# Development of an Automatic Elastic Torque Control System Based on a Two-Mass Electric Drive Coordinate Observer

^{*}

## Abstract

**:**

## 1. Introduction

- shock (impact) loads that result in fatigue destruction of mechanical equipment;
- vibrations that occur in the shaft lines of stands and in the electromechanical systems that are resiliently connected by the strip;
- instability of control systems due to torque limitations being applied to the electric drives.

## 2. Problem Formulation

_{M}and M

_{S}, the reference speed n

_{ref}and the actual speeds n

_{UMD}, n

_{LMD}of the upper roll drive (UMD) and lower roll drive (LMD), respectively. Before nipping, no-load motor speeds are controlled within the range 3–3.5 mps. Rolls grip the strip as the drives are decelerating, whereby the angular gaps are fully open. Motor torques enter the domain limited at M

_{Mmax}, which is set equal to 4200 kN·m. Torque fluctuations attenuate due to dissipative forces. However, the damping effects of the motor are mitigated by the short loss of drive control. Shaft torques have a ratio k

_{Sm}= 2.2. This is the ratio of the maximum dynamic torque to the running torque (steady-state torque) (M

_{Smax}= 6500 kN∙m, M

_{st}= 3000 kN∙m).

- Earlier papers present research into electromechanical systems with two electric machines installed on the ends of an elastic shaft [42,43]. This enables control over the coordinates of both electric machines. These parameters include speeds and currents, which are used to calculate the torques of these machines as well as the elastic shaft torques. These coordinates are used in control algorithms.

- 2.

- Dynamic processes should be presented in the form of continuously measured physical parameters yet require minimum computing.
- Controllable coordinates are motor speed and torque as well as the applied load torque. The output coordinates are the roll speed and the elastic torque of the spindle.
- The developed solutions should not require complex mathematics (computing) and should be implementable in the controllers already used by the rolling mill APCS (automatic process control system).

## 3. Materials and Methods

_{S1}. Block 2 represents the closed torque control loop.

_{m}is the electromechanical time constant of the motor.

_{2}must be reconstructible from the reference torque M

_{RT}. Motor torque M

_{M}and first mass speed ω

_{1}are directly measurable, whereas the load torque M

_{ST}functions as the disturbance. Configuring the model is difficult because the observer must be able to reconstruct transients when the static torque M

_{ST}increases abruptly. In a closed uncontrolled mass coordinate control system, such an operation could not be implemented without overshooting and oscillations unless high controller gain (measured in thousands of units) was applied, which would be impractical at best.

_{M(PDA)}and motor speed n

_{M}as acquired from the process data acquisition (PDA) system the mill is equipped with. Using the observer (digital twin), transients of the elastic torque M

_{R(DT)}and roll speed n

_{R}were reconstructed from this data. The figure also shows actual curves of the elastic torque M

_{R(PDA)}as acquired from the mill sensors. They demonstrate that the static torque signal increases over the time Δt ≈ 0.07 s. This facilitates configuring the observer in development.

_{R(DT)}and M

_{R(PDA)}match in quality and amplitude alike. The error does not exceed 5%. Speed curves were only verified by modeling because the mill does not measure roll speeds. These dependencies were derived for the case of pre-closed angular gaps in the spindle joints. Thus, the dynamic shock (impact) caused by closing the gaps when gripping the strip was not there and would not induce errors in reconstructed torque. As shown below, the actual object has arrangements in place to ensure that the gaps be closed at the moment of gripping. Because the transients under consideration coincide, the conclusion is that the model and the object are adequate.

## 4. Implementation

#### 4.1. Development of the Closed Coordinate Control System for the Uncontrolled Mass

_{2}; elastic torque controller TC

_{12}; external controller (SC1) of the first mass speed J

_{1}, whose output is the desired motor torque. Gear ratios of the controllers were synthesized from the configuration rules of the slave coordinate control loops [51].

_{2}:

_{f}is connected in series with it to ensure astatism, see Figure 10b. This eliminates static disturbance error.

#### 4.2. Simulation Results

_{1}in the motor acceleration section with the angular gaps closed.

_{S}) reaches 120% at time instant t

_{2}and is kept there by the control system. This effectively limits the elastic torque of the spindle. As a result, the second mass speed control loop is open within the interval t

_{2}–t

_{3}. Therefore, the transient of the motor torque M

_{M}becomes complex, and the torque varies within the range from (−60%) to (+130%). In the second figure, the second mass speed control loop does not open. Transients of the first mass speed and second mass speed (n

_{MT}, n

_{S}) last about the same (t

_{1}–t

_{4}) in either case. Dynamic speed deviations also fall within identical ranges: 19–35 rpm at the gripping speed of 30 rpm; therefore, speed deviations from the target vary from (−37%) to (+17%).

_{M}does reach the limit. Elastic torque of the shaft M

_{S}changes similarly to the curve in Figure 11b and increases without overshooting. Speed transients in these figures differ; however, both last ~0.5 s.

- the amplitude margin is 10 dB, which is sufficient per Nyquist criterion;
- the phase plot is beak-shaped, no crossing through −180° to the left of the cutoff frequency; the phase margin is about 50°, which is also sufficient.

## 5. Experimental Results

- The system under consideration is a two-mass system where the first mass (the motor’s rotor) does not change its inertia.
- The stiffness of the elastic shaft (spindle) does not change either. This parameter depends on the length, diameter, and the properties of the metal that the shaft is made of. These parameters do not change even when the spindle is replaced.
- Second mass inertia depends on the mass of work and backup rolls, which is constant as well. The inertia of the rolled bar is 5% to 15% of the total second mass inertia. For configuration, use the mean bar inertia that deviates by 1.7% to 5%. Such deviations are commensurate with the error of the instrumentation sensors. Thus, they do not cause significant error in the configuration of the control algorithms.

#### 5.1. Accuracy of Reconstructed Two-Mass Coordinates

_{R(PDA)}from the sensor is amplified, whereas the reconstructed torque M

_{R(DT)}remains unchanged. This increases the error of reconstruction to 8–10%. The reconstructed torque has a greater amplitude than what is shown in the actual oscillogram.

- Greater gauge is associated with greater overshoot. Thus, at 9 mm, i.e., in the later passages, the ‘gapless’ amplitude is 35%, and the ‘gapped’ amplitude is 45%. In case of 30 mm gauge (the earlier passages), it is 240% and 212%, respectively.
- The torque component that stems directly from the gap-closing shock (impact) is relatively insignificant at 6.9% at 9 mm (1.55 and 1.45 p.u. of difference in amplitudes), 8.6% at 30 mm (2.4 and 2.21 p.u., respectively). This indirectly confirms that the ‘elastic’ torque component has the maximum impact on torque overshoot [54].
- Relative difference between actual and reconstructed values decreases as the gauge increases: 6.9% at 9 mm, 5.6% at 18 mm, and 4.1% at 30 mm.

#### 5.2. Testing the Electric Drive Coordinates in the Developed Three-Loop System

_{ST}differed due to the difference in electric drive loads, which in turn were due to different pressures on the upper and lower rolls. Metal strip was gripped with the gaps closed. This can be seen from the ‘upsurge’ in the oscillograms of M

_{S}and M

_{M}that occurred at the times t

_{1}and t

_{2}as the gaps were closed by pre-accelerating the electric drives. The algorithm forces the electric drive to decelerate between t

_{3}and t

_{4}after gripping.

_{3}, static load M

_{ST}equal to 2500 kN∙m is applied. This load is close to nominal; therefore the motor torques rise significantly. Load is applied at steady-state speed. Thus, technological factors cause a speed drop from 3.75 to 3.25 m/s. This deceleration takes almost 1 s, i.e., the deceleration rate is about 0.5 m/s

^{2}. The resulting dynamic torque is negligible in comparison with the load torque. Thus, change in the motor torques over the timeframe t > t

_{4}is insignificant, making it barely visible on the oscillograms.

_{Sm}and K

_{MTm}. They are found as the ratio of the maximum torque M

_{Smax}or M

_{Mmax}to the running torque M

_{ST}.

- k
_{Sm}was 1.8 with the initial configuration, changed to k_{Sm}= 1.2 after implementing the new algorithm. Thus, the algorithm reduced the torque amplitude by a factor of 1.5. - k
_{Sm}was 1.4 for the UMD torque M_{MT}, 1.12 for LMD, i.e., reduced by a factor of 1.25. - Elastic oscillations attenuated at least twice as fast. k
_{Sm}was 1.4 for the UMD torque Mmt, 1.12 for LMD, i.e., reduced by a factor of 1.25.

## 6. Discussion of the Results

- in case of gripping with pre-closed angular gaps in spindle joints, difference in the amplitudes of the measured vs reconstructed torque averages at 5 to 7%;
- equivalent difference in case of open gaps hits 10%.

## 7. Conclusions

- proportional speed controller for the first mass, whose output is the target motor torque;
- proportional controller of elastic torque. This configuration is optimal because the object structure contains an integrator, and this controller provides high performance;
- proportional speed controller for the second mass, which features a corrective proportional integral element at the input. This element increases the slope of the Bode magnitude plot in the low-frequency domain and enables astatic control over the second mass speed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Kinematic diagram of the rolls: 1 is the frame; 2 and 3 are electromechanical and hydraulic screw-downs, respectively; 4 and 5 are the backup rolls and work rolls, respectively; 6 is retractable spindles; 7 is the counterbalance; 8 is motors.

**Figure 2.**Spindle of a horizontal stand, Mill 5000: 1 is the sensor; 2 is the spindle shaft; 3 is the telemetry ring; 4 is the spindle head; 5 is the upper backup roll.

**Figure 3.**Oscillograms of the motor torques M

_{M}, elastic torques M

_{S}of spindle shafts, reference speed n

_{ref}and actual speeds n

_{UMD}, n

_{LMD}of the upper/lower roll drives.

**Figure 4.**Block diagram of the UMD and LMD speed control system; S

_{ref}, V

_{ref}are the movement trajectory and the speed reference values; V(t) is the interpolator-generated electric drive speed signal.

**Figure 5.**Kinematic diagram of the transmission (

**a**) and block diagram of the two-mass electromechanical system (

**b**): T

_{µ}for the uncompensated time constant; J

_{1}, J

_{2}for the moments of inertia, first mass and second mass; C

_{12}for the elastic coefficient of the mechanical transmission; β is the natural damping ratio (viscous friction type); M

_{M}is the motor torque; M

_{12}is the elastic torque of the spindle; M

_{ST}for the load torque; ω

_{ref}for the configured angular speed of the motor; ω

_{1}, ω

_{2}for the speeds of the first mass (the motor) and the second mass (the roll), respectively; k

_{S1}for the first mass speed feedback gain; k

_{fT}for the motor torque feedback gain in the figure.

**Figure 6.**Two-mass system model: (

**a**) for observer development; (

**b**) as a matrix: M

_{RT}is the target torque of the motor.

**Figure 8.**Transients of the measured and reconstructed coordinates in the virtual system (

**a**) and in the electric drives of Mill 5000 (

**b**), gripping action.

**Figure 9.**Generalized structure of the system with limited controller output signals (

**a**) and block diagram of the developed three-loop speed ACS for the second mass (

**b**).

**Figure 10.**Block diagrams of closed-loop elastic torque control (

**a**) for elastic torque and (

**b**) for speed of the second mass: M

_{RT(el)}is the configured elastic torque; ω

_{1ref}, ω

_{12ref}are the configured speeds of the first mass and the second mass; 1 is the elastic torque controller; 2 is speed 1 closed loop; 3 is the filter; 4 is the speed 2 controller; k

_{s2}is the second mass speed feedback gain; k

_{t}is elastic torque feedback gain.

**Figure 11.**Torque and speed transients in case of shock (impact) load, elastic torque limited (

**a**) and not limited (

**b**).

**Figure 12.**Counterparts of the transients in Figure 11b at an increased second mass speed controller gain.

**Figure 13.**Bode magnitude plots of the closed speed control loops (

**a**), Bode magnitude and phase plots with an open external loop (

**b**).

Parameter | Symbol | Dimensions | Value |
---|---|---|---|

Moment of inertia of the first moving mass (the motor) | J_{1} | kg∙m^{2} | 125,000 |

Moment of inertia of the second moving mass (the roll) | J_{2} | kg∙m^{2} | 114,571 |

Elastic coupling rigidity | c_{12} | N∙m/rad | 5,934,842 |

Eigenfrequency of elastic oscillations | ω_{12} | rad/s | 9.96 |

Electric drive acceleration | ε_{0} | rad/s^{2} | 1–3 |

Transmission gap | δ | rad | 0.017–0.051 (1–3°) |

Mean elastic torque | M_{12} | MN∙m | 1.9 |

Damping ratio | β | - | 2.817 |

Attenuation decrement | ξ | - | 0.172 |

Speed controller gain | k_{sc} | - | 19.5 |

Speed controller time constant | T_{sc} | s | 0.0041 |

Gauge | Measured Values | Observer Values | |||||||
---|---|---|---|---|---|---|---|---|---|

M_{max}(Gapped) | M_{max}(Gapless) | ΔM_{max}(Gap/Gapl.) | M_{max}(Obs.) | |ΔM_{max}|(Obs./Gap) | |ΔM_{max}|(Obs./Gapl.) | ||||

mm | p.u. | p.u. | p.u. | % | p.u. | p.u. | % | p.u. | % |

9 | 1.55 | 1.45 | 0.1 | 6.9 | 1.35 | 0.2 | 13.8 | 0.1 | 6.9 |

12 | 1.65 | 1.54 | 0.11 | 7.1 | 1.43 | 0.22 | 14.3 | 0.11 | 7.1 |

18 | 1.92 | 1.77 | 0.15 | 8.5 | 1.67 | 0.25 | 14.1 | 0.1 | 5.6 |

24 | 2.15 | 2 | 0.15 | 7.5 | 1.89 | 0.26 | 12.1 | 0.11 | 5.5 |

30 | 2.4 | 2.21 | 0.19 | 8.6 | 2.12 | 0.28 | 11.7 | 0.09 | 4.1 |

Parameter | Oscillogram Value | |||||
---|---|---|---|---|---|---|

UMD | LMD | |||||

Shaft torque | M_{st} | M_{Smax} | k_{Sm} | M_{st} | M_{Smax} | k_{Sm} |

kN∙m | - | kN∙m | - | |||

1400 | 2500 | 1.8 | 2250 | 2700 | 1.2 | |

Motor torque | M_{st} | M_{Smax} | k_{Sm} | M_{st} | M_{Smax} | k_{Sm} |

kN∙m | - | kN∙m | - | |||

1400 | 2000 | 1.4 | 2250 | 2550 | 1.12 |

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**MDPI and ACS Style**

Radionov, A.A.; Karandaev, A.S.; Gasiyarov, V.R.; Loginov, B.M.; Gartlib, E.A.
Development of an Automatic Elastic Torque Control System Based on a Two-Mass Electric Drive Coordinate Observer. *Machines* **2021**, *9*, 305.
https://doi.org/10.3390/machines9120305

**AMA Style**

Radionov AA, Karandaev AS, Gasiyarov VR, Loginov BM, Gartlib EA.
Development of an Automatic Elastic Torque Control System Based on a Two-Mass Electric Drive Coordinate Observer. *Machines*. 2021; 9(12):305.
https://doi.org/10.3390/machines9120305

**Chicago/Turabian Style**

Radionov, Andrey A., Alexandr S. Karandaev, Vadim R. Gasiyarov, Boris M. Loginov, and Ekaterina A. Gartlib.
2021. "Development of an Automatic Elastic Torque Control System Based on a Two-Mass Electric Drive Coordinate Observer" *Machines* 9, no. 12: 305.
https://doi.org/10.3390/machines9120305