Abstract
According to the hydraulic principle diagram of the subgrade test device, the dynamic pressure cylinder electrohydraulic servo pressure system math model and AMESim simulation model are established. The system is divided into two parts of the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem. On this basis, a RBF neural network backstepping sliding mode adaptive control algorithm is designed: using the double sliding mode structure, the two RBF neural networks are used to approximate the uncertainties in the two subsystems, provide design methods of RBF sliding mode adaptive controller of the dynamic pressure cylinder displacement subsystem and RBF backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem, and give the two RBF neural network weight vector adaptive laws, and the stability of the algorithm is proved. Finally, the algorithm is applied to the dynamic pressure cylinder electrohydraulic servo pressure system AMESim model; simulation results show that this algorithm can not only effectively estimate the system uncertainties, but also achieve accurate tracking of the target variables and have a simpler structure, better control performance, and better robust performance than the backstepping sliding mode adaptive control (BSAC).
1. Introduction
The track subgrade dynamic response test device is mainly used to simulate the comprehensive impact of high-speed running trains on the subgrade. The constant pressure of static pressure cylinder is set by the pilot type electrohydraulic proportional pressure reducing valve to simulate the static load generated by the train's own weight on the subgrade; the alternating hydraulic pressure is applied to the dynamic pressure cylinder through the servo valve to simulate the dynamic load on the subgrade during the train high-speed running [1–3]. The hydraulic schematic diagram of the track subgrade test device is shown in Figure 1. The dynamic pressure cylinder piston rod outputs an alternating dynamic load, obtaining the resultant load force by superimposing the static load of the static pressure piston rod, and finally, the loading force is loaded on the tested subgrade through the sensor and the excitation block. Therefore, the dynamic pressure cylinder system is a typical electrohydraulic servo pressure system.
The control performance of the composite loading force depends on the precise control of the dynamic pressure cylinder electrohydraulic servo pressure system, because the dynamic pressure cylinder electrohydraulic servo pressure system has the parameter uncertainty and flow nonlinearity, which increase the difficulty of the control system design. The backstepping control constructs the Lyapunov function at all levels, selects the intermediate virtual control quantity at each level according to the design goals, and obtains the control law of the system by step backward recursion; it is a feedback control method based on the Lyapunov stability theory [4, 5]. Sliding mode variable structure control has the advantages of high control precision and simple structure, can greatly reduce the influence of system nonlinearity, and has strong robustness [6, 7]. Adaptive control is often used to reduce the impact of parameter uncertainty on system performance [8–10]. Therefore, backstepping sliding mode adaptive control has been widely used in electromechanical servo control [11–13], electrohydraulic servo control [14–16], flight navigation control [17, 18], and other fields, achieving good control effects.
In the actual system, the external interference is unknown, and the system still has modeling errors. Therefore, the upper bounds of uncertainties in the system are often difficult to determine. The uncertainty boundary problem has become an important part of controller design, which directly affects the performance of the whole control system. In recent years, with the development of intelligent control theory, neural networks with their good approximation characteristics have been widely used in the estimation of unknown parts of the system and have achieved good results. Xu Chuanzhong [19] designed the RBF neural network adaptive law to estimate the upper bound of uncertain factors in the backstepping sliding mode control system, thus improving the robustness of the system to factors such as modeling errors and uncertain disturbances. Chen Ziyin [20] compensated the model uncertainty in the pitch motion of underwater vehicles through a neural network controller and designed an adaptive robust controller to eliminate the approximation error of the neural network.
In order to achieve rapid and accurate pressure tracking control of dynamic pressure cylinder electrohydraulic servo pressure system, this paper designed a RBF neural network backstepping sliding mode adaptive control method, which can effectively reduce the influence of system uncertainties and nonlinearities, so that the system output pressure has good tracking performance and robust performance.
2. Model of Dynamic Pressure Cylinder Electrohydraulic Servo Pressure System
2.1. Mathematical Model
The dynamic pressure cylinder electrohydraulic servo pressure control system mainly includes control signal, servo amplifier, servo valve, dynamic pressure cylinder, sensor, and load.
The servo valve system includes the spool equation and the flow equation:
where XV is the servo valve spool displacement, KS is the servo valve system overall gain, GSV is the servo valve transfer function at unity gain, Ue is the servo amplifier input voltage signal, QL is the servo valve output flow, Cd is the servo valve port flow coefficient, ω is the servo valve main spool area gradient, PS is the system supply pressure, PL is the load pressure, and ρ is the oil density.
Since the natural frequency of the servo valve is close to the hydraulic frequency of the dynamic hydraulic cylinder, this paper uses the second-order oscillation element to describe the servo valve transfer function [21] and retain the flow nonlinear part of the servo valve. The description of the load flow is as follows:
where a6, a7, a81 are the servo coefficients and is the flow nonlinear part.
Dynamic pressure cylinder can be described as
where m is the mass of dynamic pressure cylinder vibration system, Bm is the load damping coefficient, K is the subgrade elastic stiffness, FL is the static load of static pressure cylinder, Ap is the effective area of dynamic pressure cylinder piston, Ctp is the dynamic pressure cylinder total leakage coefficient, Vm is the system pipe total compression volume, and βe is the effective volumetric elastic modulus of hydraulic oil.
Combining (4) and (5), using static load FL and servo valve output flow QL as input variables, and selecting dynamic pressure cylinder displacement, speed, and output pressure PL as state variables, the state equation of the dynamic pressure cylinder can be obtained as follows:
where X1 is the dynamic pressure cylinder displacement, X2 is the dynamic pressure cylinder speed, X3 is the dynamic pressure cylinder output pressure, a1 = K/m, a2 = B/m, a3 = Ap/m, a4 = 4Apβe/Vm, a5 = 4Ctpβe/Vm, af = 1/m, b1 = 4βe/Vm.
Substituting (3) into the third item of (6), introducing the target variable Pr into the state variable, and letting ξ1 = X1, ξ2 = X2, ξ3 = Pr-X3, , , (6) can be transformed to
where a8 = a81b1, a9 = a5+a6, a10 = a7+a5a6, a11 = a5a7, a12 = a4a6, a13 = a4a7, a14 = a12-a2a4, a15 = a13-a1a4, a16 = a3a4, a17 = afa4, a18 = a10-a16, a19 = a11 + a3a14, a21 = a1a14, a22 = afa14, a20 = a15-a2a14, , , .
The external disturbance is much smaller than the static load FL(150KN). Therefore, ignoring the influence of external interference, the static load FL is equivalent to an external disturbance, being constant and bounded.
2.2. AMESim and Simulink Cosimulation Model
It can be seen from the hydraulic schematic diagram Figure 1 of the track subgrade test device that the dynamic pressure cylinder electrohydraulic servo pressure control system mainly includes dynamic pressure cylinder, flow servo valve, and sensor. The dynamic pressure cylinder electrohydraulic servo pressure system AMESim and Simulink cosimulation model is established as Figure 2.
3. Backstepping Sliding Mode Controller Design
3.1. System Decomposition
The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be divided into two parts: the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem.
Dynamic pressure cylinder displacement subsystem:
Dynamic pressure cylinder output pressure subsystem:
3.2. Dynamic Pressure Cylinder Displacement Subsystem Sliding Mode Control
According to (8) description, is set as the expected displacement of the dynamic pressure cylinder displacement subsystem; define the displacement tracking error as e1 = ξ1-ξd1, and construct the sliding mode switch function of displacement subsystem as follows:
where c1, c2 are switching function coefficients, positive real numbers.
Taking the derivative of sliding mode switching functions S1 and substituting (8) into , we can get
where aa1 = c2a1, aa2 = c2a2 -c1, aa3 = c2a3, aa4 = c2af, .
Let ξd3 be the expected variable of the displacement subsystem variable ξ3, and then its tracking error e31 = ξ3-ξd3; the expectation ξd3 of this paper is ξd3 = 0, so e31 = ξ3, and ξ3 is replaced by a virtual output control variable e31. Assuming that the above parameters and uncertainties are known, we can obtain the virtual controller as follows [22].
3.3. Dynamic Pressure Cylinder Output Pressure Subsystem Backstepping Sliding Mode Control
Set ξd3 as the expected output pressure of the dynamic pressure cylinder output pressure subsystem, and the tracking error of the output pressure is e3 = ξ3-ξd3; use backstepping algorithm, combined with (9), to gradually derive the virtual control variables at all levels as follows.
Step 1. Construct Lyapunov function as and derivativeLet the derivative of tracking error e3 be e4 = ξ4-ξ4d, and take the virtual control variable ξ4d asSubstituting (14) into (13), we can get
Step 2. Construct the Lyapunov function as and derivativeLet the derivative of be e5 = ξ5-ξ5d, and take the virtual control variable ξ5d asSubstituting (17) into (16), we can getwhere K3, K4, K5 are Lyapunov function coefficients, positive real numbers.
Step 3. Design the sliding mode switching function of the dynamic pressure cylinder output pressure subsystem aswhere c3, c4, c5 are switching function coefficients, positive real numbers.
Substituting (9) into , we can getwhere aa5 = c5a21, aa6 = c5a20, aa7 = c5a19, aa8 = c5a18 − c3, aa9 = c5a9 − c4, aa10 = c5a22, aa11 = c5a8, .
Let ; the expression of the backstepping sliding mode controller of the dynamic pressure cylinder output pressure subsystem can be obtained:where .
3.4. The Selection of the Expected Displacement of the Displacement Subsystem
When the dynamic pressure cylinder displacement subsystem is stable, the displacement tracking error e1 is very small, at this time, ξ1 ≈ ξd1. Since , , and , according to the virtual controller (12), combined with the expected output pressure ξd3, the desired displacement ξd1 of the dynamic pressure cylinder can be expressed approximately as follows:
The virtual control variable e31 is used to implement the tracking control of (22); with the premise of good displacement tracking performance, we expect e31 to be as small as possible. However, e31 may be relatively large in actual operation, resulting in a large difference in displacement ξ1 between (8) and (7); thus it has some influence on the dynamic pressure cylinder output pressure subsystem. Because the two subsystems independently carry out the stability design, the above mentioned differences between the e31 and e3 will not affect the stability of the whole system, and the final output pressure tracking performance is only related to the design of the virtual controller (12) and the backstepping sliding mode controller (21).
4. RBF Neural Network Backstepping Sliding Mode Adaptive Controller Design
4.1. Dynamic Pressure Cylinder Displacement Subsystem RBF NN Sliding Mode Adaptive Control
The dynamic pressure cylinder displacement subsystem described by (8) constructs the displacement subsystem sliding mode switching function such as (10); let f1 = C2Δ1, and (11) can be expressed as
4.1.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Displacement Subsystem
Using the good approximation performance of the RBF neural network, estimate the uncertainty term f1 of the dynamic pressure cylinder displacement subsystem, which can effectively solve the problem that the upper bound of the uncertain term is difficult to determine.
where is the weight vector of the RBF, ; is the radial basis vector of the RBF, , is the number of hidden layer nodes.
And is a Gaussian function with the following expression:
where is the central vector of the jth network node; bj is the base width parameter of the jth network node.
Assumption 1. Using the RBF neural network to approximate the uncertain term , there is an optimal weight to make the neural network approximation error to satisfy , and , where is the upper bound of the uncertainty of ; i.e., .
The uncertain term in (23) is estimated by the RBF neural network of (24); the adaptive virtual controller of the sliding mode RBF neural network of the dynamic cylinder displacement subsystem can be obtained:
4.1.2. Design of RBF NN Sliding Mode Adaptive Controller
The boundary layer method is introduced to reduce chattering near the sliding surface [23, 24], and the adaptive virtual controller is modified to
where K1 is the switching gain, and its adaptive law is designed as , K11 is a positive real number;is the boundary function.
Furthermore, the weight vector adaptive law of the displacement subsystem RBF neural network is
where δ1 is the weight vector correction coefficient, which can reduce the weight vector size and prevent the controller gain saturation, thus improving the robustness of the neural network approximation error [25] and satisfying .
4.2. Dynamic Pressure Cylinder Output Pressure Subsystem RBF NN Backstepping Sliding Mode Adaptive Control
Let ; (20) can be expressed as
4.2.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Output Pressure Subsystem
f2 is the uncertainty term of the dynamic pressure cylinder output pressure subsystem, and its RBF neural network approximator is as follows:
where m is the number of hidden layer nodes; is input vector of the RBF; is the weight vector of the RBF, ; is the radial basis vector of the RBF, .
And is a Gaussian function with the following expression:
where is the central vector of the nth network node; bpn is the base width parameter of the nth network node.
Assumption 2. Using the RBF neural network to approximate the uncertain term , there is an optimal weight ; make the neural network approximation error to satisfy , and , where is the upper bound of the uncertainty of ; i.e., .
Then we can obtain the RBF neural network backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem:
4.2.2. Design of RBF NN Backstepping Sliding Mode Adaptive Controller
Using the boundary layer method, the controller is as follows:
where K2 is the switching gain, and its adaptive law is designed as , K22 is a positive real number;is the boundary function.
The weight vector adaptive law of the output pressure subsystem RBF neural network is
where δ2 is the weight vector correction coefficient, satisfying .
4.3. Design and Stability Analysis of RBF Neural Network Backstepping Sliding Mode Adaptive Control for the Dynamic Pressure Cylinder Electrohydraulic Servo Pressure System
4.3.1. Design of RBF Neural Network Backstepping Sliding Mode Adaptive Control
Figure 3 is the control structure block diagram of the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control. In Figure 3, the dynamic pressure cylinder system consists of the displacement subsystem described by (8) and the output pressure subsystem described by (9); two RBF neural networks ( and ) and their adaptive laws are used to approximate the subsystem uncertainties and and realize the tracking control of the output pressure of the dynamic pressure cylinder by separately constructing the virtual controller and the pressure controller .
Furthermore, the dynamic pressure cylinder RBF neural network backstepping sliding mode adaptive control system can be constructed by Theorem 3.
Theorem 3. The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be decomposed into the dynamic pressure cylinder displacement subsystem described in (8) and the dynamic pressure cylinder output pressure subsystem described in (9); the dynamic pressure cylinder displacement subsystem adopts the sliding mode switching function of (10), uses RBF neural network described by (24) to approximate the uncertain term , selects the adaptive law of (29) used to update the RBF neural network weight vector , and constructs a sliding mode virtual controller of formulas (26) and (27); the dynamic pressure cylinder output pressure subsystem adopts the sliding mode switching function of (19), uses RBF neural network described by (31) to approximate the uncertain term , selects the adaptive law of (36) used to update the RBF neural network weight vector , and constructs a backstepping sliding mode controller of formulas (33) and (34); both of the above subsystems can be consistently bounded at the end, so that the dynamic pressure cylinder electrohydraulic servo pressure system is gradually stabilized, and finally the output pressure tracking error of the system is converged.
4.3.2. Stability Analysis
Discuss the stability of the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem separately, and then we can evaluate the stability of the entire dynamic pressure cylinder electrohydraulic servo pressure system.
Proof. Stability of the dynamic pressure cylinder displacement subsystem
Substituting the sliding mode adaptive virtual controller described in (27) for ξ3 in (23), we can getIt can be known from Assumption 1 that (37) can be simplified towhere is the RBF neural network weight vector estimation error and ε1 is the approximation error of the RBF neural network for the uncertainty term .
Select the Lyapunov function:Taking the derivative of V1 and substituting (39) into ,Substituting the RBF neural network weight vector adaptive law (29) into (41),From the Young inequality , λab is the normal number, and we can deriveDiscuss with the boundary function:
The adaptive law of switching gain K1 is , and the coefficient K11 is a positive real number; we can know , so that K1 ≥ 0 can be obtained. aa3 = c2a3 > 0; is positive real number.
(a) When , where , , , .
Select the parameters and being nonnegative real numbers, multiply by both sides of (45), and obtain the definite integral over the interval :When , converges to , all signals in the closed-loop system are uniformly bounded, and the tracking error is made as small as possible by selecting appropriate design parameters [26, 27].
(b) When , where , .
we can obtainAll signals in the closed-loop system are consistently bounded.
To sum up,The dynamic pressure cylinder displacement subsystem uses the adaptive RBF neural network of (24) and (29) to approximate the uncertain term , constructs the sliding mode virtual controller of (27), and selects the appropriate parameters; the system tracking error and the parameter approximation error can be ultimately bounded, and the closed-loop system eventually converges to a small neighborhood of zero.
Stability of the dynamic pressure cylinder output pressure subsystem
Substituting the backstepping sliding mode adaptive controller (34) into (30), we getIt is known by Assumption 2 that (50) can be simplified towhere is the RBF neural network weight vector estimation error and ε2 is the approximation error of the RBF neural network for the uncertainty term .
Design the Lyapunov function by referring to (13) to (21):Taking the derivative of V51 and substituting (53) into ,Substituting the RBF neural network weight vector adaptive law (36) into (54),We can get by Young inequality thatLet ; we can get , combined with (18); we knowDiscuss the following according to the definition of boundary function:
is the adaptive law of switching gain K2, and by the coefficient K22 being a positive real number, we know , so that K2 ≥ 0 can be obtained. It is known by (3) that ; ; is positive real number.
(a) When , where , , , .
Select the parameters Kr3 and Kp1 being nonnegative real numbers, multiply by both sides of (59), and obtain the definite integral over the interval :When , V51 converges to , all signals in the closed-loop system are uniformly bounded, and designing ensures that the closed-loop system eventually converges to a small neighborhood of zero.
(b) When , where , .
we can getAll signals in a closed-loop system are consistently bounded.
To sum up,The dynamic pressure cylinder output pressure subsystem uses the adaptive RBF neural network of (31) and (36) to approximate the uncertain term and, through constructing the backstepping sliding mode controller of (34), selects the appropriate parameters to make the system tracking error and the parameter approximation error ultimately bounded, thus ensuring that the closed-loop system eventually converges to a small neighborhood of zero.
Stability of the dynamic pressure cylinder electrohydraulic servo pressure system
Based on the discussion of the stability of the above two subsystems, the final Lyapunov function of the design system is expressed asThe derivative of V6 iswhere , .
Further, we can getIt can be seen that Theorem 3 can make the tracking error and parameter approximation error of the dynamic pressure cylinder electrohydraulic pressure system bounded, thus ensuring the stable convergence of the closed-loop system.
Proof completed.
5. Simulation Research
Based on the dynamic pressure cylinder servo pressure system described in (7), according to Theorem 3, the AMESim and Simulink cosimulation block diagram of RBF neural network backstepping sliding mode adaptive control is constructed as in Figure 4.
In Figure 4, AS1 is the AMESim model of the dynamic pressure cylinder electrohydraulic servo pressure system, and its parameters settings are shown in Table 1. C1 is the sliding mode adaptive controller of the dynamic pressure cylinder displacement subsystem, and S3 is the RBF neural network approximator of the displacement subsystem uncertainty term. C1 and S3 constitute the dynamic pressure cylinder displacement subsystem RBF neural network sliding mode adaptive control. C2 is the backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem, S4 is the RBF neural network approximator of the output pressure subsystem uncertainty item. C2 and S4 constitute the dynamic pressure cylinder output pressure subsystem RBF neural network backstepping sliding mode adaptive control. Finally, through the virtual controller e31 and the dynamic pressure cylinder output pressure controller u, realize the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control.
Select the target variable , and the expected deviation of the output pressure deviation ξ3 is ξd3 = 0; we can refer to (22) to derive the approximate dynamic pressure cylinder expected displacement ξd1, and set the parameters of the backstepping sliding mode adaptive controller according to Table 2.
The dynamic pressure cylinder displacement subsystem RBF neural network is designed as a 2-11-1 structure, containing 11 neurons; i.e., . The first set of 11 network node center vectors of the input variable ξ1 are evenly distributed in the region, and the other set of 11 network node center vectors of the input variable ξ2 are evenly distributed in the region. The network node base width parameter is .
The dynamic pressure cylinder output pressure subsystem RBF neural network is designed as a 3-16-1 structure, containing 16 neurons; i.e., m=16. The first set of 16 network node center vectors of the input variable ξ3 are evenly distributed in the region, the second set of 16 network node center vectors of the input variable ξ4 are evenly distributed in the region, and the third set of 16 network node center vectors of the input variable ξ5 are evenly distributed in the region. Network node base width parameters are , , .
Carry out the AMESim and Simulink cosimulation of the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control. The performance simulation curves are shown in Figure 5.
(a) Output pressure and set pressure curve
(b) Output pressure and set pressure deviation contrast curve
(c) Controller output comparison curve
(d) Virtual control variable curve
(e) Uncertainty term RBFNN approximation curve
(f) Uncertainty term RBFNN approximation curve
Figures 5(a) and 5(b) are, respectively, the contrast curves of the dynamic pressure cylinder AMESim model RBF neural network backstepping sliding mode adaptive control (RBFNNBSAC) and backstepping sliding mode adaptive control (BSAC) output pressure and their deviations. Compared with the backstepping sliding mode adaptive control (BSAC), the RBF neural network backstepping sliding mode adaptive control (RBFNNBSAC) has a short dynamic response time and no overshoot, and the output pressure deviation amplitude is only 3.1e-3 (about 5.3e4Pa) of the set pressure amplitude, about 70% of the BSAC output pressure deviation amplitude (about 7.4e4Pa), showing that the RBF neural network backstepping sliding mode adaptive control has better dynamic and static performance.
The comparison curves of the outputs of RBFNNBSAC and BSAC controller are shown in Figure 5(c). Compared with the output u of the BSAC controller, the output of the RBFNNBSAC controller u1 has a short adjustment time, fast convergence, and smooth curve, so that better control performance can be achieved.
In Figure 5(d), the virtual control variable e31 is much larger than the output pressure e3, although there is a large deviation, because the stabilities of the two subsystems are independent of each other, and therefore the whole system is still stable.
Figures 5(e) and 5(f) are, respectively, the RBF neural network adaptive estimation curves for the dynamic pressure cylinder displacement subsystem uncertainty item and the dynamic pressure cylinder output pressure subsystem uncertainty item , the approximation curves are stable and bounded, and the output of the controller u1 can be adjusted in real time to reduce the influence of parameter uncertainty on the tracking performance of the dynamic pressure cylinder output pressure.
Further, at 1.5s, a sinusoidal interference signal (0.2 sin (20 pit), lasting 1 s) is applied to the RBF neural network backstepping sliding mode adaptive controller output u1, and the interference response curve is as shown in Figure 6.
(a) Output pressure and set pressure curve
(b) Output pressure and set pressure deviation contrast curve
(c) The ratio of output pressure deviation and set pressure
(d) NNBSAC controller output
(e) Uncertainty term f1 RBFNN approximation curve
(f) Uncertainty term f2 RBFNN approximation curve
Figure 6(a) shows the good anti-jamming performance of the RBFNNBSAC control system. Figures 6(b) and 6(c) show more directly the changes in output pressure tracking deviation during the whole process of interference generation and disappearance: although the interference makes the amplitude of the output pressure deviation larger, its max amplitude is only 7.9e-3 (about 1.3e5Pa) of the set pressure amplitude, still having high tracking accuracy, and the output pressure deviation can be rapid return to the pre-interference level after the interference disappears.
The output of the RBFNNBSAC controller in Figure 6(d) can be adjusted according to the interference signal, and after the interference disappears, the output size of controller can be restored. Uncertainty terms f1 and f2 RBFNN approximation of the interference response curves are shown in Figures 6(e) and 6(f); the interference still has no effect on the uncertainty f1 RBFNN approximation curve, but the uncertain term f2 RBFNN approximation curve can quickly and sensitively respond to the interference signal, adjusting the compensation of the RBFNN approximation network to the interference signal in real time.
The target variable Pr is set to triangle wave and square wave signal with amplitude 1.7e7Pa and frequency 10Hz, respectively, and modifies some parameters of RBF neural network backstepping sliding mode adaptive controller; the simulation curves of the RBF neural network backstepping sliding mode adaptive control based on dynamic pressure cylinder AMESim model are, respectively, shown in Figures 7 and 8.
(a) Output pressure and set pressure curve
(b) Output pressure and set pressure deviation contrast curve
(c) RBFNNBSAC controller output
(d) Uncertainty term f2 RBFNN approximation curve
(a) Output pressure and set pressure curve
(b) Output pressure and set pressure deviation contrast curve
(c) RBFNNBSAC controller output
(d) Uncertainty term f2 RBFNN approximation curve
From Figures 7(a)–7(c) and Figures 8(a)–8(c), it can be seen that the RBF neural network backstepping sliding mode adaptive control (RBFNNBSAC) can also effectively track triangular and square wave signals. There are some certain tracking errors; however, compared with the backstepping sliding mode adaptive control (BSAC), the algorithm has good dynamic and static control performances (fast response, small overshoot, small steady-state error, etc.), and the demand of control performance of dynamic pressure cylinder electrohydraulic servo pressure system can be satisfied.
6. Conclusion
Based on the backstepping sliding mode adaptive control of the dynamic pressure cylinder, the RBF neural networks are introduced to approximate the uncertain terms f1 and f2. According to the double sliding surface, the RBF neural network weight vector adaptive laws of the displacement subsystem and the output pressure subsystem are, respectively, constructed, thus realizing the automatic updates of the displacement subsystem virtual controller e31 and the output pressure subsystem backstepping sliding mode controller u, reducing the difficulty of controller design.
Target variable Pa is set, the RBF neural network backstepping sliding mode adaptive algorithm is applied to the dynamic pressure cylinder AMESim model, and the control performances of the algorithm are simulated and analyzed. The results show that, compared with the backstepping sliding mode adaptive control (BSAC), the RBFNNBSAC algorithm has better dynamic and static performances and tracking performances, and it can effectively track the target expected variable Pr. Further, an interference signal is applied to the dynamic pressure cylinder, and the uncertainty term f2 RBFNN can quickly respond to the change of the interference signal, continuously adjusting the compensation amount of the RBFNN to the interference signal, so that the controller output u adaptive responded to the change of the interference signal, greatly reducing the influence of the interference signal on the tracking error, and had better anti-interference ability.
Finally, the triangular and square wave signals with amplitude 1.7e7Pa and frequency 10Hz are applied to the dynamic pressure cylinder AMESim model; the algorithm (RBFNNBSAC) and the backstepping sliding mode adaptive (BSAC) are simulated by contrast curves. It is found that RBFNNBSAC has better dynamic and static performances, and the control output is unsaturated and smoother, which can better track the desired pressure signal.
In future, we plan to apply the RBFNN backstepping sliding mode adaptive control algorithm to experimental platform of the track subgrade test device, and further optimize the control algorithm to improve the control performance of the device.
Data Availability
The readers can access the data used in this paper by contacting the corresponding author.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (51027002) and Wuhan Branch of Baosteel Central Research Institute (R&D Center of Wuhan Iron & Steel Co., Ltd.) of China Baowu Steel Group Corporation Limited [grant number K18BWBCA50].