
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)02022-4
10.1016/j.isci.2024.110797
110797
Article
Enhanced energy conservation and response accuracy of a pneumatic control system
Lin Zhonglin 1
Wang Haitao 1
Zhang Xinglong 2
Liu Wenchao 1
Gan Jinyu 1
Huang Feng huangf@fzu.edu.cn
13∗
Zhang Tianhong 2
1 School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350108, China
2 Jiangsu Province Key Laboratory of Aerospace Power System, College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
∗ Corresponding author huangf@fzu.edu.cn
3 Lead contact

26 8 2024
20 9 2024
26 8 2024
27 9 11079723 5 2024
5 8 2024
20 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Summary

The energy consumption of pneumatic systems is occupying an increasingly considerable proportion in the industrial systems. However, due to the response characteristics of actuators, the pneumatic control system generally has a low energy utilization efficiency. How to improve the response accuracy of the pneumatic system while reducing energy consumption remains a key problem to be solved. In this paper, a three-voltage acceleration waveform and its generation method are proposed, and the acceleration circuit is designed. A multi-mode acceleration switching strategy and backstepping sliding mode controller (BSMC) are applied. The test results show that compared to the traditional methods, BSMC respectively saves 26.27% of the air consumption, as well as 32.35% of the valve group power consumption. It also achieves the lowest root-mean-square error (RMSE), of 4.8421 kPa. All the experiments prove that the controller proposed can effectively improve the energy utilization efficiency while maintaining high tracking precision.

Graphical abstract

Highlights

• A waveform generation circuit of three-voltage acceleration has been designed

• A multi-mode switching strategy has been proposed to enhance tracking accuracy

• The pressure tracking and energy-saving performance have been verified by trials

Engineering; Control Systems

Subject areas

Engineering
Control systems
Published: August 26, 2024
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pmcIntroduction

With the development of global industrialization, the demands for energy and environmental pollution have emerged as imperative issues that must be addressed. Pneumatic systems, characterized by low cost, pollution-free operation, and easy maintenance, play an increasingly significant role in various industries,1 including steel, automotive, semiconductor,2 biopharmaceuticals, textiles, and food. They have become one of the primary energy-consuming systems in modern industrial enterprises. The energy consumption of compressed air systems in countries like the United States and South Africa accounts for approximately 9% of their national total industrial electricity consumption, whereas in Europe, it constitutes around 10% of the overall industrial electricity usage. In the present scenario of escalating crude oil prices and pressing energy challenges, concerns have been raised about issues such as low efficiency and significant waste in pneumatic systems. Therefore, energy saving in pneumatic systems has become an important research topic for academic institutions and industrial companies. The compressed air industry possesses substantial energy-saving potential, capable of reducing current consumption by 40%–75%.3,4 This underscores the importance and urgency of developing more efficient pneumatic systems.

Energy-saving control algorithms for on-off valve

For pneumatic systems in industry, fast response, high energy efficiency, high accuracy, and high stability are required. The combination of a fixed volume chamber and servo valves or on-off valves for simulation system design has the advantages of simple structure and high reliability.5 The characteristic of the servo valve is that there is a linear relationship between the mass flow or pressure output and the control signal. The on-off valve has only two states of opening and closing, and its control accuracy is much lower than that of the servo valve.6 The simple structure of the on-off valve brings the advantages of low price and easy maintenance. The main problem that restricts the control accuracy of the on-off valve is its inevitable dead zone. The dead zone is the limited interval in which the output quantity does not have any perceptible change when the input signal of the on-off valve changes. The width of the dead zone is closely related to the switching frequency of the valve.7

In order to minimize the dead zone effect of the on-off valve, an optimized control algorithm can be applied for switching time calculation. Du et al.8 use a nonlinear dynamic optimization method combining finite element polynomial configurations and a reduced space sequential quadratic programming (SQP) algorithm. The objective is to compute the optimal switching time sequence for a bridge circuit valve based on four switching valves, which can significantly improve the energy efficiency of a pneumatic drive system. Ren et al.9 optimizes a fractional-order proportional-integral-derivative (FPID) controller for a servo system using an online multivariate multi-objective genetic algorithm (MMGA), and the experimental results showed better performance in terms of tracking accuracy and control energy consumption as compared to other existing methods. The optimized control algorithms are usually combined with pulse width modulation (PWM) techniques. There are two states of 0 and 1 in the PWM waveform, which correspond to the closing and opening states of the on-off valve. When the on-off valve under PWM control transmits the airflow to an actuator, it is equivalent to continuously sending independent airflow packets. Appropriate improvements to the PWM waveform can improve the performance of the valve. A modified differential PWM (M-D-PWM) that has a differential time in the switching process of an on-off valve is proposed in Shih and Ma.10 On-off valves will always cause considerable temperature increases when driven at high voltages, which will result in a noticeable rise in power consumption. Therefore, this subsequently influences the response precision of on-off valves and the energy utilization efficiency of the pneumatic system. A driving control scheme including a soft switch to reduce the energy losses during the switching process is proposed by Cai et al.11 In Su et al.,12 the frequency and duty cycle of the PWM waveform are optimized to reduce the power consumption of the solenoid valve coil while ensuring reliable pull-in of the spool. In Zhang et al. and Zhong et al.,13,14 a special PWM waveform including positive and negative voltage is proposed. While reducing valve opening and closing time, it also brings lower temperature rise and energy consumption for the on-off valve. In Zhao et al.,15 an optimal boost voltage of the on-off valve is found to reach the highest energy utilization efficiency. Seslija et al.16 adopts a pneumatic actuator system that combines pulse width modulation and bypass valve control, achieving a 30% energy saving.

The previous studies are based on the characteristics of the on-off valve itself, but no in-depth research has been carried out on the temperature and power consumption of the on-off valve in the process of the pressure, force, or position servo control. The existing on-off valve control system has the problems of inefficiency and high energy consumption, which seriously restricts the performance improvement of the pneumatic system. It is of great academic and practical application value to study how to improve the energy efficiency and response accuracy of pneumatic systems by optimized control algorithms.

Industry-leading innovations on pneumatic servo control algorithms

In pneumatic servo systems where multiple valves exist, the switching strategy of the valves is critical, which can have an impact on the system control accuracy. For example, in a servo system that uses four switching valves to adjust the cylinder outlet rod position, there are sixteen different combinations of switching valve states, of which only seven are considered to be practically useful.17 Jiang et al.18 proposed a control strategy by establishing a mathematical model for a unique type of internal combustion engine circuit and employing SQP-based optimization methods. This strategy effectively reduces energy consumption during the working stroke of the cylinder and achieves a savings of over 50% in air consumption. A nonlinear dynamic optimization model with air consumption as the objective function was established by Du et al.19 The simultaneous configuration method was employed to obtain the switch time sequence, introducing the concept of optimal performance evaluation. A thorough analysis of the energy-saving performance and stability of the bridge-type pneumatic system was conducted. Additionally, a digital control valve based on a bridge circuit was developed, effectively addressing the issue of low energy efficiency in traditional circuit-controlled pneumatic systems. A reasonable switching strategy can effectively reduce system energy consumption and improve system tracking accuracy.

In order to apply the algorithm to a pressure servo system, an error-based controller is also required. The most commonly used algorithm in industrial servo control is proportional-integral-differential (PID) control, which has the advantages of simple principles and is easy to implement. However, the actual effect of PID control is not ideal in situations where the system is highly nonlinear. In past research, sliding mode controller (SMC) has achieved good performance in pneumatic servo systems.20 One of the drawbacks of SMC is the chattering phenomenon. In order to overcome this problem, the backstepping sliding mode controller (BSMC) can be used. In Carbonell et al. and Lu et al.,21,22 BSMC is applied in the air motor and pneumatic muscle control systems. Compared with other methods, the BSMC achieves less chattering, and the final control accuracy is higher. The importance of valve switching strategy in multi-valve pneumatic servo systems is addressed. This paper aims to design a reasonable on-off switching strategy to optimize the system performance and explore the potential application of control algorithms such as the BSMC to improve energy efficiency and accuracy.

This paper focuses on improving system accuracy and reducing energy consumption at low cost. Building on our previous work,23 which utilized a commercial acceleration circuit board for on-off valves, we have designed a three-voltage acceleration method along with universal acceleration circuit board for waveform generation. The proposed multi-mode switching strategy is designed for pneumatic systems composed of on-off valve groups. Compared to the traditional five-mode24 and three-mode strategy, the proposed strategy enables more accurate control of system output and input. By incorporating the BSMC algorithm, this system significantly enhances precision and energy efficiency at both on-off valves and the overall system, compared to traditional methods such as PID25 and SMC.20

The innovations and main research contents of the paper are as follows:• Based on the modeling and simulation analysis of the on-off valve, a three-voltage acceleration waveform and its generating circuit are designed to reduce the dead zone effect and lower the energy consumption.

• A multi-mode switching strategy is designed to address the relatively poor tracking accuracy of the traditional three-mode and five-mode switching strategies. This approach refines system modes, thereby enhancing system tracking accuracy.

• The pressure tracking and energy-saving performance are intended to be improved by integrating the three-voltage acceleration waveform, multi-mode switching strategy, and BSMC algorithm.

The outline of the paper is as follows: the second half of the introduction presents the modeling of the on-off valves and air chambers, along with the methods for accelerated waveform generation, the design and implementation of the accelerated circuit, and the multi-mode accelerated switching strategy. This section also delves into the design of the BSMC algorithm and the proof of system stability. The results section discusses the hardware and software setup for the pneumatic servo system, followed by the experimental results. Finally, the discussion and limitations of the study sections are presented. A more detailed and complementary explanation of the three-voltage acceleration waveform and the multi-mode switching strategy is provided in the STAR★Methods section.

Mathematical modeling of the pneumatic system

A rodless cylinder with two air holes is selected as the simulation air chamber. Two on-off valves are chosen to regulate its pressure, and the basic structure of the pneumatic system is shown in Figure 1. The left on-off valve is configured as the inlet valve, whereas the right one is configured as the outlet valve. Before the modeling, the following system assumptions are made: the air in the system is ideal; the pressure and temperature in the air chamber are homogeneous; the air leakage in the pneumatic system is negligible. The pressure of the air supply and the atmosphere is constant. The thermodynamic process is adiabatic.Figure 1 Basic structure of the pneumatic system

Modeling of the on-off valve

The type of the chosen on-off valve shown in Figure 1 is direct-acting, two-position, and two-way. Its response time is 5 ms for opening and 2 ms for closing. The on-off valve modeling is composed of four subsystems: the electrical, magnetic, mechanical, and fluid subsystems.26 According to the equations in McCloy et al.,27 the mass flow rate of the gas as it flows through the valve orifice is defined as(Equation 1) dm(Pi,Po)dt=m˙(Pi,Po)={0.0405CdPiST,Pc≤PcrCdPiST2γR(γ−1)((Pc)2γ−(Pc)γ+1γ),Pc>Pcr

where Pi is the inlet pressure, Po is the outlet pressure, Cd is the valve discharge coefficient, S is the effective cross-sectional area of the valve orifice, T is the air temperature, R is the ideal gas constant, m is the gas mass, Pc is the ratio of outlet pressure to inlet pressure, γ=1.2 is the specific heat ratio of air, and Pc=0.528 is the critical pressure ratio of outlet pressure to inlet pressure. When Pc≤Pcr, the gas mass flow rate is linear with the inlet pressure, and the gas is in the sonic state; when Pc>Pcr, the gas mass flow rate is nonlinear with the inlet pressure and the outlet pressure, and the gas is flowing at subsonic speed.

During the opening and closing of an on-off valve, the current is generated with a hysteresis when the coil is energized or deenergized due to the inductance. The number of turns of the on-off valve coil used in this paper is 450, and the coil resistance is about 4.50 Ω. The mathematical model of the valve coil is represented as(Equation 2) U=RI+LdIdt

where U is the excitation voltage, R is the equivalent resistance, I is the coil current, and L is the equivalent inductance. The instantaneous current is expressed as(Equation 3) I=Ii+(UR−Ii)(1−e−tRL)

where Ii is the initial coil current. According to Equations 2 and 3, the hysteresis time can be expressed as(Equation 4) td=LRlnU−IiRU−IoR

where td is the hysteresis time and Io is the current required to open the valve. The valve opening and closing hysteresis time is expressed as(Equation 5) tdo=L1RlnU−IiRU−IoR

(Equation 6) tdc=L2RlnU−IiRU−IcR

where tdo is the opening hysteresis time, tdc is the closing hysteresis time, Ic is the current required to close the valve, L1 is the equivalent inductance at the initial position at the opening stage, and L2 is the equivalent inductance at the initial position at the closing stage.

Modeling of the air chamber

The air chamber used in this paper has only one inlet hole and one exhaust hole, according to the ideal gas law:(Equation 7) PV=mRT

where P is the pressure inside the air chamber, and V is the volume of the air chamber. By deriving Equation 7, the following equation can be obtained:(Equation 8) m˙=ddt(PVRT)

Equation 8 can also be expressed as(Equation 9) m˙i−m˙o=VRTP˙

where m˙i is the mass flow rate of the gas flowing into the air chamber, and m˙o is the mass flow rate of the gas flowing out of the air chamber. According to the law of energy conservation, it can be obtained that(Equation 10) Qi−Qo+γCv(m˙iTi−m˙oT)−W˙=U˙

where Qi is the input heat, Qo is the output heat, Cv is the specific heat capacity for constant volume, Ti is the temperature of the inflowing gas, W˙ is the rate of work change, and U˙ is the rate of internal energy change. Then W˙ and U˙ can be expressed as(Equation 11) U˙=ddt(CvmT)=1γ−1ddt(PV)=Vγ−1P˙

(Equation 12) W˙=PV˙=0

Substituting Equations 11 and 12 into 10, the following equation can be obtained:(Equation 13) Qi−Qo+Rγγ−1(m˙iTi−m˙oT)=Vγ−1P˙

Since assuming the adiabatic process, Qi–Qo=0 and Ti=T, then Equation 13 can be rewritten as(Equation 14) P˙=γRTV(m˙i−m˙o)=γRTVm˙(Pi,Po)

Equation 14 shows that the pressure inside the air chamber is proportional to the mass flow rate into and out of the air chamber.

Acceleration waveform generation and multi-mode switching strategy

A three-voltage acceleration waveform generation method and a multi-mode switching strategy are proposed to decrease the influence caused by the dead zone of the on-off valve. The main principle of acceleration waveform is to increase the voltage and power of the on-off valve during the opening period. During the mode switching, the opening and closing states of the on-off valve are chosen using the system error and other factors.

Acceleration waveform generation method

The parameters of the commercially available on-off valves are established following factory assembly. However, the response characteristics can be enhanced by tuning the driving waveform. The on-off valve model chosen for this paper is the MX821.103C224 from the Italian manufacturer Matrix. It has a flow rate of 100 L/min, a single input and output, a pressure range of 0–8 bar, and is powered by a 24 V voltage. The equations in preceding Section are used to construct the simulation model in MATLAB/Simulink, which simulates the response characteristics of this on-off valve model. As shown in Figure 2, the current and spool position response are reproduced using a PWM driving waveform with an amplitude of 24 V, a duty cycle of 50%, and a frequency of 20 Hz. The on-off valve has an opening time of 5 ms, a closing time of 2 ms, and a maximum working frequency of 142.86 Hz. The three-voltage acceleration waveform designed for the valve is shown in Figure 3. In STAR★Methods, there is a more detailed explanation of the three-voltage acceleration waveform.Figure 2 Valve response characteristics under the action of traditional PWM waveform

Figure 3 Valve response characteristics under the action of the three-voltage acceleration waveform

Acceleration circuit design and implement

Figure 4 illustrates the conventional drive circuit of the on-off valve. The power N-type metal-oxide-semiconductor field-effect transistor (MOSFET), model IRF640N, serves as the primary component. The control signal is 0 or 5 V and can be in the PWM waveform. This signal is amplified by the power N-MOSFET to switch the valve on and off.Figure 4 Drive circuit of the traditional PWM waveform

In order to generate the acceleration waveform shown in Figure 3, the conventional drive circuit cannot be used directly, and the acceleration circuit needs to be redesigned. The acceleration circuit of the three-voltage acceleration waveform is shown in Figure 5, which includes schematic diagram and three-dimensional view of the printed circuit board (PCB). The AD5726 chip is chosen as the digital-to-analog converter, and the OPA548 chip is chosen as the operational amplifier to fulfill the output range of the voltage and the driving requirements of the valve. The input signal of AD5726 follows Serial Peripheral Interface (SPI) protocol, the signal amplification of OPA548 is set to 3, and four OPA548 are used to provide four channels of outputs.Figure 5 Acceleration circuit of the three-voltage acceleration waveform

(A) Circuit schematic.

(B) Three-dimensional view of the PCB.

(C) Real PCB.

The designed acceleration circuit is controlled by SPI signals, and the FPGA is used for SPI signal generation. Figure 6 shows the control system consisting of the acceleration circuit and the FPGA module. It comprises a personal computer (PC), an embedded controller National Instruments (NI) CompactRIO (cRIO), a digital output card, and an acceleration circuit. The NI cRIO-9054 is an industrial controller with real-time (RT) and FPGA modules. The NI 9401 is a C-series card for this embedded controller and can output digital voltage signals.Figure 6 Acceleration circuit works with the FPGA module

The PC is utilized to create a LabVIEW application that communicates with the NI cRIO-9054 over Ethernet for data exchange. By using the FPGA driver code as shown in Figure 7A, SPI communication with the AD5726 on the acceleration circuit is possible, and the SPI timing is shown in Figure 7B.Figure 7 SPI driver in NI FPGA

(A) NI FPGA code.

(B) SPI timing diagram.

An open-loop inflation experiment is performed to test the acceleration waveform and circuit. The experiment is shown in Figure 8, where only one on-off valve was used to inflate the air chamber and the source pressure was set to 0.3 MPa. Three types of driving methods are compared, the first one is the Matrix official acceleration board model HSDB 990.012 with PWM waveform, the second one is the traditional drive circuit with PWM waveform, and the third one is our acceleration circuit with three-voltage acceleration waveform. The frequency and duty cycle of the PWM waveform and three-voltage acceleration waveform are all set to 20 Hz and 50%. In Figure 8A, the use of accelerated waveforms allows the pressure to reach the set value faster, indicating that the use of accelerated waveforms at the same moment allows the switching valve to pass more air, which corresponds to Figure 8C and improves the efficiency of the use of the on-off valve. In Figure 8B, which represents the energy consumed by the Charging valve, it can be concluded that the accelerated waveform uses the least energy to drive the switching valve. The combined analysis yields the lowest open-loop inflation time (5.603 s) and valve energy consumption (about 12.1 J in 10 s) when using this accelerated waveform and circuit. This indicates the shortest response time to the valve switch and a higher frequency of valve switching.Figure 8 Open-loop charging test under three drive circuits

(A) Pressure tracking results.

(B) Energy consumption of the charging valve.

(C) Air volume through the charging valve.

Multi-mode switching strategy

There are sixteen different combinations of on-off valve output states for a pneumatic servo system, which has a double-acting double-rod cylinder with two chambers and four on-off valves. This combination can also be called the modes of the system, and only seven of these modes are considered to be effective. The precision of the closed-loop control system can be increased with a suitable mode switching method. In Nguyen et al.,28 a three-mode switching method is proposed in a system with a double-acting double-outlet cylinder with two chambers and four on-off valves. Assume that the cylinder’s two chambers are designated as A and B, respectively, and that each chamber is connected to two on-off valves. One of the valves is a charging valve with the inlet port connected to the air supply and the exhaust port connected to the chamber. The other valve is a discharging valve, with the exhaust port connected to the atmosphere and the inlet port connected to the chamber. The three switching modes are as follows: (1) chamber A is connected to the air supply, and chamber B is connected to the atmosphere; (2) chamber A is connected to the atmosphere, and chamber B is connected to the air supply; and (3) chamber A and chamber B closed at the same time. However, only using these modes results in less precise locations and higher energy costs. In Ho et al.,29 the three switching modes are extended to seven switching modes. The four additional modes are as follows: (1) chamber A is connected to the air supply, and chamber B is closed; (2) chamber A is connected to the atmosphere, and chamber B is closed; (3) chamber B is connected to the air supply, and chamber A is closed; (4) chamber B is connected to the atmosphere, and chamber A is closed. The addition of several situation-specific modes allows for finer position adjustment.

We develop a multi-mode switching strategy in the pneumatic system made up of a single air chamber and two on-off valves after being inspired by the aforementioned mode switching methods. For a conventional system consisting of two on-off valves, there are theoretically only four system modes, but only three are effective. For a single on-off valve, there are only two states, off and on. With the introduction of accelerated waveforms in this paper, a single on-off valve has three states: off, not accelerated on, and accelerated on. The state that does not accelerate to open is the fully open state. The system mode theoretically has a total of nine modes, in addition to the seven modes listed in Table 1. There are three modes: (1) two on-off valves are not accelerated to open; (2) charging valves are not accelerated to open, the discharging valve is accelerated to open; (3) charging valves are accelerated to open, the discharging valve is not accelerated to open. However, these three modes are more wasteful of air or difficult to control, so we do not adopt them. As a result, only seven of the nine models were considered valid, and the specific forms of the seven models are summarized in Table 1. M1, M2, and M3 enable the system to achieve a charging effect, and M5, M6, and M7 enable the system to achieve a discharging effect. During the charging process, the three modes have inconsistent inflation volumes. When the set pressure value exceeds the actual pressure value, choosing an appropriate switching strategy can adjust the system’s charging volume, thereby improving inflation control accuracy. The discharging process of the three modes is similar to the charging process, and selecting the appropriate switching strategy can improve the accuracy of discharging control. As this study focuses on pressure control in a sealed container chamber, where the actual pressure inside the chamber continuously changes, the charging process does not directly transition to discharging but always passes through M4. Therefore, designing a reasonable seven-mode switching strategy can enhance the system’s control accuracy. In the STAR★Methods section, detailed explanations of each mode are provided.Table 1 Specific forms of the seven modes

Mode	M1	M2	M3	M4	
Fast inflation	Slow inflation	Fine inflation	Stop	
Charging valve	Opening without acceleration	Accelerated opening	Accelerated opening	Closed	
Discharging valve	Closed	Closed	Accelerated opening	Closed	
Mode	M5	M6	M7	/	
	Fine deflation	Slow deflation	Fast deflation	/	
Charging valve	Accelerated opening	Closed	Closed	/	
Discharging valve	Accelerated opening	Accelerated opening	Opening without acceleration	/	

Figure 9 shows the specific implementation of the multi-mode switching strategy, which restricts each mode in two main dimensions, one of which is the pressure difference between the setting and actual value. The pressure difference ΔP is described as(Equation 15) ΔP=z1=Pref−P

where z1 is the system error, Pref is the setting pressure, and P is the actual pressure. Only the interconversion between modes M2 and M3 and between modes M5 and M6 are affected by the pressure difference.Figure 9 Multi-mode switching strategy

The other dimension is the sliding function s, which is designed as(Equation 16) s=z¨1ζ2+2ρz˙1ζ+z1

where z˙1=P˙ref−P˙, z¨1=P¨ref−P¨, ζ(ζ>0), and ρ(ρ>0) are the parameters of the sliding function.

The sliding function and the pressure difference can be switched between seven modes to achieve precise control of the pressure in the air chamber. When |s|>Sp (Sp>0, Sp is a parameter in the mode switching strategy), it shows that there is a significant discrepancy between the pressure setting and actual value, indicating that the air chamber needs to be inflated or deflated quickly. When s>Sp, mode M1 will be used to achieve quick inflation. When s<−SP, mode M7 will be applied. When |s|<Su (Sp>Su>0, Su is a parameter in the mode switching strategy), it shows that there is little discrepancy between the pressure setting and actual value, indicating that mode M4 is required to stabilize the actual pressure value. When Sp>|s|>Su, the system error is small, so use the system error as a condition to achieve slow inflation, fine inflation, slow deflation, and slow deflation. When Sp>s>Su and ΔP>δt (ΔP>δt, δt is a parameter in the mode switching strategy), the system will be operated in mode M2 for slow inflation. When Sp>s>Su and ΔP<δt, the system will be performed in mode M3 to achieve the fine inflation. When −Sp<s<−Su and ΔP<−δt, the system will be driven in mode M6 for the slow venting. When −Sp<s<−Su and ΔP>−δt, the system will be executed in mode M5 to realize the fine venting. The black arrows in Figure 9 represent the system mode switching determined by the sliding function, and the red arrows represent the system mode switching determined by the differential pressure. By selecting the appropriate mode switching parameters, the proposed strategy can achieve the best performance.

Design of BSMC algorithm

Due to the strong nonlinearity of the pneumatic system and gas leakage problems, a robust controller needs to be designed. Based on the feedback value, as seen in Figure 10, the controller outputs a controlled quantity to the multi-mode switching strategy. According to the input values, the multi-mode switching strategy implements mode switching, and the output duty cycle controls the charging and discharging valves, which in turn regulate the pressure inside the air chamber.Figure 10 BSMC algorithm design

Algorithm design

Consider the system as a first-order nonlinear system with x1 and x2 as the state variables, y as the system output, u as the control quantity, and d(t) as the system disturbance, then(Equation 17) {x˙1=x2x˙2=(A+ΔA)x2+(B+ΔB)u+d(t)y=x1

Equation 17 can be rewritten as follows:(Equation 18) x˙2=Ax2+Bu+F

where F is the total uncertainty, and its expression is(Equation 19) F=ΔAx2+ΔBu+d(t)

where |F|≤F¯, and ΔA and ΔB are the uncertain parts of the system parameters.

The Lyapunov function is defined as(Equation 20) v1=12z12

Define(Equation 21) x2=y˙d−z2+cz1

where z2 serves as a fictitious control term and c is a positive constant, then(Equation 22) z2=y˙d−x2+cz1

As a result, z˙1 is(Equation 23) z˙1=z2−cz1

then(Equation 24) v˙1=z1z˙1=z1z2−cz12

Define the switching function as(Equation 25) σ=kz1+z2

where k>0. Equation 25 can be substituted with Equation 23:(Equation 26) σ=kz1+z2=kz1+z˙1+cz1=(k+c)z1+z˙1

Since k+c>0, it follows that if σ=0, then z1=0, z2=0, and v˙1≤0. So that the controller can be built further, another Lyapunov function is defined as(Equation 27) v2=v1+12σ2

Then(Equation 28) v˙2=v˙1+σσ˙=z1z2−cz12+σσ˙=z1z2−cz12+σ(kz˙1+z˙2)=z1z2−cz12+σ(k(z2−cz1)+x˙2−y¨d+cz˙1)=z1z2−cz12+σ(k(z2−cz1)+A(z2+y˙d−cz1)+Bu+F−y¨d+cz˙1)

Therefore the designed controller is(Equation 29) u=B−1(−k(z2−cz1)−A(z2+y˙d−cz1)−F¯sgn(σ)+y¨d−cz˙1−h(σ+βsgn(σ)))

where h and β are constants in positive.

Proof of system stability

In order to demonstrate the stability of the controller, Equation 29 is substituted into Equation 28:(Equation 30) v˙2=z1z2−cz12−hσ2−hβ|σ|+Fσ−F¯|σ|≤−cz12+z1z2−hσ2−hβ|σ|

Describe Q as(Equation 31) Q=[c+hk2hk−12hk−12h]

Due to(Equation 32) zTQz=[z1z2][c+hk2hk−12hk−12h][z1z2]T=cz12−z1z2+hk2z12+2hkz1z2+hz22=cz12−z1z2+hσ2

where zT=[z1z2]. If Q is certain to be a positive definite matrix, then v˙2≤−zTQz−hβ|σ|≤0. Due to(Equation 33) |Q|=h(c+hk2)−(hk−12)2=h(c+k)−14

It is possible to make |Q|>0 and hence Q, a positive definite matrix, by obtaining the values of h, c, and k. This in turn ensures v˙2≤0.

According to the LaSalle invariance principle, when v˙2≡0, results in z≡0 and σ≡0. When t→∞, results in z→0, σ→0, z1→0, z2→0, x1→yd, and x˙1→y˙d. As a result, the stability of this controller for the backstepping sliding mode is assured.

Results

Experimental setup of the simulation system

A pressure simulation system is established to verify the control effect of the multi-mode switching strategy with BSMC. Figure 11 illustrates the general structure of the pressure simulation system, which is made up of three parts: the pneumatic setup, electrical setup, and software.Figure 11 Architecture of the simulation system

Pneumatic setup

The pneumatic setup includes two on-off valves, an air chamber, and an oil-free air compressor. The on-off valves are Matrix model MX821.103C224, inlet, and outlet valves. Additionally, an oil-free air compressor, TUOWIN model TW7504S, equipped with an internal air tank, filter, regulator, and lubricator (SMC model AC5000-10D), is employed as the air source, generating clean and stable compressed air. More specific detailed parameters are summarized in Table 2.Table 2 Detailed parameters of the pneumatic setup

Equipment name	Model	Parameter 1	Parameter 2	Parameter 3	
On-off Valve	Matrix, MX821.103C224	Response time:
5 ms	Max frequency:
200 Hz	Max flow rate: 180 L/min	
Air Compressor	TUOWIN,
TW7504S	Discharge capacity:
180–608 L/Min	Pressure:
0.1–0.8 MPa	Storage capacity: 120 Lt	
Air Chamber	Customized	0.1–0.8 MPa: 0.8 L	Operating pressure:
0.1–0.8 MPa	–	

Electrical setup

The electrical part consists of a PC, a real-time controller model NI cRIO-9054, a digital output card model NI 9401, an acquisition card model NI 9205 (voltage acquisition range of ±10 V), an acceleration circuit, a pressure sensor (MEACON model MIK-P300), a volumetric flowmeter (SIARGO model MF4008), and a DC power supply. The NI 9401 is a bi-directional digital module with eight digital input/output channels, which is used to output five V/TTL digital voltage signals to communicate with the acceleration circuit by SPI communication protocol. The NI 9205 acquisition card is a high-precision analog input module with a voltage range of ±10 V to ±200 mV, which is used to capture feedback signals of the pressure sensor and the volumetric flowmeter. The pressure sensor is connected to the air chamber and is used to measure the pressure inside the air chamber. The volumetric flowmeter is installed between the inlet valve and the air chamber. Figure 12 shows the real simulation system.Figure 12 Real pictures of the simulation system

Software platform

The system software is developed by the LabVIEW programming environment, and the system software architecture is shown in Figure 13. The major components of the software are the bottom-layer FPGA program, the middle-layer RT program, and the upper-layer PC program. The corresponding programs are FPGA.vi, RT.vi, and HOST.vi. Figure 13 illustrates the duties of each layer. The relevant programs of the simulation system are designed and deployed in NI LabVIEW 2019, LabVIEW RT 19.0, LabVIEW FPGA 19.0, and NI cRIO driver19.0.Figure 13 Software architecture

Experimental results

By tracking complicated reference pressures, the steady-state response performance, dynamic response performance, and superiority in different controllers of the proposed method are experimentally verified to confirm the viability of the technique. Table 3 lists the system parameters and values. The parameter Sp is selected as 0.06, the parameter Su as 0.001, and the parameter δt as 0.2 in the multi-mode switching strategy.Table 3 System parameters

Symbol	System parameters	Value	
Cd	Valve discharge coefficient	0.17 × 10−3	
Pcr	Critical pressure ratio of outlet pressure to inlet pressure	0.528	
P0	Ambient pressure	0.1 MPa	
Pi	Supply pressure	0.6 MPa	
T	Air temperature	293.15 K	
V	Volume of the air chamber	1.6 × 10−4 m3	
Steady-state response.

To verify the steady-state performance of the proposed method, five groups of steady-state response tests are conducted. The reference steps in each test are 0.2 MPa–0.6 MPa, with 0.1 MPa serving as the beginning pressure and 0.1 MPa serving as the step pressure. The corresponding parameters of the BSMC are set as follows: k=15, c=10, h=20, and β=1.5. Each group’s pressure tracking results are displayed in Figure 14A, and each group’s compressed air usage is displayed in Figure 14B. The results of measuring the method’s efficacy by obtaining the average values of adjustment time, overshoot, compressed air usage, and steady-state error are displayed in Table 4. Each group of tests is performed at least 10 times.Figure 14 Steady-state experimental results

(A) Pressure tracking results.

(B) Compressed air usage.

Table 4 Analysis of steady-state experimental results

Groups (MPa)	Adjustment time (s)	Overshoot (%)	Steady-state error (MPa)	Compressed air usage (L)	
0.1–0.2	0.523	1.25	0.016	0.867	
0.1–0.3	1.034	1.02	0.013	1.311	
0.1–0.4	1.502	0.55	0.009	1.637	
0.1–0.5	2.049	0.45	0.008	2.240	
0.1–0.6	2.326	0.52	0.010	2.366	

As can be observed from Figure 14 and Table 4, the proposed algorithm allows fast-tracking of a given pressure. The steady-state error is kept within 0.016 MPa, and the average overshoot is kept within 1.25%. It can be concluded that the BSMC has a fast response time to control the intake and exhaust in time according to the change of the reference pressure, while not showing large errors at the intake and exhaust. At the same time, there is no large error at the moment when the reference pressure is reached.

Dynamic response

The dynamic tracking test aims to verify the tracking performance of the proposed multi-mode acceleration switching strategy with the BSMC under the continuous change of reference pressure. In order to simulate the complexities of pneumatic systems in industry, mainly random mutation signals, pressure tracking tests were performed. The corresponding parameters of the BSMC are set as follows: k=15, c=10, h=20, and β=1.5. The three sets of test results are displayed in Figures 15, 16, and 17.Figure 15 Experimental results of tracking the triangular wave signals

(A) Pressure tracking results.

(B) Pressure tracking error.

(C) Compressed air usage.

Figure 16 Experimental results of tracking the random step signals

(A) Pressure tracking results.

(B) Pressure tracking error.

(C) Compressed air usage.

Figure 17 Experimental results of tracking the random mutation signals

(A) Pressure tracking results.

(B) Pressure tracking error.

(C) Compressed air usage.

In the experiment of tracking a triangular wave signal shown in Figure 15, the tracking error of the system can be kept within ±0.012 MPa. Similarly, in the experiment of tracking a random step signal shown in Figure 16, the tracking response time is within 1.4 s for each step stage, and its steady-state error can be kept within the range of −0.015 MPa and 0.008 MPa. In addition, in the experiment of tracking a random mutation signal shown in Figure 17, the tracking error of the system can be kept within ±0.012 MPa. Taken together, the proposed method allows for timely tracking in the case of rapidly changing given pressures.

Comparison results of different controllers

To verify the comprehensive performance of the BSMC, multi-mode acceleration switching strategy, and acceleration waveforms combined in this paper, we compare the method of this paper with a PID controller, an SMC controller combined with a conventional mode switching strategy, and PWM modulation. Three algorithms are used respectively for the tracking of sine wave signals with a signal frequency of 0.2 Hz. The PID controller is represented by Equation 34, where Kp is the proportional gain, Ki is the integral gain, and Kd is the derivative gain. The SMC controller is represented by Equation 35. The corresponding parameters of the PID are set as follows: Kp=200, Ki=20, and Kd=0.0002. The corresponding parameters of the SMC are set as follows: k=10 and ε=0.02. The corresponding parameters of the BSMC are set as follows: k=25, c=500, h=9, and β=1. The test results are shown in Figures 18 and 19. The overshoot, RMSE, compressed air usage, and energy consumption of the charging valve of each group are analyzed to verify the effectiveness and energy saving of the proposed method in Table 5.(Equation 34) u=Kpz1+Ki∫z1(τ)dτ+Kdz˙1

(Equation 35) u=1B(−k|z˙1|−|ΔBr(t)|−F¯−ε−|ΔAy˙d|−|Ay˙|)sgn(s)

Figure 18 Comparison results of different controllers

(A) Pressure tracking results.

(B) Pressure tracking error.

Figure 19 Comparison results of different controllers

(A) Compressed air usage.

(B) Energy consumption of the valve group.

Table 5 Comparison of evaluation indicators of the three controllers

Methods	Reference	Overshoot (MPa, %)	RMSE (kPa, %)	Compressed air usage (L)	Improvement (%)	Energy (J)	Improvement (%)	
PID	Bhaskaran et al.25	0.0231, 7.10	9.7337, 0.32	1.0114	/	10.5346	/	
SMC	Ren et al.20	0.0187, 5.86	5.7466, 0.19	0.7613	/	9.5502	/	
BSMC	This work	0.0163, 5.92	4.8412, 0.16	0.5613	44.50% over PID, 26.27% over SMC	6.4610	38.67% over PID, 32.35% over SMC	

Figures 18 and 19 and Table 5 show that the proposed method performs better than the other two controllers. Firstly, compared to the PID controller, the proposed method maintains pressure tracking error at 0.017 MPa without severe misalignment, with a maximum misalignment value of 0.0163 MPa and an RMSE of just 4.8412 kPa. Secondly, the energy savings of the proposed method are also much greater than those of the other two controllers, as seen in Figure 19, achieving remarkable savings of 44.50% and 26.27% in air usage, respectively.

To further demonstrate the advantages of the proposed method in energy conservation, we measured and calculated the energy consumption of the charging valve. The formula is derived from Equation 36 and defined as(Equation 36) W=RI2+LIdIdt

where R represents the resistance of the on-off valve, which is 4.50 Ω, I denotes the current flowing through the on-off valve, and L signifies the inductance of the on-off valve, which is 5 H. The results of the calculations are presented in Table 5, where the Improvement columns indicate the enhancement of the proposed method over the current technique, showcasing the proposed method’s remarkable savings of 38.67% and 32.35% in energy consumption of the charging valve compared to PID and SMC, respectively. BSMC, compared to the two traditional controllers, can better handle nonlinear systems, offering strong robustness and high precision, effectively addressing uncertainties and external disturbances.

Discussion

In this paper, a pneumatic system is designed that can effectively save energy while improving the tracking accuracy of the system. Firstly, for the dead zone problem affecting the response accuracy of the on-off valve, a three-voltage acceleration waveform is proposed, and the comparison with the traditional waveform yields obvious advantages. Then, combining the acceleration waveform, we propose a seven-mode switching strategy for a pneumatic system with multiple on-off valves. Integrating the BSMC algorithm, we form a complete pneumatic system. To verify the tracking accuracy and energy consumption of the system, we conducted a pressure tracking experiment. The steady-state and dynamic experiments show that the proposed method achieves the minimum pressure tracking error. In the comparison experiments, BSMC respectively saves 26.27% of the air consumption, as well as 32.35% of the valve group power consumption. It also achieves the lowest RMSE, of 4.8421 kPa. In summary, this proves that the method has more potential for practical applications.

Limitations of the study

The research of pneumatic systems still faces certain challenges, which can be addressed in the future research. Specifically, efforts can be directed toward achieving energy savings through more precise negative pressure tracking.

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Feng Huang (E-mail: huangf@fzu.edu.cn).

Materials availability

This study did not generate new materials. Materials used in the study are commercially available.

Data and code availability

• All data reported in this paper will be shared by the lead contact upon reasonable request.

• All code in this paper will be shared by the lead contact upon reasonable request.

• Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon reasonable request.

Acknowledgments

This work was supported by the funding of the 10.13039/501100003392 Natural Science Foundation of Fujian Province of China (grant number 2021J05113 ), funding of the 2020 Fujian Province Young and Middle-aged Teacher Education Research Project (Technology) (grant number JAT200030 ), funding of the 10.13039/501100008859 Fuzhou University Research Start-up Funding (grant number GXRC-20051 ), and funding of the Crosswise Project of “Research on the Air Pressure Simulation System Design of Aero-engine Surge” (grant number 2021011902 ).

Author contributions

Z.L.: conceptualization, methodology, software, writing—original draft preparation, writing—review & editing, funding acquisition. H.W.: conceptualization, software, writing—original draft preparation, writing—review & editing. X.Z.: data curation, visualization, investigation, resources. W.L.: validation, visualization. J.G.: validation, formal analysis. F.H.: writing—review & editing, supervision, project administration, funding acquisition. T.Z.: supervision, project administration, funding acquisition.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Chemicals, peptides, and recombinant proteins	
	
Polyethylene	FESTO	N/A	
Steel	FESTO	N/A	
	
Software and algorithms	
	
LabVIEW	National Instruments Corporation	N/A	
MATLAB	Mathworks, Inc.	N/A	
Origin	OriginLab Corporation	N/A	

Experimental model and study participant details

This study did not involve or contain any experimental models and study participants (animals, human participants, plants, microbe strains, cell lines, primary cell cultures).

Method details

Accelerated waveforms further explained

At the initial state, a voltage of 24 V is used for excitation so that the current increases rapidly. As the on-off valve opens, the voltage is reduced to 5 V, while the voltage is maintained, at which the current drops but does not fall below the closing current of the valve. When the on-off valve enters the closing stage, the voltage becomes −12 V, and the current drops rapidly to the closing current to achieve the effect of accelerated closing.

The acceleration waveform reduces the opening time of the valve to 0.9 ms and the closing time to 1 ms, resulting in a maximum operating frequency of 526.32 Hz, which is 3.68 times higher than the operating frequency without the acceleration. This method also stops the current from rising continuously after the valve opens, thereby reducing heat generation, lowering energy consumption, and increasing the service life of the valve. For closed-loop control, the controllable time within a control cycle is improved, reducing the effect of the dead zone.

Specific implications of each model

The multi-mode switching strategy from mode M1 to M7 represents fast inflation, slow inflation, fine inflation, stop, fine deflation, slow deflation, and fast deflation, respectively. In mode M1, it is necessary to disable the discharging valve so that the charging valve can fully open without the need for the acceleration waveform because the set pressure value is significantly higher than the actual pressure value. In mode M2, the actual pressure value is gradually approaching the setting value. Using mode M1 will cause the actual pressure value to be greater than the setting value, leading to a significant overshoot. As a result, it is essential to control the charging valve using the acceleration waveform. Under specific circumstances, the system operates in mode M3 for more precise pressure control. Both the discharging and charging valves are in the accelerated opening state, and the charging valve will have a slightly larger PWM modulation duty cycle than the discharging valve. The duty cycle difference between the charging and discharging valve will be determined by the subsequent controller. When the pressure difference between the setting and actual value is within the minimum error requirement, both valves will be closed in mode M4. Similar to mode M3, mode M5 achieves a fine deflation. Mode M6 achieves slow deflation and is comparable to mode M2. For fast deflation, mode M7 is similar to mode M1.

Quantification and statistical analysis

The raw experimental data in the part of results and discussion was measured by an industrial controller model NI cRIO-9054 with a digital output card model NI 9401 and an acquisition card model NI 9205. Figures were produced by Origin 2022 from the raw data.
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