Open AccessArticle

Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator

Zanwei Chen

¹,

Hao Yan

^1,*,

Peng Zhang

²,

Jiefeng Shan

¹ and

Jiafeng Li

School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China

Beijing Institute of Precision Mechatronics and Controls, Beijing 100076, China

Author to whom correspondence should be addressed.

Actuators 2024, 13(12), 471; https://doi.org/10.3390/act13120471

Submission received: 16 October 2024 / Revised: 17 November 2024 / Accepted: 21 November 2024 / Published: 22 November 2024

(This article belongs to the Section Control Systems)

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Figure 1
The schematic structure of the electro-hydraulic load simulator. "> Figure 2
A block diagram of the proposed control scheme. "> Figure 3
Test platform of electro-hydraulic load simulator. "> Figure 4
Force-tracking curve for large load. "> Figure 5
Control input for large load. "> Figure 6
Variation curve of <math display="inline"><semantics> <mover accent="true"> <mi>W</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> for large load. "> Figure 7
Variation curve of <math display="inline"><semantics> <mover accent="true"> <mi>ϑ</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> for large load. "> Figure 8
Force-tracking curve for small load. "> Figure 9
Control input for small load. "> Figure 10
Variation curve of <math display="inline"><semantics> <mover accent="true"> <mi>W</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> for small load. "> Figure 11
Variation curve of <math display="inline"><semantics> <mover accent="true"> <mi>ϑ</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> for small load. ">

Versions Notes

Abstract

To address the issues of derivative explosion in traditional backstepping control and the strong nonlinearity of hydraulic systems, this paper develops an adaptive neural network control method tailored for electro-hydraulic load simulators. Neural networks are employed to handle external disturbances, modeling uncertainties, and the derivatives of virtual control inputs. First, the precise state-space equations of the system are derived. Next, the approximation property of neural networks is used to design an adaptive backstepping controller, and the symmetric barrier Lyapunov function is used to prove the boundedness of the controller and control parameters. Finally, experiments are conducted to verify the effectiveness and reliability of the control algorithm. The results demonstrate that the proposed control algorithm exhibits excellent tracking performance and effectively reduces control errors.

Keywords:

electro-hydraulic load simulator; adaptive neural network control; barrier Lyapunov function (BLF)

1. Introduction

Electro-hydraulic load simulators with high reliability, fast response frequency, and large output capacity are widely used in the development of aerospace and naval control systems, particularly for replicating external forces and torque disturbances [1,2], such as in ship steering mechanisms and missile launching systems. Given their critical role in the defense sector, extensive research has been dedicated to electro-hydraulic load simulators. A significant amount of academic work has been previously conducted on structural improvements, which have enhanced the performance of electro-hydraulic load simulators. Notable advancements include dual-valve parallel control [3] and synchronized motor compensation [4]. As research in this area has reached its limitations, attention has shifted towards the optimization of control algorithms. Due to hydraulic systems being inherently complex and nonlinear, issues such as parameter uncertainty and load disturbances substantially impact the accuracy of load simulation. Therefore, resolving these uncertainties within control algorithms to improve loading precision remains a central focus of current research [5].

To enhance the tracking performance of nonlinear and uncertain load simulation systems, researchers focus on optimizing control algorithms, among which PID control [6] and ADRC control [7,8] are the most commonly used methods in engineering applications. In references [9,10], an improved PID algorithm which utilizes Particle Swarm Optimization (PSO) to adaptively optimize PID parameters was proposed. In [11,12], an Extended State Observer (ESO) was designed to observe load disturbances and uncertain nonlinear factors in load simulators, and combining backstepping control to improve tracking accuracy. In [13,14], an adaptive RBF neural network (RBFNN) was used to approximate and compensate model uncertainties and load disturbances in electro-hydraulic servo systems and was conjoined with adaptive terminal sliding mode control to effectively improve tracking performance. In [15,16], a multi-layer neural adaptive controller which was directly used to control force in hydraulic systems and improve the system’s disturbance resistance was proposed. In [17,18], command filter control was applied to hydraulic systems to address the “explosion of complexity” in backstepping control and was combined with nonlinear observers to achieve control objectives. In recent years, the barrier Lyapunov function has been widely used to ensure that nonlinear control systems can effectively maintain state and output constraints under preset conditions [19]. Due to its ease of integration with other control strategies, many researchers have combined it with backstepping control to achieve the desired tracking objectives [20,21,22].

To enhance robustness and improve the force-tracking capability, a disturbance rejection control strategy based on a state-and-disturbance observer which effectively achieves force tracking was proposed in [23]. However, it does not fully address the “explosion of complexity” issue inherent in backstepping control. In [18,24], the authors employed an observer approach to estimate the nonlinearity of the force system and combined it with command filtering control to compensate for filter errors. However, in practical applications, determining the appropriate bandwidth settings for the filter can be challenging. In [25,26], sliding mode control showed good performance in force control, but it was prone to chattering under disturbance impacts. Based on the above analysis, this article investigates an adaptive neural network tracking control algorithm based on a barrier Lyapunov function. The algorithm leverages the approximation capability of neural networks to identify nonlinear uncertainties caused by modeling errors and external disturbances acting on the control system and compensates for them. By combining the BLF with the neural network, the approach prevents constraint violations of tracking errors and the explosion of differentiation calculations in the backstepping iteration. Experimental results demonstrate the effectiveness and practicality of the proposed method.

The main contributions of this article are summarized as follows.

(1) By employing the symmetric barrier Lyapunov function in the control design, the adaptive neural network force control proposed in this study ensures that the system state remains free from disturbances and faults within any predefined boundary.

(2) Multi-layer neural networks are employed for fitting disturbance errors, friction, system leakage, and other uncertainties in motion processes and reducing the complexity of backstepping-based adaptive control.

2. System Modeling

The electro-hydraulic load simulator mainly consists of the testing system and the loading system, and the working principle is shown in Figure 1. The latter is the focus of this study; it primarily includes a servo valve and a hydraulic cylinder. The force sensor can provide a direct observation of the output of the loading system. The testing system outputs a displacement signal to drive the movement of the loading system. The loading system controls its servo valve spool moving to adjust the pressure and flow rates in both chambers of the hydraulic cylinder, thereby achieving force control while ensuring that the desired motion trajectory is followed.

The force balance equation for the hydraulic cylinder in motion can be expressed as follows:

F_{P} = P_{1} A_{1} - P_{2} A_{2} = m {\ddot{x}}_{L} + B {\dot{x}}_{L} + F_{f} + F_{s}

(1)

where

F_{P}

is the hydraulic cylinder force;

P_{1}

and

P_{2}

are the rodless cavity pressure and the rod cavity pressure;

A_{1}

and

A_{2}

are the areas of the rodless cavity and the rod cavity; m is the equivalent total mass;

x_{L}

is the piston rod displacement; K and B are the equivalent spring stiffness and viscous damping coefficients; and

F_{f}

and

F_{S}

are the friction and the output of the force sensor.

The actual output force of the system is measured by the force transducer and can be expressed mathematically as

F_{s} = K (x_{L} - x_{R})

(2)

where

x_{R}

is the displacement of the loaded system, which also means the positional turmoil of the loaded system.

For servo valves with high response frequencies, the spool displacement can be approximated as being proportional to the control volume:

x_{v} = K_{s v} u

(3)

where

K_{s v}

is the gain of the servo valve.

The load flow controlled by the spool displacement can be described by the following equation:

Q_{1} = C_{d} w K_{sv} u \sqrt{\frac{2}{ρ} (\frac{(1 + ϕ (x_{v})) P_{s}}{2} - ϕ (x_{v}) (P_{1} - P_{r}))}

(4)

Q_{2} = C_{d} w K_{s v} u \sqrt{\frac{2}{ρ} (\frac{(1 - ϕ (x_{v})) P_{s}}{2} + ϕ (x_{v}) (P_{2} - P_{r}))}

(5)

where

P_{s}

is the supply pressure of the hydraulic system,

ρ

is the oil density, and

ϕ (•)

ϕ (•) = \{\begin{matrix} 1, & i f • > 0 \\ - 1, & i f • < 0 \end{matrix}

(6)

The asymmetric cylinder flow continuity equation can be written as

{\dot{P}}_{1} = \frac{β_{e}}{V_{1}} (Q_{1} - C_{i p} (P_{1} - P_{2}) - A_{1} {\dot{x}}_{L} - d_{1} (t))

(7)

{\dot{P}}_{2} = \frac{β_{e}}{V_{2}} (- Q_{2} + C_{ip} (P_{1} - P_{2}) - A_{2} {\dot{x}}_{L} + d_{2} (t))

(8)

where

β_{e}

is the modulus of elasticity of the hydraulic fluid;

C_{i p}

is the internal leakage coefficient;

d_{1} (t)

and

d_{2} (t)

are unmodeled errors; and

V_{1}

and

V_{2}

are the volumes of the rodless cavity and the rod cavity, which can be calculated by

\{\begin{matrix} V_{1} = V_{10} + A_{1} x_{L} \\ V_{2} = V_{20} - A_{2} x_{L} \end{matrix}

(9)

where

V_{10}

and

V_{20}

are the initial volumes.

By assuming that the return pressure is zero and defining the system state variable as

x = {[x_{1}, x_{2}, x_{3}]}^{T} = {[F_{s}, {\dot{x}}_{L}, F_{P}]}^{T}

, the state equation can be expressed as

\{\begin{matrix} {\dot{x}}_{1} = K x_{2} - K {\dot{x}}_{R} \\ {\dot{x}}_{2} = \frac{1}{m} x_{3} - \frac{B}{m} x_{2} - \frac{1}{m} x_{1} + F_{f} \\ {\dot{x}}_{3} = ψ u - f_{3} \end{matrix}

(10)

where

\begin{matrix} f_{3} = - (\frac{A_{1}^{2}}{V_{1}} + \frac{A_{2}^{2}}{V_{2}}) β_{e} x_{2} - (\frac{A_{1}}{V_{1}} + \frac{A_{2}}{V_{2}}) β_{e} C_{i p} (P_{1} - P_{2}) + \frac{β_{e} A_{1}}{V_{1}} d_{1} (t) + \frac{β_{e} A_{2}}{V_{2}} d_{2} (t) \end{matrix}

ψ = (\frac{A_{1} R_{1}}{V_{1}} + \frac{A_{2} R_{2}}{V_{2}}) β_{e} C_{d} ω \sqrt{\frac{2}{ρ}} K_{s v}

Assumption 1.

The existing states are constrained and can be expressed as

|x_{i}| \leq k_{b i}

Lemma 1

([27]). If there exist nonnegative real numbers a and b, c > 1, d is a nonzero real number, and (1/c) + (1/d) = 1, then the following expression holds:

a b \leq \frac{a^{c}}{c} + \frac{b^{d}}{d} .

(11)

In solving parameter uncertainties and positional perturbations in electro-hydraulic load simulators, a neural network is an effective tool for adaptive control design. The following neural network [28] will be used to approximate the unknown smooth function:

U (Z) = W^{T} σ (ϑ^{T} x)

(12)

where

σ (x) = 1 / (1 + e^{- 10 x})

Lemma 2

([29]). If a neural network can be described as (12), then the estimation error can be descripted as

{\hat{W}}^{T} σ ({\hat{ϑ}}^{T} \bar{z}) - W^{* T} σ (ϑ^{* T} \bar{z}) = {\tilde{W}}^{T} (σ ({\hat{ϑ}}^{T} \bar{z}) - {\hat{σ}}^{'} {\hat{ϑ}}^{T} \bar{z}) + {\hat{W}}^{T} {\hat{σ}}^{'} {\tilde{ϑ}}^{T} \bar{z} + d_{r}

(13)

where the boundary of

d_{r}

can be defined as

|d_{r}| \leq {∥ϑ^{*}∥}_{F} {∥\bar{z} {\hat{W}}^{T} {\hat{σ}}^{'}∥}_{F} + ∥W^{*}∥ ∥{\hat{σ}}^{'} {\hat{ϑ}}^{T} \bar{z}∥ + ∥W^{*}∥ ∥σ ({\hat{ϑ}}^{T} \bar{z}) - σ (ϑ^{* T} \bar{z})∥

(14)

In order to understand the design process of control systems in Section 3 better, the definitions of some parameters are provided in Table 1.

3. Adaptive Neural Network Controller Design

In this section, an adaptive neural network control method combined with backstepping is proposed for the electro-hydraulic load simulator in (10). The structures of virtual controller

α_{i}

, actual controller u, and adaptive law

W_{i}

ϑ_{i}

are designed, and Figure 2 shows the control flow chart.

To facilitate controller design, it is crucial to clearly define the following variables:

z_{1} = x_{1} - y_{d}

(15)

z_{i} = x_{i} - α_{i - 1} .

(16)

Step 1: Design of first-order virtual control law

α_{1}

The derivative of

z_{1}

can be expressed as

{\dot{z}}_{1} = K x_{2} - K {\dot{x}}_{R} - {\dot{y}}_{d}

(17)

To incorporate the unknown positional turmoil function related to

x_{R}

and

y_{d}

, we utilize a multi-layer neural network approximation, which is defined as follows:

F_{1} ({\bar{z}}_{1}) = W_{1}^{* T} σ_{1} (ϑ_{1}^{* T} {\bar{z}}_{1}) - δ_{1} ({\bar{z}}_{1})

(18)

where

{\bar{z}}_{1} = [{\dot{x}}_{r}

{\dot{y}}_{d}]

and

δ_{1} ({\bar{z}}_{1})

represents the approximation error and satisfies

|δ_{1} ({\bar{z}}_{1})| \leq {\bar{δ}}_{1}

The virtual control law

α_{1}

can be designed as

\frac{α_{1}}{K} = - k_{1} z_{1} + {\dot{y}}_{d} + {\hat{W}}_{1}^{T} σ_{1} ({\hat{ϑ}}_{1}^{T} {\bar{z}}_{1}) - φ_{1} (‖ {\bar{z}}_{1} {\hat{W}}_{1}^{T} {\hat{σ}}_{1}^{'} ‖_{F}^{2} + ‖ {\hat{σ}}_{1}^{'} {\hat{ϑ}}_{1}^{T} {\bar{z}}_{1} ‖^{2} + 1 + \frac{1}{2 a_{1}})

(19)

where

φ_{1} = \frac{z_{1}}{k_{b 1}^{2} - z_{1}^{2}}

The construction of NN weight adaptive update laws is designed as

{\dot{\hat{W}}}_{1} = - F_{1} [φ_{1} ({\hat{σ}}_{1} - {\hat{σ}}_{1}^{'} {\hat{ϑ}}_{1}^{T} {\bar{z}}_{1}) + o_{1} {\hat{W}}_{1}]

(20)

{\dot{\hat{ϑ}}}_{1} = - Γ_{1} (φ_{1} {\bar{z}}_{1} {\hat{W}}_{1}^{T} {\hat{σ}}_{1}^{'} + γ_{1} {\hat{ϑ}}_{1})

(21)

where

o_{1}

and

γ_{1}

are designed normal numbers, and

Γ_{1} = Γ_{1}^{T} > 0

and

F_{1} = F_{1}^{T} > 0

are parameters that need to be designed.

The time-varying symmetric barrier Lyapunov function is defined as

V_{1} = \frac{1}{2} l n (\frac{k_{b_{1}}^{2}}{k_{b_{1}}^{2} - z_{1}^{2}}) + \frac{1}{2} {\tilde{W}}_{1}^{T} F_{1}^{- 1} {\tilde{W}}_{1} + \frac{1}{2} t r \{{\tilde{ϑ}}_{1}^{T} Γ_{1}^{- 1} {\tilde{ϑ}}_{1}\}

(22)

where

{\tilde{W}}_{1} = {\hat{W}}_{1} - W_{1}^{*}

and

{\tilde{ϑ}}_{1} = {\hat{ϑ}}_{1} - ϑ_{1}^{*}

represent the estimation errors of the network weights.

Bringing (19)–(21) into the derivation of (22) yields

{\dot{V}}_{1} \leq φ_{1} (k z_{2} + k α_{1} - W_{1}^{* T} σ_{1} (ϑ^{*} {\bar{z}}_{1})) - \frac{φ_{1}^{2}}{2 a_{1}} + \frac{1}{2} α_{1} {\bar{δ}}_{1}^{2} + {\tilde{W}}_{1}^{T} F_{1}^{- 1} {\dot{\hat{W}}}_{1} + tr \{{\tilde{ϑ}}_{1}^{T} Γ_{1}^{- 1} {\dot{\hat{ϑ}}}_{1}\} .

(23)

Step 2: Design of virtual control law

α_{2}

The derivative of

z_{2}

can be expressed as

{\dot{z}}_{2} = \frac{x_{3}}{m} - \frac{B}{m} x_{2} - \frac{1}{m} x_{1} + F_{f} - {\dot{α}}_{1}

(24)

where

{\dot{α}}_{1} = \frac{\partial α_{1}}{\partial x_{1}} (K x_{2} - K {\dot{x}}_{r}) + \sum_{j = 0}^{1} \frac{\partial α_{1}}{\partial y_{d}^{(j)}} y_{d}^{(j + 1)} + \frac{\partial α_{1}}{\partial {\hat{W}}_{1}} {\dot{\hat{W}}}_{1} + \frac{\partial α_{1}}{\partial {\hat{\hat{ϑ}}}_{1}} {\dot{\hat{ϑ}}}_{1} .

We use the multi-layer neural network approximation external disturbance:

F_{2} ({\bar{z}}_{2}) = W_{2}^{* T} σ_{2} (ϑ_{2}^{* T} {\bar{z}}_{2}) + δ_{2} ({\bar{z}}_{2})

(25)

where

{\bar{z}}_{2} = [x_{r}, y_{d}, {\dot{y}}_{d}, x_{2}, x_{1}, F_{f}]

and

δ_{2} ({\bar{z}}_{2})

represents the approximation error and satisfies

|δ_{2} ({\bar{z}}_{2})| \leq {\bar{δ}}_{2}

The second-order virtual control law

α_{2}

can be designed as

m α_{2} = - k_{2} z_{2} + {\hat{W}}_{2}^{T} {\hat{σ}}_{2} ({\hat{ϑ}}_{2}^{T} {\bar{z}}_{2}) - \frac{ν_{1}}{ν_{2}} z_{1} - φ_{2} (‖ {\bar{z}}_{2} {\hat{W}}_{2}^{T} {\hat{σ}}_{2}^{'} ‖_{F}^{2} + ‖ {\hat{σ}}_{2}^{'} {\hat{F}}_{2}^{T} {\bar{z}}_{2} ‖^{2} + 1 + \frac{1}{2 a_{2}})

(26)

where

ν_{1} = \frac{K}{k_{b 1}^{2} - z_{1}^{2}}

ν_{2} = \frac{1}{m (k_{b 2}^{2} - z_{2}^{2})}

, and

φ_{2} = \frac{z_{2}}{k_{b 2}^{2} - z_{21}^{2}}

The construction of weight adaptive update laws is designed as

{\dot{\hat{W}}}_{2} = - F_{2} [φ_{2} ({\hat{σ}}_{2} - {\hat{σ}}_{2}^{'} {\hat{ϑ}}^{T} {\bar{z}}_{2}) + o_{2} {\hat{W}}_{2}]

(27)

{\dot{\hat{ϑ}}}_{2} = - Γ_{2} (φ_{2} {\bar{z}}_{2} {\hat{W}}_{2}^{T} {\hat{σ}}_{2}^{'} + γ_{2} {\hat{ϑ}}_{2})

(28)

where

o_{2}

and

γ_{2}

are designed normal numbers, and

Γ_{2} = Γ_{2}^{T} > 0

and

F_{2} = F_{2}^{T} > 0

are parameters that need to be designed.

We have the second-order barrier Lyapunov function as

V_{2} = V_{1} + \frac{1}{2} l n (\frac{k_{b 2}^{2}}{k_{b 2}^{2} - z_{2}^{2}}) + \frac{1}{2} {\tilde{W}}_{2}^{T} F_{2}^{- 1} {\tilde{W}}_{2} + \frac{1}{2} t r \{{\tilde{ϑ}}_{2}^{T} Γ_{2}^{- 1} {\tilde{ϑ}}_{2}\}

(29)

where

{\tilde{W}}_{2} = {\hat{W}}_{2} - W_{2}^{*}

and

{\tilde{ϑ}}_{2} = {\hat{ϑ}}_{2} - ϑ_{2}^{*}

represent the estimation errors of the network weights.

Substituting (26) and (28) into the equation after deriving (29) yields

{\dot{V}}_{2} = {\dot{V}}_{1} + φ_{2} (z_{3} / m - W_{2}^{* T} σ_{2} (ϑ_{2}^{* T} {\bar{z}}_{2}) + δ_{2} ({\bar{z}}_{2}) + α_{2} / m) + {\tilde{W}}_{2}^{T} F_{2}^{- 1} {\dot{\hat{W}}}_{2} + tr \{{\tilde{ϑ}}_{2}^{T} Γ_{2}^{- 1} {\dot{\hat{ϑ}}}_{2}\} .

(30)

Step 3: Design of virtual control law

α_{3}

The derivative of

z_{3}

can be expressed as

{\dot{z}}_{3} = ψ u - f_{3} - {\dot{α}}_{2}

(31)

where

{\dot{α}}_{2} = \sum_{j = 1}^{2} \frac{\partial α_{2}}{\partial x_{j}} {\dot{x}}_{j} + \sum_{j = 0}^{2} \frac{\partial α_{1}}{\partial y_{d}^{(j)}} y_{d}^{(j + 1)} + \sum_{j = 1}^{2} \frac{\partial α_{2}}{\partial {\hat{W}}_{j}} {\dot{\hat{W}}}_{j} + \sum_{j = 1}^{2} \frac{\partial α_{2}}{\partial {\hat{ϑ}}_{j}} {\dot{\hat{ϑ}}}_{j} .

The unknown function

F_{3} ({\bar{z}}_{3})

is defined by

F_{3} ({\bar{z}}_{3}) = W_{3}^{* T} σ_{3} (ϑ_{3}^{* T} {\bar{z}}_{3}) + δ_{3} ({\bar{z}}_{3})

(32)

where

{\bar{z}}_{3} = [x_{r}, y_{d}, {\dot{y}}_{d}, {\ddot{y}}_{d}, x_{2}, x_{1}, p_{2}, p_{1}]

and

δ_{3} ({\bar{z}}_{3})

represents the estimation error of the network weights

|δ_{3} ({\bar{z}}_{3})| \leq {\bar{δ}}_{3}

The final control law u can be designed as follows:

\frac{u}{ψ} = - k_{3} z_{3} + {\hat{W}}_{3}^{T} {\hat{σ}}_{3} - \frac{ν_{2}}{ν_{3}} z_{3} - φ_{3} (‖ {\bar{z}}_{3} {\hat{W}}_{3}^{T} {\hat{σ}}_{3}^{'} ‖_{F}^{2} + ‖ {\hat{S}}_{3}^{'} {\hat{V}}_{3}^{T} {\bar{z}}_{3} ‖^{2} + 1 + \frac{1}{2 a_{3}})

(33)

where

v_{3} = \frac{ψ}{k_{b 3}^{2} - z_{3}^{2}}

φ_{3} = \frac{z_{3}}{k_{b 3}^{2} - z_{3}^{2}}

The adaptation laws are constructed as

{\dot{\hat{W}}}_{3} = - F_{3} [φ_{3} ({\hat{σ}}_{3} - {\hat{σ}}_{3}^{'} {\hat{ϑ}}^{T} {\bar{z}}_{3}) + o_{3} {\hat{W}}_{2}]

(34)

{\dot{\hat{ϑ}}}_{3} = - Γ_{3} (φ_{3} {\bar{z}}_{3} {\hat{W}}_{3}^{T} {\hat{σ}}_{3}^{T} + γ_{3} {\hat{ϑ}}_{3})

(35)

where

o_{3}

and

γ_{3}

are designed normal numbers, and

Γ_{3} = Γ_{3}^{T} > 0

and

F_{3} = F_{3}^{T} > 0

are parameters that need to be designed.

We consider a barrier Lyapunov function candidate as

V_{3} = V_{2} + \frac{1}{2} l n (\frac{k_{b 3}^{2}}{k_{b 3}^{2} - z_{3}^{2}}) + \frac{1}{2} {\tilde{W}}_{3}^{T} F_{3}^{- 1} {\tilde{W}}_{3} + \frac{1}{2} t r \{{\tilde{ϑ}}_{3}^{T} Γ_{3}^{- 1} {\tilde{ϑ}}_{3}\}

(36)

where

{\tilde{W}}_{3} = {\hat{W}}_{3} - W_{3}^{*}

and

{\tilde{ϑ}}_{3} = {\hat{ϑ}}_{3} - ϑ_{3}^{*}

represent the estimation errors of the network weights.

Substituting (33)–(35) after deriving (36) yields

{\dot{V}}_{3} = {\dot{V}}_{2} + φ_{3} (ψ u - W_{3}^{* T} σ_{3} (ϑ_{3}^{* T} {\bar{z}}_{3}) + δ_{3} ({\bar{z}}_{3}) - {\dot{α}}_{2}) + {\tilde{W}}_{3}^{T} F_{3}^{- 1} {\dot{\hat{W}}}_{3} + tr \{{\tilde{ϑ}}_{3}^{T} Γ_{3}^{- 1} {\dot{\hat{ϑ}}}_{3}\} .

(37)

This can be derived by applying Young’s inequality:

φ_{i} δ_{i} ({\bar{z}}_{i}) \leq \frac{φ_{i}^{2}}{2 a_{i}} + \frac{1}{2} a_{i} {\bar{δ}}_{i}^{2} .

(38)

Substituting (38) into (37) gives

\begin{matrix} {\dot{V}}_{3} \leq & - \sum_{i = 3}^{3} η_{i} φ_{i} z_{i} - \sum_{i = 3}^{3} d_{r i} + \sum_{i = 3}^{3} \frac{1}{2} a_{i} {\bar{δ}}_{i}^{2} - \sum_{i = 3}^{3} o_{i} {\tilde{W}}_{i}^{T} {\hat{W}}_{i} - \sum_{i = 3}^{3} γ_{i} tr \{{\tilde{ϑ}}_{i}^{T} {\tilde{ϑ}}_{i}\} \\ - \sum_{i = 3}^{3} φ_{i} (‖ {\bar{z}}_{i} {\hat{W}}_{i}^{T} {\hat{σ}}_{i}^{'} ‖_{P}^{2} + ‖ {\hat{σ}}_{i}^{'} {\hat{ϑ}}_{i}^{'} {\bar{z}}_{i} ‖^{2} + 1) \end{matrix}

(39)

where

η_{1} = K k_{1}, η_{2} = \frac{k_{2}}{m},

and

η_{3} = ψ k_{3}

This can be obtained by using Young’s inequality:

- o_{i} {\tilde{W}}_{i}^{T} \hat{W} = - o_{i} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} - o_{i} {\tilde{W}}_{i}^{T} W_{i}^{*} \leq - \frac{1}{2} o_{i} {\tilde{W}}_{i}^{2} + \frac{1}{2} o_{i} W_{i}^{* 2}

(40)

\begin{matrix} - γ_{i} tr \{{\tilde{ϑ}}_{i}^{T} {\hat{ϑ}}_{i}\} = - γ_{i} tr \{{\tilde{ϑ}}_{i}^{T} {\tilde{ϑ}}_{i} + {\tilde{ϑ}}_{i}^{T} ϑ_{i}^{*}\} \\ \leq - γ_{i} ‖ {\tilde{ϑ}}_{i} ‖_{F}^{2} - γ_{i} ‖ {\tilde{ϑ}}_{i} ‖_{F} ‖ ϑ_{i}^{*} ‖_{F} \leq - \frac{1}{2} γ_{i} ‖ {\tilde{ϑ}}_{i} ‖_{F}^{2} + \frac{1}{2} γ_{i} {‖ ϑ_{i}^{*} ‖}_{F}^{2} \end{matrix}

(41)

φ_{i} ‖ ϑ_{i}^{*} ‖_{F} ‖ {\bar{z}}_{i} {\hat{W}}_{i}^{T} {\hat{σ}}_{i}^{'} ‖_{F} \leq φ_{i}^{2} ‖ {\bar{z}}_{i} {\hat{W}}_{i}^{T} {\hat{σ}}_{i}^{'} ‖_{F}^{2} + \frac{1}{4} {‖ ϑ_{i}^{*} ‖}_{F}^{2}

(42)

φ_{i} ‖ W_{i}^{*} ‖ ‖ {\hat{σ}}_{i}^{'} {\hat{ϑ}}_{i}^{T} {\bar{z}}_{i} ‖ \leq φ_{i}^{2} ‖ {\hat{σ}}_{i}^{'} {\hat{ϑ}}_{i}^{T} {\bar{z}}_{i} ‖^{2} + \frac{1}{4} {‖ W_{i}^{*} ‖}^{2}

(43)

φ_{i} ‖ W_{i}^{*} ‖ ‖ σ ({\hat{ϑ}}^{T} \bar{z}) - σ (ϑ^{* T} \bar{z}) ‖ \leq φ_{i} ‖ W_{i}^{*} ‖ \leq φ_{i}^{2} + \frac{1}{4} {‖ W_{i}^{*} ‖}^{2} .

(44)

Substituting (40)–(44) into (37) gives

{\dot{V}}_{3} \leq - \sum_{i = 1}^{3} η_{i} φ_{i} z_{i} + \sum_{i = 1}^{3} \frac{1}{2} α_{i} {\bar{δ}}_{i}^{2} + \sum_{i = 1}^{3} \frac{1}{2} ∥ W_{i}^{*} ∥^{2} + \sum_{i = 1}^{3} \frac{1}{4} ∥ ϑ_{i}^{*} ∥_{F}^{2} + \sum_{i = 1}^{3} \frac{1}{2} o_{i} W_{i}^{* 2} + \sum_{i = 1}^{3} \frac{1}{2} γ_{i} {∥ ϑ_{i}^{*} ∥}_{F}^{2} .

(45)

Then, the following inequality holds:

{\dot{V}}_{3} \leq - A V_{3} + B

(46)

where

B = \sum_{i = 3}^{3} \frac{1}{2} a_{i} {\bar{δ}}_{i}^{2} + \sum_{i = 3}^{3} \frac{1}{2} ∥ W_{i}^{*} ∥^{2} + \sum_{i = 3}^{3} \frac{1}{4} ∥ ϑ_{i}^{*} ∥_{F}^{2} + \sum_{i = 3}^{3} \frac{1}{2} o_{i} W_{i}^{* 2} + \sum_{i = 3}^{3} \frac{1}{2} γ_{i} {∥ ϑ_{i}^{*} ∥}_{F}^{2}

and

A = min \{2 η_{i}, o_{i} λ (F_{i}), γ_{i} λ (Γ_{i})\}

By solving (46), we can obtain

V_{3} \leq \frac{B}{A} + C e^{- A t} .

(47)

Inequality (47) can be rewritten as an equality, and we can obtain

C = V_{3} (0) - \frac{B}{A}

by substituting t=0 into the equality. Then, (47) can be described as

V_{3} \leq (V_{3} (0) - \frac{B}{A}) e^{- A t} + \frac{B}{A} \leq V_{3} (0) + \frac{B}{A} .

(48)

By combining Equations (22) and (29), (36) can be rewritten as

V_{3} = \sum_{i = 1}^{3} \frac{1}{2} ln (\frac{k_{b i}^{2}}{k_{b i}^{2} - z_{i}^{2}}) + \sum_{i = 1}^{3} \frac{1}{2} {\tilde{W}}_{i}^{T} F_{i}^{- 1} {\tilde{W}}_{i} + \sum_{i = 1}^{3} \frac{1}{2} {\tilde{ϑ}}_{i}^{T} Γ_{i}^{- 1} {\tilde{ϑ}}_{i}

(49)

Then, the following inequality holds:

\frac{1}{2} log (\frac{k_{b i}^{2}}{k_{b i}^{2} - z_{i}^{2}}) \leq V_{3} (0) e^{- A t} + \frac{B}{A}

(50)

\frac{1}{2} {\tilde{W}}_{i}^{T} F_{i}^{- 1} {\tilde{W}}_{i} \leq V_{3} (0) e^{- A t} + \frac{B}{A}

(51)

\frac{1}{2} {\tilde{ϑ}}_{i}^{T} Γ_{i}^{- 1} {\tilde{ϑ}}_{i} \leq V_{3} (0) e^{- A t} + \frac{B}{A}

(52)

Thus, the bounds of the error as well as the bounds of the weight matrix can be obtained:

|\begin{matrix} z_{i} \end{matrix}| \leq k_{b i} \sqrt{1 - e^{- 2 V_{3} (0) e^{- A t} + \frac{2 B}{A}}}

(53)

∥{\tilde{W}}_{i}∥ \leq \sqrt{2 λ_{max} (F_{i}) (V_{3} (0) e^{- A t} + \frac{B}{A})}

(54)

∥{\tilde{ϑ}}_{i}∥ \leq \sqrt{2 λ_{max} (Γ_{i}) (V_{3} (0) e^{- A t} + \frac{B}{A})} .

(55)

At this point, we have completed the design of the controller and proved its stability by a barrier Lyapunov function. From the entire design process, it is evident that using neural networks to approximate the system’s nonlinearities and the derivative terms in traditional backstepping control is a feasible approach. Moreover, this method significantly reduces the complexity of controller design.

4. Experimental Verification

In the above sections, we have completed the design and stability analysis of the algorithm. To validate the effectiveness and reliability of the control algorithm in an actual hydraulic system, a test platform was constructed, as illustrated in Figure 3. The testing system was tasked with applying various position commands, while the loading system was responsible for force control in the presence of position turmoil.

The test platform consisted of a hydraulic cylinder, a servo valve (with a maximum flow rate of 100 L/min), a pump station (with a maximum oil supply pressure of 35 MPa), a force sensor (Interface 1060-1500), two pressure sensors (KYB2003 P05VP4C2-I), a control cabinet, and a testing system. The measurement and control system included an Advantech IPC-610L, control software, an Advantech PCI-1742U multifunction data acquisition card, an AITI CPCI-9840 CAN communication card, and a voltage conversion module. The control software was developed in the QT 5.12.12 environment, with the system sampling rate fixed at 0.001 s. The platform parameters are shown in Table 2.

4.1. Experimental Scenario of Large Load

Under this loading condition, the force command for the loading system was set to

F_{s} = 200 sin (0.25 π t)

, and the testing system operated at a velocity of

- 0.01 m / s

. For comparison, a PID control algorithm based on velocity feedforward was employed, denoted by “PID” in the result figures. During the experiment, multiple trials were conducted, and the best experimental results were selected for presentation along with the recorded control parameters. The PID control parameters were as follows:

P = 0.036

I = 0.003

, and

D = 0.01

, with a feedforward coefficient of 0.081. The integral limit for the PID algorithm was set to 200. The parameters for the control algorithm were defined as

k_{b 1} = k_{b 3} = 400 + 100 e^{- 2 * t} - 10 e^{- sin (π t / 4)}

k_{b 2} = 0.5 + e^{- 5 t}

o_{1} = 0.35

o_{2} = 0.3

o_{3} = 0.4

γ_{1} = 0.3

γ_{2} = 0.3

γ_{3} = 0.3

a_{1} = 0.3

a_{2} = 0.5

a_{3} = 0.4

{\hat{W}}_{1} (0) = 1.16

{\hat{W}}_{2} (0) = 0.6

{\hat{W}}_{3} (0) = 0.91

{\hat{ϑ}}_{1} (0) = {[0.2 - 0.2]}^{T}

{\hat{ϑ}}_{2} (0)

= [0.2 −0.15 0.1 0.1 0.0001 0.3 −0.002

]^{T}

{\hat{ϑ}}_{3} (0)

= [0.2 −0.15 0.1 0.1 0.0001 0.3 −0.002 0.0002 0.0002

]^{T}

, and

Γ_{i} = 1.5 I

F_{i} = 2.5 I

. This condition is considered a high-output loading condition. Figure 4 illustrates the tracking curve in the upper part and the error curve in the lower part. It can be observed that the proposed control algorithm achieves higher control accuracy, with a maximum error of 19.362 kN, compared with 20.164 kN with “PID” control. The presence of a dead zone in the servo valve leads to a discontinuous displacement of the valve core when the force direction changes, resulting in force impact phenomena. The proposed control algorithm effectively reduces control errors, demonstrating the effectiveness of the adaptive neural network in the control algorithm. Ultimately, the proposed algorithm achieves a control error within 7%, while the PID control based on the velocity feedforward does not exhibit comparable performance. In Figure 5, the control signal remains within the range of 1.5 V to 4 V. The variation curves of the weighting parameters

\hat{W}

and

\hat{ϑ}

are shown in Figure 6 and Figure 7.

4.2. Experimental Scenario of Small Load

Under this loading condition, the force command for the loading system was set to

F_{s} = 100 sin (0.25 π t)

, and the testing system operated at a velocity of

- 0.01 m / s

. The PID control parameters were as follows:

P = 0.03

I = 0.003

, and

D = 0.01

, with a feedforward coefficient of 0.076. The integral limit for the PID algorithm was set to 200. The parameters for the control algorithm were defined as

k_{b 1} = k_{b 3} = 200 + 100 e^{- 2 * t} - 10 e^{- sin (π t / 4)} - 5 e^{- sin (π t / 2)}

k_{b 2} = 0.5 + e^{- 5 t}

o_{1} = 0.55

o_{2} = 0.42

o_{3} = 0.35

γ_{1} = 0.2

γ_{2} = 0.2

γ_{3} = 0.2

a_{1} = 0.3

a_{2} = 0.35

a_{3} = 0.35

{\hat{W}}_{1} (0) = 1.07

{\hat{W}}_{2} (0) = 0.58

{\hat{W}}_{3} (0) = 0.91

{\hat{ϑ}}_{1} (0) = {[0.2 - 0.2]}^{T}

{\hat{ϑ}}_{2} (0)

= [0.2 −0.15 0.1 0.1 0.00015 0.3 −0.0015

]^{T}

{\hat{ϑ}}_{3} (0)

= [0.2 −0.15 0.1 0.1 0.00015 0.3 −0.0018 0.0002 0.0002

]^{T}

Γ_{i} = 2 I

, and

F_{i} = 2 I

. This condition is considered a low-output loading condition. Figure 8 shows the tracking curve in the upper part and the error curve in the lower part. It can be observed that the proposed control algorithm achieves high control accuracy, with a maximum error of 7.924 kN, compared with 16.834 kN with PID control. Compared with high-output conditions, the controller demonstrates superior performance under low-output conditions. Specifically, it exhibits faster response to directional impact and smoother control curves. In Figure 9, the control volume is maintained between 2.2 V and 3 V throughout the test. The variation curves of the weight parameters

\hat{W}

and

\hat{ϑ}

are depicted in Figure 10 and Figure 11.

5. Conclusions

In this study, an adaptive neural network (NN) control algorithm is proposed for force control in an electro-hydraulic load simulator, addressing the issues of derivative explosion in traditional backstepping control and the strong nonlinearity of hydraulic systems. Compared with conventional state-constrained control theories, the constraint boundaries considered in this work are time-varying functions related to state variables. To address this, a symmetric Barrier Lyapunov Function (BLF) is constructed to prevent state violations of constraint bounds. A multilayer neural network is employed to approximate the unknown model dynamics arising from lumped uncertainties, such as uncertain hydraulic parameters and unknown external loads. Furthermore, Lyapunov stability theory is used to analyze system stability, ensuring that all signals in the closed-loop system remain bounded. The designed controller achieves system performance such that the output closely tracks the target trajectory, with the tracking error converging to a small bounded range. Experimentally validated, the proposed control algorithm effectively suppresses force-commutation-induced shocks, ensuring stable operation of the electro-hydraulic load simulator. For future research, designing adaptive controllers and adaptive laws for multi-input multi-output nonlinear systems to ensure stability will be one of our key directions.

Author Contributions

Conceptualization, Z.C. and J.L.; methodology, Z.C. and H.Y.; experimental Z.C., P.Z. and J.S.; review and editing, Z.C. and H.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available upon request from the corresponding author.

Acknowledgments

We thank all the authors for their contributions and efforts and the anonymous reviewers for their valuable comments.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The schematic structure of the electro-hydraulic load simulator.

Figure 2. A block diagram of the proposed control scheme.

Figure 3. Test platform of electro-hydraulic load simulator.

Figure 4. Force-tracking curve for large load.

Figure 5. Control input for large load.

Figure 6. Variation curve of

\hat{W}

for large load.

Figure 6. Variation curve of

\hat{W}

for large load.

Figure 7. Variation curve of

\hat{ϑ}

for large load.

Figure 7. Variation curve of

\hat{ϑ}

for large load.

Figure 8. Force-tracking curve for small load.

Figure 9. Control input for small load.

Figure 10. Variation curve of

\hat{W}

for small load.

Figure 10. Variation curve of

\hat{W}

for small load.

Figure 11. Variation curve of

\hat{ϑ}

for small load.

Figure 11. Variation curve of

\hat{ϑ}

for small load.

Table 1. Nomenclature.

Parameter	Definition	Parameter	Definition
$k_{s v}$	Servo valve gain	$F_{f}, F_{S}$	Friction and output of force sensor
u	Control voltage	$F_{P}$	Loading force
$ω$	Area gradient of servo valve	$C_{d}$	Flow coefficient
$P_{1}, P_{2}$	Pressure of rodless cavity and rod cavity	$x, B, K$	Load mass, damping coefficient, and load spring constant
$A_{1}, A_{2}$	Area of rodless cavity and rod cavity	$x_{L}, y_{d}$	Actual and desired force of loading cylinder
$x_{v}$	Spool displacement	$x_{R}$	Displacement of tested cylinder
$k_{i}, γ_{i}, o_{i} (i = 1, 2, 3)$	Positive constants of control	$k_{b i} (i = 1, 2, 3)$	Positive function of states
$z_{i} (i = 1, 2, 3)$	State errors	$α_{i} (i = 1, 2)$	ith-Order virtual control
$Γ_{i}, F_{i} (i = 1, 2, 3)$	Constants of control	$W_{i}, ϑ_{i} (i = 1, 2, 3)$	Weight matrix

Table 2. Test platform parameters.

Parameter	Value	Unit
$P_{s}$	21	MPa
$β_{e}$	$6.9 \times 10^{8}$	Pa
B	2000	$N \cdot s / m$
$C_{i p}$	$2 \times 10^{- 15}$	$m^{5} / (N \cdot s)$
D	0.348	m
d	0.180	m
L	0.7	m
m	300	kg
K	$1 \times 10^{8}$	$N / m$
$K_{s v}$	$1.25 \times 10^{- 5}$

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Chen, Z.; Yan, H.; Zhang, P.; Shan, J.; Li, J. Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator. Actuators 2024, 13, 471. https://doi.org/10.3390/act13120471

AMA Style

Chen Z, Yan H, Zhang P, Shan J, Li J. Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator. Actuators. 2024; 13(12):471. https://doi.org/10.3390/act13120471

Chicago/Turabian Style

Chen, Zanwei, Hao Yan, Peng Zhang, Jiefeng Shan, and Jiafeng Li. 2024. "Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator" Actuators 13, no. 12: 471. https://doi.org/10.3390/act13120471

APA Style

Chen, Z., Yan, H., Zhang, P., Shan, J., & Li, J. (2024). Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator. Actuators, 13(12), 471. https://doi.org/10.3390/act13120471

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Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator

Abstract

1. Introduction

2. System Modeling

3. Adaptive Neural Network Controller Design

4. Experimental Verification

4.1. Experimental Scenario of Large Load

4.2. Experimental Scenario of Small Load

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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