The fundamental concepts of servo motion control have not changed significantly in the last 50
years. The basic reasons for using servo systems in contrast to open loop systems include the need
to improve transient response times, reduce the steady state errors and reduce the sensitivity to
load parameters.
Improving the transient response time generally means increasing the system bandwidth. Faster
response times mean quicker settling allowing for higher machine throughput. Reducing the steady
state errors relates to servo system accuracy. Finally, reducing the sensitivity to load parameters
means the servo system can tolerate fluctuations in both input and output parameters. An example
of an input parameter fluctuation is the incoming power line voltage. Examples of output
parameter fluctuations include a real time change in load inertia or mass and unexpected shaft
torque disturbances.
Servo control in general can be broken into two fundamental classes of problems. The first class
deals with command tracking. It addresses the question of how well does the actual motion follow
what is being commanded. The typical commands in rotary motion control are position, velocity,
acceleration and torque. For linear motion, force is used instead of torque. The part of servo
control that directly deals with this is often referred to as “Feedforward” control. It can be thought
of as what internal commands are needed such that the user’s motion commands are followed
without any error, assuming of course a sufficiently accurate model of both the motor and load is
known.
The second general class of servo control addresses the disturbance rejection characteristics of the
system. Disturbances can be anything from torque disturbances on the motor shaft to incorrect
motor parameter estimations used in the feedforward control. The familiar “P.I.D.” (Proportional
Integral and Derivative position loop) and “P.I.V. ” (Proportional position loop Integral and
proportional Velocity loop) controls are used to combat these types of problems. In contrast to
feedforward control, which predicts the needed internal commands for zero following error,
disturbance rejection control reacts to unknown disturbances and modeling errors. Complete servo
control systems combine both these types of servo control to provide the best overall performance.
We will examine the two most common forms of disturbance rejection servo control, P.I.D. and
P.I.V. After understanding the differences between these two topologies, we will then investigate
the additional use of a simple feedforward controller for an elementary trapezoidal velocity move
profile.
P.I.D. 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.
T » Kt I (1)
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).
G(s) » 1 (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. It is also assumed
that the load is rigidly coupled such that the torsional rigidity moves the natural mechanical
resonance point well 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.
Figure
Figure 1. Basic P.I.D. Servo Control Topology.
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 P.I.D. controller. The
trajectory generator typically provides only position setpoint commands labeled in Fig.1 as q*(s).
The P.I.D. controller operates on the position error and outputs a torque command that is
sometimes scaled by an estimate of the motor's torque constant, ˆ
Kt . If the motor’s torque constant
is not known, the P.I.D. 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 cont roller. 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.
ˆ Kt » Kt (3)
There are three gains to adjust in the P.I.D. controller, Kp, Ki and Kd. These gains all act on the
position error defined in (4). Note the superscript * refers to a commanded value.
error(t) = q* (t) - q(t) (4)
We now look at how one selects the gains, Kp, Ki and Kd.
P.I.V. Control
In order to be able to better predict the system response, an alternative topology is needed. One
example of an easier-to-tune topology is the P.I.V. controller shown in Fig. 3. This controller
basically combines a posit ion loop with a velocity loop. More specifically, the result of the
position error multiplied by Kp becomes a velocity correction command. The integral termKi now
operates directly on the velocity error instead of the position error as in the P.I.D. case and finally,
the Kd term in the P.I.D. position loop is replaced by a Kv term in the P.I.V. velocity loop. Note,
however, they have the same units, Nm/(rad/sec).
Figure 2. Basic P.I.V. Servo Control Topology.
P.I.V. control requires the knowledge of the motor velocity, labeled velocity estimator in Figure 2.
This is usually formed by a simple filter; however, significant delays can result and must be
accounted for if truly accurate responses are needed. Alternatively, the velocity can be obtained by
use of a velocity observer. This observer requires the use of other state variables in exchange for
providing zero lag filtering properties. In either case, a clean velocity signal must be provided for
P.I.V. control.
As an example of this tuning approach, we investigate the response of a Parker Gemini series servo
drive and built- in controller using the same motor from the previous example. Again, we begin
with observing the response to a step input command with no external disturbance torque (Td = 0).
Tuning the P.I.V. Loop
To tune this system, only two control parameters are needed, the bandwidth (BW) and the damping
ratio (z). An estimate of the motor’s total inertia, ˆ J , and damping, ˆb , are also required at set-up
and are obtained using the motor/drive set up utilities. Figure 3 illustrates typical response plots for
various bandwidths and damping ratios.
a) Damping fixed(? ? ?), Bandwidth varies
b) Bandwidth fixed(BW=20Hz), Damping varies
Figure 3. Step Response curves for P.I.V. tuning.
Disturbance Rejection Characteristics
If we consider the case where the motor is holding a final position and an unknown disturbance at
some frequency is applied to the motor shaft, can we predict how the shaft will respond? To graphically quantify the effect of the P.I.V. gains on this disturbance torque, disturbance rejection
plots are drawn showing the magnitude of disturbance needed at a given frequency to result in a
specified shaft motion. Fig. 4 illustrates the two cases of constant damping and varying bandwidths
(Fig. 4a) and constant bandwidth and varying damping (Fig. 4b).
a) Constant Damping (z=1), Bandwidth varies
b) Constant Bandwidth (BW=20Hz), Damping varies
Figure 4. Disturbance rejection plots for the P.I.V. system.
In Fig. 5a) the low frequency rejection gain increases as the bandwidth value goes up. Notice for
high- frequency rejection, only the motor's inertia is effective regardless of the bandwidth of the
servo controller. This is why users occasionally request "high inertia" motors even at the expense
of requiring more overall torque to make their move. A similar disturbance rejection trend is seen
in Fig. 5 b). Here, the mid-frequency rejection gain increases slightly as the damping ratio is
raised.
In general, the higher the disturbance rejection, the stiffer the system is and the more likely it will
provide repeatable moves in the presence of unknown shaft disturbances.
Trapezoidal Motion Profiles
Typically, servo systems are first tuned with a step input in order to get a feel for the system
response. Once this is done, the user now is often interested in how their actual mo tion will
behave. At this point, the user must decide on the nature of the velocity profile. By far the most
common velocity profile is the trapezoid. This is due to the relative ease of calculating all the state
variables needed for motion: position, velocity and acceleration. As the need for smoother
accelerations and decelerations becomes greater, either "S" profiles or cubic splines are often
employed.
For the purposes of our investigation, we will focus on the use of a simple trapezoidal velocity
profile. The test move is 2 revolutions in a quarter of a second, with equal times for acceleration,
constant velocity and deceleration. Fig. 5 shows the position and velocity profiles for this move
using P.I.V. control. Again notice how by increasing the bandwidth, the position response
improves (Fig. 5a) as does the velocity response (Fig. 5b).
a) Position Response
b) Velocity Response
Figure 5. Response Curves using P.I.V. control with z=1?, and varying BW
As a rule of thumb, the bandwidth should be increased as high as possible while still maintaining
stable and predictable operation. If some overshoot can be tolerated, the damping ratio can be
lowered to further reduce the rise time. Fig. 6 depicts the case where the bandwidth is held
constant at 20 Hz and the damping ratio is lowered from 1 (critically damped) to .5 (under
damped). Figs. 6 a) and b) show the position and ve locity responses respectively. Notice that even
with the damping ratio as low as .5, there is very little overshoot. This is because the trapezoidal
profile does not greatly excite this damped resonance.
a) Position Response
b) Velocity Response
Figure 6. Response Curves using P.I.V. control with BW=20Hz?, and varying z
Also notice in Fig. 6 the amount of following error in both the position and velocity profiles. The
fundamental requirement of any disturbance rejection control is the need to have error in order for
the system to respond. Clearly, if we want to achieve near zero following error, another solution is
needed.
Feedforward Control
Feedforward control goes a long way towards reducing settling times and minimizing overshoot;
however, there are several of assumptions that ultimately limit its effectiveness. For example,
servo amplifiers all have current limits and finite respons e times. For motion bandwidths in the sub
50 Hz range, the current loops can be safely ignored; however, as the need to push the motion
bandwidths higher, the current loops ne ed to be accounted for as well. In addition, the single most
limiting factor in servo motion control is the resolution and accuracy of the feedback device. Lowresolution
encoders contribute to poor velocity estimations that lead to either limit cycling or
velocity ripple problems. Finally, compliant couplers that connect the load to the servomotor must
also be accounted for as they too limit the useable motion bandwidths.
In summary, disturbance rejection control can be obtained by one of a number of ways, the two
most common are P.I.D. and P.I.V. control. The direct use of P.I.D. control can often meet lowperformance
motion control loops and are generally set by either the Ziegler Nichols or by trialand-
error methods. Overshoot and rise times are tightly coupled, making gain adjustments
difficult. P.I.V. control, on the other hand, provides a method to significantly decouple the
overshoot and rise time, allowing for easy set up and very high disturbance rejection
characteristics. Finally, feedforward control is needed in addition to disturbance rejection control
to minimize the tracking error.
