Estimation and Control of a Resonant Plant Prone to Stick-Slip Behavior

• US 20070148007A1
• Filed: 11/29/2006
• Published: 06/28/2007
• Est. Priority Date: 11/29/2005
• Status: Active Grant

• Current Assignee: Unico, LLC (Regal Rexnord Corp.)
• Original Assignee: Unico, Inc.
• Inventors: Garlow, Mark
• Granted Patent: US 7,645,124 B2
• Time in Patent Office: Days
• US Class Current: 417/44.110
• CPC Class Codes: F04B 2203/0209 Rotational speed
• CPC Class Codes: F04B 49/20 by changing the driving speed

Abstract

A method and apparatus are provided for estimating and/or precluding stick-slip, or other oscillatory or resonant behavior, through use of a virtual transducer, which precludes the need for having sensors located adjacent to a driven element of the system, or adjacent contact surfaces at which the stick-slip relative motion may occur. Parameters measurable at a drive mechanism are utilized for controlling a system in a manner which precludes stick-slip, or other oscillatory or resonant behavior, of the driven element. Relative motion between contacting surfaces in the driven element, prone to stick-slip behavior, is controlled such that, after sufficient force is applied by the drive element to overcome static friction forces between the contacting surfaces and break them free from one another, relative motion between the surfaces is maintained at a high enough relative speed that the surfaces are precluded from statically contacting one another, so that stick-slip behavior is precluded.

Claims

1. A method to control a system exhibiting resonant behavior and/or stick-slip behavior and having unmeasurable states comprising the steps of:

receiving an electrical torque parameter, a crank angle parameter, and a crank speed parameter;

estimating the unmeasurable states;

sending estimates of the unmeasurable states to a regulator; and

regulating the system to minimize differences between reference states and the estimates.

2. The method of claim 1 wherein the regulator is one of a linear quadratic regulator, a binomial full state feedback regulator, a Bessel full state feedback regulator, and an ITAE ((integral of time multiplied by the absolute value of error) full state feedback regulator.

3. The method of claim 2 wherein the system is a down-hole pump system and the unmeasurable states are pump angle and pump speed and the regulator structure has a gain [k1;

k2;

k3;

k4] that corresponds to a gain for errors of a reference vector

x*=[θ

c*, ω

c*, θ

p*, ω

p*] minus four system states {circumflex over (x)}=[{circumflex over (θ

)}_c, {circumflex over (ω

)}_c, {circumflex over (θ

)}_p, {circumflex over (ω

)}_p] where θ

c* is a crank angle command, ω

c* is a crank speed command, θ

p* is a pump angle command, ω

p* is a pump speed command, {circumflex over (θ

)}_c is a crank angle position, {circumflex over (ω

)}_c is a crank speed, {circumflex over (θ

)}_p is a pump angle estimate, and {circumflex over (ω

)}_p is a pump speed estimate.

4. The method of claim 1 wherein the step of estimating the unmeasurable states comprises the step of estimating the unmeasurable states with a finite difference state estimator.

5. The method of claim 4 wherein the system is a down-hole pump system having a motor driving a pump via a gearbox and at least one rod and the unmeasurable states are pump angle and pump speed, wherein the step of estimating the unmeasurable states with a finite difference state estimator comprises the steps of estimating the pump angle in accordance with the equation

θ

p(z)=θ

c(z)−

Ng/Kr(T_e(z)−

(b2+br/Ng)*ω

c(z)−

(J2/Ng*T)*(ω

c(z)−

ω

c(z−

1))

where θ

c is a crank angle parameter, Te is an electrical torque parameter, ω

c is a crank speed parameter, Ng is an estimate of the gear reduction ratio, Kr is an estimate of the rod spring stiffness constant, b2 is an estimate of the motor damping, br is an estimate of the rod damping, J2 is an estimate of motor and gearbox inertia, and θ

p is the estimated pump angle.

6. The method of claim 5 wherein the step of estimating the unmeasurable states with a finite difference state estimator further comprises the steps of estimating the pump speed in accordance with the equation

ω

p(z)=1/T(θ

p(z)−

θ

p(z−

1))

where ω

p(z) is the estimated pump speed.

7. The method of claim 1 wherein the step receiving an electrical torque parameter, a crank angle parameter, and a crank speed parameter comprises the steps of:

receiving a voltage measurement and a current measurement;

estimating the electrical torque parameter, the crank angle parameter, and the crank speed parameter based upon the voltage measurement and the current measurement.

8. The method of claim 1 wherein the system has unmeasurable states in a plurality of sections connected to each other and the step of estimating the unmeasurable states comprises the step of estimating the unmeasurable states with a multi-section finite difference state estimator having a plurality of nodes, wherein each of the plurality of nodes estimates the angle and speed of each section in the multi-section state estimator.

9. The method of claim 8 wherein the system is a down-hole pump system having a plurality of rods connected to a pump, the unmeasurable states are pump angle and pump speed, and a first stage node in the plurality of nodes estimates an intermediate angle θ

(2) estimate and speed ω

(2) estimate based upon the electrical torque parameter, the crank angle parameter, and the crank speed parameter.

10. The method of claim 9 wherein each of the remaining nodes estimates an angle and a speed with inputs of previous estimates.

11. A computer-readable medium having computer executable instructions to control a system exhibiting resonant behavior and/or stick-slip behavior and having unmeasurable states, the computer executable instructions, when executed, performing the steps of:

receiving an electrical torque parameter, a crank angle parameter, and a crank speed parameter;

estimating the unmeasurable states;

sending estimates of the unmeasurable states to a regulator; and

regulating the system to minimize differences between reference states and the estimates.

12. The computer readable medium of claim 11 wherein the regulator is one of a linear quadratic regulator, a binomial full state feedback regulator, a Bessel full state feedback regulator, and an ITAE ((integral of time multiplied by the absolute value of error) full state feedback regulator.

13. The computer readable medium of claim 12 wherein the system is a down-hole pump system and the unmeasurable states are pump angle and pump speed and the regulator structure has a gain [k1;

k2;

k3;

k4] that corresponds to a gain for errors of a reference vector

x*=[θ

c*, ω

c*, θ

p*, ω

p*] minus four system states {circumflex over (x)}=[{circumflex over (θ

)}_c, {circumflex over (ω

)}_c, {circumflex over (θ

)}_p, {circumflex over (ω

)}_p] where θ

c* is a crank angle command, ω

c* is a crank speed command, θ

p* is a pump angle command, ω

p* is a pump speed command, {circumflex over (θ

)}_c is a crank angle position, {circumflex over (ω

)}_c is a crank speed, {circumflex over (θ

)}_p is a pump angle estimate, and {circumflex over (ω

)}_p is a pump speed estimate.

14. The computer readable medium of claim 11 wherein the step of estimating the unmeasurable states comprises the step of estimating the unmeasurable states with a finite difference state estimator.

15. The computer readable medium of claim 14 wherein the system is a down-hole pump system having a motor driving a pump via a gearbox and at least one rod and the unmeasurable states are pump angle and pump speed, wherein the step of estimating the unmeasurable states with a finite difference state estimator comprises the steps of estimating the pump angle in accordance with the equation

θ

p(z)=θ

c(z)−

Ng/Kr(T_e(z)−

(b2+br/Ng)*ω

c(z)−

(J2/Ng*T)*(ω

c(z)−

ω

c(z−

1))

where θ

c is a crank angle parameter, Te is an electrical torque parameter, ω

c is a crank speed parameter, Ng is an estimate of the gear reduction ratio, Kr is an estimate of the rod spring stiffness constant, b2 is an estimate of the motor damping, br is an estimate of the rod damping, J2 is an estimate of motor and gearbox inertia, and θ

p is the estimated pump angle.

16. The computer readable medium of claim 15 wherein the step of estimating the unmeasurable states with a finite difference state estimator further comprises the steps of estimating the pump speed in accordance with the equation

ω

p(z)=1/T(θ

p(z)−

θ

p(z−

1))

where ω

p(z) is the estimated pump speed.

17. The computer readable medium of claim 11 wherein the step receiving an electrical torque parameter, a crank angle parameter, and a crank speed parameter comprises the steps of:

receiving a voltage measurement and a current measurement;

estimating the electrical torque parameter, the crank angle parameter, and the crank speed parameter based upon the voltage measurement and the current measurement.

18. The computer readable medium of claim 11 wherein the system has unmeasurable states in a plurality of sections connected to each other and the step of estimating the unmeasurable states comprises the step of estimating the unmeasurable states with a multi-section finite difference state estimator having a plurality of nodes, wherein each of the plurality of nodes estimates the angle and speed of each section in the multi-section state estimator.

19. The computer readable medium of claim 18 wherein the system is a down-hole pump system having a plurality of rods connected to a pump, the unmeasurable states are pump angle and pump speed, the computer readable medium having further computer executable instructions for performing the step of a first stage node in the plurality of nodes estimating an intermediate angle θ

(2) estimate and speed ω

(2) estimate based upon the electrical torque parameter, the crank angle parameter, and the crank speed parameter.

20. The computer readable medium of claim 19 the computer readable medium having further computer executable instructions for performing the step of each of the remaining nodes estimating an angle and a speed with inputs of previous estimates