Thursday, March 19, 2009

Servo Motion Control - PID Control

PID position loops

The velocity loop is the most basic servo control loop. However,
since a velocity loop cannot ensure that the machine stays in
position over long periods of time, most applications require
position control. There are two common configurations used for
position control: the cascaded position-velocity loop, as discussed
last month, and the PID position controller, as shown below.

Block diagram of PID position loop
The position loop compares a position command to a position
feedback signal, and calculates the position error, PE. In a PID
controller, current command is generated with three gains: PE is
scaled by the proportional gain (KPP), the integral of PE is scaled by
the integral gain (KPI), and the derivative of PE is scaled by the
derivative gain (KPD).

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Servo Motion Control - PID Control

The basic components of a typical servo motion system are
depicted in Fig.1 using standard LaPlace notation. In this figure,
the servo drive closes a current loop and is modeled simply as
a linear transfer function G(s). Of course the servo drive will
have peak current limits, so this linear model is not entirely
accurate, however it does provide a reasonable representation
for our analysis. In their most basic form, servo drives receive
a voltage command that represents a desired motor current.
Motor shaft torque, T is related to motor current, I by the torque
constant, Kt. Equation (1) shows this relationship.

For the purposes of this discussion the transfer function of
the current regulator or really the torque regulator can be
approximated as unity for the relatively lower motion frequencies
we are interested in and therefore we make the following
approximation shown in (2).

The servomotor is modeled as a lump inertia, J, a viscous damping
term, b, and a torque constant, Kt. The lump inertia term is
comprised of both the servomotor and load inertia. I t is also
assumed that the load is rigidly coupled such that the torsional
rigidity moves the natural mechanical resonance point well
out beyond the servo controller’s bandwidth. This assumption
allows us to model the total system inertia as the sum of the
motor and load inertia for the frequencies we can control.
Somewhat more complicated models are needed if coupler
dynamics are incorporated.

The actual motor position, q(s) is usually measured by either an
encoder or resolver coupled directly to the motor shaft. Again the
underlying assumption is that the feedback device is rigidly
mounted such that its mechanical resonant frequencies can be
safely ignored. External shaft torque disturbances, Td are added
to the torque generated by the motor's current to give the torque
available to accelerate the total inertia, J.

Around the servo drive and motor block is the servo controller that
closes the position loop. A basic servo controller generally contains
both a trajectory generator and a PID controller. The trajectory
generator typically provides only position setpoint commands labeled
in Fig.1 as q* (s). The PID controller operates on the position error
and outputs a torque command that is sometimes scaled by an
estimate of the motor's torque constant, ˆt K . I f the motor's torque
constant is not known, the PID gains are simply re-scaled accordingly.
Because the exact value of the motor's torque constant is generally
not known, the symbol "^ " is used to indicate it is an estimated value
in the controller. In general, equation (3) holds with sufficient accuracy
so that the output of the servo controller (usually + / - 10 volts) will
command the correct amount of current for a desired torque.

There are three gains to adjust in the PID controller, Kp, Ki and Kd.
These gains all act on the position error defined in (4). Note the
superscript "* " refers to a commanded value.

The output of the PID controller is a torque signal. I ts mathematical
expression in the time domain is given in (5).

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