ControlExperiments - Precision Modular Servo

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33-927S I

PRECISION MODULAR SERVO

Control Experiments

Precision Modular Servo

Control Experiments

33-927S(For use with MATLAB

R2006b version 7.3)

CONFIDENTIAL

Feedback Instruments Ltd., Park Road, Crowborough, East Sussex, TN6 2QR, UK

Telephone: +44 (0) 1892 653322, Fax: +44 (0) 1892 663719

E-mail: [email protected], Website: www.feedback-instruments.com

Manual: 33-927S Ed01 122006

Feedback Part No. 1160-33927S


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THE HEALTH AND SAFETY AT WORK ACT 1974

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While we provide the fullest possible user information relating to the proper use of this equipment, ifthere is any doubt whatsoever about any aspect, the user should contact the Product Safety Officer at

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PREFACE


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OPERATING CONDITIONS

This equipment is designed to operate under the following conditions:

Operating Temperature 10C to 40C (50F to 104F)

Humidity 10% to 90% (non-condensing)

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PREFACE

WARNING:

This equipment must not be used in conditions of condensing humidity.


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Table of Contents

Manual overview ..................................................................................................... 1

Introduction ............................................................................................................. 2

Precision Modular Servo set description .............................................................. 3

DC Motor model ...................................................................................................... 5

Exercise 1 Model testing....................................................................... 8

PMS model identification ...................................................................................... 11Static friction compensation ................................................................................. 12

Exercise 2 Static friction compensation ................................................... 12

Model 1 identification ........................................................................................... 14

Exercise 3 Model 1 identification ........................................................... 14

Model 2 identification ........................................................................................... 15

Exercise 4 Model 2 identification ........................................................... 15PMS setup control ................................................................................................. 17

Plant control ........................................................................................................ 17

PID controller ...................................................................................................... 18

PMS velocity control ............................................................................................ 19

Exercise 5 PID control of motor velocity .................................................. 19Exercise 6 Real time PID control of motor velocity ..................................... 21

Control signal saturation ...................................................................................... 25

Exercise 7 Anti-windup design .............................................................. 26Exercise 8 Anti-windup for PMS ............................................................. 28

PMS position control ........................................................................................... 29

Exercise 9 PID control of motor position .................................................. 29 Exercise 10 Real time PID control of motor position .................................... 31Exercise 11 Position tracking ................................................................ 33

PMS control under variable load conditions ......................................................... 34

Simple gain scheduling algorithm ..................................................................... 35

Exercise 12 Gain scheduling ................................................................. 35

TABLE OF CONTENTS


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Manual overview

The following manual refers to the Feedback Instruments Precision Modular Servo

Control application. It serves as a guide for the control tasks and provides useful

information about the physical behaviour of the system. A linear model is proposed

then identification algorithms are introduced. The models obtained are compared with

the phenomenological model and the Modular Servo setup. Control algorithms are

developed, tested on the Modular Servo models and then implemented in a real time

application.

Throughout the manual various exercises are proposed to bring the user closer to the

Modular Servo control problem. Depending on the knowledge level of the user some

of the sections and exercises can be skipped. The more advanced users can try to

model, identify and control the Modular Servo on their own from the beginning.

The relative difficulty of each exercise is indicated using the following icons:

easy level,

medium level,

expert level.

If any of the identification or controller design exercises appear to be too difficult or

the results are not satisfactory you may go straight to the examples that are supplied

and test them by changing the parameters of the controllers.

MANUAL OVERVIEW


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Introduction

The Precision Modular Servo (PMS) workshop serves as a model of a very popular

device, the DC Servo Motor. This is often used in robotic applications. The word

servo comes from the Latin word servus meaning servant or slave. Thus the servo

motor is intended to react to a given command, for example a desired position or

velocity. In order for the motor to be called a servo motor it has to be equipped with a

velocity and position measurement unit and motor driver. It may also include a simple

servo motor controller.

The main features distinguishing servo motors from normal motors are:

minimized rotor inertia,

minimized armature inductance,

improved withstand voltage and magnet saturation,

designed to withstand sudden acceleration and high frequency operation.

The PMS unit allows for the design of different servo motor controllers and to test

them in real time using the Matlab and Simulink environment.

INTRODUCTION


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Precision Modular Servo set description

The description of the PMS setup in this section refers mostly to the mechanical part

and the control aspect. For details of the mechanical and electrical connection, the

interface and an explanation of how the signals are measured and transferred to the

PC, refer to the Installation & Commissioning manual.

As shown in Figure 1, the PMS mechanical unit consists of a DC Motor,

Tachometer/Gearbox, Digital Encoder, Input and Output Potentiometers andMagnetic Brake.

Figure 1 Precision Modular Servo mechanical unit

Apart from the mechanical units, the electrical units play an important role for motor

control. They allow measured signals to be transferred to the PC via an I/O card. The

amplifiers are used to transfer control signals from the PC to the DC Motor. The

mechanical and electrical units provide a complete control system setup presented in

Figure 2.

PRECISION MODULAR SERVO SET DESCRIPTION


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Figure 2 PMS control system

In order to design any control algorithms one must first understand the physical

background behind the process and carry out identification experiments. The next

section explains the modelling process of the PMS.

PRECISION MODULAR SERVO SET DESCRIPTION


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DC Motor model

Every control project starts with plant modelling, so as much information as possible

is given about the process itself. The mechanical-electrical model of the servo motor

is presented in Figure 3.

Figure 3 DC motor phenomenological model

Usually, phenomenological models are nonlinear, that means at least one of the

states (i current, motor position) is an argument of a nonlinear function. In order

to present such a model as a transfer function (a form of linear plant dynamics

representation used in control engineering), it has to be linearised. However for the

DC motor model, nonlinearities are so small that they can be neglected.

According to the electrical-mechanical diagram presented in Figure 3 the linear

model equations can be derived.

DC MOTOR MODEL


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The torque generated by the DC motor is proportional to the motor current i:

iKt =

where Kt is the torque constant. The induced electromotive force, vemf, is a voltage

proportional to the angular shaft velocity & :

&= bemf Kv

where Kbis a motor property dependant constant related to physical properties of the

motor. Using Newtons law the mechanical dynamics can be described by:

iKdJ ti +== &&&

where dis a linear approximation of the viscous friction.

The equations governing the behaviour of the electrical model are:

iRdtdiLvu emf +=

where uis the control signal. Including equation (2) transforms (4) into:

&++= bKiRdt

diLu

Equations (3) and (5) constitute the 2 main differential equations. Their

transformation leads to state space form:

+=

+=

uLL

Ki

L

R

dt

di

iJ

K

J

d

b

t

1

,

&

&&&

[ ] .01

,10

=

+

=

iy

u

Li

L

R

L

KJ

K

J

d

idt

d

b

t

&

&&

For angular velocity & notation simplification we may use , where &= .

(1)

DC MOTOR MODEL

(2)

(3)

(4)

(5)

(6) (7)


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To complete the model given by motion equations (6) and (7), we must introduce the

value of all parameters. The motor used for the experiments is a 24V DC brushed

motor with a no-load speed of 4050rpm. The following table gives the values of the

motor parameters supplied by the manufacturer:

Table 1. DC Motor parameters

The bound for the control signal is set to [2.5V .. +2.5V].

The DC motor is a SISO plant single input single output (Figure 4). However the

output choice of the model changes depending on the control scheme we use

(velocity or position control).

Figure 4 DC Motor models for velocity and position control

The difference between these two models is trivial. Since the velocity &= is the

derivative of the position and we have the velocity as an output in the model

described by (6) and (7), we can integrate it to acquire the position.

DC MOTOR MODEL


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Exercise 1 Model testing

Introduction

For the initial exercise the user has been provided with the DC Motor model

described by equations (6). The model can be opened in Simulink -

DCmotor_ManModel.mdl. As you will see in the later exercises, the basic DC

Motor model responds differently from the actual motor when it is used within

the PMS. This is because of the dynamics of the additional mechanical and

electrical PMS components that are not included in the basic model. However

these differences can easily be detected during the course of identification.

One can also modify the phenomenological model by adding gain or by certain

parameter tuning. Such modifications are implemented in the model called

DCmotor_TunModel.mdl. An amplification is added thus the PMS model is

created and controlled with the signal of 2.5 [V] amplitude in Simulink just as

the real plant is and responds similarly to the real plant.

Load Matlab and the menu system using the Windows Start menu (Figure 5).

Figure 5 Start menu

Matlab will run and the Simulink model menu will open (Figure 6). This contains

links to the simulation-only models (Modular Servo Simulation Models), the

real-time models (Modular Servo Real-Time Models) and the manual

documents.

DC MOTOR MODEL

DC MOTOR MODEL


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Figure 6 Simulink model menu

Double click on the Modular Servo Simulation Models block in the main menu.

A sub-menu containing all of the simulation-only models will open (Figure 7).

Figure 7 Simulation models menu


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Double click the required model DCmotor_ManModel.mdl. The Simulink model

window will open.

Task

To begin with the user is advised to check the responses of the DC motor

model alone (DCmotor_ManModel.mdl) with zero, constant and sinusoidal

voltages applied (u). Try applying 24V and observe the motor runs at full speed

(approximately 4000rpm). You may change the values of the parameters to see

how the response varies. Try changing the moment of inertia value J(double

click on the motor block to change its parameters).

The tuned model (DCmotor_TunModel.mdl) includes an extra gain term that

models the gain provided by the PA150 preamplifier and SA150 Servo

Amplifier. This model can be run with a 2.5 [V] control signal amplitude (as

used by Simulink to drive the DAC output), which corresponds to the 24 [V]

amplitude signal applied to the motor.

Example results and comments

Figure 8 presents voltage step responses for the tuned PMS model with

minimum load.

Figure 8 Tuned PMS model results

DC MOTOR MODEL


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PMS model identification

As mentioned in the previous section where the phenomenological model was

derived, two control aspects are taken into account: position control or velocity

control. Thus two plant models can be identified, one for which the rotor velocity will

be the output and another for which the rotor position is the output. Both of them

should only differ by an extra integration action in the model relating the control

voltage uand the rotor position. These models will be used for controller design and

tests in the Precision Modular Servo Controlsection.

There are a few important things that the control system designer has to keep in

mind when carrying out an identification experiment:

Stability problem if the plant that is identified is unstable, the

identification has to be carried out with a working controller, which

introduces additional problems that will be discussed further on. If the

plant is stable and does not have to work with a controller the

identification is much simpler.

Structure choice a very important aspect of the identification. For

linear models it comes down to choice of the numerator anddenominator order of the transfer function. It applies both for continuous

and discrete systems.

As far as the discrete models are concerned the structures are also

divided in terms of the error term description: ARX, ARMAX, OE, BJ1.

Sampling time the sampling time choice is important both for

identification and control. It cannot be too short nor can it be too long.

Too short sampling time might influence the identification quality

because of the quantisation effect introduced by the AD. Furthermore

the shorter the sampling time the faster the software and hardware has

to be and more memory is needed. However short sampling time willallow for elimination of aliasing effects and thus anti aliasing filters2will

not have to be introduced. Long sampling times will not allow for

including all of the dynamics.

Excitation signal for the linear models the excitation choice is simple.

Very often designers use white noise however in industrial applications

it is often disallowed. It is attractive however because of the fact that it

1More information about these structures can be obtained during the System Identification

courses.2These are the basics of the Digital Signal Processing course. For more insight the user is

asked to study more on signal processing and digital control.

PMS MODEL IDENTIFICATION


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holds very broad frequency content thus the whole dynamics of the

plant can be identified. If the dynamics are not too complex several

sinusoids with different frequencies can be summed to produce a

satisfactory excitation signal.

Identification method usually two methods are used, the Least Mean

Square (LMS) method and the Instrumental Variable method. The LMS

method is the most popular and is implemented in Matlab. This method

minimizes the error between the model and the plant output. The

optimal model parameters, for which the square of the error is minimal

is the result of the identification.

Static friction compensation

Before any of the identification and control experiments are carried out we need to

compensate the static friction (stiction) as best as possible. The problem with static

friction compensation is that when it is applied at the instant of start-up it

compensates well, but then provides too large an offset value when the DC Motor is

turning and the friction forces are smaller. Furthermore it is strongly dependant on

the position of the Zero Set potentiometer on the PA150C preamplifier unit.

However to keep it simple, only a voltage offset will be added to the control signal to

provide compensation and further nonlinear effects as friction will be dealt with by the

controller. Friction is influenced by the mechanical setup of the PMS system, thus

every time you change its mechanical properties the friction compensation should be

repeated.

Exercise 2 Static friction compensation

IntroductionIn order to define the voltage offset values that need to be applied a friction

application has been designed PMSFriction.mdl.

Task

To get the best results from this test, connect a DVM across the Pre-amp

PA150C output terminals 2 and 3 and finally adjust the zero set pot until the

output is zero. If a DVM is not available, adjust the zero set pot very carefully

so that the motor is not turning and the pot is in the middle of the adjustment

range over which the motor does not turn.



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Open the PMSFriction.mdland run the application (if it is the first time you are

running a real time application refer to the Installation & Commissioning

manual for guidance). This model slowly increases the drive signal to the motor

until static friction is overcome and the motor just starts to move, then records

the voltage offset value. It then repeats in the opposite direction. Record the

values of the voltage offsets for positive and negative movement. These are the

values that have to be inserted in the Friction compensation block (Figure 9,

Figure 10) placed in the Feedback DAC block in every application. However if

the result of the test is significantly greater than 0.2V it may appear to be

biased by other nonlinearities and should be investigated further. You shouldtry to adjust the mechanical setup to reduce the friction forces.

Figure 9 Friction compensation block

Figure 10 Friction compensation block menu (double click the block to open)


Depending on the mechanical setup of the PMS you can obtain different values

of the voltage offsets for each direction of rotation in the automatic friction

detection procedure, for example offn = 0.015 [V]for negative direction and offp

= 0.010 [V] for positive direction.



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Model 1 identification

The following exercise includes all of the above facts and provides an identification

experiment, which results in a discrete model of the DC Motor. The first model to be

identified describes the relation between the control voltage u and the angular

velocity of the DC motor, &= .

Exercise 3 Model 1 identification

IntroductionAll of the control real time simulations are carried out with a sampling time of

Ts= 0.001 [s] therefore the same sampling time is used with the identification

experiments to provide full compatibility. For the identification the Matlab

System Identification Toolbox is used.

The identification experiment is carried out using the model called

VelocityModel_Ident.mdl. This model excites the motor system and records its

response. The excitation signal is composed of several sinusoids. The

experiment lasts 20 seconds and two signals are collected in the form of

vectors and are available in the Workspace.

Task

Before running the model adjust the zero set pot on the PA150C Pre-amplifier

very carefully so that the motor is not turning and the pot is in the middle of the

adjustment range over which the motor does not turn.

Carry out the identification experiment and collect the data. Identify a discrete

model using the Matlab identification interface (type identat the command line

- the identification interface will open as in Figure 9).

For instructions on identification experiments refer to the Matlab Guide

manual.


The step response of the identified system should be similar to the one

presented in Figure 11 but there will be significant differences because the

basic motor model does not take into account the properties of the real system

such as the loading effect of the gearbox, tachometer, output potentiometer etc.

If the model is transferred into the Matlab Workspace it can be comparedagainst the discrete equivalent of the continuous transfer function. In order to



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obtain the discrete form use the c2d command. Make sure you specify the

proper sampling time.

Figure 11 Step response of the model

Compare the step responses or bode plots of the two systems (use the step

and bode commands). You can also transform the discrete models into their

continuous equivalent with the means of d2c command.

The model obtained is used in the first PID control exercise.

Model 2 identification

A similar identification experiment can be carried out when identifying the transfer

function between the control voltage uand the rotor position/angle .

Exercise 4 Model 2 identification

Introduction

This identification experiment is carried out with the PositionModel_Ident.mdl.

The excitation signal is composed of several sinusoids. The experiment lasts

20 seconds and two signals are collected in a form of vectors and are available

in the Matlab Workspace.



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Task

Carry out the identification experiment and collect the data. Use the Matlab

identification interface to identify a discrete model. For simplicity assign the

input and output data to the uand y vectors:

u = simout(:,1); y = simout(:,2);

Upload the signals for the identification from the Workspace. Make sure you

specify the proper sampling time (1ms). Select the proper structure of the

model (e.g. OE 2 2 1). Press estimate. You can check the quality of the

response of the identified model by the step response analysis, transient

response, pole and zeros map, frequency response and model residuals.


The response of the model is presented in Figure 12. If the model is transferred

into the Workspace it can be compared against the discrete equivalent of the

continuous system (equations 6 and 7). In order to obtain the discrete form use

the c2d command. Make sure you specify the proper sampling time.

Note that the models you obtain may strongly depend on the magnetic brake

position, which simulates different load. Every time you change the magnet

position the model should be updated.

Figure 12 Identified model step response


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PMS setup control

The PMS control aspect covers three main control areas:

position control,

velocity control,

control under different load conditions.

Each of these control problems has its own characteristic. For position control thesteady state control signal value should be 0 whereas for the velocity control in

steady state the control signal should be constant. Different loads will require some

kind of an adaptive control solution.

Matlab provides various analysis methods for linear systems as far as dynamics are

concerned (root locus, frequency analysis tools Bode diagrams, Nyquist plots, pole

and zero maps etc.). With the information that Matlab provides about the dynamics of

the system, controllers can be designed. The following sections explain how the PID

controller works, and how it can be tuned.

Plant control

There are numerous control algorithms however the PID control is the most popular

because of its simplicity. A general schematic of a simple control closed loop system

is presented in Figure 13.

Figure 13 Simple control system closed loop

Assuming that the plant is represented by its linear model its transfer function can be

described as:

)(

)()(

sA

sBsG = (8)

PMS SETUP CONTROL

C(s) G(s)


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where s is the Laplace operator. The idea of control algorithms is to find such a

controller (transfer function, discrete transfer function, any nonlinear), which will fulfil

our requirements (certain dynamic response, certain frequency damping, good

response to the dynamic changes of the desired value etc.). Every controllers input

is the e(t) error signal. Sometimes disturbance signals are also measured.

Depending on the present and past values of the error signal, the controller performs

such an action (changes the u(t)control signal) that the y(t)is as close to the ydesired(t)

value as possible at all times.

There are a lot of controller design and tuning methods. All of them consider thebehaviour of the closed loop system (plant with a controller Figure 13) and provide

controller parameters according to the assumed system characteristics. With the

known plant transfer function G(s)it is possible to find satisfactory parameters of the

C(s)controller such that the closed loop system will have the desired characteristics

described by the transfer function Tc(s):

)()(1

)()()(

sGsC

sGsCsTc

+

=

PID controller

A PID controller consists of 3 blocks: Proportional, Integral and Derivative. The

equation governing the PID controller is as follows:

)()()(

)()()()(

tytyte

dt

tdeDdtteItePtu

desired =

++=

With the means of the Laplace transform such a structure can be represented as a

transfer function:

s

IPsDssD

s

IP

sE

sUsC

sEsDs

IPsU

++=++==

++=

2

)()(

)()(

)()()(

Each of the PID controller blocks (P, Iand D) plays an important role. However for

some applications, the Integral or Derivative part has to be excluded to give

satisfactory results. The Proportional block is mostly responsible for the speed of the

system reaction. However for oscillatory plants it might increase the oscillations if thevalue of Pis set to be too large.

(12)

(13)

(10)

(11)

(9)

PMS SETUP CONTROL


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The Integral part is very important and assures zero error value in the steady state,

which means that the output will be exactly what we want it to be. Nevertheless the

Integral action of the controller causes the system to respond slower to the desired

value changes and for systems were very fast reaction is very important it has to be

omitted. Certain nonlinearities will also cause problems for the integration action.

The Derivative part has been introduced to make the response faster. However it is

very sensitive to noise and may cause the system to react very nervously. Thus very

often it is omitted in the controller design. Derivative part output filtering may reduce

the nervous reaction but also slows the response of the controller down andsometimes undermines the sense of using the Derivative part at all. Proper filtering

can help to reduce the high frequency noise without degrading the control system

performance in the lower frequency band.

There are several PID tuning techniques. Most often Ziegler-Nichols rules are used

or a relay experiment is undertaken. Very often the closed loop system roots are

analysed and set in the desired position by proper choice of P, I and D values.

Matlab delivers a root locus tool, which helps in such designs. To gain an

understanding of root locus controller design and its effects on the system, a motor

velocity control exercise is proposed.

PMS velocity control

As a first control task, velocity control of the PMS is chosen. The following exercises

will guide you through the controller design, testing on the model and on the real time

application.

Exercise 5 PID control of motor velocity

Introduction

To design a PID controller a model of the plant is needed. For this purpose we

can use a discrete model and design a discrete PID controller, or use a

continuous model (equations 6 and 7) and design a continuous PID controller,

however when a controller is implemented on some kind of a control unit it has

to be used in a discrete form.

For the beginning the linear continuous model of the DC motor will be used:

PMS SETUP CONTROL


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tb

tadd

KKdRsdLJRsJL

KK

su

ssG

++++

==

)()(

)()(

21

,

where Kaddis the additional gain value introduced by the amplifiers.

Task

Design a PID controller for the above transfer function. Input the transfer

function in Matlab using the tf command. You may also use the model that you

have identified. The discrete model obtained in PMS Model Identification

section can be transformed into a continuous model before the controller

design.

For instructions on PID controller design refer to the Matlab Guide

manual.

The first controller tests have to be done on the models. The file

Model_PID_velocity.mdlhas been created for the test purpose.


The root locus of the identified model with the PID control with P = 0.00124,I =0.0241, D = 0is presented in Figure 14.

Figure 14 Root locus with PID controller

(14)

PMS SETUP CONTROL


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You can move the poles, zeros and the gain to obtain for example faster step

response of the closed loop system. Then you can export the controller into

Workspace and test it on the model (equation 14) of the DC Motor with

Model_PID_velocity.mdlor on the model that you have identified.

Use a sinusoidal signal for the set value of the motor velocity desired(t).Change

the frequency and see how the output follows the desired value. Decrease and

increase the values of the proportional, integral and derivative gains in the PID

controller. See how that influences the tracking of the desired value. You

should observe the influence that is described in the beginning of this section:

the Ipart assures zero steady state error but slows the system reaction, the D

part makes the system faster but causes the controller to behave nervously

observe the controller output.

The controller that has been obtained in the above exercise can be tested on the

PMS setup. The following exercise will guide you through it.

Exercise 6 Real time PID control of motor velocity

Introduction

Just as in exercise 5 the motor velocity will be controlled. This time it will be

tested in real time. For this exercise use the PMS_PID_velocity.mdlpresented

in Figure 15.

Figure 15 Real time PMS velocity control

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Control Experiments

As you open the PMS_PID_velocity.mdlyou will notice that this simulation is

directed to an external module, which is indicated in the upper part of the

simulation window. The PID controller has been already designed, however

you can change its parameters according to the results obtained in exercise 5.

Before you run the real time simulation make sure the PMS setup is properly

connected and that the Power Source is turned off. Refer to the Installation &

Commissioning manual for more instructions on real time simulations.

Task

The task is to change the sinusoidal desired(t)signal frequency and see how the

cart reacts to it. Change the controller P, Iand Dvalues to see how the signal

desired(t) is tracked by the system output. Experiment with the position of the

magnetic brake.

Make sure the changes of the P, Iand Dgains will not have a destructive effect

on the DC Motor. You should already gain some experience in exercise 5 on

how the values of these gains can be changed.

Example results and commentsFigure 16 and Figure 17 present real time simulation results of the velocity PID

control. Two I and D gain values have been selected. The simulation shows

that the I value increase may slow the system down and cause a larger

overshoot. You should also notice that the bigger the D value the faster the

response but the more nervous the control signal u. Very energetic control

signal changes can often lead to control unit brake down.

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Control Experiments

Figure 16 PID control with P = 0.0012, I =0.0241, D =0

The result presented in Figure 16 proves how efficient controllers can be when

an identification experiment is incorporated in their design. Even load variation

has little influence on the control system performance. Figure 17 shows how

wrong controller tuning may degrade the system performance.

Figure 17 Real time control simulation results, with greater I gain and smaller P gain

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Control Experiments

Just like every other actuator, the DC motor has its limits as far as the control signal

and output variable value is concerned. Figure 18 and Figure 19 show what happens

whenever excessive angular velocity value is demanded under PID control. The

saturation effect can be easily achieved with the aid of the magnetic load. It is

explained in detail in the following section.

Figure 18 Control signal saturation, motor velocity

Figure 19 Control signal saturation, controller output

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Control Experiments

Control signal saturation

Every plant and actuator has its limits. As far as the PMS is concerned anything

above +2.5 [V] or 2.5 [V] will be limited to the boundary value. This control signal

bound is transferred to the maximal angular velocity value that can be reached under

different load values. The greater the load, the lower the maximum achievable

angular velocity. One could identify a static characteristic of the motor for different

loads to get the full picture of the achievable output variable values for the PMS.

The saturation itself is not that troublesome but together with PID control andespecially the integration of the error signal, it can cause big trouble. Of course

nothing can be done as far as the control algorithms are concerned to overcome the

problem of angular velocity bounds. But one must make sure that whenever the

bound is crossed the integration action is turned off. If it is not, the controller will

integrate the error and will be continuously increasing the value of the control signal

(windup effect), even though the system cannot respond due to the saturation.

Whenever that happens and the desired velocity value is later changed back to the

permissible value, because of the previous error integration, it will take some time

until the integrated value is reduced by the opposite sign error value. This will result

in an observable control signal switch delay, the length of which depends on how

long the windup effect lasts.

To overcome this problem anti-windup circuits are designed so that fast controller

reaction is ensured. There are several methods to design them. The following steps

lead to the design of one of the simplest methods:

measure the control signal saturation value (it is given as 2.5 V for PMS)

put a saturation block with the appropriate saturation values after the output

of the PID controller

subtract the saturation block output value from its input

multiply that signal by a gain (AWgain) and subtract it from the integration

block input error

The anti-windup can be tested on a DC motor model in AntiW_model.mdl.

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Control Experiments

Exercise 7 Anti-windup design

Introduction

A linear continuous model of the DC motor will be used for the anti-windup test.

You can use a model to design that algorithm on your own or use the

AntiW_model.mdlfor tests and analysis. The desired value of angular velocity

is first switched to a value exceeding the limit and then it is reduced to a

permissible value.

TaskCheck the anti-windup algorithm. Change the AWgain value and observe the

system reaction. Use large, small and zero values to see how fast the controller

switches the control signal when returning to the permissible angular velocity

desired value range.


The following figures (Figure 20) present the results for three situations, where

the anti-windup is switched off, for AWgain = 50 (medium value) and for

AWgain = 1000 (high value). Observe the value of the output to see how the

anti-windup improves the system performance.

Figure 20 Anti-windup performance

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Control Experiments

Without the anti-windup the controller has to compensate for the error integration

before it switches to the next desired velocity value. Areas A and B are the same

so one can actually tell how long the controller will need to compensate for the error

integration during saturation.

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Exercise 8 Anti-windup for PMS

Introduction

This time the anti-windup algorithm will be tested on the real plant and under

full load conditions. Use the AntiW_PMS.mdlfor tests and analysis on the PMS

setup. Before the experiment is started you can check the maximum angular

velocity for the boundary voltage value of 2.5 [V] with the

PMS_PID_velocity.mdl. Remember to lower the magnet before any

experiments are run.

Task

Check the anti-windup algorithm. Change the AWgain value and observe the

system reaction. Use large, small and zero values to see how fast the controller

switches the control signal when returning to the permissible angular velocity

desired value range.


Figure 21 presents the results for two situations, where the anti-windup is

switched off and on. Observe the value of the output to see how the anti-

windup improves the system performance.

Figure 21 Anti-windup for PMS

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Control Experiments

PMS position control

The position control task is similar to velocity control however there is one significant

difference: the position control model has an additional integration action.

To acquire a model suitable for position control we can use the model prepared for

velocity control and simply transform it:

tb

tadd

KKdRsdLJRsJL

KK

su

ssG

++++

==

)()(

)()(

21

where

.)()(

)(

)(

)()(

,)(

)(

)(

)()(

,

23

1

2

1

sKKdRsdLJRsJL

KK

s

sG

su

ssG

su

ss

su

ssG

tb

tadd

++++

===

==

=

&

where Kaddis the additional gain value introduced by the amplifiers.

For position control you can use a discrete model acquired in the identification task of

Model 2 and transform it to continuous form through the d2c function. The position

controller is designed in a similar way to that shown in the velocity control exercise.

Exercise 9 PID control of motor position

Introduction

A linear continuous model of the DC motor will be used:

sKKdRsdLJRsJL

K

su

ssG

tb

t

++++==

)()()(

)()(

232

which means that our goal will be to follow a certain rotor position/angle

desired(t) profile.

Task

Design a PID controller for the above transfer function. Input the transfer

function in Matlab using the tf command. You may also use the model that you

have identified. The discrete model obtained in the PMS Model Identification

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Control Experiments

section can be transformed into a continuous model before the controller

design.

Open the root locus tool. Just as in the previous exercise, design a PID

controller. Observe how the closed loop system step response changes when

you move the poles and zeros and change the gain of the controller.

Position controller tests can be carried out on the models. The file called

Model_PID_position.mdlhas been created for the purpose of the test.


The root locus of the identified model using PID control with P = 0.0357,

I =0.00714, D = 0is presented in Figure 22.

Figure 22 Root locus with PID controller

You can move the poles, zeros and the gain to obtain, for example, faster step

response of the closed loop system. Then you can export the controller into

Workspace and test it on the model (equation 15) of the DC Motor with

Model_PID_position.mdlor on the model that you have identified.

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The controller that has been obtained in the above exercise can be tested on the

PMS setup. The following exercise will guide you through it.

Exercise 10 Real time PID control of motor position

Introduction

Just as in exercise 9 the motor position will be controlled. This time it will be

tested in real time. For this exercise use the PMS_PID_position.mdl.

Before you run the real time simulation make sure the PMS setup is properly

connected and that the Power Source is turned off. Refer to the Installation &

Commissioning manual for more instructions on real time simulations.

Task

The task is to change the sinusoidal desired(t)signal frequency and see how the

motor reacts to it. Change the controller P, Iand Dvalues to see how the signal

desired(t)is tracked by the system output.

Make sure the changes of the P, Iand Dgains will not have a destructive effect

on the DC Motor.


Figure 23 presents real time simulation results of the position PID control.

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Figure 23 PID control with P = 0.0357, I =0.00714, D = 0

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Exercise 11 Position tracking

Introduction

The PMS setup IP150H potentiometer can serve as an external set point

signal. Its output is an analogue signal ranging from around 15 [V] to +15 [V].

For the position control application we can read that analogue voltage and

transform this signal into radians where half a turn gives a rotation of and a

corresponding signal of 15 [V]. Furthermore to use the OP150K as a

visualisation of the control system performance we can multiply the IP150H

signal, being the corresponding desired position, by 30 (gearbox ratio). Thus if

the algorithm works well the OP150K will track the movement of the IP150H.

The model PMS_PositionTracking.mdl has been created to test this PMS

feature.

Task

Use the PMS_PositionTracking.mdlto perform the tests. Vary the rate at which

the IP150H is turned to check the dynamic properties of the control system.

Vary the gains to see how the system performs. Use the step/invert switch on

the IP150H to see how the controller reacts.


Figure 24 presents real time experiment results. The IP150H has been rotated

with different frequencies. The step/invert switch has also been used.

Figure 24 Position tracking

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Control Experiments

PMS control under variable load conditions

Control algorithms are the solution for following a certain path under different types of

disturbances. A variable load could be considered as a disturbance if its change is

relatively small. Thus once designed, a linear controller should be able to handle

small load changes. For best performance, large load variations require some kind of

controller adaptation to the new plant dynamics. If the load varies often we say that

the plant is non-stationary. We could use an advanced adaptive controller to

suppress the effect of load changes, however the nature of a servo motor application

is such that the device is kept in steady state most of the time. With small changes ofthe control and output signals the adaptation would have to be frozen not to cause

estimator instability3. With the complexity of adaptive algorithms this could prove to

be a difficult and ineffective solution.

A robust adaptive solution could be utilized with a gain scheduling scheme if the

position of the magnet could be measured. Gain scheduling is a very well known and

often used approach in adaptive control. The idea of gain scheduling is to bound

usually one parameter of the controller (which is designed for the initial condition)

with the parameter determining the character of the disturbance. That initial condition

in the case of DC Motor and load simulation through eddy current could be a

situation when no extra load is applied with the magnet. For this position a linear PID

controller can be designed as in the previous sections. Then the controller design is

repeated with the magnet lowered simulating full load. However only one controller

parameter is changed with respect to the initial condition controller, this usually is the

controller gain. So depending on the angle of the magnet the gain of the controller is

changed according to a certain function, which we have to determine. If angle

information is fed electronically to the control system the gain scheduling is then

automatic. However if such a measurement is not available we can provide it on our

own restricting the magnet position to certain areas. For the beginning two areas

could only be considered: magnet low, magnet high. The performance of such

system will certainly be better comparing with the situation where the gain of the

controller is not changed at all with different load conditions.

We could think of other load estimation methods, which depending on current,

voltage and angular velocity measurement would give an estimate of the load. This

however will not be considered. The following section explains how the design of a

simple gain scheduling algorithm is carried out.

3In order to get a full insight of Adaptive Control the user is advised to refer to publications on

that topic or attend a suitable course.

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Control Experiments

Simple gain scheduling algorithm

The gain scheduling algorithm design can be divided into the following steps:

Design a PID controller for zero load conditions the model identified in the

modelling section.

Carry out an identification experiment to find a model for the full load

condition.

Design a new PID controller for a full load situation (full load model) by

changing only the Pgain of the controller that was designed for the zero loadconditions. Try to obtain the same closed loop system response as for the

zero load closed loop system.

Test the performance of the two controllers on the real time DC Motor

application.

Combine two of the controllers into one. Change the P gain of the PID

controller depending on the magnet position.

Of course a fully automatic gain scheduling algorithm should perform the gain

changing on its own depending on the value of the disturbance variable. Due to the

fact that such a measurement is not available we can only test a manual version ofthe gain scheduling algorithm.

Exercise 12 Gain scheduling

Introduction

As Figure 24 shows the system performance has degraded when the magnet

was lowered. The reason for that is the change of plant dynamics. With the

load applied the controller designed for the zero load conditions is

underperforming. By changing its gain according to the magnet position its

performance can be improved.

Task

According to the steps presented above, design a position tracking and gain

scheduling algorithm and test it on the DC motor. Carry out the position model

identification experiment under full load conditions. Use the model and the

previously designed controller and change its gain for optimal performance.

The gain scheduling has already been implemented for you in the

PMS_GainScheduling.mdl. A manual switch is provided which doubles the P

gain when set to position P2. Double click the switch to change its position.

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Run the model and investigate the controller performance when the load is

applied with and without the extra P gain.


As a result of the identification experiment a new model is obtained under full

load conditions. Using the Matlab rltool an observation can be made that

doubling the P gain to a value of P=0.0714ensures a similar step response

under full load conditions to that achieved by the original position controller

under no load conditions. Figure 25 shows example results using thePMS_GainScheduling.mdl model with gain scheduling. After 25 seconds the

magnet is lowered and the controller gain is doubled by changing the position

of the Manual Switch. The performance of the gain scheduling controller is

compared with the fixed gain controller in Figure 26. Although the experiments

are not identical the difference in performance can clearly be seen.

Figure 25 Controller with gain scheduling

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Figure 26 Controller without gain scheduling

If gain scheduling is not applied at the moment the magnet is lowered the

performance of the control system deteriorates. Changing the P gain clearly

improves the performance.

PMS SETUP CONTROL

ControlExperiments - Precision Modular Servo

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