# Automatic Current-Constrained Double Loop ADRC for Electro-Hydrostatic Actuator Based on Singular Perturbation Theory

^{*}

## Abstract

**:**

## 1. Introduction

- 1.
- A novel cascade double-loop ADRC control architecture, including a reduced order position control loop and an integrated speed–current control loop, is presented, which has a simpler structure and fewer tuning parameters compared with the existing architecture.
- 2.
- For the position control loop, a reduced-order ADRC controller (ROADRC) is synthesized based on singular perturbation theory. ROADRC does not need the acceleration information. Moreover, the noise sensitivity of ESO is significantly weakened. Hence, the control output signal of ROADRC is smoother, and the practical application difficulty of ROADRC is easier.
- 3.
- An effectively integrated speed–current ADRC with automatic current-constrained (CACADRC) is designed based on the barrier function. CACADRC not only solves the problem of excessive bandwidth of the current loop but also effectively solves the problem that the current cannot be constrained automatically after the integrated design. Furthermore, the detailed stability proof of CACADRC is given according to the Lyapunov theory.

## 2. System Description

#### 2.1. Basic Principle

#### 2.2. Mathematical Modelling

#### 2.2.1. Equations for Voltage and Motion of PMSM

_{q}and u

_{d}are q, d axis voltage components; R

_{h}is the stator resistance; i

_{q}and i

_{d}are q, d axial current components; L

_{q}and L

_{d}are the equivalent inductance of q, d axis; ω

_{m}is the mechanical angular speed of PMSM; p is the number of pole pairs; and ψ

_{f}is the flux linkage of the rotor permanent magnet.

_{p}is the rotor inertia of PMSM; B

_{p}is the viscous friction coefficient of the reversible plunger pump; D

_{p}is the displacement of the reversible plunger pump; and p

_{a}and p

_{b}are the oil outlet and inlet pressures of the pump, respectively.

#### 2.2.2. Flow Equation of Reversible Plunger Pump

_{a}and Q

_{b}are the oil outlet and inlet flow of the pump, respectively; Q

_{ip}, Q

_{opa}, and Q

_{opb}are the internal and external leakage flow of the pump, respectively; β

_{e}is the effective oil bulk modulus; and V

_{a}and V

_{b}are respectively the volumes of oil outlet chamber and oil return chamber of the pump.

#### 2.2.3. Flow Equation of Hydraulic Cylinder

_{1}and Q

_{2}are, respectively, the inflow and outflow flow of the two chambers of the hydraulic cylinder; A is the effective working area of the hydraulic cylinder piston; x is the displacement of the piston rod of the hydraulic cylinder; p

_{1}and p

_{2}are respectively the pressure of the two chambers of the hydraulic cylinder; Q

_{ic}is the internal leakage flow in the hydraulic cylinder; and V

_{10}and V

_{20}are the volumes of the closed chamber on both sides of the hydraulic cylinder, respectively.

#### 2.2.4. Motion Equation of the Hydraulic Cylinder

_{t}is the total mass of the piston, piston rod, and load; B

_{t}is the total viscous friction coefficient of the hydraulic cylinder and the load; and FL is the external load force applied to the piston rod.

#### 2.2.5. Equation of Pressure Dynamics

_{c}

_{1}, Q

_{c}

_{2}, Q

_{r}

_{1}, and Q

_{r}

_{2}are the flow through the check valve and the relief valve, respectively.

_{a}= p

_{1}, p

_{b}= p

_{2}. In addition, V

_{a}= V

_{10}+ Ax, V

_{b}= V

_{20}−Ax, considering Equations (3), (4), and (6), we can obtain the following:

_{v}

_{1}= −Q

_{c}

_{1}+ Q

_{r}

_{1}; Q

_{v}

_{2}= −Q

_{c}

_{2}+ Q

_{r}

_{2}.

_{10}≈ V

_{20}= V

_{0}, and considering that the hydraulic cylinder is symmetric, it is approximately ${\dot{p}}_{1}=-{\dot{p}}_{2}$. Equation (7) is taken into consideration to obtain:

_{L}= p

_{1}−p

_{2}is the load pressure; C

_{t}is the total leakage coefficient of the pump and the hydraulic cylinder; and Q

_{un}= −(Q

_{opa}−Q

_{opb})/2 + (Q

_{v}

_{1}+ Q

_{v}

_{2})/2.

**X**= (x

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6})

^{T}= (x, $\dot{x}$, p

_{L}, ω

_{m}, i

_{q}, i

_{d})

^{T}, the state equation of EHA can be depicted as follows:

_{Lm}= −F

_{L}; d

_{Lp}= −D

_{p}p

_{L}.

_{d}, u

_{q}as the direct control quantity, EHA is a 5-order, 2-input single-output system. For the purpose of simplifying the controller design process, the whole system is divided into three subsystems; namely, the position subsystem composed of state variables x, $\dot{x}$, p

_{L}, the speed subsystem composed of state variable ω

_{m}, and the current subsystem composed of state variables i

_{d}, i

_{q}.

#### 2.3. Reduced Order Model of EHA Based on Singular Perturbation Theory

_{e}is a numerically sizeable physical quantity in Equation (8); usually, its value is (7~15) × 10

^{8}Pa, so ɛ

_{s}= β

_{e}

^{−1}V

_{0}is chosen as the singular perturbation parameter. Defining the state variables

**X**

_{s}= (x, $\dot{x}$)

^{T}and Z

_{s}= p

_{L}, Equations (5) and (8) can be written in the singularly perturbed standard form:

_{s}= 0, the following formula of Equation (10) can degenerate into an algebraic equation:

_{L}can be obtained as follows:

_{τ}= p

_{L}− ${\overline{p}}_{L}$, then:

_{s}= 0, τ

_{s}= t/ɛ

_{s}, then Equation (13) can be converted into the new timescale τ

_{s}framework, then the boundary layer model equation can obtain the following:

_{τ}= 0. According to Tikhonov theorem, it can be known that:

**X**= (x

_{1}, x

_{2})

^{T}= (x,$\dot{x}$)

^{T}, then Equation (15) can be written as the following state space form:

_{x}

_{0}is the nominal value of g

_{x}= AD

_{p}/(m

_{t}C

_{t}); d

_{x}= Δg

_{x}ω

_{m}−[A

^{2}(m

_{t}C

_{t})

^{−1}+ m

_{t}

^{−1}B

_{t}]‧$\dot{x}$ + A(m

_{t}C

_{t})

^{−1}Q

_{un}−m

_{t}

^{−1}F

_{L}; and Δg

_{x}is the perturbation value of parameter g

_{x}.

## 3. Design of Novel ADRC for EHA

#### 3.1. ROADRC Design

_{x}, converting it into a corresponding control quantity to cancel the disturbance, and then considering it as a nominal model without any disturbance to design the controller, the difficulty of controller design will be significantly reduced.

_{mc}represents the disturbance compensation control quantity and ω

_{m}

_{0}represents the nominal control quantity, then the output of the ROADRC controller is ω

_{m}

^{*}= ω

_{m}

_{0}+ ω

_{mc}.

_{x}, and the total disturbance d

_{x}is expanded into an additional state denoted by x

_{3}, and its derivative satisfies this condition: ${\dot{d}}_{x}$ = h

_{x}. Then, the ESO can be constructed as:

_{x}

_{1}, l

_{x}

_{2}and l

_{x}

_{3}> 0 are satisfying the Hurwitz condition, and ${\widehat{x}}_{1}$, ${\widehat{x}}_{2}$ and ${\widehat{x}}_{3}$ are the estimated values of x

_{1}, x

_{2,}and x

_{3}, respectively.

_{m}

^{*}can be designed as:

_{d}is the position reference, and k

_{x}

_{1}and k

_{x}

_{2}are the control parameters of ROADRC.

_{x}

_{1}~ l

_{x}

_{3}need to be tuned, and the characteristic equation s

^{3}+ l

_{x}

_{1}s

^{2}+ l

_{x}

_{2}s + l

_{x}

_{3}of ESO can be integrated into the mode of multiple poles, namely (s + ω

_{ox})

^{3}. In this way, l

_{x}

_{1}~ l

_{x}

_{3}is only related to the ESO bandwidth ω

_{ox}, so ESO only needs to set one parameter ω

_{ox}. Then, l

_{x}

_{1}~ l

_{x}

_{3}can be easily calculated as [26]:

^{2}+ k

_{x}

_{2}s + k

_{x}

_{1}is integrated into the mode of multiple poles, namely (s + ω

_{cx})

^{2}. ω

_{cx}is the control bandwidth, so one only needs to adjust the ω

_{cx}parameter. In this case, k

_{x}

_{1}~ k

_{x}

_{2}can be easily calculated as:

#### 3.2. CACADRC Design

_{ω}= −J

_{p}

^{−1}B

_{p}ω

_{m}−J

_{p}

^{−1}D

_{p}P

_{L}.

_{ωq}

_{0}is the nominal value of g

_{ωq}= 3/2∙pψ

_{f}J

_{p}

^{−1}L

_{q}

^{−1}; d

_{ωq}= Δg

_{ωq}u

_{q}+ g

_{ω}[−L

_{q}

^{−1}R

_{h}i

_{q}−L

_{q}

^{−1}pω

_{m}(L

_{d}i

_{d}+ ψ

_{f})] $+{\dot{a}}_{\omega}$, and Δg

_{ωq}is the perturbation value of parameter g

_{ωq}.

_{m}= (ω

_{m}

_{1}, ω

_{m}

_{2})

^{T}= (ω

_{m}, ${\dot{\omega}}_{m}$)

^{T}and rewriting Equation (22) as the following state space form:

_{qc}represents the disturbance compensation control quantity and u

_{q}

_{0}represents the nominal control quantity, the output of the integrated speed–current control is u

_{qi}

^{*}= u

_{q}

_{0}+ u

_{qc}.

_{ωq}, and the total disturbance d

_{ωq}is expanded into a state, namely ω

_{m}

_{3}= d

_{ωq}, denoted as ${\dot{d}}_{\omega q}$ = h

_{ωq}, ESO needs to be designed as order 3, and the ESO is constructed as:

_{ωq}

_{1}, l

_{ωq}

_{2,}and l

_{ωq}

_{3}> 0 are satisfying Hurwitz condition, and ${\widehat{\omega}}_{m1}$, ${\widehat{\omega}}_{m2}$ and ${\widehat{\omega}}_{m3}$ are the estimated values of ω

_{m}

_{1}, ω

_{m}

_{2,}and ω

_{m}

_{3}, respectively.

_{ωq}

_{1}= ω

_{md}−${\widehat{\omega}}_{m1}$, e

_{ωq}

_{2}=${\dot{\omega}}_{md}$− ${\widehat{\omega}}_{m2}$, then u

_{qi}

^{*}is designed as:

_{md}is the speed reference, and k

_{ωq}

_{1}and k

_{ωq}

_{2}are the control parameters of the integrated speed–current loop controller.

_{m}and i

_{q}is processed, i

_{q}is in an uncontrollable state, which means the amplitude of i

_{q}cannot be limited as the traditional ADRC architecture; namely, i

_{q}may be larger than i

_{qmax}. To deal with this problem, a CACADRC controller that can automatically constrain i

_{q}is developed here. Therefore, Equation (25) is amended as follows:

_{qmax}is the limiting value of i

_{q}.

**Remark**

**1**

**:**

_{ωq2}of speed tracking error e

_{ωq1}characterizes the damping or angular acceleration of the system to some extent. The larger the damping is, the slower the tracking speed of the rotating speed is, but the more stable the tracking transient is. As can be seen fromFigure 3, when i

_{q}approaches i

_{qmax}, the nonlinear current penalty term increases, the damping of the system increases, and the angular acceleration decreases. It is noted that the angular acceleration is proportional to the current, so i

_{q}decreases. As i

_{q}moves away from i

_{qmax}, the nonlinear current penalty term decreases, the damping of the system decreases, the angular acceleration increases, and i

_{q}begins to increase again. Finally, i

_{q}fluctuates around i

_{qmax}.

_{oωq}. At the same time, since the ESO of CACADRC is also of order 3, similar to the ESO’s parameter tuning process of the position loop, l

_{ωq}

_{1}~ l

_{ωq}

_{3}can be easily calculated as:

_{cωq}. In this case, it is easy to calculate k

_{ωq}

_{1}~ k

_{ωq}

_{2}as:

## 4. Stability Proof of the System

#### 4.1. Stability Proof of ROADRC

#### 4.1.1. Proof of Convergence of ESO

_{ox}> 0, according to the binomial theorem:

_{i}= x

_{i}−${\widehat{x}}_{i}$,i = 1, 2, 3, subtract Equation (17) from Equation (16), and we have:

_{i}= e

_{i}/ω

_{ox}

^{i}

^{−1}, i = 1, 2, 3, then:

**A**satisfies Hurwitz’s condition, there exists a positive definite matrix

**P**, such that

**A**

^{T}

**P**+

**PA**= −

**I**. Defining the Lyapunov function V(

**ɛ**) =

**ɛ**

^{T}

**Pɛ**, and taking the derivative about time:

_{x}is bounded, i.e., |h

_{x}| < M, when $\Vert \epsilon \Vert >2\Vert PB\Vert {h}_{x}/{\omega}_{ox}^{2}$, $\dot{V}(\mathsf{\epsilon})<0$ is always satisfied, then $\Vert \mathsf{\epsilon}\Vert $ will converge to the range $\Vert \mathsf{\epsilon}\Vert \le 2\Vert PB\Vert {h}_{x}/{\omega}_{ox}^{2}$, so $\Vert \mathsf{\epsilon}\Vert $ is bounded. Considering ɛ

_{i}= e

_{i}/ω

_{ox}

^{i}

^{−1}, i = 1, 2, 3, and inequality (35), we can obtain the following:

_{x}= 0,

**e**is asymptotically stable, namely $\underset{t\to \infty}{\mathrm{lim}}e=0$.

#### 4.1.2. Stability Proof of Controller

_{x}

_{1}and k

_{x}

_{2}satisfy Hurwitz’s condition, there exists a positive definite matrix

**P**

_{x}, such that ${A}_{x}^{T}{P}_{x}+{P}_{x}{A}_{x}=-I$. Defining the Lyapunov function $V={E}^{T}{P}_{x}E$, taking the derivative of V about time:

_{max}(

**P**

_{x}) is the largest eigenvalue of

**P**

_{x}.

#### 4.2. Stability Proof of CACADRC

#### 4.2.1. Proof of Convergence of ESO

#### 4.2.2. Stability Proof of Controller

**Theorem**

**1.**

_{q}(0) ϵ (-i

_{qmax}, i

_{qmax}), (e

_{wq1}, e

_{wq2}) asymptotically converges to a bounded closed set${\mathsf{\Omega}}_{E}=\left\{{e}_{\omega q}\left|{\beta}_{k\omega q}V({e}_{\omega q})\le \frac{1}{2}{\overline{d}}_{\omega q}^{2}\right.\right\}$, where β

_{kωq}is indicated below. In the meantime, i

_{q}(t) ϵ (-i

_{qmax}, i

_{qmax}) always hold.

**Proof:**

_{ωq}):

_{ωq}

_{1}> 1 and ${k}_{\omega q2}>1-\frac{{k}_{\omega q1}}{2}$, then:

_{kωq}= min {k

_{ωq}

_{1}−1, k

_{ωq}

_{1}+ 2k

_{ωq}

_{2}−2 + 2l/tan

^{−1}(i

_{qmax}

^{2}−i

_{q}

^{2})} is a positive number.

_{wq1}, e

_{wq2}) asymptotically converges Ω

_{E}.

_{ωq}

_{1},e

_{ωq}

_{2}) is a continuous function when (−i

_{qmax}, i

_{qmax}). If i

_{q}approaches ±i

_{qmax}, $F({e}_{\omega q1},{e}_{\omega q2})\to -\infty $. There is a constant ${\overline{i}}_{qmax}\in (0,{i}_{qmax})$, such that there must be ${e}_{\omega q2}{\dot{e}}_{\omega q2}<0$ when ${i}_{q}\in [{\overline{i}}_{qmax},{i}_{qmax})\cup (-{i}_{qmax},$ $-{\overline{i}}_{qmax}]$, which means $\left|{i}_{q}\right|<{i}_{q\mathrm{max}}$ always hold. Thus, the inequality (45) is always true for any time.

_{q}(t) ϵ (−i

_{qmax}, i

_{qmax}) always hold. □

## 5. Simulation Results

- 2PI: the conventional PI controller is employed in the speed and current double-loop.
- 2ADRC: the conventional ADRC is employed in the speed and current double-loop.
- 3PI: the conventional PI controller is employed in the position, speed, and current three-loop.
- 3ADRC: the conventional ADRC is employed in the position, speed, and current three-loop.
- 3ADRC1: the position loop ADRC of 3ADRC.
- Proposed method: ROADRC is employed in the position loop, and CACADRC is adopted for the integrated speed–current loop.

#### 5.1. Performance Simulation Analysis of CACADRC

_{m}

_{d}is 500 rad/s.

_{cωq}= 85 rad/s, ω

_{oωq}= 5ω

_{cωq}; the control bandwidth of the current loop in d-axis is ω

_{cd}= 1257 rad/s; and the ESO bandwidth is ω

_{od}= 10ω

_{cd}.

**Case 1:**Figure 4a–c shows ω

_{m}, i

_{q}and u

_{q}response curves under the action of different current penalty coefficients l, respectively. It can be seen that i

_{q}is constrained within the amplitude limit of 16A when l = 0.1k

_{ωq}

_{2}, k

_{ωq}

_{2}, and 2k

_{ωq}

_{2}, which means that the CACADRC can effectively realize automatic current-constraint control, and by adjusting the current penalty coefficient l in the barrier function, the current limiting intensity can be effectively adjusted. When l increases, the current i

_{q}will decrease, resulting in a decrease in the response speed of ω

_{m}. It should be noted that if the value of l is too large, the transient performance of the system will suffer a certain loss. On the contrary, if the value of l is too small, the current i

_{q}will exceed the amplitude limit, which makes the circuit safety threatened. Therefore, a compromise should be considered in practical applications.

_{m}, i

_{q}and u

_{q}response curves. To quantitatively analyze the performance of the different methods, the performance evaluation indices are employed, which are listed in Table 2, including overshoot (OS), settling time (ST), peak current (PC), speed drop (SD), recovery time (RT) and the number of parameters (PN).

_{Pω}= 0.05, K

_{Iω}= 0.3; q-axis current loop: K

_{Pq}= 2.2, K

_{Iq}= 564.8; d-axis current loop: K

_{Pd}= 20, K

_{Id}= 2011.

_{cω}= 21.4 rad/s, and the ESO bandwidth is ω

_{oω}= 10ω

_{cω}; the control bandwidth of the current loop in the d-axis, q-axis is ω

_{cd}= 1257 rad/s, ω

_{cq}= 1257 rad/s, respectively; and the ESO bandwidth is ω

_{od}= ω

_{cd}, ω

_{oq}= ω

_{cq}, respectively.

_{ωq}

_{2}, ω

_{cωq}= 72 rad/s, ω

_{oωq}= 10 ω

_{cωq}; ω

_{cd}= 1257 rad/s, ω

_{od}= 10 ω

_{cd}.

**Case 2:**In the startup phase, it is observed in Figure 5a and Table 2 that, when compared with 2PI and 2ADRC, both CACADRC and 2ADRC can realize tracking without overshoot, while 2PI has a large overshoot. CACADRC and 2ADRC have the similar OS and ST, i.e., comparable tracking performance, which indicates that CACADRC does not need an excessive inner loop control bandwidth and requires fewer tuning parameters to have a similar dynamic performance as 2ADRC. In addition, in accordance with the requirement of i

_{q}< 16A, it can be seen from Figure 5b and Table 2 that the i

_{q}of 2PI, 2ADRC, and CACADRC are all kept within the current restricted range. The reason is that 2PI and 2ADRC constrain the current by limiting the control output i

_{q}

^{*}, while CACADRC constrains the current by adding the barrier function to the controller, which has the same current limiting effect as well as 2ADRC and 2PI.

#### 5.2. Performance Simulation Analysis of ROADRC

_{Px}= 12661.58, K

_{Ix}= 48091.89. K

_{Pω}= 0.05, K

_{Iω}= 0.3; K

_{Pq}= 2.2, K

_{Iq}= 564.8; K

_{Pd}= 20, K

_{Id}= 2011.

_{cx}= 94 rad/s, and the ESO bandwidth is ω

_{ox}= 10 ω

_{cx}. ω

_{cω}= 628 rad/s, ω

_{oω}= 5 ω

_{cω}; ω

_{cd}= 1257 rad/s, ω

_{cq}= 1257 rad/s; ω

_{od}= ω

_{cd}, ω

_{oq}= ω

_{cq}.

_{ωq}

_{2}, ω

_{cx}= 94 rad/s, ω

_{ox}= 10 ω

_{cx}; ω

_{cωq}= 628 rad/s, ω

_{oωq}= 5 ω

_{cωq}; ω

_{cd}= 1257 rad/s, ω

_{od}= ω

_{cd}.

**Case 1:**The position reference x

_{d}is 0.1m, and a uniformly distributed noise signal with an amplitude of 1 × 10

^{−5}is added to the position signal x. The comparison curves of position step performance are shown in Figure 6. As shown in Figure 6a, it is seen that although 3PI has the fastest response speed, it produces a 10% overshoot, while the proposed method and 3ADRC achieve almost the same tracking performance without overshoot.

_{x}of ROADRC:

_{x}

_{1}of 3ADRC1:

_{m}

^{*}of 3ADRC1 is more seriously polluted by noise than that of ROADRC. The reason is the ESO order of 3ADRC1 is one order higher than that of ROADRC, so it is sensitive to noise, and the 3ADRC1 introduces the actuation acceleration information, which has an amplification effect on noise. However, the control output curve ω

_{m}

^{*}of the position loop PI is the smoothest, because only the proportional term of the position loop PI introduces noise, and the integral term can suppress the noise to a certain extent.

**Case 2:**The position reference x

_{d}is 0.001 sin (4πt) m, and a uniformly distributed noise signal with an amplitude of 1 × 10

^{−5}is added to the position signal x. In order to verify the influence of interference force and parameter mutation on system performance, it is assumed that external load F

_{L}= 500 sin (2πt) N is applied at t = 0 s, while total mass of the piston, piston rod and load m

_{t}and total leakage coefficient of the pump and the hydraulic cylinder C

_{t}become twice the original value, respectively.

_{m}

^{*}of 3ADRC1 is more seriously polluted by noise than that of ROADRC.

## 6. Conclusions

- The barrier function introduced in CACADRC can effectively achieve automatic current-constraint control, and by adjusting the current penalty coefficient l in the barrier function, the current limiting intensity can be effectively adjusted. Compared with 2PI and 2ADRC, both CACADRC and 2ADRC can realize tracking without overshoot, while 2PI has a large overshoot. In terms of anti-disturbance ability, CACADRC and 2ADRC have a more excellent anti-disturbance performance than 2PI, and CACADRC has a slightly better anti-disturbance ability than 2ADRC.
- When the position reference is a step signal, both the proposed method and 3ADRC can achieve tracking without overshoot, while 3PI has a large overshoot. Moreover, the disturbance rejection performance of the proposed method is similar to that of 3ADRC and superior to that of 3PI when subjected to step disturbance. In addition, when the position reference is a sinusoidal time-varying signal, the disturbance rejection ability of the proposed method is slightly better than that of 3ADRC and significantly better than that of 3PI.
- Due to the reduced order processing of the position subsystem, the order of the ESO of ROADRC is lower than that of the ESO of 3ADRC1, and the noise sensitivity is effectively weakened, which makes the disturbance estimation of the ROADRC’s ESO smoother than that of 3ADRC1. Moreover, in the process of ROADRC design, the use of acceleration information is avoided, so the control output signal ω
_{m}^{*}of ROADRC is smoother than that of 3ADRC1. - The proposed novel cascaded double-loop ADRC control architecture is simpler than the traditional cascaded three-loop ADRC control architecture and requires fewer parameters to be tuned, which is more conducive to the application in practical engineering.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Comment |

EHA | electro-hydrostatic actuator |

ADRC | active disturbance rejection control |

ESO | extended state observer |

ROADRC | reduced-order ADRC controller |

CACADRC | automatic current-constrained ADRC controller |

PI | proportional integral controller |

2DOF | two-degree-of-freedom |

RISE | the robust integral of the sign of error |

PMSM | permanent magnet synchronous motor |

u_{q}, u_{d} | q,d axis voltage components |

R_{h} | stator resistance |

i_{g}, i_{d} | q,d axial current components |

L_{q}, L_{d} | the equivalent inductance of q, d axis |

ω_{m} | the mechanical angular speed of PMSM |

p | the number of pole pairs |

ψ_{f} | flux linkage of the rotor permanent magnet |

J_{p} | rotor inertia |

B_{p} | the coefficient of viscous friction of the reversible plunger pump |

D_{p} | displacement of the reversible plunger pump |

p_{a}, p_{b} | oil outlet and inlet pressures of the pump |

Q_{a}, Q_{b} | oil outlet and inlet flow of the pump |

Q_{ip}, Q_{opa}, Q_{opb} | internal and external leakage flow of the pump |

β_{e} | effective oil bulk modulus |

V_{a}, V_{b} | volumes of the oil outlet chamber and oil return chamber of the pump |

Q_{1}, Q_{2} | inflow and outflow flow of the two chambers of the hydraulic cylinder |

A | the effective working area of the hydraulic cylinder piston |

x | displacement of the piston rod of the hydraulic cylinder |

p_{1}, p_{2} | the pressure of the two chambers of the hydraulic cylinder |

Q_{ic} | internal leakage flow in the hydraulic cylinder |

V_{10}, V_{20} | volumes of the closed chamber on both sides of the hydraulic cylinder |

m_{t} | the total mass of the piston, piston rod, and load |

B_{t} | the total viscous friction coefficient of the hydraulic cylinder and the load |

F_{L} | the external load force applied to the piston rod |

Q_{c}_{1}, Q_{c}_{2} | flow through the check valve |

Q_{r}_{1}, Q_{r}_{2} | flow through the relief valve |

p_{L} | load pressure |

C_{t} | total leakage coefficient of the pump and the hydraulic cylinder |

${\overline{p}}_{L}$ | quasi-steady state quantity |

ω_{mc} | disturbance compensation control quantity of ROADRC |

ω_{m}_{0} | nominal control quantity of ROADRC |

ω_{m}_{*} | control output of ROADRC |

l_{x}_{1}, l_{x}_{2}, l_{x}_{3} | ESO gains of ROADRC |

X_{d} | position reference |

k_{x}_{1}, k_{x}_{2} | control parameters of ROADRC |

ω_{ox} | ESO bandwidth of the position loop |

ω_{cx} | control bandwidth of the position loop |

u_{qc} | disturbance compensation control quantity of the integrated speed–current control |

u_{q}_{0} | nominal control quantity of the integrated speed–current control |

u_{qi}_{*} | the output of the integrated speed–current control |

l_{ωq}_{1}, l_{ωq}_{2}, l_{ωq}_{3} | ESO gains of the integrated speed–current control |

ω_{md} | speed reference |

k_{ωq}_{1}, k_{ωq}_{2} | control parameters of the integrated speed–current controller |

u_{q}_{*} | control output of CACADRC |

f_{br} | barrier function |

l | current penalty coefficient |

i_{qmax} | limiting value of i_{q} |

ω_{oωq} | ESO bandwidth of CACADRC |

ω_{cωq} | control bandwidth of CACADRC |

2PI | the conventional PI controller is employed in the speed and current double-loop |

2ADRC | the conventional ADRC is employed in the speed and current double-loop |

3PI | the conventional PI controller is employed in the position, speed, and current three-loop |

3ADRC | the conventional ADRC is employed in the position, speed, and current three-loop |

3ADRC1 | the position loop ADRC of 3ADRC |

Proposed method | ROADRC is employed in the position loop, and CACADRC is adopted for the integrated speed–current loop |

ω_{cd} | control bandwidth of the current loop in the d-axis |

ω_{od} | ESO bandwidth of the current loop in the d-axis |

D | piston diameter of the hydraulic cylinder |

d | piston rod diameter of hydraulic cylinder |

L | piston stroke |

V | the total volume of the hydraulic cylinder |

V_{0} | the initial one-sided volume of the hydraulic cylinder |

u_{N} | PMSM-rated voltage |

I_{N} | PMSM-rated current |

T_{N} | PMSM-rated torque |

ω_{mN} | PMSM-rated rotating speed |

OS | overshoot |

ST | settling time |

PC | peak current |

SD | speed drop |

RT | recovery time |

PN | number of parameters |

K_{Pω}, K_{Iω} | proportion and integral gains of speed loop of 2PI |

K_{Pq}, K_{Iq} | proportion and integral gains of the q-axis current loop of 2PI |

K_{Pd}, K_{Id} | proportion and integral gains of the d-axis current loop |

ω_{cω} | control bandwidth of the speed loop of 2ADRC |

ω_{oω} | ESO bandwidth of the speed loop of 2ADRC |

ω_{cq} | control bandwidth of the current loop in the q-axis of 2ADRC |

ω_{oq} | ESO bandwidth of the current loop in the q-axis of 2ADRC |

K_{Px}, K_{Ix} | proportion and integral gains of the position loop of 3PI |

ω_{cx} | control bandwidth of the position loop |

ω_{ox} | ESO bandwidth of the position loop |

d_{x} | disturbance quantity of ROADRC |

d_{x}_{1} | disturbance quantity of 3ADRC1 |

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**Figure 4.**Performance comparisons under CACADRC with different current penalty coefficients l. (

**a**) Speed response curves; (

**b**) q-axis current response curves; and (

**c**) q-axis voltage response curves.

**Figure 5.**Speed step response comparison curves with ω

_{md}= 500 rad/s. (

**a**) Speed response curves; (

**b**) q-axis current response curves; and (

**c**) q-axis voltage response curves.

**Figure 6.**Position step response comparison curves with x

_{d}= 0.1m. (

**a**) Comparison curves of position tracking errors of 3ADRC, the proposed method and 3PI; (

**b**) control output ω

_{m}

^{*}response curves; (

**c**) disturbance estimation error by ESO of ROADRC; and (

**d**) disturbance estimation error by ESO of 3ADRC1.

**Figure 7.**Position time-varying signal response comparison curves under complex disturbance when x

_{d}= 0.001 sin (4πt) m. (

**a**) Comparison curves of position tracking errors of 3ADRC, the proposed method and 3PI; (

**b**) control output ω

_{m}

^{*}response curves; (

**c**) disturbance estimation error by ESO of ROADRC; and (

**d**) disturbance estimation error by ESO of 3ADRC1.

Parameters | Value |
---|---|

piston diameter of hydraulic cylinder (m) | 0.066 |

piston rod diameter of hydraulic cylinder (m) | 0.045 |

the piston stroke (m) | 0.2 |

total leakage coefficient of hydraulic pump and cylinder(m^{3}/s‧Pa^{−1}) | 2 × 10^{−11} |

effective oil bulk modulus (Pa) | 6.86 × 10^{8} |

the total volume of the hydraulic cylinder (m^{3}) | 3 × 10^{−4} |

the initial one-sided total volume of the hydraulic cylinder(m^{3}) | 1.5 × 10^{−4} |

the total viscous friction coefficient of the hydraulic cylinder and the load (N/m‧s^{−1}) | 100 |

the total mass of the piston, piston rod, and load (kg) | 60 |

displacement of the reversible plunger pump (m^{3}/rad) | 1.5 × 10^{−6} |

the viscous friction coefficient of the reversible piston pump (N‧m/rad‧s^{−1}) | 0.002 |

PMSM-rated voltage (V) | 380 |

PMSM-rated current (A) | 7.6 |

PMSM-rated torque (N‧m) | 8 |

rotor flux linkage (Wb) | 0.25 |

rotor inertia (kg·m^{2}) | 0.0012 |

the number of pole pairs | 2 |

stator resistance ($\mathsf{\Omega}$) | 1.1 |

the equivalent inductance (H) | 0.0817 |

PMSM-rated rotating speed (rad/s) | 550 |

Performance | OS (%) | ST (s) | PC (A) | SD (rad/s) | RT (s) | PN |
---|---|---|---|---|---|---|

2PI | 9 | 0.78 | 16 | 32 | 0.64 | 6 |

2ADRC | 0 | 0.45 | 16 | 11 | 0.27 | 6 |

CACADRC | 0 | 0.4 | 16 | 5.5 | 0.13 | 5 |

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## Share and Cite

**MDPI and ACS Style**

Yang, R.; Ma, Y.; Zhao, J.; Zhang, L.; Huang, H.
Automatic Current-Constrained Double Loop ADRC for Electro-Hydrostatic Actuator Based on Singular Perturbation Theory. *Actuators* **2022**, *11*, 381.
https://doi.org/10.3390/act11120381

**AMA Style**

Yang R, Ma Y, Zhao J, Zhang L, Huang H.
Automatic Current-Constrained Double Loop ADRC for Electro-Hydrostatic Actuator Based on Singular Perturbation Theory. *Actuators*. 2022; 11(12):381.
https://doi.org/10.3390/act11120381

**Chicago/Turabian Style**

Yang, Rongrong, Yongjie Ma, Jiali Zhao, Ling Zhang, and Hua Huang.
2022. "Automatic Current-Constrained Double Loop ADRC for Electro-Hydrostatic Actuator Based on Singular Perturbation Theory" *Actuators* 11, no. 12: 381.
https://doi.org/10.3390/act11120381