Robust Sliding Mode Control of Air Handling Unit for Energy Efficiency Enhancement

Abstract

In order to achieve feasible and copacetic low energy consuming building, a robust and efficient air conditioning system is necessary. Since heating ventilation and air conditioning systems are nonlinear and temperature and humidity are coupled, application of conventional control is inappropriate. A multi-input multi-output nonlinear model is presented. The temperature and humidity 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 in the presence of uncertainties. A simple proportional-integral-derivative (PID) controller is used as a comparison template to highlight the efficiency of the proposed controller. To accomplish tracking targets, a variety of desired temperature and relative humidity commands (including ramp 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 makes SMC more energy efficient.

1. Introduction

With the progression in lifestyle, Heating, Ventilation and Air Conditioning (HVAC) systems are widely employed to provide better comfort and indoor air quality (IAQ). The important factor that needs to be handled in HVAC systems is energy loss. Around 50% of total energy consumed by a modern building is due to HVAC [1]; therefore, optimal control of HVAC system is crucial. Air handling unit (AHU), which regulates and conditions (warm or cool) airflow to thermal zones, is the main part of HVAC systems. For AHU design, a mathematical model of components is pressingly needed to analyze the system performance. Furthermore, it is very difficult to design an exact mathematical model because of its time-varying characteristics and complex nonlinear behavior [2]. The most common parameters of AHU are temperature and humidity. Airflow and water flow control is used inside 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, dynamic equations are achievable for AHU by combining mass balance equations with engineering models [3,4,5]. For the control of AHU, the sliding mode design is perceived as the most effective techniques to layout robust controllers for complex higher order nonlinear systems working under uncertainties. Controller configuration gives a deliberate way to deal with the issue of keeping up consistent performance and stability. It also has the probability of balancing out some nonlinear systems that are not stabilizable by continuous state input laws. Robustness is the system is on a sliding surface. Then, it creates minimum variations in parameters.

Researchers have carried out several investigations in recent years for efficient energy management of air conditioning system [6]. One control strategy of HVAC indoor temperature and humidity is maintained by proportional-integral-derivative (PID)-fuzzy controller in the presence of uncertainties [7]. Khan et al. presented multivariable controller based on adaptive fuzzy and genetic algorithm for AHU [8]. Wang et al. presented a control strategy for indoor environments by using a number of occupants and applying predictive control technique [9]. Prediction is done by video surveillance and concentration of room carbon dioxide. A multi-input multi-output (MIMO) controller is designed to enhance indoor air quality (IAQ) by changing air supply fan and compressor speed in a direct expansion (DX) air conditioning system [10]. Mba et al. presented prediction of indoor temperature 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 combination system of air handling units, the full-order observer is a tool that can be utilized for satisfactory calculation of the state variables. Design of robust optimal control for buildings based on chilled water systems and considering it with minimum life cycle cost and quantified uncertainties is presented in [13]. The proposed study upgrades the plan of chilled water pump frameworks with the uncertainities of configuration inputs. This proposed outline strategy limits the yearly aggregate cost under various control techniques. A physical statistical based approach, which considers distributed air conditioning control for commercial building, is discussed in [14]. Control designs are produced for zone cooling and indicated by inhabitance designs at zone levels, and a novel reaction model is created in this paper. The control designs show endeavors to improve cooling system planning on the 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 disturbance rejection, is discussed in [15]. In this study, a multivariable control technique in view of feedback linearization technique is executed. The air and water stream rates are the inputs to the controller and thermal space temperature and humidity are the outputs. Nevertheless, it is difficult to implement such type of approach because optimal realization controller is complex at run time, which is caused by inversion and posses of nonlinear state space model.

Temperature and humidity are decoupled for a hot and humid atmosphere in [16]. Three distinct techniques are planned in a hot and sticky atmosphere area for air handling units and decoupling assessments 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-cooling curl and a dry fundamental cooling loop to dehumidify natural air that enables the controller to deal with the coupling issue. $H_{\infty }$ controller is designed for AHU and its comparison is done with pole placement method in [17]. An observer and a regulator are intended for the estimation of state factors and unsettling influence dismissal, respectively.

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

Pre cooling coil is added in a VAV system with a residential load factor (RLF) in [20]. The double cooling 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 on better control of humidity only. Prediction of fan speed control based on wavelet decomposition and neural networks is discussed in [21]. Predictive control of heating and cooling of office building with cooperative fuzzy mode (CFMPC) is discussed in [22]. In [23], predictive control based on fuzzy-neural technique of heating buildings is proposed. Cascaded HVAC control of superheat temperature 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 utilization of the day-ahead price (DAP) and keeping in mind the end goal to foresee the energy utilization related with the HVAC is presented. The study presented in [26] models the HVAC system in TRNSYS software (v.16, Thermal Energy System Specialists, Madison, WI, USA), the principle components of which are a reference building, cooling tower, and a water chiller. The cooling tower display is approved utilizing test information in a pilot plant. In [27], a point by point enhancement of plan parameters of warmth wheels is performed with a specific end goal to amplify sensible viability and to limit pressure drop. The investigation is brought out through a one-dimensional lumped parameters heat wheel model, which settles heat and mass exchange conditions, and through suitable relationships to 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 been compared for the MIMO AHU dynamic model. Advantages and disadvantages of both strategies are discussed from different point of views such as (a) energy consumption, and (b) meeting control objectives. SMC is completely insensitive to external disturbances and parametric uncertainties during the sliding mode [28]. Since HVAC system is highly nonlinear and uncertainties based, SMC plays a vital role in HVAC control.

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

The AHU description and its dynamic model are discussed in Section 2. In Section 3, control laws are developed. Section 4 presents the simulation results achieved from the system. Finally, this article is 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 supply air fan, duct work, filters and air mixing dampers. In our study, we consider the cooling mode of the HVAC 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.

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 differential equations [29,30]:

$$\begin{array}{cc}\hfill \dot{T_{sa}}& =\frac{\dot{f_{ar}}}{V_{cu}}\left(T_{ia}-T_{sa}\right)+\frac{0.25\dot{f_{ar}}}{V_{cu}}(T_{oa}-T_{ia})-\frac{\dot{f_{ar}}{h}_s}{C_a{V}_{cu}}(0.25w_{oa}+0.75w_{ia}-w_{sa})-\dot{f_{wr}}\left(\frac{{\rho }_{wd}{C}_w\delta T_{cu}}{{\rho }_{ad}{C}_a{V}_{cu}}\right),\hfill \\ \hfill \dot{T_{ia}}& =\frac{1}{{\rho }_{ad}{C}_a{V}_{ia}}(\dot{H_l}-h_v\dot{M_l})+\frac{\dot{f_{ar}}{h}_v}{C_a{V}_{ia}}(w_{ia}-w_{sa})-\frac{\dot{f_{ar}}}{V_{ia}}(T_{ia}-T_{sa}),\hfill \\ \hfill \dot{w_{ia}}& =\frac{\dot{M_l}}{{\rho }_{ad}{V}_{ia}}-\frac{\dot{f_{ar}}}{V_{ia}}\left(w_{ia}-w_{sa}\right),\hfill \end{array}$$

(1)

where $T_{sa}/w_{sa},T_{ia}/w_{ia}$ and $T_{oa}/w_{oa}$ are the supply conditioned air temperature and humidity ratios, thermal zone air (indoor), and outdoor fresh air, respectively. $\delta T_{cu}$ is cooling unit temperature gradient, $\dot{f_{wr}}$, and $\dot{f_{ar}}$ are the cold water and air flow rates, respectively. $\dot{H_l},\dot{M_l}$ are heat and humidity load strengths, respectively. ${\rho }_{wd}/C_a,{\rho }_{ad}/C_a$ are the cold water, and air mass density/specific heat. $h_v,h_s$ are enthalpy of vaporization and saturated water (parameters of thermofluids and operating point vales are presented in

Table 1. Air handling unit thermo-fluid parameters.

Parameters of Thermofluids
$V_{ia}$	Thermal space volume	$w_{oa}$	Humidity ratio of outdoor fresh air
$V_{cu}$	Cooling unit volume	$w_{sa}$	humidity proportion of Supply air
${\rho }_{wd}$	Water mass density	$w_{ia}$	Thermal zone humidity ratio
${\rho }_{ad}$	Air mass density	$T_{oa}$	Outdoor fresh air temperature
$h_v$	Enthalpy of vapors	$T_{sa}$	conditioned air supply temperature
$h_s$	Enthalpy of saturated water	$T_{ia}$	Thermal zone air temperature
$c_w$	Specific heat of water	$\delta T_{cu}$	Cooling unit temperature gradient
$c_a$	Specific heat of air	${\dot{M}}_l$	Humidity source strength
${\dot{f}}_{wr}$	water flow rate	${\dot{H}}_l$	Heat load
${\dot{f}}_{ar}$	air flow rate

and

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

Operating Point Values
$V_{ia}=400$ m${}^3$	$w_{oa}=0.0082$ kg H${}_2$O/kg dry air
$V_{cu}=1$ m${}^3$	$w_{sa}=0.0080$ kg H${}_2$O/kg dry air
${\rho }_{wd}=1000$ kg/m${}^3$	$w_{ia}=0.00804$ kg H${}_2$O/kg dry air
${\rho }_{ad}=1.18$ kg/m${}^3$	$T_{oa}=32$${}^{\circ }$C
$h_v=2500$ kJ/kg	$T_{sa}=17$${}^{\circ }$C
$h_s=80$ kJ/kg	$T_{ia}=20$${}^{\circ }$C
$c_w=4180$ J/kg ${}^{\circ }$C	$\delta T_{cu}=6$${}^{\circ }$C
$c_a=1000$ J/kg ${}^{\circ }$C	${\dot{M}}_l=0.000115$ kg/s
${\dot{f}}_{wr}=0.0009$ m${}^3$/s	${\dot{H}}_l=20$ KW
${\dot{f}}_{ar}=2.6$ m${}^3$/s

).

Equation (1) formulates the AHU dynamics, humidity ratio ($w_{ia}$) of indoor air and temperature of indoor air and supply air ($T_{ia},Tsa$), in the form of differential equations. The following definitions are made to simplify Equation (1):

$$\begin{array}{cc}\hfill {\alpha }_1& =\frac{1}{V_{ia}},{\alpha }_2=\frac{1}{{\rho }_{ad}{V}_{ia}},{\alpha }_3=\frac{1}{V_{cu}},\hfill \\ \hfill {\beta }_1& =\frac{h_v}{C_a{V}_{ia}},{\beta }_2=\frac{{\rho }_{wd}{C}_a\delta T_{cu}}{{\rho }_{ad}{C}_a{V}_{cu}},\hfill \\ \hfill {\gamma }_1& =\frac{1}{{\rho }_{ad}{C}_{p}a{V}_{ia}},\gamma 2=\frac{h_s}{C_a{V}_{c}u}.\hfill \end{array}$$

(2)

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

$$\begin{array}{cc}\hfill u_1& =\dot{f_{ar}},u_2=\dot{f_{wr}},\hfill \\ \hfill y_1& =w_{ia},y_2=T_{ia},\hfill \\ \hfill x_1& =T_{ia},x_2=w_{ia},x_3=Tsa.\hfill \end{array}$$

(3)

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

$$\begin{array}{cc}\hfill \dot{x_1}& ={\gamma }_1(\dot{H_l}-h_s\dot{M_l})+{\beta }_1{u}_1(x_2-w_{sa})-{\alpha }_1{u}_1(x_1-x_3),\hfill \\ \hfill \dot{x_2}& ={\alpha }_2\dot{M_l}-{\alpha }_1{u}_1(x_2-w_{sa}),\hfill \\ \hfill \dot{x_3}& ={\alpha }_3{u}_1(x_1-x_3)+0.25{\alpha }_3{u}_1(T_{oa}-x_1)-{\gamma }_2{u}_1[0.25w_{oa}+0.75x_2-w_{sa}]-{\beta }_2{u}_2,\hfill \\ \hfill y_1& =w_{ia},\hfill \\ \hfill y_2& =T_{ia}.\hfill \end{array}$$

(4)

Simplifying the Equation (4) yields

$$\begin{array}{cc}\hfill \dot{x_1}& =f_1\left(x\right)+g_1\left(x\right)u_1,\hfill \\ \hfill \dot{x_2}& =f_2\left(x\right)+g_2\left(x\right)u_1,\hfill \\ \hfill \dot{x_3}& =f_3\left(x\right)+g_3\left(x\right)u_2,\hfill \end{array}$$

(5)

where

$$\begin{array}{cc}\hfill f_1\left(x\right)& ={\gamma }_1[\dot{H_l}-h_s\dot{M_l}],\hfill \\ \hfill g_1\left(x\right)& ={\beta }_1(x_2-w_{sa})-{\alpha }_1(x_1-x_3),\hfill \\ \hfill f_2\left(x\right)& ={\alpha }_2\dot{M_l},\hfill \\ \hfill g_2\left(x\right)& ={\alpha }_1(x_2-w_{sa}),\hfill \\ \hfill f_3\left(x\right)& ={\alpha }_3{u}_1(x_1-x_3)+0.25{\alpha }_3{u}_1(T_{oa}-x_1)-{\gamma }_2{u}_1[0.25w_{oa}+0.75x_2-w_{sa}],\hfill \\ \hfill g_3\left(x\right)& =-{\beta }_2.\hfill \end{array}$$

(6)

3. Controller Design

In this section, controller design for the nonlinear dynamic AHU is discussed. It is observed from 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 represents the block diagram of the general AHU feedback controller:

3.1. PID Controllers

PID controller is the most popular type of controllers utilized in industrial process. PID controller contains 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 simple feedback control compares a measured variable with a setpoint and generates an error. The output to track the setpoint is based on the error and input settings. The general form of PID controller can be presented as [31]:

$$\begin{array}{c}\hfill u\left(t\right)=k_p·e\left(t\right)+k_i·\int e\left(t\right)d\left(t\right)+k_d·\frac{de\left(t\right)}{dt}\end{array}$$

(7)

Tuning of PID Controllers

The Ziegler and Nichols method is used to select the values of $K_p$, $K_i$ and $K_d$ for the proposed study. The Ziegler–Nichols technique depends on tests executed on a set up control loop. The following steps are involved in selecting the gains [32]:

• Transform the PID controller into a P by adjustment of $K_i=\infty$ and $K_d=0$. At first, set pick up to $K_p=0$. Close the loop in programmed mode for the controller.
• Increase $K_p$ unless there are maintained motions in the signs in the control system. This $K_p$ esteem is signified a definitive gain, $K_{pu}$.
• Measure a specific period $P_u$ of the maintained oscillations.
• Figure the controller parameter esteems as per

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

  Controller	${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$
  proportional (P) controller	$0.5K_{pu}$	∞	0
  proportional-integral (PI) controller	$0.45K_{pu}$	$\frac{P_u}{2}$	0
  proportional-integral-derivative (PID) controller	$0.6K_{pu}$	$\frac{P_u}{2}$	$\frac{P_u}{8}=\frac{P_i}{4}$

  , and utilize these parameter esteems in the controller.

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 control system that is less sensitive to variations and disturbances in plant parameters. The feedback control law 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 organized strategy. The sliding mode is the specific working method of the framework, as it slides towards the predetermined limits of the control scheme. Furthermore, the geometrical locus, fundamentally comprising of the limits, is referred as the sliding surface of the framework. Figure 3 delineates an occurrence of the direction of a specific framework related to the SMC procedure. This scheme represents 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 $s=cx+\dot{x}=0$.

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 over linearization of the system. Then again, a sliding mode control doesn’t overlook nonlinearity.
• 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 addition the coveted reliable performance.
• The sliding mode control implementation on hardware is simple. In the microcontroller, it needs less numerical and computational calculations. The standard protocols are promptly perfect with it, for example, Modbus and the Ethernet/IP, RS-232.
• For the situations, where accuracy and stability are required, SMC requires altogether less maintenance 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 $s\left(t\right)$ to represent the presence of the displaying perturbations and additionally imprecision. The chattering phenomena is added as the outcome of the blemish of related control switching. Practically speaking, chattering is totally unwanted for the framework because it requires a unique control design. Other than that, this may present higher order frequency elements that were disregarded in the modelling scenario. For a nonlinear system, we have

$$\begin{array}{cc}\hfill \ddot{x}& =f\left(x\right)+g\left(x\right)u+d\left(t\right),\hfill \end{array}$$

(8)

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

$$\begin{array}{c}\hfill s\left(x\right)=A^T\left(x\right)=\sum _{i=1}^n{a}_i{x}_i=\sum _{i=1}^{n-1}{a}_i{x}_i+x_n,\end{array}$$

(9)

where x denotes a state vector and $x_i=x_i^{n-1},i=1,\dots ,n,A={[a_1\dots ,a_{n-1},n]}^T.$ If s(x) → 0, to ensure $x_i\to 0$, $a_1,a_2,\dots a_n$ are selected in such a way that $c^{n-1}+a_{n-1}{c}^{n-2}+\dots a_{2}c+a_1$, the polynomial is Hurwitz and c is a Laplace operator.

Define the sliding mode function as

$$\begin{array}{c}\hfill s\left(t\right)=je\left(t\right)+\dot{e}\left(t\right),\end{array}$$

(10)

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

$$\begin{array}{cc}\hfill e& =x_d-x_1,\hfill \\ \hfill \dot{e}& =\dot{x_d}-\dot{x_1},\hfill \end{array}$$

(11)

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

$$\begin{array}{cc}\hfill \dot{s}& =j\dot{e}+\ddot{e}\hfill \\ & =j(\dot{x_d}-\dot{x_1})+(\ddot{x_d}-\ddot{x})\hfill \\ & =j(\dot{x_d}-\dot{x_1})+(\ddot{x_d}-f-g.u-d).\hfill \end{array}$$

(12)

Define the exponential reaching law

$$\begin{array}{c}\hfill \dot{s}=-\zeta sgns-ms,\zeta ,m>0.\end{array}$$

(13)

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

$$\begin{array}{c}\hfill j(\dot{x_d}-\dot{x_1})+(\ddot{x_d}-f-g.u-d)=-\zeta sgns-ms.\end{array}$$

(14)

Now, the control law can be defined as

$$\begin{array}{c}\hfill u\left(t\right)=\frac{1}{g}\left(\zeta sgns\right)+ms+j(\dot{x_d}-\dot{x_1}+\ddot{x}-f-d).\end{array}$$

(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

$$\begin{array}{c}\hfill u\left(t\right)=\frac{1}{g}\left(\zeta sgns\right)+ms+j(\dot{x_d}-\dot{x_1}+\ddot{x}-f-Dsgns),\end{array}$$

(16)

where $d=-Dsgns$.

After substituting Equation (16) into Equation (12) and simplifying, the following expression is obtained

$$\begin{array}{c}\hfill \dot{s}=-\zeta sgns-ms-Dsgns-d.\end{array}$$

(17)

With further simplification, we get

$$\begin{array}{cc}\hfill s\dot{s}& =s(-\zeta sgns-ms-Dsgns-d)\hfill \\ & =-m{s}^2-\zeta log\left|s\right|-Dlog\left|s\right|-ds.\hfill \end{array}$$

(18)

The following Lyapunov candidate function is adopted for stability analysis

$$V=\frac{1}{2}{s}^2.$$

Therefore, we have

$$\begin{array}{c}\hfill \dot{V}=s\dot{s}\left(t\right)=-m{s}^2-\zeta log\left|s\right|-Dlog\left|s\right|-ds\le -2mV.\end{array}$$

(19)

Ultimately, the expression becomes [34].

$$\begin{array}{c}\hfill V\left(t\right)\le e^{-2mt}V\left(0\right).\end{array}$$

(20)

It is clear that sliding mode function exponentially goes to zero with m value. The control laws $u_1$ and $u_2$ are designed in a similar way. The detailed formulations of control laws are given in Appendix A:

$$\begin{array}{cc}\hfill u_1& =\frac{1}{{\alpha }_1(x_2-w_{sa})}[-\dot{x_{2r}}+{\alpha }_2\dot{M_l}+a_1{e}_2+\upsilon ],\hfill \\ \hfill u_2& =\frac{1}{{\beta }_2}[a\dot{x_{1r}}+\dot{x_{3r}}+ah\left(x\right)u_1+a{A}_1+g\left(x\right)+\upsilon ],\hfill \end{array}$$

(21)

where $\upsilon =\dot{V}=-\beta \left|s\right|$. We use it as negative discontinuous control to ensure stability.

4. Results

Three different setpoints including multilevel steps, steps and ramps, and their combination are used for investigation of tracking temperature and humidity ratios. The three random setpoints for comfort 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 $k_p$ term diminishes the rise time but builds the overshoot and decreases the steady state error. The $k_i$ term reduces the rise time and overshoot but enlarges the settling time. The $k_d$ term will have the impact of enhancing the stability of the system, decreasing the overshoot and improving the transient response. The output temperature response by SMC and PID on multilevel step input is presented 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, whereas PID controller lags in tracking the setpoint for both temperature and humidity. The desired setpoint (case (b) of Figure 4), multilevel steps and ramp for temperature and humidity tracking response are presented in Figure 7 and Figure 8, respectively, PID based controller shows more overshoot in both temperature and humidity setpoints, but response time for convergence is almost the same as those of SMC in temperature setpoint cases, whereas tracking of humidity setpoint for PID control is unsatisfactory. Figure 9 and Figure 10 present the temperature and humidity output (for the reference signal presented in case (c) of Figure 4), respectively. It is clear that SMC outperforms PID in overshoot and time response for convergence.

With respect to SMC responsiblity for AHUs in practice, the frequency of control move could be a critical factor for support of actuators and valves. The high frequency of control action can damage the valves and actuators, and it will also increase the maintenance cost. Figure 11 represents this performance 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 both temperature 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 performance index is a significant measure of system performance and is chosen to emphasize important system specifications. The system is the best control system, when settings are adapted to allow the index to reach the minimum value. Two types, (a) integral of the 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) are used to analyse the controller performance. The errors are calculated by the following equations:

$$\begin{array}{c}\hfill ISE={\int }_0^T{e}^2\left(t\right)dt,\\ \hfill ITSE={\int }_0^{T}t{e}^2\left(t\right)dt,\\ \hfill IAE={\int }_0^T\left|ϵ\right|dt,\\ \hfill ITAE={\int }_0^{T}t\left|ϵ\right|dt.\end{array}$$

(22)

To improve readability, the performance indices graphs are presented in Appendix B. Performance indices graph of controller output on multilevel steps reference for temperature and humidity are presented in Figure A1 and Figure A2, respectively. The numeric values of these indices are presented in

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	$5.5\times {10}^{14}$	614.9	$2.55\times {10}^{19}$	$6.63\times {10}^{20}$	49.28	0.203
SMC	0.00043	0.058	$2.79\times {10}^7$	167	$1.208\times {10}^{19}$	$2.51\times {10}^{20}$	5.959	0.028
	INPUT 2
PID	$2.38\times {10}^{-5}$	0.0113	$5.44\times {10}^3$	1.29	$7.42\times {10}^{16}$	$7.1\times {10}^{18}$	10.15	1.102
SMC	$8.8\times {10}^{-6}$	0.0036	8.57	0.159	$2.73\times {10}^{10}$	$4.57\times {10}^{15}$	8.5	0.159
	INPUT 3
PID	0.016	5.735	$7.6\times {10}^8$	$8.9\times {10}^5$	$1.114\times {10}^{18}$	$5.488\times 1020$	128.8	1.252
SMC	0.0018	0.0177	$5.5\times {10}^8$	$6.7\times {10}^5$	$4.51\times {10}^{17}$	$1.4\times 1020$	16.07	0.183

. SMC values of all four errors are extensively less as compared to PID. Similarly, Figure A3 and Figure A4 represents performance indices of PID and SMC for temperature and humidity on ramp 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 Figure A5 and Figure 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 values for all four errors (ISE, ITSE, IAE, ITAE). Unmistakably, the performance indices values of SMC are significantly smaller than those of PID controller for each of the three reference signals.

5. Conclusions

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

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

• Sliding mode control ensures robustness by tracking the setpoints perfectly (low overshoot and less time for convergence) in the presence of uncertainties, but PID shows high oscillation as compared to SMC in tracking the setpoint objectives.
• Sliding mode based controller design for AHU contributes to lower energy consumption due to less time for convergence and less overshoot contributing to lowering the air and the water flow requirement.
• The oscillatory behavior of PID controller is disadvantageous for air and water flow actuating dampers.

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