Estimation and control of a resonant plant prone to stick-slip behavior
View Patent ↗An apparatus is 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 a driven element of the system. Relative motion between contacting surfaces in the driven element, prone to stick-slip behavior, is controlled 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 contacting one another, so that stick-slip behavior is precluded.
1. A tangible, non-transitory computer readable medium storing a set of instructions executable to control a system exhibiting stick-slip behavior and having unmeasurable states, the computer executable instructions, when executed, performing 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, 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, and, wherein the system is a down-hole pump system and the unmeasurable states are pump angle and pump speed and the regulator has a gain [k 1 ;k 2 ;k 3 ;k 4 ] that corresponds to a gain for errors of a reference vector x* =[Ac*, Wc*, Ap*, Wp*] minus four system states {circumflex over (x)}=[Â c , Ŵ c , Â p ,Ŵ p ] where Ac* is a crank angle command, Wc* is a crank speed command, Ap* is a pump angle command, Wp* is a pump speed command, Â c is a crank angle position, Ŵ c is a crank speed, Â p is a pump angle estimate, and Ŵ p is a pump speed estimate.
2. The computer readable medium 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.
3. The computer readable medium of claim 2 , 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.
Ap
(
z
)
=
Ac
(
z
)
-
Ng
Kr
(
T
e
(
z
)
-
b
2
+
b
Ng
*
Wc
(
z
)
-
(
J
2
Ng
*
T
)
*
(
Wc
(
z
)
-
Wc
(
z
-
1
)
)
where T is the sampling period, Ac is the crank angle parameter, Te is the electrical torque parameter, We is the crank speed parameter, Ng is an estimate of gear reduction ratio, Kr is an estimate of rod spring stiffness constant, b 2 is an estimate of drive damping, b is an estimate of the pump damping, J 2 is an estimate of the inertia of the pump, and Ap is the estimated pump angle.
4. The computer readable medium of claim 3 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
Wp
=
1
T
(
Ap
(
z
)
-
Ap
(
z
-
1
)
)
where Wp is the estimated pump speed.
5. The computer readable medium of claim 1 wherein the step of 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.
6. The computer readable medium 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 a angle and speed of each section in the multi-section state estimator.
7. The computer readable medium of claim 6 wherein the system having a plurality of rods connected to the pump, 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 a( 2 ) estimate and speed w( 2 ) estimate based upon the electrical torque parameter, the crank angle parameter, and the crank speed parameter.
8. The computer readable medium of claim 7 wherein the computer readable medium having further computer executable instructions for performing the step of each of nodes estimating an angle and a speed of each section with inputs of previous estimates.