signal. Once again, SMC outperforms PID in performance,especially performance indices for temperature have a bigger difference than humidity value.Temperature and humidity values of all performance indices are presented in Figures A5 and A6,respectively. The input signal is a combination of ramp and multilevel steps with a sequence of steps.By inspection of numeric values and graphical presentation, it is clear that SMC have minimum valuesfor all four errors (ISE, ITSE, IAE, ITAE). Unmistakably, the performance indices values of SMC aresignificantly smaller than those of PID controller for each of the three reference signals.

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(a)

(b)

Figure 11. Output response of proportional-integral-derivative (PID) and sliding mode control (SMC)for multilevel steps input for (a) temperature and (b) humidity. The performance comparison for lessfrequent actuator moves.

5. Conclusions

In this article, PID (simple) controller and an optimal sliding mode control are evaluated for airhandling unit (AHU) operational control. The nonlinear model of AHU that is based on thermal massand heat transfer equations is presented. The desired values of the humidity ratio and temperature areachieved by manipulation of flow rate of air and water within the AHU. The results of both approachesare discussed with different perspectives, such as energy consumption and control goals.

Three feasible commands of the desired humidity ratio and temperature setpoints: (a) sequence ofsteps, (b) steps with ramp, and (c) integration of both are taken to study the tracking objectives. Withthe comparison of different tracking responses for SMC and PID approaches, the following conclusionsare drawn.

• Sliding mode control ensures robustness by tracking the setpoints perfectly (low overshoot andless time for convergence) in the presence of uncertainties, but PID shows high oscillation ascompared to SMC in tracking the setpoint objectives.

• Sliding mode based controller design for AHU contributes to lower energy consumption dueto less time for convergence and less overshoot contributing to lowering the air and the waterflow requirement.

• The oscillatory behavior of PID controller is disadvantageous for air and water flowactuating dampers.

This research can be further extended for HVAC in multizone buildings, where each zone canhave different setpoints of temperature and relative humidity.

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Acknowledgments: This work was financially aided by the Natural Science Foundation of China (Nos. 61640310,61603316, 61773323).

Author Contributions: All authors contributed to this work. Particularly, Awais Shah carried out most of theresearch work under the supervision of Deqing Huang.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. SMC Design

Formulation of control law u1 and u2 are given as

x2 = α2Ml − α1u1(x2 − wsa). (A1)

The tracking error and its derivative in sliding mode are defined, respectively:

e2 = x2 − x2r,

e2 = x2 − ˙x2r.(A2)

Then, consider the augmented system and we combine it with Equation (A1):

˙e20 = e2,

e2 = x2 − ˙x2r

= α2Ml − α1u1(x2 − wsa)− ˙x2r.

(A3)

Now, define the sliding surface in SMC

s = a1e20 + e2. (A4)

The derivative of sliding surface will be

s = a1 ˙e20 + e2

= a1e2 + α2Ml − α1u1(x2 − wsa)− ˙x2r.(A5)

The following Lyapunov candidate function is adopted for stability analysis

V =12

s2,

V = ss

= s(a1e2 + α2Ml − α1u1(x2 − wsa)− ˙x2r).

(A6)

Now, define the control law for the input u1 as

u1 =1

α1(x2 − wsa)[− ˙x2r + α2Ml + a1e2 + υ]. (A7)

By using Equation (5) to design control input u2

x1 = f1(x) + g1(x)u1,

x3 = f2(x)− g2(x)u2.(A8)

Energies 2017, 10, 1815 16 of 21

We calculate the tracking error and its derivative for u2 and consider augmented system as:

e1 = x1 − x1r,

e1 = x1 − ˙x1r,

e3 = x3 − x3r,

e3 = x3 − ˙x3r,

˙e10 = e1,

˙e30 = e3.

(A9)

Expand the values of x1 and x3, and we have

e1 = f1(x) + g1(x)u1 − ˙x1r,

e3 = f2(x)− f2(x)u2 − ˙x3r.(A10)

The sliding surface and its derivative are designed as the following:

s = ae10 + e3,

s = ae1 + e3.(A11)

After expanding the values of e1 and e3, we have

s = a[ f1 + g1u1 − ˙x1r] + f3(x)− g2(x)u2 + ˙x3r. (A12)

The Lyapunov candidate function V = 12 s2 is adopted for stability analysis, and then we have

V = ss

= s(a[ f1(x) + g1(x)u1 − ˙x1r] + f2(x)− g2(x)u2 + ˙x3r).(A13)

The control law for input u2 can be defined as:

u2 =1

g2(x)[a ˙x1r + ˙x3r + ag1(x)u1 + a f1(x) + f2(x) + υ]. (A14)

Appendix B. Performance Indices Graphs

(a) (b)

(c) (d)

Figure A1. Performance indices, (a) square of the error (ISE), (b) integral of time into squared error(ITSE), (c) integral absolute error (IAE) and (d) integral time absolute error (ITAE) graph for temperatureoutput of the first reference signal, sequence of steps.

Energies 2017, 10, 1815 17 of 21

(a) (b)

(c) (d)

Figure A2. Performance indices (a) ISE, (b) ITSE, (c) IAE, (d) ITAE graphs of controllers for humidityoutput of the first reference signal, sequence of steps.

(a) (b)

(c)

h

(d)

Figure A3. Performance indices (a) ISE, (b) ITSE, (c) IAE, (d) ITAE graph of controllers for temperatureoutput of the second reference signal, ramp with steps.

Energies 2017, 10, 1815 18 of 21

(a) (b)

(c) (d)

Figure A4. Performance indices (a) ISE, (b) ITSE, (c) IAE, (d) ITAE graph of controllers for humidityoutput of the second reference signal, ramp with steps.

(a) (b)

(c) (d)

Figure A5. Performance indices (a) ISE, (b) ITSE, (c) IAE, (d) ITAE graph of controllers for temperatureoutput of the third reference signal, combination of ramp with steps and multilevel steps.

Energies 2017, 10, 1815 19 of 21

(a) (b)

(c) (d)

Figure A6. Performance indices (a) ISE, (b) ITSE, (c) IAE, (d) ITAE graph of controllers for humidityoutput, combination of ramp with steps and multilevel steps.

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