#### Abstract

An output constrained control with input delay is proposed for a central heating system. Due to the delay of signal transmission and valves opening time, an input delay is considered into the system and an auxiliary system is employed to handle this issue by converting the delayed input into a delay-free one. Moreover, to ensure the output supply water temperature within a limited range, Barrier Lyapunov algorithm is involved to achieve desired control accuracy. Finally, external disturbance and model uncertainty are incorporated into the dynamic system and neural networks (NN) are trained in an online fashion for the compensation. The stability of the control system is guaranteed through rigorous Lyapunov analysis and the excellent control performance over traditional PID control is demonstrated via numerical simulation study.

#### 1. Introduction

Currently, central heating system is playing an essential role to human’s daily life and the boast of economy in everywhere all over the world. Due to the development of technology, advanced and intelligent central heating system has been widely researched and deployed to the municipal service system. This kind of system has the advantages of energy saving and contributes a lot to the less-pollution environment. Typically, this system measures the indoor/outdoor temperature and other information to control the valve opening degree in primary pip network which indirectly controls the supply water temperature in secondary network and provides heating service to the users.

In literature, many researches have been done to estimate or calculate the heating load which subsequently can help to adjust the heating control scheme [1–3]. However, it is rare research which focuses on the automatic control of central heating system. The representative works that can be tracked include [4] in which an indirect connection of district heating system architecture is built to achieve an optimal control. In [5], a regional automatic control strategy is proposed to designate temperature monitor and valve on the user end for the heating quality enhancement. A load prediction based multiple constrained control is put forward in [6] to handle the accuracy of water temperature supply and input water volume constraints. However, none of the above mentioned research considers the input delay due to the signal transmission inside the system and the valve operating time. It brings additional challenges for the control design since the time-lag will easily cause the instability of the whole system. For linear system, many researches in the literature can be found to cope with this problem [7, 8]. But for the complicated nonlinear system, it becomes more difficult to design a proper control. In [9], a virtual observer is developed to act as an auxiliary system that can convert the input delay system into a nondelay one. The target control system has a multi-input and multioutput (MIMO) nonlinear structure. A predictor of an uncertain nonlinear system is developed for the compensation of delay in the input to upgrade the control performance of a PID control. In this paper, inspired by the aforementioned researches, we will convert the original input delay heating system into a delay-free one similar to the Artstein model [10] together with the output constrained control which will be introduced in the following contents. This design will enhance to the stability of the delay system via providing designed accumulated compensation. The excellent performance will be verified in the simulation section.

Apart from input delay, the accuracy of supply water temperature in the secondary network is also of significance to the heating quality. Therefore, in this work, the output tracking error constraints are novelly taken into consideration for the control design. Some of the existing researches offer good results such as the ones using artificial potential field [11], prescribed performance control [12, 13], model predictive control [14] as well as reference governor [15]. Additionally, Barrier Lyapunov Function (BLF) is introduced in [16–18]. This method requires less restrictive initial conditions and does not rely on explicit system solution. Due to its advantage, BLF is tentatively incorporated into the target heating system with some customized modifications to guarantee the limited tracking error of the supply water temperature. Finally, the external disturbance, model uncertainty error, and other unknown factors are modeled as a disturbance term complemented into the system dynamic model. Different from a robust control [11, 19], in this paper, we design an NN [20] observer to estimate the disturbance to achieve more accurate control and attenuate the noise. The overall system diagram of the proposed central heating control system is summarized in Figure 1.

The contributions of this paper are threefold:(i)Different from existing heating system control research, input delay due to the action of valve and signal transmission is novelly considered in the nonlinear system. The delay system is converted into a delay free one through an axillary updating system for the compensation.(ii)The input delay control is proposed in combination with output constraints that can regulate the tracking accuracy of the supply water temperature in the secondary loop with specifically designed BLF method.(iii)The external disturbance of the system and the model error are considered and handled with an NN approximator. Weights updating law is developed with rigorous mathematical verification. And the outstanding performance is demonstrated through simulation study.

The organization of this paper is as follows. Section 2 introduces the birdview of the overall system and dynamic model. The detailed mathematical deduction of the proposed control is presented in Section 3. Section 4 conducts a numerical simulation study to demonstrate the efficiency of the proposed scheme. Finally, some concluding remarks are reached in Section 5.

#### 2. System Overview and Dynamic Model

In this work, the target heat supply system is an indirectly connected central heating system with excellent efficiency and energy conservation capability. The whole picture of this system can be depicted as Figure 2 where primary and secondary network are connected together via heat exchanger for the purpose of heat transformation. The heat transformation process follows the heat balance principle. In the following contents, we focus on the secondary network since this loop influences directly the users’ well-being and heating quality. The mathematical model is put forward as [21]where represents design building space-heating load, denotes heat from radiator, and is the heat provided to the users through heat supply loop. , , , , , and stand for the heating index, building volume, scalar parameter, heat radiating area of the radiator, design flow of secondary network, and water specific heat capacity, respectively. is fixed as for hot water floor radiation heating. and denote indoor and outdoor temperature. and represent supply and return water temperature in secondary network. Following the heat balance principle, the mentioned three heat loads , , and are supposed to be equal to each other as presented in (1).

*Assumption 1. *Secondary network usually has 3-5 % heat loss. For simplicity purpose, in this work, the loss is neglected.

*Assumption 2. *In this paper, the daily real-time heating load and desired secondary network supply water temperature are known.

#### 3. NN Based Adaptive Control with Input and Output Constraints

##### 3.1. Preliminaries on NN Approximation

Radial Basis Function Neural Network (RBFNN) has been demonstrated to have outstanding function approximation and learning capabilities. If we have a continuous nonlinear function defined on a compact set , it can be approximated by RBFNN with following expression:where represents the input vector. stands for the adjustable weight matrix and is the number of neuron. is the basis function vector, where . are the centers of respective field and denotes Gaussian function width. Concretely, (5) can be described aswhere is the ideal weight and denotes the approximation error and is subject to , where should be a positive constant. The ideal weight matrix can be defined asIt should be noted that is only an “artificial" quantity for the purpose of analysis. In control design, the ideal values will be estimated by with developed updating law [20]. Moreover, NN approximation error indicates the minimum deviation between optimal approximation solution, i.e., and the unknown function . It has been proved that approximation with neural networks can theoretically have any expected accuracy if the neuron number is large enough; i.e., is able to be arbitrarily small if is sufficiently large [20].

##### 3.2. Adaptive Control with Multiple Constraints

###### 3.2.1. Mathematical Model Formulation

If we are already given the supply water temperature in secondary network, our main aim is to control the valve opening degree in primary network to guarantee the supply water temperature following the desired values. Concretely, the given temperature value acts as the desired signal symbol as . The transfer function in this heat exchange process can be expressed as [22]where denotes a scalar coefficient, and represent the constant inertial time, and is the lagging time. This mathematical model depicts the transformation from system input , i.e., opening degree of the valve in primary pipe network, to the system output , i.e., actual water temperature in secondary pipe network. For the purpose of control design, we convert the model above into a time-domain formulation.Equation (9) can be concisely written aswhere , , and . In order to extend the generalization of our control, we consider the following remarks and assumptions.

*Remark 3. *Even though the parameters , , and in [22] are assumed as constants, in practical application, it is apparent that the heat exchange process is time-varying. In other words, the aforementioned coefficients should be state-dependent and time-varying. In this regard, , , and can be more accurately considered as , , and .

*Remark 4. *In actual system, external disturbance as well as the model uncertainty should be considered in model (10). For simplicity purpose, they are merged into single term as .

Besides the remarks and assumption made above, one more variable is considered which is actually denotes the temperature variation in secondary loop. Thus, the system model of (10) can be modified as

*Assumption 5. *In many researches, only single primary/secondary pipe network is considered; i.e., aforementioned model is defined as an SISO system. In this paper, system is extended to be an MIMO system which can produce multiple water temperatures in secondary pipe network to satisfy different operating requirements.

*Assumption 6. *The inertia matrix is invertible and is bounded. The upper bound will be denoted as , where is a positive constant bound.

With Assumption 5, the input and output variables become a second order vector and . Additionally, since the disturbance term is going to be estimated with an NN observer, it is neglected temporarily. The resultant system model can be concisely expressed as follows. Herein, we slightly abuse the use of , to ,

###### 3.2.2. Adaptive Control Design without Disturbance

In this paper, the output constraints, i.e., the tracked secondary loop water temperature errors, will also be considered to regulate within a desired range. In order to handle the output constraint problem, Symmetry Barrier Lyapunov Function (SBLF) [23] is employed together with a backstepping approach for the control development.

*Step 1. *Denoteandwhere the desired trajectory satisfies . refers to stabilizing function. Select a positive definite and continuous SBLF candidate aswhere denotes the tracking error constraint such that should be satisfied through the control.

*Remark 7. *In practical application, the initial conditions of water temperature and temperature variation are the same with the desired values. Hence, should be satisfied.

Time derivative of givesDifferentiating with respect to time givesSubstituting (18) into (17), we obtainDesign the stabling function asSubstituteing (20) into (19), the following can be derived:

*Step 2. *Invoke an auxiliary state to cope with the input delay which has the expression following formulation:where is developed with the following adaptive law.where are positive coefficients. Multiplying both sides of (22) with and involving , the derivative of giveswhere without loss of generality is considered as a constant matrix since the inertial time constants change very slowly. The time-varying part can be added to the disturbance term . Moreover, is designed as follows with the consideration of Mean Value Theorem [24]:where the bounding function denotes a globally positive function. is defined as , where refers toAfter the introduction of auxiliary state , the input-delay system is transferred into a delay-free system as expressed in (24). In order to prove the stability of the resultant system, a quadratic form Lyapunov-Krasovskii candidate function can be designated as [25]Differentiate and incorporate (21), (22), (23), and (24); this yieldsThe control law can be developed as follows:Substituting (29) into (28) and invoking (25) and Assumption 6, we haveTo proceed to the following analysis, Young’s inequality is considered:where and denote vectors and represents positive constant. With this inequality, the term in (30) becomesSimilar property is held for other terms in (30). In addition, considering the requirement of , the following inequalities yield:For , the similar transformation can be derived. DefineInvoking (31), (32), (33), and (34), (30) can be deducted asCauchy-Schwarz inequality can help to regulate the upper bound of asMoreover, the following fact can be proven:Considering (36) and (37), (35) giveswhere and they satisfy − , , and and the tuning parameters are selected , , , , and . .

Lemma 8 (see [26, 27]). *Given bounded initial conditions, if there exists a continuous and positive definite Lyapunov function satisfying , such that , where , :, are class functions and , then the solution is uniformly bounded.*

*Remark 9. *Combining Lemma 8, Remark 7 and (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37), and (38), the Semiglobally Uniform Boundness (SGUB) of all the close-loop signals is validated with the involvement of input delay. Moreover, the tracking error can be constrained within .

###### 3.2.3. Control Design with System Disturbance

Due to the system parameter uncertainty and external disturbance, the dynamic model is augmented with a disturbance term in the following format.Under the disturbance case, the first step of the control design is totally similar to Step 1. And the whole verification before (23) remains the same; (24) will be adjusted as

*Assumption 10. *The disturbance considered in this paper is bounded such that a global approximator can be employed to estimate this signal.

In order to approximate the unknown disturbance, a RBF neural network is employed.Define as the estimated weights, optimal weights, and approximation error, respectively. stands for the input vector of the neural network. The specific expression on will be introduced in the simulation section. Develop the update law of the NN weights asControl input under this condition should be augmented intoThe control law in (43) will be guaranteed with SGUB of all the close-loop system signals.

*Proof. *Denote the optimal weights as . The Lyapunov candidate function in the presence of disturbance can be selected asConsidering the weight updating law (42), time derivative of becomesIncorporating the inequality becomeswhere and they satisfy , , , and , the tuning parameters are selected , , , , and . .

With similar deduction of Remark 9, the SGUB can still be guaranteed after the augmentation of NN control.

#### 4. Simulation Study

In this simulation, a numerical case study is conducted on a dual subnetwork heating system in the presence of output constraints and input delay.

##### 4.1. Constrained Control with Input Delay

In this subsection, the control performance of proposed control is investigated first. As mentioned above, the complicated nonlinear system model is employed and the mass matrix is separated and the major constant part is considered explicitly in the model. The specific details of the system are considered as , , , and the disturbance and model uncertainty error . The whole simulation time is set as 40s and the desired temperature trajectory of secondary network is defined as . This design tracking trajectory describes the increment of temperature in secondary network from 20.3/21.7°C to 22.49/22.8°C in 40s, respectively. The constant input delay is fixed at 2.1s which is quite challenging for the control system. The control output temperature constraints are . It should be noted that the output constraints can be selected in an arbitrary fashion; however, too narrow constraints will cause the system instability.

As for the control parameters, the main coefficients of , , , and are selected as , , , , respectively. In the neural networks, the learning rate and coefficients are = = 0.1, = = 0.01. After trial and error, the number of neural nodes are set as 32. The input includes , , , and . The corresponding centres are distributed within -5,5], -2,2], -1,1], and -2,2]. The control performance is presented in Figures 3–5.

**(a)**Temperature tracked of in subnetwork 1

**(b)**Temperature tracked of in subnetwork 2**(a) Tracking error of subnetwork 1**

**(b) Tracking error of subnetwork 2**

**(a) Control input of proposed control**

**(b) Temptation variation speed**

From the result, it can be observed that the overall control performance is excellent with proposed control in the presence of output constraints and input delay as well as disturbance and model uncertainty. In Figure 3, the control trajectory can follow the desired temperature trajectory well. In the tail few seconds of the simulation, the control error increases become during this period and the increment of desired temperature becomes faster. Even this, the proposed control can still successfully regulate the tracking error within the desired constraints as shown in Figure 4. The error goes up in Figure 4(a) and before crossing the constraint limit, the control regulates it to reduce the error still within the desired area, which verifies the efficiency of proposed control. Finally, Figure 5 shows the control input and the temperature variation speed, i.e., . The result reveals that all the close-loop signals are bounded under the proposed control scheme.

##### 4.2. Control Performance with PID

To further illustrate the advantages of proposed control, a comparative study using PID control is conducted under the same system settings with input delay and disturbance. Due to the involvement of these challenging factors, the stability of the PID control can hardly be achieved. After many tuning trials, the best performance that PID control can achieve is as Figures 6 and 7 show. The corresponding PID parameters are , , and .

**(a) PID tracking control of subnetwork1**

**(b) PID tracking control of subnetwork1**

**(a) PID tracking error**

**(b) PID control input**

Although all the close-loop signals including control error and control input are bounded, the errors are very large compared to the proposed control. It demonstrates that, under such challenging situation, traditional simple control cannot provide satisfactory performance and the proposed control manages to handle the issues efficiently.

#### 5. Conclusion

In this paper, a novel adaptive NN based constrained control has been proposed to a city central heating system in the presence of input delay. The valve opening degree in the primary loop is controlled such that the supply water temperature in secondary loop can follow the desired values. In order to handle output tracking error constraints problem, BLF method is involved with designed Lyapunov candidate function. The input delay due to the slow action of valves and system signal transmission is resolved with an auxiliary system which facilitates to convert the delay system into a delay-free one. Finally, the unknown external disturbance and model uncertainty are approximated through neural network with developed adaptation law. Thorough simulation study is carried out to demonstrate that the proposed control can outperform traditional control scheme in such operation conditions.

#### Data Availability

The simulation results data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work is supported by the Natural Science Guide Foundation of Liaoning Province under Project no. 20170540747.