Position Control of a Single-Rod Electro-Hydrostatic Actuator Experiencing a Leaky Piston Seal

Ren, Guangan; Mou, Xiangwei; Wen, Xiaohao; Chen, Lintao

doi:https://doi.org/10.1155/2022/3166926

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3166926 | https://doi.org/10.1155/2022/3166926

Position Control of a Single-Rod Electro-Hydrostatic Actuator Experiencing a Leaky Piston Seal

Guangan Ren,¹Xiangwei Mou,²Xiaohao Wen,²and Lintao Chen²

Academic Editor: Xingling Shao

Received28 May 2022

Revised30 Jun 2022

Accepted17 Aug 2022

Published15 Sept 2022

Abstract

Single-rod cylinders are generally employed in electro-hydrostatic actuators (EHAs). A condition that is difficult to detect and could degrade the performance in single-rod EHAs, is the faulty cylinder piston seal. It causes internal leakage from one chamber of the actuator to another. In this work, a position controller with tolerance to actuator internal leakage is synthesized for a single-rod EHA using quantitative feedback theory (QFT). The controller is also robust to different loading and environmental stiffnesses. The ability of the controller is compared with another QFT controller that is synthesized without considering leakage fault. The simulation results show that the QFT fault-tolerant controller can meet prescribed specifications despite internal leakage up to 8.6 L/min.

1. Introduction

An electro-hydrostatic actuator (EHA) is pump-controlled that has already been used for aircrafts [1], vehicles [2], and manipulators [3, 4]. In the literature, a double-rod cylinder is often used in an EHA system and its circuit structure is simple. However, single-rod EHAs with four possible circuit configurations have more potential applications [5, 6]. Single-rod EHAs can suffer from various faults. In particular, a leaky piston seal can cause internal leakage from one chamber of the actuator to another that can degrade system performance. In order to compensate for the adverse effects caused by faults, fault-tolerant control (FTC) methods are widely utilized. FTC schemes can be divided into two groups, namely, active and passive FTC. In the former, the controller is adaptive and alters online as faults happen. In the latter, a robust controller is designed that is insensitive to faults in closed-loop performance [7–11]. This type of controllers is preferable due to its simple structure and easy application.

Quantitative feedback theory (QFT) is a robust linear controller design method. During the controller design process, performance specifications, parametric uncertainties, and controller structure can be balanced [12, 13]. QFT controllers have been successfully applied to deal with internal leakage fault. Karpenko and Sepehri [14] developed an active QFT-based FTC scheme, and they further designed passive QFT fault-tolerant controllers [15, 16]. All these controllers were implemented on valve-controlled systems. Ren et al. [17–20] synthesized QFT position and QFT actuating pressure controllers despite leakage. These controllers were developed for double-rod EHAs. Apart from QFT, an adaptive backstepping technique developed by Chen and Liu [21] has been proposed to deal with internal leakage. This system is not an EHA because directional valves were used to control the flow. Maddahi et al. [22] developed a fractional-order PID fault-tolerant controller for a valve-controlled system. Moghaddam et al. [23] combined fractional-order PID controllers and a fuzzy inference system to accommodate internal leakage for a single-rod EHA. However, this active FTC strategy necessitates a fault detection algorithm. The contribution of the work is to design of a passive FTC scheme for a single-rod EHA that is tolerant to internal leakage fault et al. l four quadrants. The system includes various load masses, environmental stiffnesses, and other uncertainties. The novelties are (1) the development of a robust fault-tolerant controller despite cross port leakage for a single-rod EHA, (2) the establishment of the mathematical model for the system considering leakage fault, and (3) the performance comparison of the fault-tolerant controller and the one designed for normal operation (no leak).

The remainder of this paper is organized as follows: mathematical model of the system is described in Section 2. Section 3 shows the development of a QFT fault-tolerant controller and QFT normal controller. Section 4 examines the performances of the two QFT controllers in simulations. Conclusions are provided in Section 5.

2. Modeling

The novel single-rod EHA circuit developed by Costa and Sepehri [6] is used for this study. Its schematic is shown in Figure 1. The system includes a bidirectional main pump, a servomotor, an auxiliary pump, a relief valve, a single-rod cylinder, a load mass, a spring, a one-directional flow control valve, and a three-position four-way directional valve. The settling pressure of a relief valve (item 4) is 5.5 × 10⁵ Pa. The working principle of the system is described in [6, 24]. When the load pressure p_L (p_L = p_a-A_bp_b/A_a) > 0, the solenoid y of V2 is energized, and when p_L < 0, the solenoid z is energized. The four quadrants of the system is shown in Figure 2.

From Figure 1, the flow equation of the main pump is

The flows out of and into the pump are represented by Q₁ and Q₂, respectively; is the speed of the servo motor; the pump displacement is represented by V_d. The speed of the servo motor is [17].

Here, K_m represents the servomotor gain; τ_m represents the time constant of the motor; u is used as input voltage to the motor. The actuator area ratio iswhere A_a and A_b are the area of the piston at the cab side and the rod side, respectively. The load pressure iswhere p_a and p_b are the chamber pressures of the actuator. The continuity equations arewhere Q_ac and Q_bc are flows from the auxiliary circuit; Q_a and Q_b are the flows into and out of the actuator, respectively; x_p and are the actuator position and velocity, respectively; β_e is the effective bulk modulus of the fluid; V_oa and V_ob are actuator chamber volumes at the two sides, respectively. The internal leakage Q_l is constructed as follows [17]:

In (7), K_i is the coefficient of the internal leakage. For single-rod actuators, the following assumption can be used [25,26].where C is the hydraulic compliance. The dynamic equation of the piston is

In (9), is actuator acceleration; m_rod is the piston and the rod mass; m_L is the load mass; f is the viscous damping coefficient; the load force F_L iswhere is the gravitational acceleration; k is the stiffness of the environment.

When the actuator is extending ( >0), the auxiliary pump provides flow into one side of the actuator through V1 (Quadrants I and II in Figure 2). The following equation can be obtained:

From Equations (3) to (8) and (11), the following equations can be obtained:

In Quadrant I, p_L > 0 and Q_ac = 0. Performing Laplace transformation of Equations (1), (2), (5), (9), (10), and (12), the following plant transfer function P₁(s) can be obtained:where the constant A₁ and B₁ are

In Quadrant II, p_L < 0 and Q_bc = 0. The plant model P₂(s) can be got using Equations (1), (2), (6), (9), (10), and (13):where the constant A₂ and B₂ are

When the actuator is retracting ( <0), the oil flows from one chamber of the actuator to the tank (Quadrants III and IV in Figure 2).

In Quadrant III, p_L < 0 and Q_bc = 0, Q_ac is [24].where K_a is the pressure sensitivity gain. Using Equations (1) to (10) together with (18), the plant model P₃(s) can be obtained as follows:where the constant A₃, B_3, and C₃ are

In Quadrant IV, p_L > 0 and Q_ac = 0, Q_bc is [24].where K_b is the pressure sensitivity gain. Using Equations (1) to (10) together with (21), the plant model P₄(s) iswhere the constant A₄, B_4, and C₄ are

The system includes the above four cases and is hereby the system model is expressed as P(s)∈{P₁(s), P₂(s), P₃(s), P₄(s)}. The internal leakage coefficient K_i and spring stiffness k change the plant type. Table 1 lists the parameter values [24] of the system. The minimum value of K_i is 0, which represents a healthy piston seal. The maximum value of K_i is prescribed to represent the most severe piston faulty condition. Note that, the uncertainty range of m_L and k are also considered to ensure that both resistive and assistive load forces can be generated. These parameters are used in the simulations.

3. QFT Controller Design

Figure 3 shows the schematic of the control system [13]. As per prescribed specifications, a controller G and a prefilter F have to be synthesized despite uncertainties in the plant P.

3.1. Plant Templates

Parametric uncertainties of the plant (shown in Table 1) are captured by templates in the frequency domain on the Nichols chart. Template sizes are influenced by the effects of uncertainties on the plant. The templates of the plant P(s) for normal operation (K_i = 0) and the ones considering internal leakage (K_i ≥ 0) are shown in Figures 4(a) and 4(b), respectively. Note that internal leakage increases templates sizes at low frequencies. It introduces phase variation with a maximum value of 90 degree and a magnitude variation. The larger templates make it hard to design the controller.

(a)

(b)

3.2. Prescribed Specifications

3.2.1. Tracking Specification

The uncertain plant P(s) is expressed as P(s, β), where the vector β = [τ_m, K_m, f, C, m_rod, m_L, K_i, k, K_a, K_b]^T. With reference to Figure 3, the transfer function of the closed-loop system T(s, β) iswhere G(s) represents the QFT controller for normal operation G_N(s) or the QFT fault-tolerant controller G_FTC(s); F(s) is the prefilter designed for G_N(s) or G_FTC(s). The tracking requirement is shown as follows:where T_U(s) is the upper tracking bound and T_L(s) is the lower tracking bound. According to (25), the frequency responses of T(s, β) should be within the above-given two tracking bounds.

3.2.2. Stability Specification

The stability specification is [13]

(26) ensures a gain margin of 4.22 dB and a phase margin of 36.42^o [17, 18].

3.2.3. Sensitivity Specification

The following equation needs to be satisfied for disturbance rejection:

3.3. Loop Shaping and Prefilter Design

A nominal plant P(jω, β₀) is selected by using a set of parameters in P(jω, β). It is then employed to calculate QFT bounds together with the above-prescribed specifications and plant templates on the Nichols chart. The controller is designed by shifting the nominal plant P(jω, β₀) until the nominal loop transmission L(jω, β₀) = G(jω) P(jω, β₀) satisfies QFT bounds et al.l selected frequencies. The bounds are either open or closed. In order to satisfy these bounds, L(jω, β₀) should be above the open bounds and outside closed bounds at the corresponding frequencies on the Nichols chart.

Figure 5(a) shows the QFT bounds and a suitable loop transmission for normal operation (K_i = 0). In order to satisfy open bounds, a small gain is employed in the controller. The next two poles are also added to satisfy closed bounds at high frequencies. The normal controller is shown in the following equation:

(a)

(b)

When the internal leakage is considered (K_i ≥ 0), the QFT bounds and a suitable loop transmission are shown in Figure 5(b). An integrator is added in the fault-tolerant controller to make L(jω, β₀) satisfy QFT bound requirements with a smaller controller bandwidth. Next, the open-loop gain is increased to meet open bounds. Finally, two zeros and two poles are used in the controller to satisfy closed bounds at intermediate frequencies and high frequencies, respectively. The designed fault-tolerant controller is shown in the following equation:

By observing (28), an integrator, a high open-loop gain, and two zeros are needed in the fault-tolerant controller to cope with internal leakage fault. The ratio of controller gains |G_FTC(s)|/| G_N(s)| is also calculated to further ascertain the price, as shown in Figure 6. It is seen that the integrator part in G_FTC(s) introduces extra gain at low frequencies (ω < 1 rad/s) to remove static errors caused by leakage. At the intermediate-frequency band (1 rad/s≤ ω ≤ 10 rad/s), two zeros of G_FTC(s) leads to over 8 dB ratio, that is required to satisfy its QFT bounds. Although the magnitude of G_FTC(s) is much higher than that of G_N(s) (ratio >25 dB) at high frequencies (ω > 10 rad/s), which indicates G_FTC(s) is more susceptible to noise and unmodelled high-frequency dynamics, the prescribed specifications are still satisfied for both controllers.

Loop shaping just ensures the satisfactions of Equations (26), (27), and (30), therefore a prefilter was synthesized to meet (25). The prefilter can make closed-loop frequency responses within upper and lower QFT tracking bounds. The prefilter designed for normal operation and the one synthesized considering internal leakage are given by (31), respectively. Both prefilters have the same number of zeros and poles.

4. Simulation Studies

The designed two QFT controllers were examined under normal operation (no leak) and leaky operation, respectively. Their ability to satisfy tracking bounds was shown in simulations. The ranges of parameters listed in Table 1 were also considered.

In the first test, the nominal system was operated under normal operation. A load of 300 kg and a spring of 130 kN/m was chosen as the load force. Simulation results for the normal controller G_N and fault-tolerant controller G_FTC are shown in Figures 7 and 8, respectively. As is seen, the actuator position responses of the two controllers are within tracking bounds. In addition, when the quadrant switches, the control signal of G_FTC is more oscillatory than that of G_N. This is because the bandwidth of G_FTC is higher, making it more sensitive to the changes and disturbances of the system.

(a)

(b)

(c)

(a)

(b)

(c)

Next, internal leakage was gradually introduced (K_i increases from 0 to its maximum value) to evaluate the performance of the two controllers in leaky operation. The responses of G_N and G_FTC to a 100-mm square-wave input are shown in Figures 9 and 10, respectively. It is seen that the steady-state error of G_N increases with leakage and finally the tracking specification is violated. On the other hand, the position response of G_FTC satisfies tracking bounds, even when leakage increases to 4.4 L/min.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Finally, G_FTC was tested under various leakage levels (K_i increases from 0 to 2.4 × 10⁻¹¹m³/(s·Pa)), step inputs (50 mm, 100 mm, and 150 mm), load masses (0 kg, 150 kg, and 300 kg), and environmental stiffnesses (0 kN/m, 65 kN/m, and 130 kN/m). Parametric uncertainties in Table 1 were also considered. With reference to Figure 11, tracking bounds are satisfied even if leakage increases to 8.6 L/min.

(a)

(b)

(c)

(d)

5. Conclusions

A fault-tolerant controller was synthesized for a single-rod EHA. The controller required an integrator, a high open-loop gain, and two zeros to compensate for leakage flow. Another QFT controller was also designed under normal operation (no leak). Simulation results demonstrated that the QFT fault-tolerant controller was capable of maintaining actuator responses within tracking bounds despite internal leakage up to 8.6 L/min. However, the QFT normal controller could not satisfy the prescribed specifications if internal leakage occurred.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

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

Acknowledgments

This work was supported in part by Hunan Provincial Natural Science Foundation of China under Grant no. 2022JJ40550, in part by the National Natural Science Foundation of China under Grant no. 52105077, and in part by the Guangxi Natural Science Foundation under Grant no. 2018GXNSFAA050026.

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Copyright © 2022 Guangan Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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