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
2.2.2. Flow Equation of Reversible Plunger Pump
2.2.3. Flow Equation of Hydraulic Cylinder
2.2.4. Motion Equation of the Hydraulic Cylinder
2.2.5. Equation of Pressure Dynamics
2.3. Reduced Order Model of EHA Based on Singular Perturbation Theory
3. Design of Novel ADRC for EHA
3.1. ROADRC Design
3.2. CACADRC Design
4. Stability Proof of the System
4.1. Stability Proof of ROADRC
4.1.1. Proof of Convergence of ESO
4.1.2. Stability Proof of Controller
4.2. Stability Proof of CACADRC
4.2.1. Proof of Convergence of ESO
4.2.2. Stability Proof of Controller
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
5.2. Performance Simulation Analysis 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 ωm* 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 |
uq, ud | q,d axis voltage components |
Rh | stator resistance |
ig, id | q,d axial current components |
Lq, Ld | the equivalent inductance of q, d axis |
ωm | the mechanical angular speed of PMSM |
p | the number of pole pairs |
ψf | flux linkage of the rotor permanent magnet |
Jp | rotor inertia |
Bp | the coefficient of viscous friction of the reversible plunger pump |
Dp | displacement of the reversible plunger pump |
pa, pb | oil outlet and inlet pressures of the pump |
Qa, Qb | oil outlet and inlet flow of the pump |
Qip, Qopa, Qopb | internal and external leakage flow of the pump |
βe | effective oil bulk modulus |
Va, Vb | volumes of the oil outlet chamber and oil return chamber of the pump |
Q1, Q2 | 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 |
p1, p2 | the pressure of the two chambers of the hydraulic cylinder |
Qic | internal leakage flow in the hydraulic cylinder |
V10, V20 | volumes of the closed chamber on both sides of the hydraulic cylinder |
mt | the total mass of the piston, piston rod, and load |
Bt | the total viscous friction coefficient of the hydraulic cylinder and the load |
FL | the external load force applied to the piston rod |
Qc1, Qc2 | flow through the check valve |
Qr1, Qr2 | flow through the relief valve |
pL | load pressure |
Ct | total leakage coefficient of the pump and the hydraulic cylinder |
quasi-steady state quantity | |
ωmc | disturbance compensation control quantity of ROADRC |
ωm0 | nominal control quantity of ROADRC |
ωm* | control output of ROADRC |
lx1, lx2, lx3 | ESO gains of ROADRC |
Xd | position reference |
kx1, kx2 | control parameters of ROADRC |
ωox | ESO bandwidth of the position loop |
ωcx | control bandwidth of the position loop |
uqc | disturbance compensation control quantity of the integrated speed–current control |
uq0 | nominal control quantity of the integrated speed–current control |
uqi* | the output of the integrated speed–current control |
lωq1, lωq2, lωq3 | ESO gains of the integrated speed–current control |
ωmd | speed reference |
kωq1, kωq2 | control parameters of the integrated speed–current controller |
uq* | control output of CACADRC |
fbr | barrier function |
l | current penalty coefficient |
iqmax | limiting value of iq |
ωoωq | ESO bandwidth of CACADRC |
ωcωq | 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 |
ωcd | control bandwidth of the current loop in the d-axis |
ωod | 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 |
V0 | the initial one-sided volume of the hydraulic cylinder |
uN | PMSM-rated voltage |
IN | PMSM-rated current |
TN | PMSM-rated torque |
ωmN | PMSM-rated rotating speed |
OS | overshoot |
ST | settling time |
PC | peak current |
SD | speed drop |
RT | recovery time |
PN | number of parameters |
KPω, KIω | proportion and integral gains of speed loop of 2PI |
KPq, KIq | proportion and integral gains of the q-axis current loop of 2PI |
KPd, KId | proportion and integral gains of the d-axis current loop |
ωcω | control bandwidth of the speed loop of 2ADRC |
ωoω | ESO bandwidth of the speed loop of 2ADRC |
ωcq | control bandwidth of the current loop in the q-axis of 2ADRC |
ωoq | ESO bandwidth of the current loop in the q-axis of 2ADRC |
KPx, KIx | proportion and integral gains of the position loop of 3PI |
ωcx | control bandwidth of the position loop |
ωox | ESO bandwidth of the position loop |
dx | disturbance quantity of ROADRC |
dx1 | disturbance quantity of 3ADRC1 |
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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(m3/s‧Pa−1) | 2 × 10−11 |
effective oil bulk modulus (Pa) | 6.86 × 108 |
the total volume of the hydraulic cylinder (m3) | 3 × 10−4 |
the initial one-sided total volume of the hydraulic cylinder(m3) | 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 (m3/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·m2) | 0.0012 |
the number of pole pairs | 2 |
stator resistance () | 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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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
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 StyleYang, 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