Figure 1.
Buck converter connected to an electric DC motor and controlled by zero average dynamics-fixed-point induced control (ZAD-FPIC).
Figure 1.
Buck converter connected to an electric DC motor and controlled by zero average dynamics-fixed-point induced control (ZAD-FPIC).
Figure 2.
Diagram of the buck converter connected to a DC motor.
Figure 2.
Diagram of the buck converter connected to a DC motor.
Figure 3.
Electromechanical system with the switch ON.
Figure 3.
Electromechanical system with the switch ON.
Figure 4.
Electromechanical system with the switch OFF.
Figure 4.
Electromechanical system with the switch OFF.
Figure 5.
Buck-motor system working in discontinuous conduction mode (DCM).
Figure 5.
Buck-motor system working in discontinuous conduction mode (DCM).
Figure 6.
ZAD-FPIC controller simulation blocks.
Figure 6.
ZAD-FPIC controller simulation blocks.
Figure 7.
Numerical and experimental results for the system in closed loops: (a) mechanical speed and in closed loop and (b) error of in closed loop.
Figure 7.
Numerical and experimental results for the system in closed loops: (a) mechanical speed and in closed loop and (b) error of in closed loop.
Figure 8.
Numerical and experimental results in closed loops: (a) mechanical speed and reference speed in a closed-loop system and (b) error of in a closed-loop system.
Figure 8.
Numerical and experimental results in closed loops: (a) mechanical speed and reference speed in a closed-loop system and (b) error of in a closed-loop system.
Figure 9.
Behavior of the buck-motor system in closed-loop while changing with : (a) output for the simulation; (b) for the simulation; (c) output for the experiment; and (d) d for the experiment.
Figure 9.
Behavior of the buck-motor system in closed-loop while changing with : (a) output for the simulation; (b) for the simulation; (c) output for the experiment; and (d) d for the experiment.
Figure 10.
Bifurcation diagram of the variable when the parameter is changed (, and ) and the values of and = 1 are kept constant: (a) bifurcation diagram vs. for the experimental test; (b) in the experimental test when ; (c) in the experimental test when ; and (d) in the experimental test when .
Figure 10.
Bifurcation diagram of the variable when the parameter is changed (, and ) and the values of and = 1 are kept constant: (a) bifurcation diagram vs. for the experimental test; (b) in the experimental test when ; (c) in the experimental test when ; and (d) in the experimental test when .
Figure 11.
Bifurcation diagram of d vs. and behavior of , and when the values of and : (a) bifurcation diagram of d vs. in the experimental test; (b) behavior of d when in the experimental test; (c) behavior of d when in the experimental test; and (d) behavior of d when in the experimental test.
Figure 11.
Bifurcation diagram of d vs. and behavior of , and when the values of and : (a) bifurcation diagram of d vs. in the experimental test; (b) behavior of d when in the experimental test; (c) behavior of d when in the experimental test; and (d) behavior of d when in the experimental test.
Figure 12.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter and maintaining fixed values of : (a) vs. for the simulation test; (b) vs. for the experimental test; (c) error vs. for the simulation test; and (d) error vs. for the experimental test.
Figure 12.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter and maintaining fixed values of : (a) vs. for the simulation test; (b) vs. for the experimental test; (c) error vs. for the simulation test; and (d) error vs. for the experimental test.
Figure 13.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter and maintaining fixed values of : (a) d vs. for the simulation test; (b) d vs. for the experimental test; (c) vs. for the simulation test; and (d) vs. for the experimental test.
Figure 13.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter and maintaining fixed values of : (a) d vs. for the simulation test; (b) d vs. for the experimental test; (c) vs. for the simulation test; and (d) vs. for the experimental test.
Figure 14.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter with and a 2T delay period: (a) vs. for the simulation test; (b) vs. for the experimental test; (c) error vs. for the simulation test; and (d) error vs. for the experimental test.
Figure 14.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter with and a 2T delay period: (a) vs. for the simulation test; (b) vs. for the experimental test; (c) error vs. for the simulation test; and (d) error vs. for the experimental test.
Figure 15.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter with and a 2T delay period: (a) d vs. for the simulation test; (b) d vs. for the experimental test; (c) vs. for the simulation test; and (d) vs. for the experimental test.
Figure 15.
Bifurcation diagrams for the simulation and experimental test of the system with ZAD () and FPIC, changing the parameter with and a 2T delay period: (a) d vs. for the simulation test; (b) d vs. for the experimental test; (c) vs. for the simulation test; and (d) vs. for the experimental test.
Figure 16.
Behavior of the buck-motor system when perturbations in are presented for the experimental test with , , and = 1: (a) variation of over time; (b) regulated signal over time; (c) error in the controlled variable; and (d) phase diagram of the error vs. .
Figure 16.
Behavior of the buck-motor system when perturbations in are presented for the experimental test with , , and = 1: (a) variation of over time; (b) regulated signal over time; (c) error in the controlled variable; and (d) phase diagram of the error vs. .
Figure 17.
Experimental behavior of the buck-motor system after perturbations in when , and = 1: (a) variation in over time and (b) phase diagram of error vs. .
Figure 17.
Experimental behavior of the buck-motor system after perturbations in when , and = 1: (a) variation in over time and (b) phase diagram of error vs. .
Figure 18.
Experimental behavior of the buck-motor system after perturbations in when , , and = 1: (a) variation of over time and (b) phase diagram of the error vs. .
Figure 18.
Experimental behavior of the buck-motor system after perturbations in when , , and = 1: (a) variation of over time and (b) phase diagram of the error vs. .
Table 1.
Rated values of the DC motor.
Table 1.
Rated values of the DC motor.
Parameter | Description | Value |
---|
Pr | Rated power | 250 W |
Vr | Rated voltage | 42 VDC |
Ir | Rated current | 6 A |
| Rated speed | 4000 RPM |
Table 2.
Parameters of the buck-motor system.
Table 2.
Parameters of the buck-motor system.
Parameter | Description | Value |
---|
| Internal resistance of the source | 0.84 Ω |
| Input voltage | 40.086 V |
| Diode forward voltage | 1.1 V |
| Inductance | 2.473 mH |
| Internal resistance of the inductor | 1.695 Ω |
| Capacitance | 46.27 µF |
| Armature resistance | 2.7289 Ω |
| Armature inductance | 1.17 mH |
| Viscosity friction coefficient | 0.000138 (N·m/rad/s) |
| Inertia moment | 0.000115 (kg·m2) |
| Motor torque constant | 0.0663 (N·m/A) |
| Voltage constant | 0.0663 (V/rad/s) |
| Friction torque | 0.0284 (N·m) |
| Load torque | Variable (N·m) |
| Speed of the motor [rad/s] | 28 bits |
| Armature current [A] | 12 bits |
| Voltage of the motor [V] | 12 bits |
| Current in the inductor [A] | 12 bits |
Table 3.
Parameters of the ZAD-FPIC controller.
Table 3.
Parameters of the ZAD-FPIC controller.
Parameter | Description | Value |
---|
| Input voltage | 40.086 V |
| Reference speed | Variable (rad/s) |
| Control parameter of FPIC | 1 |
Fc | Commutation frequency | 6 kHz |
Fs | Sample frequency | 6 kHz |
1T_p | 1 delay period | 166.6 µs |
| Bifurcation parameter | Variables |
| Duty cycle | 10 bits |
Table 4.
Transient responses of the buck-motor system in closed-loop with ZAD-FPIC.
Table 4.
Transient responses of the buck-motor system in closed-loop with ZAD-FPIC.
Controller | Mp (%) | ts (s) | Error (%) |
---|
Closed-loop system in the simulation test | 0.5715 | 0.1473 | −0.1645 |
Closed-loop system in the experimental test | Overdamped | 0.1859 | 0.4245 |
Table 5.
Transient results obtained from the simulation in a closed loop with ZAD-FPIC.
Table 5.
Transient results obtained from the simulation in a closed loop with ZAD-FPIC.
Parameter | Mp (%) | ts (s) | Error (%) |
---|
( = 1) | 0.5717 | 0.1701 | −0.0173 |
( = 3) | Overdamped | 0.1701 | −0.0173 |
( = 5) | Overdamped | 0.1711 | −0.0173 |
( = 7) | Overdamped | 0.1737 | −0.1473 |
( = 9) | Overdamped | 0.1802 | −0.0173 |
Table 6.
Transient results for the experimental test when the system works in closed loop with ZAD-FPIC.
Table 6.
Transient results for the experimental test when the system works in closed loop with ZAD-FPIC.
Parameter | Mp (%) | ts (s) | Error (%) |
---|
( = 1) | Overdamped | 0.2311 | 0.2775 |
( = 3) | Overdamped | 0.2381 | 0.2772 |
( = 5) | Overdamped | 0.2414 | −0.3118 |
( = 7) | Overdamped | 0.2455 | −0.3118 |
( = 9) | Overdamped | 0.2505 | −0.4590 |