Sampled data adaptive backstepping control of unsymmetrical air handling unit in heating ventilation air conditioning system

Abstract

The invention discloses sampling data self-adaptive backstepping control of an unsymmetrical air handling unit in a heating, ventilation and air conditioning system, which is used for an unsymmetrical fractional order air handling unit in a building heating, ventilation and air conditioning system so as to realize indoor temperature regulation. The method specifically comprises the following steps: constructing an unsymmetrical fractional order model for describing the dynamic behavior of the air handling unit system; designing a sampling data self-adaptive controller based on the constructed unsymmetrical fractional order model so as to realize self-adaptive fractional order control by adopting a sampling data backstepping technology; the invention aims to enhance indoor temperature control even in the presence of system uncertainty and external interference.

Description

Technical Field

[0001] The present invention relates to the technical field of heating, ventilation and air conditioning (HVAC) systems, and particularly to an adaptive backstepping control of sampled data for a disproportionate air handling unit in an HVAC system. Background Art

[0002] Today, the energy consumption of contemporary buildings accounts for about 40% of the global energy consumption, and more than half of which is consumed by heating, ventilation and air conditioning (HVAC) systems. As a basic component of HVAC systems, air handling units (AHUs) play a key role in providing air at a suitable temperature. Therefore, implementing appropriate control strategies for air handling units is necessary to ensure a comfortable indoor environment and improve the energy efficiency of buildings. An accurate mathematical model of the air handling unit system derived from building thermodynamics is crucial for designing an effective controller. Many researchers have developed dynamic models of air handling units for HVAC systems with integer orders. However, the air handling unit models established using integer nonlinear models usually need to sacrifice the accurate description of the actual complex air handling unit system to balance the complexity of the model. Otherwise, the control design based on the integer-order air handling unit model will be very complex, resulting in excessive computational overhead and energy costs. In contrast, fractional-order dynamics is an inherent characteristic of heat transfer problems, and non-local fractional-order operators can effectively capture real-world phenomena through their memory effects. This ability enables them to simplify and compactly represent complex real-world models. Therefore, an instinctive approach is to adopt a nonlinear fractional-order model to accurately describe the thermodynamics of air handling units in buildings. Compared with integer-order models, fractional-order models have fewer parameters. However, existing research related to fractional-order air handling units only focuses on modeling, and temperature control based on a concise and accurate fractional-order air handling unit model (which is particularly important for building energy-saving control) has not been developed.

[0003] In the past few decades, various heating, ventilation, and air conditioning (HVAC) control strategies have been developed based on integer-order mathematical models for building temperature regulation and energy efficiency improvement. To enhance energy efficiency, some scholars have implemented a reset control strategy based on outdoor temperature for the temperature control of air handling units (AHUs). Other scholars have developed a neural network-based controller to effectively control temperature and maintain a comfortable indoor environment. Among different AHU temperature control methods, proportional-integral-derivative (PID) control has become one of the most popular ones due to its simple algorithm and easy implementation. A PID control scheme for AHU temperature regulation was proposed using a self-tuning method with a radial basis function neural network. Meanwhile, some scholars have also explored a deep reinforcement learning algorithm to create a flexible PID controller with enhanced parameter optimization. However, due to the nonlinear and time-varying characteristics of the room temperature in air conditioners, improper setting of PID control parameters can lead to severe overshoot, steady-state error, and long adjustment time, resulting in low energy efficiency and increased operating costs. In addition, in practical situations, the system operating conditions may vary due to factors such as customized settings, environmental changes, and the presence of sensor faults, which can lead to uncertainties and unknown operating parameters. However, there are few control research results for AHUs with various uncertainties. Some scholars have studied an uncertain AHU model with time-varying unknown humidity source intensity and heat load, where fuzzy logic systems were used to approximate the uncertainties, resulting in only locally valid results. The adaptive backstepping control scheme, which is well-known for effectively handling nonlinear systems with uncertainties, was used to achieve effective temperature tracking when only the humidity source intensity was unknown. The backstepping-based adaptive dynamic surface control scheme was used for fractional-order nonlinear uncertain systems, where a semi-global result was established by using a neuro-fuzzy network system to approximate the fractional-order derivative of the composite function. However, there is currently no global adaptive backstepping control method that can address the unpredictable changes in humidity intensity, sensible heat load, outdoor temperature, and outdoor air humidity ratio while effectively regulating the temperature of integer-order AHUs, let alone precise fractional-order AHUs.

[0004] With the progress of digital computer technology, compared with the above-mentioned traditional continuous control schemes that require continuous transmission of control signals and system states, the sampled-data control scheme appears to be more advantageous and economical in practical applications because it requires fewer transmission and control resources, thereby improving the energy-saving efficiency. The adaptive iterative learning control strategy aims to regulate the temperature of the heating, ventilation, and air conditioning (HVAC) system based on a periodic sampling interval, thereby improving the practical application effect and control performance. Currently, there have been a large number of studies on the sampled-data control technology for integer-order systems. On the contrary, the study of sampled-data control in fractional-order systems is a relatively new research field. Existing schemes for sampled-data control in fractional-order systems strictly consider the hereditary and infinite memory characteristics of fractional calculus, but these studies mainly focus on linear systems and ignore the effects of uncertainties, disturbances, and nonlinearities, which actually exist in practical systems including air handling unit (AHU) systems. The stabilization control problem of a class of uncertain nonlinear fractional-order systems is studied by the sampled-data backstepping method, where periodic sampling is adopted, but the effects of the hereditary and infinite memory characteristics of fractional calculus still need to be further studied. On the other hand, due to the unique properties of fractional calculus and the inapplicability of the traditional chain rule, it is challenging to develop an adaptive sampled-data backstepping control method for nonlinear fractional-order systems. Summary of the Invention

[0005] To solve the problems existing in the prior art, the object of the present invention is to provide a sampled-data adaptive backstepping control for a disproportionate air handling unit in a heating, ventilation, and air conditioning system. The present invention aims to strengthen the indoor temperature control even in the presence of system uncertainties and external disturbances.

[0006] To achieve the above object, the technical solution adopted by the present invention is: A sampled-data adaptive backstepping control for a disproportionate air handling unit in a heating, ventilation, and air conditioning system, which is used for a disproportionate fractional-order air handling unit in a building heating, ventilation, and air conditioning system to achieve indoor temperature regulation, specifically including the following steps:

[0007] Step 1: Construct a disproportionate fractional-order model that describes the dynamic behavior of the air handling unit system;

[0008] Step 2: Based on the constructed disproportionate fractional-order model, design a sampled-data adaptive controller to achieve adaptive fractional-order control using the sampled-data backstepping technique.

[0009] As a further improvement of the present invention, the specific content of Step 1 is as follows:

[0010] The disproportionate fractional-order model that describes the dynamic behavior of the air handling unit system is as follows:

[0011]

[0012] where γ is the fractional order, T i , T s and T o are the indoor temperature, the supply air temperature and the outdoor temperature respectively, and △T h represents the temperature gradient in the heat exchanger; V i and V h represent the volumes of the indoor space and the heat exchanger respectively; h w and h v represent the enthalpies of saturated water and vaporized water respectively; C p and C pw represent the specific heat capacities of air and water respectively; ρ and ρ w are the air mass density and the water mass density respectively; H o represents the sensible heat load, M o is the moisture content; f represents the air flow rate; gm represents the chilled water flow rate; W i , W s , W o are the moisture contents of the indoor space, the supply air and the outdoor air respectively; μ e represents the air exchange rate;

[0013] Respectively represent T i and T s as the system states w1 and w2, and represent gm as the control input u a , and rewrite the fractional-order state-space form of the air handling unit system as:

[0014]

[0015] where, y a is the system output, and are known constants, and the continuous known functions Φ1(w1) and Φ2(w1, w2) are defined as and And A d and B d represent the time-varying unknown bounded disturbances of the system, which satisfy and In addition, the unknown disturbance A d is bounded by an unknown non-negative constant , and the 1-γ order fractional derivative of the unknown disturbance B d is bounded, that is where is an unknown non-negative constant.

[0016] As a further improvement of the present invention, the specific steps of step 2 are as follows:

[0017] Suppose the sampling data control signal will be updated at the sampling time t p , where p = 0, 1, 2, …; and t p ≥0;

[0018] Define the error variable as:

[0019] x1 = w1 - Tr(5)

[0020] x2 = w2 - a1(6)

[0021] where a1 represents the virtual control signal to be designed;

[0022] Define the Lyapunov candidate function as Then calculate its time derivative as:

[0023]

[0024] According to 's property, where η > 0, is an unknown non - negative upper bound of A d , we get:

[0025]

[0026] For t ∈ [tp , t p+1 ), design a1 as follows:

[0027]

[0028] where the design parameters p1 > 0, and respectively represent the estimated values of f and , then substitute (9) into (8) to get:

[0029]

[0030] where, and

[0031] Define another Lyapunov candidate function as where β > 0 and κ > 0, then its derivative is:

[0032]

[0033] For t ∈ [t p , t p+1 ), design the adaptation law as follows:

[0034]

[0035] where \(x_1(t\) p ) represents the value of \(x_1\) at the sampling time \(t\) p , and and Then, according to Equation (12), Equation (13), and Young's inequality, Equation (11) can be further expressed as:

[0036]

[0037] where \(b_1 > 1\) and \(c_1 > 1\);

[0038] Define the Lyapunov candidate function as and calculate its derivative as Based on Equation (6), we have From Equation (5) and Equation (9), \(a_1\) is a function of \(w_1\), Tr, and . Therefore:

[0039]

[0040] According to the semigroup property of the Caputo fractional derivative, for \(\gamma\in(0,1)\), the time derivative of \(w_2\) is expressed as:

[0041]

[0042] From (15) and (16), we get The calculation result of is:

[0043]

[0044] where, and Under the properties of and , where is an unknown non - negative constant, The time derivative of is calculated as:

[0045]

[0046] where, \(u\) c represents a virtual control signal;

[0047] are used to design the virtual control signal \(u\) p , \(t\) p+1 ) and the control signal \(u\) c based on the actual sampling data as follows: a as follows:

[0048]

[0049] and

[0050]

[0051] where p2 > 0, and respectively represent the estimated values of f, and ; then, according to Young's inequality, Equation (18) is further calculated as follows:

[0052]

[0053]

[0054] where and 0 < q1 < 2p2;

[0055] Define the Lyapunov candidate function Ω2 as where β1 > 0, κ1 > 0, and χ > 0, and design the adaptation law for t ∈ [t p , t p+1 ) as follows:

[0056]

[0057] where Therefore, according to Young's inequality and the designed adaptation laws (22)-(24), the time derivative of Ω2 is calculated as follows:

[0058]

[0059]

[0060] where b2 > 1, c2 > 1, and d1 > 1.

[0061] The beneficial effects of the present invention are:

[0062] The present invention develops a sampled-data adaptive backstepping control for a disproportionate air handling unit in a heating, ventilation, and air conditioning (HVAC) system for a fractional-order nonlinear air handling unit (AHUs) system, aiming to enhance indoor temperature control even in the presence of system uncertainties and external disturbances; the main innovations and contributions of the present invention are summarized as follows:

[0063] 1. To achieve efficient control using an accurate model, the present invention first attempts to solve the indoor temperature tracking control problem based on an accurate fractional-order air handling unit (AHUs) model to improve building energy efficiency;

[0064] 2. In the fractional-order air conditioner unit system studied in the present invention, the actually existing time-varying unknown sensible heat load, moisture content, outdoor temperature, and outdoor air humidity ratio are all taken into consideration. In order to handle the system uncertainties and external unknown disturbances and make the designed controller more applicable and more suitable for actual operating conditions, the adaptive backstepping temperature control strategy based on sampled data in the present invention ensures that the resulting closed-loop system has global stability, that is, the indoor temperature can perfectly track the target value while other closed-loop signals remain bounded.

[0065] 3. The present invention proposes an adaptive backstepping sampled-data control method for a non-linear uncertain mismatched fractional-order air conditioner unit and conducts a rigorous theoretical analysis by fully considering the unique characteristics of fractional-order calculus. Description of the Drawings

[0066] Figure 1 Schematic diagram of the air handling unit (AHUs) in the embodiment of the present invention;

[0067] Figure 2 Schematic diagram of the indoor space temperature and the fixed reference indoor temperature T under the sampled-data fractional-order (FO) backstepping control scheme and the traditional PID control scheme in the embodiment of the present invention; r of;

[0068] Figure 3 Schematic diagram of the integral of the square of the indoor temperature tracking error under the fractional-order (FO) control scheme and the traditional PID control scheme in the embodiment of the present invention;

[0069] Figure 4 Schematic diagram of the chilled water flow rate under the fractional-order (FO) control scheme and the traditional PID control scheme in the embodiment of the present invention;

[0070] Figure 5 Schematic diagram of the integral of the chilled water flow rate under the fractional-order (FO) control scheme and the traditional PID control scheme in the embodiment of the present invention;

[0071] Figure 6 Schematic diagram of the estimated value of the system uncertainty under the fractional-order (FO) control scheme in the embodiment of the present invention;

[0072] Figure 7 Schematic diagram of the indoor space temperature under the fractional-order (FO) control scheme and the traditional PID control scheme in the embodiment of the present invention, and the time-varying reference indoor temperature T; r2 of;

[0073] Figure 8 Schematic diagram of the integral of the square of the indoor temperature tracking error under the fractional-order (FO) control scheme and the traditional PID control scheme in the embodiment of the present invention under the condition of time-varying reference indoor temperature;

[0074] Figure 9 Schematic diagram of the estimated value of system uncertainty under the fractional-order (FO) control scheme of the embodiments of the present invention under time-varying reference indoor temperature conditions;

[0075] Figure 10 Schematic diagram of the chilled water flow rate under the fractional-order (FO) control scheme and the traditional PID control scheme of the embodiments of the present invention under time-varying reference indoor temperature conditions. Detailed implementation manners

[0076] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0077] Embodiment

[0078] This embodiment proposes a novel sampled-data adaptive backstepping control for a mismatched air handling unit in a heating, ventilation, and air conditioning (HVAC) system, which is used for a mismatched fractional-order air handling unit (AHU) in a building HVAC system to achieve indoor temperature regulation. This embodiment studies for the first time the control problem of a mismatched fractional-order air handling unit. Compared with the integer-order case, due to the concise and accurate system model form, this control method can reduce the computational cost of control implementation, thereby improving building energy efficiency. By strictly considering the infinite memory and genetic characteristics of the fractional-order system, this embodiment proposes a temperature control scheme that innovatively combines the sampled-data scheme with the backstepping technique to reduce control and transmission resources. The effectiveness of this scheme is proved based on Lyapunov stability analysis. Even in the case where the air handling unit system has time-varying uncertainties and external disturbances in practice, the indoor temperature can accurately track the target temperature, and at the same time all closed-loop signals remain globally bounded. Simulation studies verify the effectiveness of the proposed strategy and confirm the established results.

[0079] Symbol description: In this embodiment, ∥·∥ represents the Euclidean norm of a matrix or vector. The terms represent the set of positive real numbers, the set of real numbers, the set of positive integers, and the set of complex numbers, respectively. represents the function of the i-th derivative, where sup|X| represents the supremum of the absolute value of X.

[0080] The Caputo fractional-order derivative of the function is:

[0081]

[0082] where and For the sake of simplicity, in the remaining part of this embodiment, will be written as

[0083] The fractional integral of the function is:

[0084]

[0085] In this embodiment, the air handling unit (AHUs) system operates in a refrigeration mode, in which an ideal gas passes through a heat exchanger to fully mix fresh air and return air in a certain proportion, and in the case of no air leakage, the influence of wind speed change on the regional pressure is negligible. Figure 1 The studied air handling unit is shown, which consists of a supply fan and a return fan, a dehumidification / humidification unit, a heat exchanger, a filter, a chilled water piping system, a control valve, and a mixed air unit.

[0086] The asymmetric fractional order model describing the dynamic behavior of the air handling unit (AHUs) system is as follows:

[0087]

[0088] where the fractional order γ = 0.96, T i , T s and T o are the indoor temperature, the supply air temperature, and the outdoor temperature (in °C), respectively. △T h represents the temperature gradient in the heat exchanger (in °C); V i and V h represent the volumes of the indoor space and the heat exchanger (in m 3 ), respectively; h w and h v represent the enthalpies of saturated water and vaporized water (in kJ / kg), respectively; C p and C pw represent the specific heat capacities of air and water (in kJ / (kg·°C)), respectively; ρ and ρ w are the air mass density and the water mass density (in kg / m 3 ), respectively; H o represents the sensible heat load (in kW), M o is the moisture content (in kg / s); f represents the air flow rate (in m 3 / s); gm represents the chilled water flow rate (in m 3 / s); W i , W s , W o are the moisture contents of the indoor space, the supply air, and the outdoor air (in kgH2O / kg dry air), respectively; finally, μ e represents the air exchange rate, and its value is equal to 0.75.

[0089] In practical applications, the outdoor temperature T o , the sensible heat load H o , the moisture intensity M o and the moisture content W of the outdoor air o are usually continuous, time-varying, and their boundary ranges are unknown before designing the control scheme. Therefore, these parameters are regarded as continuous, unknown, time-varying bounded functions. In addition, this embodiment focuses on the case where the air velocity f is fixed, and due to sensor failures, this velocity cannot be measured. Therefore, f is an unknown system uncertainty factor, and in the subsequent proposed control strategy, this problem will be addressed by designing an adaptive law.

[0090] It can be seen from equation (3) that the dynamic model of the air handling unit (AHUs) system has an incommensurate fractional-order characteristic, where the derivative order of T i is equal to 1, while the derivative order of T s is the fraction γ. The objective of this embodiment is to construct an adaptive control law for the incommensurate fractional-order air handling unit (AHUs) system described by equation (3) based on the sampled-data backstepping control technique, so as to reasonably design the flow rate gm of the chilled water, such that the indoor temperature T i can track the given reference temperature T r well, while ensuring that other closed-loop signals are globally bounded.

[0091] Let T i and T s be represented as the system states w1 and w2 respectively, and let gm be represented as the control input u a . Then, according to equation (3), the fractional-order state-space form of the air handling unit (AHUs) system can be rewritten as:

[0092]

[0093] where y a is the system output, and are known constants, and the continuous known functions Φ1(w1) and Φ2(w1,w2) are defined as and And A d and B d represent the time-varying unknown bounded disturbances of the system, which satisfy and

[0094] Assumption 1: The unknown disturbance A d is bounded above by an unknown non-negative constant , and the unknown disturbance Bd The 1-γ order fractional derivative of is bounded, that is where is an unknown non-negative constant.

[0095] Assumption 2: The reference indoor temperature T r is known, smooth and bounded.

[0096] Since in reality, physical quantities are bounded with continuous changes, Assumption 1 is reasonable. In addition, Assumption 2 also conforms to the physical change phenomenon, and this assumption is required for backstepping tracking control whether it is integer-order or fractional-order.

[0097] Controller design and main results:

[0098] A new adaptive fractional-order control scheme using sampled-data backstepping technique:

[0099] A. Sampled-data adaptive controller design:

[0100] In this work, the sampled-data control signal will be updated at the sampling time t p where p = 0, 1, 2,... and t p ≥ 0.

[0101] First, the error variables are defined as:

[0102] x1 = w1 - T r (5)

[0103] x2 = w2 - a1 (6)

[0104] where a1 represents the virtual control signal to be designed.

[0105] Step 1: Define the Lyapunov candidate function as Then calculate its time derivative as

[0106]

[0107] According to (where η > 0), there is:

[0108]

[0109] Design a1 as follows for t ∈ [t p , t p+1 )

[0110]

[0111] where the design parameter p1 > 0, and respectively represent the estimated values of f and , and substituting (9) into (8) gives

[0112]

[0113]

[0114] where and

[0115] Define another Lyapunov candidate function as where β > 0 and κ > 0, then its derivative is:

[0116]

[0117] For t ∈ [t p , t p+1 ), design the adaptation law as follows:

[0118]

[0119] where x1(t p ) represents the value of x1 at the sampling time t p , and and Then, according to equations (12), (13) and Young's inequality, equation (11) can be further expressed as

[0120]

[0121] where b1 > 1 and c1 > 1.

[0122] Step 2: Define the Lyapunov candidate function as and calculate its derivative as Based on equation (6), we have From equations (5) and (9), it can be seen that a1 is a function of w1, T r , and , so we have:

[0123]

[0124] According to the semigroup property of the Caputo fractional derivative, for γ ∈ (0, 1), the time derivative of w2 can be expressed as:

[0125]

[0126] Therefore, from (15) and (16), it can be seen that The calculation result of

[0127]

[0128] where and Under the and properties, The time derivative of

[0129]

[0130]

[0131] where u c represents a virtual control signal.

[0132] Design the virtual control signal u p ,t p+1 ) and the control signal u c based on the actual sampling data as follows: a as follows:

[0133]

[0134] and

[0135]

[0136] where p2 > 0, and represent the estimated values of f, and respectively. Then, according to the Young's inequality, further calculate Equation (18) as follows:

[0137]

[0138]

[0139] where and 0 < q1 < 2p2.

[0140] Define the Lyapunov candidate function Ω2 as where β1 > 0, κ1 > 0, and χ > 0, and for t ∈ [t p , t p+1 ), design the adaptation law as follows:

[0141]

[0142] where Therefore, according to Young's inequality and the designed adaptation laws (22)-(24), the time derivative of Ω2 is calculated as follows

[0143]

[0144]

[0145] where b2 > 1, c2 > 1, and d1 > 1.

[0146] B. Stability analysis:

[0147] The main results of the sampled-data adaptive fractional-order control design strategy for mismatched air handling units (AHUs) can be summarized as follows:

[0148] Considering the closed-loop system that includes the uncertain mismatched fractional-order nonlinear air handling unit (AHU) (4) and the sampled-data adaptive control scheme (20) with the adaptation laws (12), (13), (22), (23), and (24), it can ensure the global uniform boundedness of all closed-loop signals, and the indoor temperature can track the reference temperature well, and the tracking error w1 - T r asymptotically converges to an adjustable small bound whose upper bound is always strictly included in the range where G3, G4, and Ω will be defined in the subsequent proof process.

[0149] Proof: According to Young's inequality, the following inequality can be obtained:

[0150]

[0151] For the term in equation (25) According to the definition of the Caputo fractional-order derivative, we have:

[0152]

[0153] where * represents the convolution operator. From equations (19) and (20), for t ∈ [t p , t p+1 ),

[0154]

[0155]

[0156] Since the rate of change of the physical value within the sampling interval is limited in practice, it can be inferred from (28) that is bounded. In addition, when γ ∈ (0, 1), t γ-1 tends to 0 as time increases, and when γ = 1, t γ-1 = 1. Therefore, from (27), it can be seen that (which is a function of x1, x2, and ) is bounded. Thus, combining (26a)-(26e), (25) becomes:

[0157]

[0158] where and Then, combining ι > 0 and ν > 0, for t ∈ [t p , t p+1 ), we get:

[0159]

[0160] Therefore, for t ∈ [tp, tp +1 ), (29) can be further expressed as:

[0161]

[0162] where:

[0163]

[0164] Define Then for t ∈ [t p , t p+1 ), we have Let the Lyapunov candidate function be And define the Lyapunov function as Ω = Ω2 + Ω I , then its derivative is calculated as:

[0165]

[0166] According to and Ω I 's definition, we get This means that

[0167]

[0168] Combined with this equation, we can obtain

[0169]

[0170]

[0171] The sampling interval satisfies and

[0172]

[0173] According to (34), when the result is Integrating both sides of (34) gives

[0174]

[0175] indicating that Ω is uniformly bounded. Therefore, x1, x2 and are bounded, which means that w1, w2, a1 and are bounded.

[0176] It can be seen from (19) that

[0177]

[0178] From the boundedness of and the fact that t -γ will monotonically decrease to zero as t → ∞, it can be inferred from equation (36) that m c is continuously bounded. Therefore, u c and u a are also bounded. In addition, it can be observed from equation (35) that for t ≥ 0, the tracking error x1 = w1 - T r is always bounded by a limit and since as t → ∞ This means that by reasonably choosing the design parameters and sampling interval, the indoor temperature can perfectly track the reference temperature with an adjustable small error bound, as shown.

[0179] It should be noted that if the value of γ in Equation (3) is equal to 1, the results derived in this embodiment will be transformed into the results in the traditional integer-order case, which indicates that the sampled-data adaptive backstepping control strategy proposed in this embodiment has universal applicability in various studied systems.

[0180] The following further illustrates this embodiment through simulation verification:

[0181] Without loss of generality, consider the case where the summer air handling unit (AHUs) is in the cooling mode. The parameters in Equation (3) for simulation are as follows: ρ = 1.19 kg / m 3 、ρ w = 1000 kg / m 3 , h v = 2528.59 kJ / kg, h w = 53 kJ / kg, C p = 1 kJ / (kg·°C), C pw = 4.18 kJ / (kg·°C), △T h = 8 °C, f e = 8.024 m 3 / s, V i = 165.56 m 3 , V h = 1.72044 m 3 、W s = 0.007 kgH2O / kg dry air, and W i = 0.0092 kgH2O / kg dry air.

[0182] A. Constant indoor reference temperature case:

[0183] The goal of simulation verification is to ensure that even in the presence of system uncertainties and unknown external disturbances, the indoor temperature y a = w1 can perfectly track the desired reference indoor temperature T r = 27 °C, while all other closed-loop signals are bounded. The initial values of the system variables and estimates are: T i (0) = 34 °C, T s (0) = 16 °C, m a (0) = m c (0) = 0 °C / s, and The sampling interval is selected as t p+1 -t p = 0.1 s, and the design parameters are selected as: β = 0.01, κ = 0.1, ζ1 = ζ2 = ζ3 = 0.51, p1 = 5, p2 = 10, η = 0.0005, β1 = 10 -3 , κ1 = 10-5 and χ = 0.1. In addition, considering the upper limit of the chilled water flow rate in practice, in the simulation study, the upper limit of the control input u a is set to 0.41 m 3 / s. The unknown time-varying M o , H o , T o and W o are: M o = 0.02 + 0.0002cos(0.05t) kg / s, H o = 20 + 0.2593cos(0.1t) + 20U(t - 15) kW, where U(t - X) is the unit step function at t = X s, T o = 33 + 0.05sin(0.2t) °C, and W o = 0.018 + 0.001cos(0.008t) kgH2O / kg dry air, which is in line with the actual situation. In addition, the uncertainty f is selected as f = 9 m 3 / s. However, this value is not known during the controller design.

[0184] To highlight the efficiency and advantages of the developed sampled-data adaptive fractional-order backstepping control scheme for asymmetric air handling units (AHUs), a comparative study was carried out by comparing it with the widely used traditional PID control strategy. For the case of a fixed reference indoor temperature, the control parameters of the PID controller are designed as follows: k P = 5, k I = 0, and k D = 1.

[0185] It can be seen from Figure 2 that even under the traditional PID control scheme, the indoor temperature T i can eventually track T r , but there will be obvious overshoot during the transient state, which will have a great negative impact on human comfort. On the other hand, under the proposed control strategy based on sampled-data fractional-order adaptive backstepping, the indoor temperature can quickly approach and stabilize at the desired temperature T r , and no overshoot will occur. Figure 3 shows that compared with the PID control method, under the control scheme designed in this embodiment, the accumulation of the indoor temperature tracking error is much smaller, which also proves the superiority of the proposed method. In addition, it can be observed from Figure 4 and Figure 6 that when the proposed control strategy is adopted, all closed-loop signals, including the control signal, can be guaranteed to be bounded. In addition, Figure 4 and Figure 5It is emphasized that, compared with the traditional PID controller, the proposed control scheme is effective and superior in reducing the control quantity and chattering phenomenon, highlighting the advantage of this method in saving control energy.

[0186] B. Time-varying indoor reference temperature condition:

[0187] With other simulation and design parameters remaining unchanged, the sensible heat load becomes H o2 = 20 + 0.2593cos(0.1t) + 20U(t - 15) + 20U(t - 40) kW, and the desired indoor temperature becomes a time-varying temperature, that is To achieve the tracking of the indoor temperature to the time-varying reference temperature, the PID controller parameters are redesigned as: k P2 = 5, k I2 = 0.5, and k D2 = 5.

[0188] From Figure 7 it can be observed that even if there are more mutations in the sensible heat load, the indoor temperature can immediately track the time-varying reference temperature under the strategy proposed in this embodiment, and the overshoot at the beginning of the transient process is negligible. However, the traditional PID control scheme cannot achieve such tracking control performance. Compared with the fractional-order backstepping adaptive control strategy of this embodiment, the time required for it to achieve indoor temperature tracking is about 10 times longer, and the transient overshoot is extremely large. Figure 8 This further confirms the effectiveness of the control scheme proposed in this embodiment, because the final cumulative indoor temperature tracking error under PID control is more than 1500 times that when using the fractional-order control scheme of this embodiment.

[0189] From Figure 9 it can be seen that when the indoor reference temperature changes with time, under the proposed sampled-data fractional-order backstepping control scheme, the closed-loop signal can still be guaranteed to be bounded. Figure 10 It shows that by adopting the control method studied in this embodiment, the flow rate of the chilled water will neither reach its actual upper limit nor cause chattering; while under the PID control law, compared with the case where the indoor reference temperature is constant, the flow rate will be saturated, and even more serious rapid switching will occur between fully closed and fully open, which may cause wear of the actual equipment and shorten its service life.

[0190] By comparing with the existing widely used PID control scheme, the effectiveness and efficiency of the developed sampled-data adaptive fractional-order backstepping control method for nonlinear uncertain non-commensurate fractional-order air handling units (AHUs) in this embodiment are confirmed, and the established theoretical results are elaborated.

[0191] In this embodiment, a new adaptive sampled-data backstepping temperature tracking control scheme is constructed for mismatched fractional-order nonlinear uncertain air handling units (AHUs) with system uncertainties and time-varying unknown external disturbances, and this scheme allows the sampling intervals to be non-uniform. By strictly considering the unique hereditary and infinite memory characteristics of fractional calculus, the proposed control strategy ensures the global stability of the closed-loop system and can accurately drive the indoor temperature to the desired reference level. Through simulation studies, including comparing the control performance of the studied control method with that of the traditional PID control method, the correctness and effectiveness of the theoretical results of this embodiment are verified.

[0192] The above-described embodiments merely represent the specific implementation manners of the present invention, and the description thereof is relatively specific and detailed, but it should not be construed as a limitation to the scope of the patent of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several variations and improvements can still be made, and these all belong to the protection scope of the present invention.

Claims

1. Adaptive backstepping control of sampled data for a disproportionate air handling unit in a heating, ventilation, and air conditioning system, characterized in that, An asymmetric fractional-order air handling unit for building heating, ventilation, and air conditioning systems to achieve indoor temperature regulation, specifically including the following steps: Step 1: Construct an asymmetric fractional-order model describing the dynamic behavior of the air handling unit system; Step 2: Based on the constructed asymmetric fractional-order model, design a sampled-data adaptive controller to achieve adaptive fractional-order control using the sampled-data backstepping technique.

2. Adaptive backstepping control of sampling data for the disproportionate air handling unit in the HVAC system according to claim 1, characterized in that, The specific content of Step 1 is as follows: The asymmetric fractional-order model describing the dynamic behavior of the air handling unit system is as follows: where γ is the fractional order, T i , T s and T o are the indoor temperature, the supply air temperature and the outdoor temperature respectively, and △T h represents the temperature gradient in the heat exchanger; V i and V h represent the volumes of the indoor space and the heat exchanger respectively; h w and h v represent the enthalpies of saturated water and vaporized water respectively; C p and C pw represent the specific heat capacities of air and water respectively; ρ and ρ w are the air mass density and the water mass density respectively; H o represents the sensible heat load, M o is the moisture content; f represents the air flow rate; gm represents the chilled water flow rate; W i , W s , W o are the moisture contents of the indoor space, the supply air and the outdoor air respectively; μ e represents the air exchange rate; Respectively represent T i and T s as system states w1 and w2, represent gm as control input u a , and re - represent the fractional - order state - space form of the air - handling unit system as: where y a is the system output, and are known constants, and the continuous known functions Φ1(w1) and Φ2(w1,w2) are defined as and and A d and B d represent the time-varying unknown bounded disturbances of the system, which satisfy and In addition, the unknown disturbance A d is bounded above by an unknown non-negative constant , and the 1-γ order fractional derivative of the unknown disturbance B d is bounded, that is where is an unknown non-negative constant.

3. The sampling data adaptive backstepping control of the disproportionate air handling unit in the HVAC system according to claim 2, characterized in that, The specific content of Step 2 is as follows: Let the sampling data control signal be updated at the sampling time t p where p = 0, 1, 2, …; and t p ≥ 0; Define the error variable as: x1 = w1 - T r (5) x2 = w2 - a1 (6) where a1 represents the virtual control signal to be designed; Define the Lyapunov candidate function as Then calculate its time derivative as follows: According to with the property that η > 0, being A d an unknown non - negative upper bound, we obtain: For \(t\in[t p ,t p+1 ), design \(a1\) as follows: where the design parameter p1 > 0, and respectively represent the estimated values of f and . Then substituting (9) into (8) gives: Among them, and Define another Lyapunov candidate function as where β > 0 and κ > 0, and its derivative is then:[ For \(t\in[t p ,t p+1 ), design the adaptation law as follows: where x1(t p ) represents the value of x1 at the sampling time t p , and and Then, according to Equation (12), Equation (13), and Young's inequality, Equation (11) can be further expressed as: where b1 > 1 and c1 > 1; Define the Lyapunov candidate function as and calculate its derivative as Based on Equation (6), we have From Equation (5) and Equation (9), we get that a1 is and a function of, so:[[]]END]] According to the semigroup property of the Caputo fractional derivative, for γ ∈ (0, 1), the time derivative of w2 is expressed as: From (15) and (16), we get The calculation result of is: Among them, and Under and properties, where is an unknown non - negative constant, The time derivative of Among them, u c represents a virtual control signal; respectively for t ∈ [t p , t p+1 ), design the virtual control signal u c and the control signal u based on the actual sampled data a as follows: and where p2 > 0, and represent the estimated values of f, and respectively; then, according to Young's inequality, Equation (18) is further calculated as follows: wherein, and 0 < q1 < 2p2; Define the Lyapunov candidate function Ω2 as where β1 > 0, κ1 > 0, and χ > 0, and for t ∈ [t p , t p+1 ) Design the adaptation law as follows: Among them Therefore, according to Young's inequality and the designed adaptation laws (22)-(24), the time derivative of Ω2 is calculated as follows: where b2 > 1, c2 > 1, and d1 > 1.

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