Robust Nonlinear Control Design for the HVAC System Based on Adaptive Sliding Mode Control

HVAC systems are the major energy-consuming devices in buildings. The building section utilizes about 74% of the electricity consumption, and more than 56% of this utilization was supposed to the HVAC systems [1]. Any bettering can result in lowering demand during peak load periods of the day. While many studies have been directed on enhancement the design of equipment and building envelopes.

Data-driven HVAC can identify strategies for system clarification and enhancement which is termed system identification (SI) in the literature [2]. Data was used to create a mathematical-based HVAC system which examine input and output variables to identify and set system parameters [3].

The impact of Neighboring zones on temperature tracking and control effort because of heat transfer was proposed by conduction through separating surfaces where heat transfer impact was reduced through feedback linearization scheme [4].

A precise model of the building is essential for the design of a complicated HVAC system. Heat dynamics of a space can be represented by the zone model, the comfortable zone which is controlled by an air-handling unit (AHU) or an air-terminal device [5].

The temperature and humidity are the most common parameters of HVAC system. Which controlled through airflow and water flow control. The capability of variable air volume (VAV) in HVAC systems are 15–40% more energy efficient than constant volume HVAC systems [6]. The wavlet artificial based neural network with infinite impulse response filter was proposed by Jahedi and Ardehali [7] for the faster and accurate identification of system dynamics to control the temperature and humidity of thermal zone. The result showed that, the used control can improve energy efficiency when compared with the proportional derivative (PD) controller. Two control strategies (gain scheduling and feedback linearization) were applied to a nonlinear dynamic model of an AHU and analyzed the strength and weakness of each approach [8]. A dynamic extension algorithm is employed to HVAC system, which its modeling including CO₂ concentration [9]. The linear quadratic regulator (LQR) was applied to optimize an stabilize the system. The unscented Kalman filter relies on scaled unscented transformation (UT) is used in the Ref. [10] to specify a state estimation of AHU which is controlled through a linear quadratic regulator, and without linearization to reduce mistakes to keep scheme features. The state dependent Riccatti Equation with the pseudo linearization was proposed, which is used for designing observer and sub-optimal controller for the AHU and compared with the extended Kalman filter [11]. A fuzzy controller was proposed by Shah et al. [12] with distinct type and number of membership functions and investigated which shape can give desired performance with energy consumption this controller is compared with variety controllers including self-tuning adaptive fuzzy controller, LQR and nonlinear controllers. A neural fuzzy structure of a parameter self-tuning decoupled fuzzy neural PID controller was proposed by Ganchev et al. [13]. A nonlinear model predictive control approach according to an optimization function is used on iterative optimization with a finite horizon and it can generate a control signal regardless of a past error [14]. Robustness of a control system is crucial because of dynamic uncertainties and external disturbances [15].

Sliding-mode control (SMC) is known as appropriate strategy for its robustness against external disturbances and insensitivity to parameter variations under matching conditions during sliding mode [16]. The SMC needs a proper control law such trajectory moves toward the sliding surface and reaches in a finite time and stays on. Sliding mode control is introduced by Shah et al. [17]. The proposed controller performance compared with performance of PID controller to prove the advantage of using SMC to improving the desired tracking was shown in the paper [18]. But the large control gains that used with conventional SMC which cause undesired chattering while the control system is in the sliding mode which is the major disadvantage of SMC [19]. Chattering can usually be attenuated by choosing appropriate control gain according to a certain adaptation law and that reducing also the control effort.

In this work, a nonlinear controller is used for controlling the humidity ratio and a controller based on an adaptive sliding mode controller to control the temperature of the thermal zone. An adaptive sliding mode control methodology is proposed by Plestan et al. [20] which guarantees a real sliding mode. Adaptive sliding mode control is introduced for many applications like for heat exchanger as in the paper [21].

The structure of this article is as follows: Firstly, the problem statement and proposed controller design is introduced in section 2, while the HVAC system description and modeling is presented in section 3. SMC is presented in section 4. Then the controllers design is explained in section 5. The simulation results are discussed in section 6. Finally, section 7 contains the conclusion.

The single thermal zone HVAC system with VAV and Variable Water Flow capabilities in cooling mode is showed in Figure 1. It consists of the following components: a heat exchanger; a chiller, which provides chilled water to the heat exchanger; a circulating air fan; the thermal space; connecting ductwork; dampers; and mixing air components [23].

1.jpg

Figure 1. HVAC drawing view with single thermal zone in the VAV system

The operation process of air conditioning system on cooling mode is consist of five stages as clarified in Figure 1:

Fresh air is allowed to enter the system by $\ell \%, \ell<1$ and mixed with circulated air, which is $(1-\ell) \%$ the mixed air has temperature T_mix. After passing through the air purifier, it passes through the heat exchanger, which allows the heat to move from the cooled water in the cooling coil to the mixed air to conditioned it. This conditioned air which called (supply air) has temperature T_sa and W_sa. Conditioned air pushed into thermal zone by fan, which by changing its speed can control the amount of air rate to the thermal zone. This conditioned air maintains the temperature and humidity at a certain level in thermal zone T_ia and W_ia. Air comes out of the thermal zone, $\ell \%$ is disposed of, and the rest is recycled and mixed with fresh air in the above-mentioned proportions.

There are some considerations that must be taken into account [23]: ideal gas behavior, mixing should be perfect in the heat exchanger and the thermal zone, constant pressure process, negligible wall and thermal storage, neglect the loss of thermal between components, Neglect the effect of filter on flow rate of the air, negligible transient effects in the flow splitter and mixer, finally, the air and the heat exchanger are assumed to have some capacitance.

Depending on the laws of mass and heat transfer, the dynamic behavior of the HVAC system can be described by the following differential equations. are given by [11, 12, 14, 17].

$\dot{W}_{i a}=\frac{\dot{M}_l}{\rho_{a d} V_{i a}}-\frac{\dot{f}_{a r}}{V_{i a}}\left(W_{i a}-W_{s a}\right)$

$\dot{T}_{ ia }=\frac{1}{\rho_{a d} C_a V_{i a}}\left(\dot{H}_l-h_v \dot{M}_l\right)+\frac{h_v \dot{f}_{a r}}{C_a V_{i a}}\left(W_{i a}-W_{ sa }\right)-\frac{\dot{f}_{a r}}{V_{i a}}\left(T_{ ia }-T_{ sa }\right)$

$\dot{T}_{s a}=\frac{\dot{f}_{a r}}{V_{c u}}\left(T_{i a}-T_{s a}\right)+\frac{\ell \dot{f}_{a r}}{V_{h e}}\left(T_{o a}-T_{i a}\right)-\frac{\dot{f}_{a r} h_s}{C_a V_{c u}}\left(\ell W_{o a}+(1-\ell) W_{i a}-W_{s a}\right)-\frac{\rho_{w d} C_w \delta T_{c u}}{\rho_{a d} C_a V_{c u}} f_{w r}$                   (1)

The parameters, variables, state and control are described in Tables 1 and 2 [7].

Table 1. Air handling unit thermo-fluid parameters

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

The HVAC system can be written as a nonlinear state-space form, as follows:

$\dot{x}_1=f_1+g_1 u_1$

$\dot{x}_2=f_2$

$\dot{x}_3=f_3-g_2 u_2$                   (2)

where,

$f_1=a_1, g_1=-a_2 x_1+a_3$

$f_2=\left(b_2 x_1-a_2 x_2+a_2 x_3-b_3\right) u_1+b_1$

$f_3=\left(-0.75 c_2 x_1+0.75 c_1 x_2-c_1 x_3+c_3\right) u_1, \quad g_2=c_4$

$u_1=\dot{f}_{a r}, u_2=\dot{f}_{w r}, y_1=W_{i a}, y_2=T_{i a}$

$x_1=W_{i a}, x_2=T_{i a}, x_3=T_{s a}$

$a_1=\frac{1}{\rho_{a d} V i a} \dot{M}_l, a_2=\frac{1}{v_{i a}}, a_3=a_2 W_{s a}$

$b_1=\frac{1}{\rho_{a d} c_a v_{i a}}\left(\dot{H}_l-h_v \dot{M}_l\right), b_2=\frac{h_v}{c_a v_{i a}}$,

$b_3=b_2 * W_{s a}, c_1=\frac{1}{v_{c u}}, c_2=\frac{h_s}{c_a v_{c u}}$,

$c_3=0.25 c_1 T_{o a}-0.25 c_2 W_{o a}+c_2 W_{s a}$,

$c_4=\frac{\rho_{w d} C_w \delta T}{\rho_{a d} C_a V_{c u}}$

Remark (1): The HVAC control system as given in Eq. (2) is non-affine since the control inputs are positive quantities only.

In this section the controller design for both thermal zone humidity ratio and air temperature are proposed. But first we refer to the following remark related to the non-affine behavior of the HVAC system:

Remark (2): from Eq. (2), both the thermal zone humidity ratio and air temperature are increased if $u_1=0$, i.e., $x_I=W_{i a} \,\,\&\,\, x_2=T_{i a} \rightarrow \infty$, as $t \rightarrow \infty$.

where, $a_1, b_1>0$.

Remark (3): from Eq. (2), to decrease the thermal zone humidity ratio, u₁ must be greater than zero (u₁>0), also to decrease the thermal zone air temperature u₂ must be positive too (u₁>0 and u₂>0).

5.1 Design for the thermal zone humidity ratio control u₁

To design the controller u₁, first the error function for the state x₁ is defined as:

$e_1=x_1-x_{d 1}$                (6)

where, x_d1 is the desired humidity. Then the thermal zone humidity ratio dynamic becomes:

$\dot{e}_1=\left[-a_2\left(e_1+x_{d 1}\right)+a_3\right] u_1+a_1$                  (7)

where, a discontinuous control law was proposed for u₁ as follows:

$u_1=k_1\left\lceil e_1\right]_{+}$                 (8)

where, [ ]₊ is the positive function defined as follows [33]:

$[z]_{+}=\left\{\begin{array}{l}1 \text { if } z >0 \\ 0 \text { if } z <0\end{array}\right.$                   (9)

and, k₁ is a positive gain.

To select a suitable value of k₁, a candidate Lyapunov function is used as follows; let the candidate Lyapunov be defined as:

$V=\frac{1}{2} e_1^2$                  (10)

To ensure the finite time stability, the inequality $\dot{V}=e_1 \dot{e}_1<0$ must be satisfied. Since $u_1=0$ for $e_1 \leq 0$, then k₁ is determined for the case when e≥0 as shown in the following steps:

$\begin{aligned} \dot{V}= & e_1 \dot{e}=e\left\{\left[-a_2\left(e_1+x_{d 1}\right)+a_3\right] k_1+a_1\right\} \\ & =-e\left\{\left[a_2\left(e_1+x_{d 1}\right)-a_3\right] k_1-a_1\right\}\end{aligned}$            (11)

Therefore, $\dot{V}<0$ if the magnitude of the gain k₁ satisfies the following inequality.

$k_1>\frac{a_1}{a_2\left(e_1+x_{d 1}\right)-a_3}$                (12)

where, (a₂ (e₁+x_d1)-a₃)>0. Since $\dot{V}<0$, the error e₁ will goes to zero in finite time due to the discontinuous control law u₁.

Now when $e<0, \dot{V}$ is given by:

$\dot{V}=e_1 \dot{e}_1=e_1\left\{a_1\right\}=-\left|e_1\right| a_1<0, \forall e_1 \neq 0$                    (13)

Therefore e₁=0 is an attractive operating point, and as a result, the control law given in Eq. (8) will ensure getting the desired thermal zone humidity ratio in finite time.

Remark (4): using the proposed controller in Eq. (8), the steady-state error for e₁ can be made arbitrarily small by using a large value of k₁.

5.2 Design for the thermal zone air temperature control u₂

In this part of controller design, the ASMC theory was used to design the controller u₂ in order to bring the temperature to the desired value. As a first step we need to define the new error state as follows:

$e_2=x_2-x_{d 2}$                    (14)

where, x_d2 is the desired indoor temperature. The relative degree with respect to e₂ as the output, is two, so the input output model is given by:

$\dot{e}_2=\dot{x}_2=e_3$

$\dot{e}_3=F(x)-G(x) u_2$                  (15)

where, $F(x)=\left(b_2 \dot{x}_1-a_2 f_2+a_2 f_3\right) u_1+\left(b_2 x_1-a_2 x_2+a_2 x_3-b_3\right) \dot{u}_1$ and $G(x)=a_2 g_2 u_1$.

As in classical SMC design, we need first to define the sliding variable s such that the relative degree is one. Accordingly, let the sliding variable be defined as:

$s=\alpha e_2+e_3$              (16)

where, α>0. Then the sliding variable derivative is given by:

$\dot{s}=\alpha \dot{e}_2+\dot{e}_3=\alpha f_2+F(x)-G(x) u_2$                 (17)

where, G(x)>0 ∀x.

Remark (5): to obtain e₃, the function f₂ must be known exactly. Because f₂ contains uncertain parameters, a robust differentiator is required to obtain e₃ (the derivative of e₂). In this work, the ACSMD is used to estimate e₃ as presented in section (5.4).

Remark (6): according to Eq. (17), and by considering that u₂ is a positive quantity, then u₂ must follow the following rule:

$u_2= \begin{cases}+ ve & \text { if } s>0 \\ 0 & \text { if } s \leq 0\end{cases}$                   (18)

This remark can be easily proved as follows:

$s \dot{s}=s\left\{\alpha f_2+F(x)\right\}-s G(x) u_2$                       (19)

Therefore, the term -sG(x)u₂<0 if and only if s>0, where both G(x) and u₂ are positive quantities.

With respect to nominal and perturbation terms, Eq. (17) can be rewritten as follows:

$\dot{s}=\alpha f_{2 o}+F_o(x)-G_o(x) u_2+\delta$                        (20)

where, f_2o, F_o(x), and G_o(x) are the nominal parts of the f₂, F(x), and G(x) respectively, while δ is the perturbation term, which is given by:

$\delta=\alpha \Delta f_2+\Delta F(x)-\Delta G(x) u_2$                     (21)

Let us now propose the following control law for u₂:

$u_2=G_o(x)^{-1}\left(u_o+u_s\right)\left[u_1 s\right]_{+}, G_o(x)=a_2 g_2 u_1$                     (22)

where, u_o is the nominal control responsible for canceling the nominal terms and u_s is the discontinues control. The function of u_s is to eliminate the effect of the perturbation term and make the sliding manifold (s=0) attractive. As a result, the proposed controller in Eq. (22) will direct the state $\left(e_2, \dot{e}_2\right)$ towards the sliding manifold (s=0) and maintain it there for all future time. Note that u₂ is taken equal to zero if s≤0 and it will act on the HVAC system when u₁=0 because of G(x)=0.

To this end, the controller parts u_o and u_s are proposed here as:

$\left.\begin{array}{l}u_o=\alpha f_{2 o}+F_o(x) \\ u_s=k_2\end{array}\right\}$                     (23)

Next, we need to show that the sliding manifold is attractive (i.e., the sliding condition is satisfied) using the proposed control law given in Eqs. (22) and (23). Firstly, we must mention that the proposed control law for u₂ works for the case where u₁≠0 as stated in Remark (3).

For s>0, and according to Eq. (22), we have:

$s \dot{s}=s\left\{\alpha f_{2 o}+F_o(x)-G_o(x) u_2+\delta\right\}=s\left\{-u_s+\delta\right\}$                      (24)

Then from Eq. (23) and for k₂>|δ|,

$s \dot{s}=s\left\{-k_2+\delta\right\} \leq-s k_2+s|\delta|=-s\left\{k_2-|\delta|\right\}<0$                (25)

This proves the attractiveness of the sliding manifold when the state initiated in the region where s>0.

But for s<0, the sliding condition $s \dot{s}$ is give by:

$s \dot{s}=s\left\{\alpha f_2+F(x)\right\}=s\left\{\alpha f_2+\left(b_2 \dot{x}_1-a_2 f_2+a_2 f_3\right) u_1+\left(b_2 x_1-a_2 x_2+a_2 x_3-b_3\right) \dot{u}_1\right\}$                        (26)

so, $s \dot{S}<0$ if and only if $\dot{s}=\left\{\alpha f_2+\left(b_2 \dot{x}_1-a_2 f_2+a_2 f_3\right) u_1+\left(b_2 x_1-a_2 x_2+a_2 x_3-b_3\right) \dot{u}_1\right\}>0$. To prove that $\dot{s}>0$, it is noted that the a₂ is very small parameter (section 3), therefore, $s \dot{s}$ can be approximated by neglecting a₂f₂+a₂f₃, and the result is:

$\dot{s} \approx \alpha f_2+b_2 \dot{x}_1 u_1+\left(b_2 x_1-a_2 x_2+a_2 x_3-b_3\right) \dot{u}_1$                  (27)

In addition, further simplification can be made by taking into account that the humidity ratio is controlled independently from the rest of the system, where u₁ is a function of x₁ only. So that when x₁ reaches the steady state value, $\dot{s}$ has the following form:

$\begin{aligned} \dot{s} \approx \alpha f_2 & =\alpha\left\{\left(b_2 x_1-a_2 x_2+a_2 x_3-b_3\right) u_1+b_1\right\} \\ & \approx \alpha\left\{\left(b_2 x_{1 s s}-b_3\right) u_{1 s s}+b_1\right\} \\ & =\alpha\left\{\left(b_2 x_{1 s s} u_{1 s s}+b_1\right)-b_3 u_{1 s s}\right\}\end{aligned}$                        (28)

where, $\dot{x}_1=\dot{u}_1=0, x_{l s s}$ and $u_{l s s}$ are the desired humidity ratio and the steady state value for control u₁. Eventually, since $\left(b_2 x_{1 s s} u_{1 s s}+b_1\right)>b_3 u_{1 s s}$ then $\dot{s}>0$, and this end the proof of the attractiveness of the sliding manifold s=0. The temperature error state will asymptotically reach the origin and the thermal zone temperature will follow the desired value.

Remark (7): To solve the chattering problem in the SMC system an approximate function is used instead of the [z]₊ function to attenuate chattering like the following approximation:

$[z]_{+} \approx \frac{2}{\pi} \tan ^{-1}(\beta z)[z]_{+}$                     (29)

where, β>0 is a design parameter that is selected such that the steady-state error is adjusted to an acceptable value.

Therefore, using the approximation in Eq. (29), u₁ and u₂ becomes:

$u_1=\frac{2 k_1}{\pi} \tan ^{-1}\left(\beta_1 e_1\right)\left[e_1\right]_{+}, \beta_1>0$

$u_2=\frac{2}{\pi} G_o(x)^{-1}\left(u_o+u_s\right) \tan ^{-1}\left(\beta_2 e_1 s\right)\left[e_1 s\right]_{+}, \beta_2>0$                     (30)

Remark (8): u₁ is a differentiable function according to Eq. (30) $\forall e_1>0$. Accordingly, its derivative can be estimated using ACSMD.

5.3 Adaptive SMC for designing the control u₂

The gain k₂ in Eq. (23) will be replaced by a time-varying gain k₂(t). Its value is adapted according to the following law:

$\dot{\mu}=\rho *|s| *[s]_{+} * \operatorname{sign}(|s|-\epsilon)$                   (31)

and k₂(t) is chosen according to the following formula:

$k_2(t)=\left\{\begin{array}{cl}\mu & \text { if } 0<\mu<k_{2 \max } \\ k_{2 \max } & \text { if } \mu \geq k_{2 \max } \\ 0 & \text { if } \mu \leq 0\end{array}\right.$                 (32)

where, k_2max is the maximum gain.

$\left|v(t)-e_3(t)\right| \leq \frac{2}{\tau \lambda} \tan \left(\frac{\pi}{2 k_3}\left|e_3\right|_{\max }\right)$                    (34)

where, the initial conditions ν(t_o)=0, γ(t_o)=-e₂(t_o), and k₃>|e₃|.

The approximate classical sliding mode differentiator will estimate e₃≈v(t) with attenuated or without chattering and the maximum error does not exceed the bound given in Eq. (34).