Control Video Lecture Industrial Automation and Control - Introduction to Automatic Control ,PID Control and PID Control Tuning

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Video Lecture PID Control Tuning

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Control Auto-Tuning Control Base on Ziegler-Nichols

Auto-Tuning Control Using Ziegler-Nichols
Automatic step testsOne of the earliest auto-tuning controllers still on the market is the 53MC5000 Process Control Station from MicroMod Automation. It uses the Easy-Tune algorithm originally developed at Fischer & Porter (now part of ABB) in the early 1980s. It automatically executes a step test similar to the open-loop Ziegler-Nichols method that forces the controller to make an abrupt change in its control effort while sensor feedback is disabled.

The amount by which the process variable subsequently changes and the time required
for it to reach 63.2% of its final value indicate the steady-state gain and time constant of the process, respectively. If the sensor in the loop happens to be located some distance from the actuator, the process’s response to such a step input may also demonstrate a deadtime between the instant that the step was applied and the instant that the process variable first began to react.

These three model parameters tell the Easy- Tune algorithm everything it needs to know about the behavior of a typical process, allowing it to predict how the process will react to any corrective effort, not just step inputs. That in
turn allows the Easy-Tune algorithm to compute tuning parameters to make the controller compatible with the process.



Closed loop tests
In 1984, Karl Åström and Tore Hägglund of the Lund (Sweden) Institute of Technology
published an improved version of Ziegler and Nichols’ closed-loop tuning method. Like the open-loop method, this technique excites the process to identify its behavior, but without disabling sensor feedback.

The Åström-Hägglund method works by forcing the process variable into a series of
sustained oscillations known as a limit cycle. The controller first applies a step input to the process and holds it at a user-defined value until the process variable passes the setpoint. It then applies a negative step and waits for the process variable to drop back below the setpoint. Repeating this procedure each time the process variable passes the setpoint in either direction forces the process variable to oscillate out of sync with the control effort, but at the same frequency. See the “Relay Test” graphic. The time required to complete a single oscillation is known as the process’s ultimate period (Tu), and the relative amplitude of the two oscillations multiplied by 4/π gives the ultimate gain (Pu). Ziegler and Nichols theorized that these two parameters could be used instead of the steady-state gain, time constant, and deadtime to compute suitable tuning parameters according to their famous tuning equations or tuning rules shown in the equation on the left.

They discovered empirically that these rules generally yield a controller that responds quickly to intentional changes in the setpoint as well as to random disturbances to the process variable. However, a controller thus tuned will also tend to cause overshoot and oscillations in the process variable, so most auto-tuning controllers offer several sets of alternative tuning rules that make the controller less aggressive to varying degrees. An operator typically only has to select the required speed of response (slow, medium, fast), and the controller chooses appropriate rules automatically.

http://www.das.ufsc.br/~aarc/ensino/posgraduacao/DAS6613/Auto-Tuning%20Control%20Using%20Ziegler-Nichols.pdf
REVISITING THE ZIEGLER-NICHOLS TUNING RULES
FOR PI CONTROL — PART II
THE FREQUENCY RESPONSE METHOD
ABSTRACT
This paper presents an analysis of the Ziegler-Nichols frequency response
method for tuning PI controllers, showing that this method has severe
limitations. The limitations can be overcome by a simple modification for
processes where the time delay is not too short. By a major modification it is
possible to obtain new tuning rules that also cover processes that are lag
dominated.

I. INTRODUCTION
II. TEST BATCH AND DESIGN METHOD
2.1 The MIGO design method
2.2 The test batch
2.3 The AMIGOs tuning rules
2.4 Parameterization
III. A FIRST ATTEMPT
3.1 Stable processes
3.2 Integrating processes
3.3 Tuning rules for balanced and
delay-dominated processes
3.4 Summary
IV. ANALYSIS
4.1 Modified tuning procedures
V. THE AMIGOF TUNING RULES
5.1 Other values of Ms
5.2 How to find the frequency ωφ?
5.3 Summary
VI. AN INTERPRETATION OF
THE RESULTS
VII. EXAMPLES
Example 1. LAG DOMINATED DYNAMICS
Example 2. BALANCED LAG AND DELAY
Example 3. DELAY DOMINATED DYNAMICS
VIII. CONCLUSION

http://www.ajc.org.tw/pages/PAPER/6.4PD/AC0604-P469-FR0371.pdf

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Control PID Controllers Auto Tuning - Relay Feedback

Relay Feedback Auto Tuning of PID Controllers

IntroductionFor a certain class of process plants, the so-called \auto tuning" procedure
for the automatic tuning of PID controllers can be used. Such a procedure
is based on the idea of using an on/off controller (called a relay controller)
whose dynamic behaviour resembles to that shown in Figure 1(a). Starting
from its nominal bias value (denoted as 0 in the Figure) the control action
is increased by an amount denoted by h and later on decreased until a value
denoted by -h.




The closed-loop response of the plant, subject to the above described ac-
tions of the relay controller, will be similar to that depicted in Figure 1(b).
Initially, the plant oscillates without a de¯nite pattern around the nominal
output value (denoted as 0 in the Figure) until a de¯nite and repeated out-
put response can be easily identi¯ed. When we reach this closed-loop plant
response pattern the oscillation period (Pu) and the amplitude (A) of the
plant response can be measured and used for PID controller tuning. In fact,
the ultimate gain can be computed as:
Having determined the ultimate gain Kcu and the oscillation period Pu
the PID controller tuning parameters can be obtained from the following
table:
Example of Relay Feedback Auto Tuning of PID Controllers

http://200.13.98.241/~antonio/cursos/control/notas/siso/atv.pdf
Relay-based PID Tuning
ABSTRACT
Relay-based auto tuning is a simple way to tune PID controllers
that avoids trial and error, and minimises the possibility
of operating the plant close to the stability limit.

http://homepages.ihug.co.nz/~deblight/AUTResearch/papers/relay_autot.pdf


An Improved Relay Auto Tuning of PID Controllers for SOPTD
Systems


Difficulties of loop tuning
When you discuss loop tuning with instrument and control
engineers, conversation soon turns to the Zeigler-Nichols
(ZN) ultimate oscillation method. Invariably the plant engineer
soon responds with ‘Oh yes, I remember the ZN tuning
scheme, we tried that and the plant oscillated itself into
oblivion — bad strategy. Moreover when it did work, the
responses are overly oscillatory’
So given the tedious and possibly dangerous plant trials
that result in poorly damped responses, it behoves one to
speculate why it is often the only tuning scheme many instrument
engineers are familiar with, or indeed ask if it has
any concrete redeeming features at all.
In fact the ZN tuning scheme, where the controller gain
is experimentally determined to just bring the plant to the
brink of instability is a form of model identification. All
tuning schemes contain a model identification component,
but the more popular ones just streamline and disguise that
part better. The entire tedious procedure of trial and error
is simply to establish the value of the gain that introduces
half a cycle delay when operating under feedback. This is
known as the ultimate gain Ku and is related to the point
where the Nyquist curve of the plant in Fig. 1(b) first cuts
the real axis.

The problem is of course, is that we rarely have the luxury
of the Nyquist curve on the factory floor, hence the
experimentation required.
Abstract Using a single symmetric relay
feedback test, a method is proposed to identify
all the three parameters of a stable second order
plus time delay (SOPTD) model with equal time
constants. The conventional analysis of relay
auto-tune method gives 27% error in the
calculation of ku,. In the present work, a method
is proposed to explain the error in the ku
calculation by incorporating the higher order
harmonics. Three simulation examples are given.
The estimated model parameters are compared
with that of Li et al. [4] method and that of
Thyagarajan and Yu [8] method. The open loop
performance of the identified model is compared
with that of the actual system. The proposed
method gives performances close to that of the
actual system. Simulation results are also given
for a nonlinear bioreactor system. The open loop
performance of the model identified by the
proposed method gives a performance close to
that of the actual system and that of the locally
linearized model. SOPTD model, symmetric relay, auto-tuning

http://ntur.lib.ntu.edu.tw/bitstream/246246/87370/1/09.pdf

DEVELOPMENT OF AN AUTO-TUNING PID AND
APPLICATIONS TO THE PULP AND PAPER INDUSTRY
Abstract
An auto-tuning industrial PID is presented. The autotuning
is realized in three steps. The process is first
adequately excited in order to generate good quality data
for the second step, the process identification. The last step
is the PID tuning based on the evaluated parametric model.
The auto-tuning PID has been implemented on two
different control systems and successful applications to
processes of the pulp and paper industry are analyzed.

http://www.iaeng.org/publication/WCECS2007/WCECS2007_pp175-181.pdf

Auto-tune system using single-run relay feedback test
and model-based controller design
Abstract
In this paper, a systematic approach for auto-tune of PI/PID
controller is proposed. A single run of the relay feedback experiment
is carried out to characterize the dynamics including the type
of damping behavior, the ultimate gain, and ultimate frequency.
Then, according to the estimated damping behavior, the process
is classified into two groups. For each group of processes,
modelbased rules for controller tuning are derived in terms of
ultimate gains and ultimate frequencies. To classify the processes,
the estimation of an apparent deadtime is required. Two artificial
neural networks (ANNs) that characterize this apparent deadtime using
the ATV data are thus included to facilitate this estimation of
this apparent deadtime. The model-based design for this auto-tuning
makes uses of parametric models of FOPDT (i.e. first-order-plus-dead-time)
and of SOPDT (i.e. second-order-plus-dead-time)
dynamics. The results from simulations show that the controllers
thus tuned have satisfactory results compared with those from
other methods.

Tuning strategy for the model-based auto-tune system.


http://w3.gel.ulaval.ca/~desbiens/publications/DevelopmentOfAnAutoTuningPID.pdf

MODIFICATION AND APPLICATION OF AUTOTUNING
PID CONTROLLER
Abstract. This contribution presents a modified autotuning algorithm of the PID controller.
The motivation for the modification of the basic autotuning algorithm is to enlarge the class
of processes to which it can be applied. The basic autotuning algorithm introduced by
Åstrom and Hägglund is extended by the preliminary identification procedure and through
the usage of the dead time compensating controller. These modifications are detailed
through the description of the algorithms’ functioning. The proposed algorithm has been
implemented in the programmable logic controller (PLC) Siemens SIMATIC S7-300. The
experimental results confirm the good robustness properties of the proposed algorithm,
which were demonstrated in a simulation study.

Structure of the modified autotuning PID controller.


http://act.rasip.fer.hr/old/papers/MED00_062.PDF

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Control What is a PID controller and Tuning

What is a PID controller?

A PID (Proportional Integral Derivative) controller is a common instrument used in industrial control applications. A PID controller can be used for regulation of speed, temperature, flow, pressure and other process variables. Field mounted PID controllers can be placed close to the sensor or the control regulation device and be monitored centrally using a SCADA system.
Example: Temperature Control using a Digital PID controller
A typical PID temperature controller application could be to continuously vary a regulator which can alter a process temperature. This may be a pulsed switching device for electrical heaters or by opening and closing a gas valve. A heat only PID temperature controller uses a reverse output action, i.e. more power is applied when the temperature is below the setpoint and less power when above. PID control for injection and extrusion applications often employ additional cooling control outputs and usually require multiple controllers.
A PID controller (sometimes called a three term controller) reads the sensor signal, normally from a thermocouple or RTD, and converts the measurement to engineering units e.g. Degrees C. It then subtracts the measurement from a desired setpoint to determine an error.
The error is acted upon by the three (P, I & D) terms simultaneously:
PID Controller Theory
The following section examines PID controller theory and provides further explanation of the question `how do PID controllers work'.
Proportional (Gain)
The error is multiplied by a negative (for reverse action) proportional constant P, and added to the current output. P represents the band over which a controller's output is proportional to the error of the system. E.g. for a heater, a controller with a proportional band of 10 deg C and a setpoint of 100 deg C would have an output of 100% up to 90 deg C, 50% at 95 Deg C and 10% at 99 deg C. If the temperature overshoots the setpoint value, the heating power would be cut back further. Proportional only control can provide a stable process temperature but there will always be an error between the required setpoint and the actual process temperature.
Integral (Reset)
The error is integrated (averaged) over a period of time, and then multiplied by a constant I, and added to the current control output. I represents the steady state error of the system and will remove setpoint / measured value errors. For many applications Proportional + Integral control will be satisfactory with good stability and at the desired setpoint.
Derivative (Rate)
The rate of change of the error is calculated with respect to time, multiplied by another constant D, and added to the output. The derivative term is used to determine a controller's response to a change or disturbance of the process temperature (e.g. opening an oven door). The larger the derivative term, the more rapidly the controller will respond to changes in the process value.
Tuning of PID Controller Terms
The P, I and D terms need to be "tuned" to suit the dynamics of the process being controlled. Any of the terms described above can cause the process to be unstable, or very slow to control, if not correctly set. These days temperature control using digital PID controllers have automatic auto-tune functions. During the auto-tune period the PID controller controls the power to the process and measures the rate of change, overshoot and response time of the plant. This is often based on the Zeigler-Nichols method of calculating controller term values. Once the auto-tune period is completed the P, I & D values are stored and used by the PID controller.
Joe Crew is the Product Manager at Data Track Process Instruments Ltd. Data Track manufactures digital panel meters, large number displays, PID controllers, signal conditioners and remote data acquisition systems for the process and control industry. Data Track can also supply HMI touchscreen operator panels and SCADA software. The Tracker 300 series of PID Controllers are fully configurable by PC software and feature universal input, single loop integrity, autotune PID, heat / cool control actions and condition monitoring features.
Article Source: http://EzineArticles.com/?expert=Joe_Crew

Model-based Tuning Methods for Pid Controllers

Author: BIN

The manner in which a measured process variable responds over time to changes in the controller output signal is fundamental to the design and tuning of a PID controller. The best way to learn about the dynamic behavior of a process is to perform experiments, commonly referred to as "bump tests." Critical to success is that the process data generated by the bump test be descriptive of actual process behavior. Discussed are the qualities required for "good" dynamic data and methods for modeling the dynamic data for controller design. Parameters from the dynamic model are not only used in correlations to compute tuning values, but also provide insight into controller design parameters such as loop sample time and whether dead time presents a performance challenge. It is becoming increasingly common for dynamic studies to be performed with the controller in automatic (closed loop). For closed loop studies, the dynamic data is generated by bumping the set point. The method for using closed loop data is illustrated. Concepts in this work are illustrated using a level control simulation.

FORM OF THE CONTROLLER

The methods discussed here apply to the complete family of PID algorithms. Examples presented will explore the most popular controller of the PID family, the Proportional-Integral (PI) controller:

In this controller, u(t) is the controller output and is the controller bias. The tuning parameters are controller gain, , and reset time, . Because is in the denominator, smaller values of reset time provide a larger weight to (increase the influence of) the integral term.

CONTROLLER DESIGN PROCEDURE

Designing any controller from the family of PID algorithms entails the following steps:

specifying the design level of operation,
collecting dynamic process data as near as practical to this design level,
fitting a first order plus dead time (FOPDT) model to the process data, and
using the resulting model parameters in a correlation to obtain initial controller tuning values.
The form of the FOPDT dynamic model is:
[2]
where y(t) is the measured process variable and u(t) is the controller output signal. When Eq. 2 is fit to the test data, the all-important parameters that describe the dynamic behavior of the process result:

Steady State Process Gain,

Overall Process Time Constant,

Apparent Dead Time,

These three model parameters are important because they are used in correlations to compute initial tuning values for a variety of controllers [1]. The model parameters are also important because:

the sign of indicates the sense of the controller (+ reverse acting; – direct acting)

the size of indicates the maximum desirable loop sample time (be sure sample time )

the ratio indicates whether a Smith predictor would show benefit (useful if )

the dynamic model itself can be employed within the architecture of feed forward, Smith predictor, decoupling and other model-based controller strategies.

DEFINING GOOD PROCESS TEST DATA

As discussed above, the collection and analysis of dynamic process data are critical steps in controller design and tuning. A "good" set of data contains controller output to measured process variable data that is descriptive of the dynamic character of the process. To obtain such a data set, the answer to all of these questions about your data should be "yes" [1]. Ultimately, it is your responsibility to consider these steps to ensure success.

Was the process at steady state before data collection started?
Suppose a controller output change forces a dynamic response in a process, but the data file only shows the tail end of the response without showing the actual controller output change that caused the dynamics in the first place. Popular modeling tools will indeed fit a model to this data, but it will skew the fit in an attempt to account for an unseen "invisible force." This model will not be descriptive of your actual process and hence of little value for control. To avoid this problem, it is important that data collection begin only after the process has settled out. The modeling tool can then properly account for all process variations when fitting the model.

Did the test dynamics clearly dominate the process noise?
When generating dynamic process data, it is important that the change in controller output cause a
response in the process that clearly dominates the measurement noise. A rule of thumb is to define a
noise band of ±3 standard deviations of the random error around the process variable during steady
operation. Then, when during data collection, the change in controller output should force the process variable to move at least ten times this noise band (the signal to noise ratio should be greater than ten). If you meet or exceed this requirement, the resulting process data set will be rich in the dynamic information needed for controller design.

Were the disturbances quiet during the dynamic test?
It is essential that the test data contain process variable dynamics that have been clearly (and in the ideal world exclusively) forced by changes in the controller output as discussed in step 2. Dynamics caused by unmeasured disturbances can seriously degrade the accuracy of an analysis because the modeling tool will model those behaviors as if they were the result of changes in the controller output signal. In fact, a model fit can look perfect, yet a disturbance that occurred during data collection can cause the model fit to be nonsense. If you suspect that a disturbance event has corrupted test data, it is conservative to rerun the test.

Did the model fit appear to visually approximate the data plot?
It is important that the modeling tool display a plot that shows the model fit on top of the data. If the two lines don't look similar, then the model fit is suspect. Of course, as discussed in step 3, if the data has been corrupted by unmeasured disturbances, the model fit can look great yet the usefulness of the analysis can be compromised.

NOISE BAND AND SIGNAL TO NOISE RATIO
When generating dynamic process data, it is important that the change in the controller output signal causes a response in the measured process variable that clearly dominates the measurement noise. One way to quantify the amount of noise in the measured process variable is with a noise band. As illustrated in Fig. 1, a noise band is based on the standard deviation of the random error in the measurement signal when the controller output is constant and the process is at steady state. Here the noise band is defined as ±3 standard deviations of the measurement noise around the steady state of the measured process variable (99.7% of the signal trace is contained within the noise band). While other definitions of the noise band have been proposed, this definition is conservative when used for controller design.

When generating dynamic process data, the change in controller output should cause the measured process variable to move at least ten times the size of the noise band. Expressed concisely, the signal to noise ratio should be greater than ten. In Fig. 1, the noise band is 0.25°C. Hence, the controller output should be moved far and fast enough during a test to cause the measured exit temperature to move at least 2.5°C. This is a minimum specification. In practice it is conservative to exceed this value.



Figure 1 – Noise Band Encompasses ± 3 Standard Deviations Of The Measurement Noise

CONTROLLER TUNING FROM CORRELATIONS
The recommended tuning correlations for controllers from the PID family are the Internal Model Control (IMC) relations [1]. These are an extension of the popular lambda tuning correlations and include the added sophistication of directly accounting for dead time in the tuning computations.

The first step in using the IMC (lambda) tuning correlations is to compute, , the closed loop time constant. All time constants describe the speed or quickness of a response. The closed loop time constant describes the desired speed or quickness of a controller in responding to a set point change. Hence, a small (a short response time) implies an aggressive or quickly responding controller. The closed loop time constants are computed as:

Aggressive Tuning: (See online version for picture of formula)

Moderate Tuning: ("")

Conservative Tuning: ("")

Final tuning is verified on-line and may require tweaking. If the process is responding sluggishly to disturbances and changes in the set point, the controller gain is too small and/or the reset time is too large. Conversely, if the process is responding quickly and is oscillating to a degree that makes you uncomfortable, the controller gain is too large and/or the reset time is too small.

EXAMPLEs: In online copy

PI Controller Tuning Map


Figure 6 – How PI controller tuning parameters impact set point tracking performance

CONCLUSIONS
Understanding the dynamic behavior of a process is essential to the proper design and tuning of a PID controller. The recommended design and tuning methodology is to: step, pulse or otherwise perturb the controller output near the design level of operation, record the controller output and measured process variable data as the process responds, and fit a first order plus dead time (FOPDT) dynamic model to this process data, use the dynamic model parameters in a correlation to compute P-Only, PI, PID and PID with Filter test your controller to ensure satisfactory performance.

LITERATURE CITED

1. Cooper, Douglas, "Practical Process Control Using Control Station," Published by Control Station,Inc, Storrs, CT (2004).

For more information about model-based tuning techniques and technologies, please see our other resources below:

PID Control – Practical Process Control Training (2 Day Workshop)

http://www.bin95.com/process_control_atlanta_training.htm

Complete list of authors:
Jeffrey Arbogast – Department of Chemical Engineering
Douglas J. Cooper, PhD – Control Station, Inc.
Robert C. Rice, PhD – Control Station, Inc.
To see the full online version with pictures, please visit http://www.bin95.com/PID_Controller_Design.htm

About the Author:

More from these authors and much more. please see ”More PID TRaining resources”...

Article Source: ArticlesBase.com - Model-based Tuning Methods for Pid Controllers

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Control Servo Motion Control Tuning the PID Loop

There are two primary ways to go about selecting the PID gains.
Either the operator uses a trial and error or an analytical approach.
Using a trial and error approach relies significantly on the
operator’s own prior experience with other servo systems. The one
significant downside to this is that there is no physical insight into
what the gains mean and there is no way to know if the gains are
optimum by any definition. However, for decades this was the
approach most commonly used. In fact, it is still used
today for low performance systems usually found in process control.
To address the need for an analytical approach, Ziegler and Nichols
[1] proposed a method based on their many years of industrial
control experience. Although they originally intended their tuning
method for use in process control, their technique can be applied to
servo control. Their procedure basically boils down to these two steps.

Step 1:
Set Ki and Kd to zero. Excite the system with a step command.
Slowly increase Kp until the shaft position begins to oscillate.
At this point, record the value of Kp and set Ko equal to this value.
Record the oscillation frequency, fo.

Step 2:

Set the final PID gains using equation (6).



Loosely speaking, the proportional term affects the overall response

of the system to a position error. The integral term is needed to force
the steady state position error to zero for a constant position
command and the derivative term is needed to provide a damping
action, as the response becomes oscillatory. Unfortunately all three
parameters are inter-related so that by adjusting one parameter will
effect any of a previous parameter adjustments. As an example of
this tuning approach, we investigate the response of a Compumotor
BE342A motor with a generic servo drive and controller.

This servomotor has the following parameters:

Motor Total Inertia J = 50E-6 kgm^2
Motor Damping b = .1E-3 Nm/ (rad/sec)
Torque Constant Kt = .6 Nm/A

We begin with observing the response to a step input command with
no disturbance torque (Td = 0).

Step 1:
Fig. 2a shows the result of slowly increasing only the proportional term.
The system begins to oscillate at approximately .5 Hz (fo = .5Hz) with
Ko of approximately 5E-5 Nm/ rad.

Step 2:

Using these values, the optimum P.I .D. gains according to
Ziegler-Nichols (Z-N) are then (using equation (6)):

Kp = 3.0E-4 Nm/ rad
Ki = 3.0E-4 Nm/ (rad/sec)
Kd = 7.4E-5 Nm/ (rad/sec)

Fig. 2b shows the result of using the Ziegler Nichols gains.
The response is somewhat better than just a straight proportional gain.
As a comparison, other gains were obtained by trial and error. One set
Of additional gains is listed in Fig. 2b. Although the trial and error gains
gave a faster, less oscillatory response, there is no way of telling if a
better solution exits without further exhaustive testing.

One characteristic that is very apparent in Fig.2 is the length of
the settling time. The system using Ziegler Nichols takes about
6 seconds to finally settle making it very difficult to incorporate
into any highperformance motion control application. In contrast,
the trial and error settings gives a quicker settling time, however
no solution was found to completely remove the overshoot.

Source ( pdf )
http://www.compumotor.com/whitepages/ServoFundamentals.pdf

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Control Servo Motion Control - PID Control

PID position loops

Theory
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).

More ( pdf )
http://apps.danahermotion.com/support/troubleshooting/
PDF_Resources/2000-08%20PID%20pos%20loops.pdf

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).

Source (pdf )

http://www.compumotor.com/whitepages/ServoFundamentals.pdf

Servo Motion Control Index

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