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CONTROL LABORATORY

SERVO-MOTOR CONTROL AND SYSTEM

SIMULATION – CONTINUOUS AND DIGITAL

TABLE OF CONTENTS

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

Objectives................................................................................................................................................................................2

Some practical tips..............................................................................................................................................................2

Assessment.............................................................................................................................................................................3

Submission ........................................................................................................................................................................3

1 Part 1: Simulation...........................................................................................................................................................4

1.1 Open loop system..................................................................................................................................................4

1.1.1 System modelling ........................................................................................................................................4

1.1.2 Creation of Transfer Functions in MATLAB .....................................................................................6

1.1.3 Simulation of the plant model................................................................................................................7

1.2 Continuous-time controller ...........................................................................................................................11

1.2.1 Controller Design......................................................................................................................................11

1.2.2 Simulation with Simulink......................................................................................................................12

1.2.3 Nyquist analysis ........................................................................................................................................13

1.3 Digital Control .....................................................................................................................................................16

1.3.1 Discrete equivalent of the plant..........................................................................................................16

1.3.2 Digital design by emulation..................................................................................................................16

1.3.3 Simulation of the Digital Controller..................................................................................................18

1.3.4 Effect of the sample time.......................................................................................................................19

2 Part 2: Evaluation with Realistic Model .............................................................................................................20

2.1 Open-loop system ..............................................................................................................................................20

2.1.1 Comparison of simulation model and real system .....................................................................20

2.2 Continuous Feedback Control of the Real Servo System...................................................................21

2.2.1 Modified System Characteristics........................................................................................................22

2.3 Real-time Digital implementation...............................................................................................................23

2.3.1 Effect of the sample time.......................................................................................................................23

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INTRODUCTION

OBJECTIVES

The aim of this exercise is to design and evaluate a controller which regulates the position of the

output shaft of a servo motor.

The exercise has two parts: in Part 1 you will model the system and assess the performance using

a linear model of the system. In Part 2 you will use a realistic model of the system which

represents aspects you will encounter with a real system, so you can compare the performance

with the linear simulation. The lab will run over two sessions, during which you are expected to

complete all two parts.

At the end of the two lab sessions, you should be able to:

• derive the mathematical model of the servo plant from first principles and how to

implement this in MATLAB,

• appreciate the differences between the system simulation and the behaviour of the real

system,

• design and implement a continuous controller,

• use the Nyquist stability criterion to estimate the robustness of the feedback control system,

• implement the continuous-time controller and evaluate its performance,

• create a digital equivalent of the plant for use in a simulation,

• design a digital controller by emulation, based on the continuous controller design,

• compare the performance of the digital controller with its continuous counterpart, and

assess the impact of the sampling time on the digital control loop,

• evaluate the controllers with a real servo motor system, and compare their performance

with simulations.

SOME PRACTICAL TIPS

You will be using MATLAB and Simulink 2020a or above throughout this lab. This software is

available on School of Engineering computers which can be accessed on the Virtual PCs (COSE

Desktop) https://rdweb.wvd.microsoft.com/arm/webclient/index.html. You can also install Matlab

/ Simulink on your own computer (create an account using your UofG email address on

www.mathworks.com and you’ll get a download link), or run the lab on the Mathwork webclient

(https://matlab.mathworks.com).

.

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ASSESSMENT

Each student must

1) answer the questions throughout this lab sheet and complete the corresponding Moodle

quizzes (one quiz for each Part), and

2) write a short individual report. The report must not exceed 3 pages (excluding plots and

figures) and should contain:

• your name and student number,

• a description of the work done in the lab, explaining the tools and methods used,

• a summary of the results obtained and the observations made, including all plots required

in the lab sheet,

• a discussion of what other methods could be used to improve the results.

SUBMISSION

You need to

• complete the two Moodle quizzes available on the course Moodle page

• submit

o the short individual report as a PDF file under Lab report submission,

o the 7 Simulink model files (extension “.slx”) which correspond to the 5 models

mentioned in this lab sheet under Simulink model submission

on the Control Systems M Moodle page,

by the submission date shown on Moodle.

The Moodle quizzes and report will be assessed and contributes 10% to your final mark.

NOTE: This assignment is part of the assessment for this course, so you are required to complete

the submission in order to gain credits. As the individual assignment is part of your summative

assessment it is subject to the University’s plagiarism policy

(https://www.gla.ac.uk/myglasgow/leads/students/plagiarism/).

While you can discuss the lab with your fellow students, you must create the Simulink

models and complete the lab individually.

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1 PART 1: SIMULATION

1.1 OPEN LOOP SYSTEM

We will start by deriving the analytical model of the open loop system from first principles by

modelling the electrical and mechanical components of the servo motor system. This plant model

will then be implemented in MATLAB and we will analyse its frequency response, using the Nyquist

plot as an analytical method and comparing this with experimental results. Finally, we will compare

the open loop response of the model with that of the real system.

1.1.1 SYSTEM MODELLING

The Quanser QUBE-Servo 2 is a direct-drive rotary servo system. Its motor armature circuit

schematic is shown in Figure 1 and the electrical and mechanical parameters are given in Table 1.

The DC motor shaft is connected to the load hub. The hub is a metal disk used to mount the disk

load, and has a moment of inertia of Jh. A disk load is attached to the output shaft with a moment of

inertia of Jd .

Figure 1: QUBE-Servo 2 DC motor and load

Here 𝑣𝑚 is the supply voltage, 𝑅𝑚 and 𝐿𝑚 are the motor resistance and inductance, respectively,

𝑒𝑒𝑚𝑓 is the back emf produced by the motor as it turns and 𝑖𝑚 is the current through the motor. The

motor torque is denoted by 𝜏𝑚, the shaft angle is 𝜃𝑚 and the motor shaft velocity is 𝜔𝑚 = 𝜃̇𝑚.

Using Kirchhoff’s voltage law we obtain the following equation:

𝑣𝑚(𝑡) − 𝑅𝑚𝑖𝑚(𝑡) − 𝐿𝑚

𝑑𝑖𝑚(𝑡)

𝑑𝑡 − 𝑒𝑒𝑚𝑓(𝑡) = 0

Since 𝐿𝑚 ≪ 𝑅𝑚, the motor inductance can be ignored, and the equation can now be simplified to

𝑖𝑚(𝑡) =

𝑣𝑚(𝑡) − 𝑒𝑒𝑚𝑓(𝑡)

𝑅𝑚

The back emf created by the motor is proportional to the motor shaft velocity, 𝜃̇𝑚 = 𝜔𝑚, i.e.

𝑒𝑒𝑚𝑓(𝑡) = 𝑘𝑚𝜃̇𝑚(𝑡). Replacing in the equation above results in

𝑖𝑚(𝑡) =

𝑣𝑚(𝑡) − 𝑘𝑚𝜃̇𝑚(𝑡)

𝑅𝑚

The torque generated by the motor is proportional to the current through the motor,

𝜏𝑚(𝑡) = 𝑘𝑡

𝑖𝑚(𝑡)

where 𝑘𝑡

is the motor-torque constant. Replacing 𝑖𝑚 with the equation above results in

𝜏𝑚(𝑡) = 𝑘𝑡

𝑣𝑚(𝑡) − 𝑘𝑚𝜃̇𝑚(𝑡)

𝑅𝑚

We now consider the mechanical aspects of the motor. Applying Newton’s 2nd law of motion to the

motor shaft results in

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(𝐽𝑚 + 𝐽ℎ + 𝐽𝑑)𝜃̈𝑚(𝑡) = 𝜏𝑚(𝑡)

where 𝐽𝑚, 𝐽ℎ and 𝐽𝑑 are the moment of inertia of the motor, the hub and the disk, respectively.

The moment of inertia of a disk about its pivot, with mass 𝑚 and radius 𝑟,

which allows us to calculate 𝐽ℎ and 𝐽𝑑, given their mass and radius. The total equivalent moment of

inertia is therefore,

𝐽𝑒𝑞𝑅𝑚𝜃̈(𝑡) + 𝑘𝑡𝑘𝑚𝜃̇(𝑡) = 𝑘𝑡𝑣𝑚(𝑡)

Q1.1: Take the Laplace transform of this equation and derive the transfer function

𝑮(𝒔) =

𝜽(𝒔)

𝑽𝒎(𝒔)

You should now implement this system in MATLAB.

Using the MATLAB editor, write a MATLAB script to calculate the numerator and

denominator coefficients of the transfer function, using the physical parameters given in

Table 1 at the end of this section.

Bring this transfer function into monic form, i.e. make sure that the coefficient associated with the

highest power of s in the denominator is equal to 1. This can be achieved by dividing numerator and

denominator by the coefficient associated with the highest power of s in the denominator.

Q1.2: Write down the resulting transfer function in monic form:

Q1.3: What are the poles and the zeros of the plant transfer function?

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1.1.2 CREATION OF TRANSFER FUNCTIONS IN MATLAB

You will use the function tf from the Control System toolbox to implement the transfer function

which you have derived, in MATLAB, using a transfer function object (TF object).

The expression P = tf(num,den) creates a continuous-time transfer function with

numerator(s) and denominator(s) specified by the vector-variables num and den. The output

variable P is a TF object storing the transfer function data.

In the Single-Input Single Output (SISO) case, num and den are the row vectors of numerator and

denominator coefficients ordered in descending powers of s. These two vectors need not have equal

length and the transfer function need not be proper.

For example,

• h = tf([1 0],1) specifies the pure derivative

h s s ( ) = ,

• g = tf([15],[6 3 2]) specifies the proper transfer function

Amend the MATLAB script which you have created above to implement the system transfer

function using the tf command and assign it to the variable P.

Symbol Description Value

DC Motor

Rm Terminal resistance 8.4Ω

kt Torque constant 0.042 N.m/A

km Motor back-emf constant 0.042 V/(rad/s)

Jm Rotor inertia 4.0 × 10−6 kg.m2

Lm Rotor inductance 1.16 mH

mh Load hub mass 0.0106 kg

rh Load hub mass 0.0111 m

Load Disk

md Mass of disk load 0.053 kg

rd Radius of disk load 0.0248 m

Table 1: QUBE Servo 2 system parameters.

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1.1.3 SIMULATION OF THE PLANT MODEL

In this part you will implement a simulation of the plant model in Simulink and evaluate this.

From within MATLAB, open Simulink and create a new model. By default, the model is set to

simulate a continuous-time system.

To increase the simulation accuracy, you need to make a change to the model settings which can be

found under the Modeling tab.

In the Configuration window, select the Solver option on the left and expand the Solver details,

then set the Relative tolerance to 1e-6, as shown in the figure below.

Hit OK to apply these settings to the model!

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Use the basic building blocks in the library to reproduce the system under analysis in

Simulink. Drag and drop the blocks from the library window to the model window, then

connect the block inputs/outputs together (see Model 1). Set the simulation time to 60

(seconds).

Model 1. Open-loop system with simulated plant

For the transfer function, you can use the block called LTI System, which encodes a transfer function

previously defined in the workspace; the block can be found in the library Control System Toolbox.

Once you have dragged the block into your model, double click on it to open its properties and

make sure that the system variable for the plant TF has the same name as the variable of your

Simulink LTI System block (e.g. P).

The system input should be a Sine Wave block (can be found in the library Simulink\Sources).

Double click on the block to set the parameters of the sine wave as following:

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Input and output of the system are plotted on a Scope block (can be found in the library

Simulink\Sinks). Double click on the Scope to open the plot and view the signals (after running the

simulation).

Scope block setup

Open the Scope and click on the Settings icon (top left). On the History (or Logging,

depending on version) tab:

• Tick the box “Save (or log) data to workspace” and choose a variable name. Set

“Format” as “Structure with time”. This will allow you to save the data generated

and plot them later for your report.

• Note: You can find the variable as part of the structure name out in the Matlab

workspace, i.e. out.ScopeData.

Plotting and saving results

You can plot the data generated using the command plot with the elements of the

ScopeData structure (saved from the Scope) as arguments. ScopeData.time is the

time, ScopeData.signals(1).values are the elements of the first Scope panel, and

ScopeData.signals(2).values are the elements of the second Scope panel. For

example, to plot the data from the lower Scope panel (i.e. the output of the real servo and

the output of the simulated model) over time, you can use the command:

plot(ScopeData.time, ScopeData.signals(2).values)

You can also save the variable to a file, so you can plot it later (for example for the

report).

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We suggest saving every Simulink model and corresponding MATLAB script you will create.

You can use a sequential number that matches model numbers in the figure in this hand-out,

e.g. model1.slx and script1.m.

Run the Simulink simulation for the two different input frequencies of 1 rad/s and 5 rad/s,

and record the output amplitude and the time shift between input and output signal at each

frequency.

Q1.4: Describe what you observe. What is the difference between the simulation at 1 rad/s

and at 5 rad/s? Does this correspond to what you would expect?

Include plots of the simulation responses in your report.

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1.2 CONTINUOUS-TIME CONTROLLER

In this part of the experiment we will use a loop-shaping approach together with the root-locus

method to select controller gains. The closed loop system will then be analysed using the Nyquist

stability criterion and simulations. In Session II, you will implement the controller with the real

servo system and evaluate its performance.

1.2.1 CONTROLLER DESIGN

Firstly, we will select a suitable compensator 𝐶(𝑠) in the continuous domain. Note that the plant

already has a pole at 0 (i.e. integrative behaviour), hence the loop gain

L s C s P s ( ) ( ) ( ) =

also has a

pole at zero, and

L j ( )

has therefore infinite gain at

= 0

. For this reason, we do not need to

include an integrator in the compensator: A PD controller structure (equivalent to a lead

compensator) can hence be used:

𝐶(𝑠) = 𝐾𝑝 + 𝐾𝑑

𝑠

𝜏𝑠 + 1

where 𝐾𝑝 and 𝐾𝑑 are the proportional and derivative gain factors, respectively. The denominator

term with the time-constant 𝜏 is required to make the derivative term realisable. This structure can

be rewritten as

𝐶(𝑠) = 𝐾

𝑇𝐷𝑠 + 1

𝜏𝑠 + 1

or

𝐶(𝑠) = 𝐾𝐷(𝑠)

with 𝐷(𝑠) =

𝑇𝐷𝑠+1

𝜏𝑠+1

.

We will design the controller in two steps: first, we setup the controller frequency response defined

by 𝐷(𝑠), using, and then we will choose a suitable controller gain 𝐾 using the root-locus method.

1.2.1.1 LOOP SHAPING FOR D(S)

We can decide to use the zero of the compensator (at − 1⁄𝑇𝐷) to cancel the high-frequency pole of

the plant (which is located at approximately −10, see your calculations earlier on). We can then add

an extra pole at −100 = −1⁄𝜏 to ensure that the compensator is proper (i.e. realisable). The

resulting compensator transfer function 𝐷(𝑠) is:

Implement the transfer function 𝑫(𝒔) in MATLAB using the tf commands (as you have done

for the plant), calling the TF system D.

Note that the overall compensator is 𝐶(𝑠) = 𝐾𝐷(𝑠), but for now we assume 𝐾 = 1.

Evaluate the effect of the compensator on the loop gain 𝑳(𝒔) by evaluating 𝑳(𝒔) = 𝑪(𝒔)𝑷(𝒔)

for 𝑲 = 𝟏.

In MATLAB, you can simply multiply the transfer functions of your compensator with the transfer

function of your plant, i.e. C=K*D; and L=C*P;.

Generate a bode plot showing

P s( ), C s( )

and

Ls( )

(using bode(P,C,L)). To add a legend

(showing which line belongs to which transfer function), use legend P C L.

Show the stability margins (right-click in the plot, select Characteristics->Minimum

Stability Margins).

Q1.5: Plot the Bode diagram and include this in your report. Explain the shape of the

magnitude and phase of P(s), C(s) and L(s), based on the poles and zeros of the transfer

functions. In which way does C(s) affect the loop gain (considering the phase and the

magnitude)?

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Q1.6: What are the cross-over frequencies for P(s) and L(s)? What are the gain and phase

margins?

1.2.1.2 CONTROLLER GAIN K

So far, we have analysed 𝐶(𝑠) = 𝐾𝐷(𝑠) with 𝐾 = 1. We will now analyse the system to select a

more suitable value for the gain K, using the root-locus analysis.

The root-locus of a feedback loop can be obtained using the MATLAB command rlocus. See doc

rlocus to get more information on how to use this command.

On the MATLAB command prompt, enter the command rlocus(L, [0:0.001:5]).

This plots the root-locus of the system for values of the feedback gain K ranging from 0 to 5. The

three ‘x’ (one at the origin, one at approx. -10, the other at -100 on the real axis) indicate the

location of the open loop poles (i.e. for a feedback gain of 0), the ‘o’ indicates the open loop zero

(from the compensator, which cancels the pole at -10). The three lines show how the locations of

the poles change as the feedback gain is increased. If you left-click on one of the lines you will see a

pop-up window showing the feedback gain at this point of the root-locus, the location of the pole,

and the corresponding characteristics of the closed loop (damping, overshoot and frequency).

Using the root-locus plot, find the feedback gains for which

A. the damping is 1.0 and the frequency of the slowest pole is 5 rad/s,

B. the overshoot is 5%.

Q1.7: Note the values for the feedback gains in the table below and determine the locations

of all three poles from the root locus plot for these gains.

Location of poles

Feedback gain K Pole 1 Pole 2 Pole 3

A

B

Table 2: Root locus gains and poles

1.2.2 SIMULATION WITH SIMULINK

Create a model similar to 1 (the open loop system with simulated plant), however this time

introduce closed-loop feedback with a controller C(s), as shown in the figure below (Model

2).

In this model, the control loop is now closed (feedback control) and the Gain block is the feedback

gain. As a signal source, you can use a Step or a Pulse Generator (sequence of steps). If you use the

pulse generator, adjust the “pulse width” to 50% of the period, to give enough time to the system to

settle at each pulse.

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Make sure the Scope block is set up as described on page 9. You can also copy-paste the Scope block

from Model 1 by drag-and-drop, preserving its properties.

Model 2. Feedback system with compensator and simulated plant

Set the Controller Gain value in the model to the values you have found above and evaluate

the responses by running the simulation for each gain value.

In the top plot of the Scope, the control signal u is shown. In the bottom plot, the reference and the

output are displayed.

Verify that the overshoot and damping specified from the root-locus analysis correspond

with those obtained in the simulation. Look also at the control signal which is shown in the

top plot of the scope.

Q1.8 Plot y_s and r against time for the two different gain values. How do the responses for

the two gain values differ?

Include the plots of the simulation responses in your report.

1.2.3 NYQUIST ANALYSIS

You will now use the Nyquist plot to analyse the stability of the system.

Generate the Nyquist plot of the loop gain for the two gain values (K_A, K_B, corresponding

to A and B in Table 2) which you have found above and include this in your report. Use the

command nyquist(K_A*L,K_B*L) where K_A and K_B are the values of the gains.

The point ‘-1’ is marked with a red + in the plot. The Nyquist stability criterion states that the closed

loop system becomes unstable if the Nyquist plot encircles ‘-1’. You can derive a phase-margin

(where the Nyquist plot crosses the unit circle) and a gain-margin (where the Nyquist plot crosses

the negative real axis) from this plot which indicate by how much your control loop can change

before it becomes unstable – in other words, how robust the control loop is.

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From the right-click menu in the Nyquist plot, select Characteristics -> Minimum Stability Margins.

This will display the unit-circle and mark the points at which the plots cross it. This indicates the

phase margins. If you move the mouse over this point you will see a pop-up showing the phasemargin,

the frequency and the resulting delay margin. Note that the gain-margin is infinite in our

system (because the Nyquist plot never crosses the negative real axis).

Q1.9: For the two gain values obtained above, determine the phase margins together with

their frequencies and the corresponding delay margin. Enter the values you find in the table

below.

Phase margin (deg) Frequency (rad/s) Delay margin (s)

A

B

Table 3: Margins

Q1.10: Generate the Bode plot of the loop gains for the two gain values using the command

bode(K_A*L, K_B*L)and include them in your report. Determine the phase margin from

this plot and compare to those in Table 3.

The phase margin

determines how much “phase” can be added to the system before it becomes

unstable. “Phase” is related to a time-delay, using the frequency

at which it occurs:

The time-delay

T

associated with the phase

(in rad) for a given period of oscillation,

T

, can be

calculated using the equation

. The frequency of oscillation,

. Inserting this into the equation above, the time-delay at a specific

frequency

can be calculated as

T

= .

This means that the system remains stable as long as the time-delay added to the loop is smaller

than the delay-margin resulting from the phase-margin.

Evaluate this in a simulation for the value of the gain for case B. In the Simulink model of

Model 2, add a Transport Delay block (from Simulink\Continuous), as shown in Model 3

below. Enter the gain value from case B (K_B) in the Controller Gain block. Then change the

Time Delay value of the Transport Delay block to a value slightly smaller than the delay

margin which you have determined from the Nyquist plot and run the simulation. Repeat

this with a value which is slightly larger than the delay margin.

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Model 3. Feedback system with compensator and simulated plant, and added feedback delay.

Q1.11 Include the plots of y_s and r against time for both cases in your report. How does the

system behave in both cases?

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1.3 DIGITAL CONTROL

The scope of this part of the laboratory is to analyse the control loop in simulations, assuming that

it is implemented using a digital computer, with a non-infinitesimal sample time. We will see that if

the sample time is small enough, then the digital controller performance is relatively similar to the

continuous one. Conversely, as the sample time increases, the performance of the digital controller

degrades until instability appears.

We will design the digital controller by emulation, i.e. by emulating the continuous controller

designed in the previous section.

1.3.1 DISCRETE EQUIVALENT OF THE PLANT

Before implementing a digital controller, we need to find the discrete equivalent model of the

(continuous) plant P(s). This is done assuming that the plant is preceded by a zero-order hold

(ZOH):

Q1.12: Given the plant transfer function P, using what you have learnt during the course, and

the transforms tables, find the discrete equivalent P(z). Assume a sample time T = 0.01 s.

The same result can be found using the MATLAB function: P_zoh = c2d(P, T, 'zoh') which

converts a continuous system to its discrete equivalent using the “zoh” rule and a sample time of T.

Verify your result.

1.3.2 DIGITAL DESIGN BY EMULATION

We will now convert the designed continuous controller D(s) into a digital one, using one of the

“emulation” techniques. Design by emulation essentially means that we an existing controller,

designed in the continuous domain, into the discrete domain. In other words, the discrete controller

“emulates” the continuous one, trying to achieve similar characteristics.

Here in particular we will use the “pole-zero matching” technique. This technique will be explained

during the class, however it consists of three steps to compute poles, zeros and gain for a discrete

transfer function, given its continuous counterpart. It relies on the correspondence between

poles/zeros in the s-plane and in the z-plane.

1. Each pole

of the digital controller is selected in such a way as to preserve the behaviour at

a critical frequency,

Q1.13: For the controller gain from B (K_B), find the zeros and poles of the discrete transfer

function of the compensator, Dd(z), and its gain (K_d) analytically:

Implement the compensator transfer function in MATLAB with the variable Dd, using the

command Dd = tf(NUM,DEN,T), where NUM and DEN need to be set according to the poles

and zeros found through the rule above, and T is your sample time (0.01s).

Define also the digital gain K_d.

Note that tf(…,…, T) now creates a discrete TF because the sample time T is specified.

Q1.14: Use your report to show the Bode diagrams of the continuous and digital

compensators using the bode function in MATLAB: bode(K_B*D, 'b', K_d*Dd, 'r--').

Describe the differences/similarities at low and high frequencies. Graphically or by other

means, estimate the phase (deg) at 200 rad/s of both continuous and digital compensators.

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1.3.3 SIMULATION OF THE DIGITAL CONTROLLER

Let’s now set up a discrete (digital) simulation in Simulink.

Create a new Simulink model. Open “Simulation->Model Configuration Parameters” and

select the “Solver” page. Set the following parameters:

• Solver options-> Type: fixed-step

• Solver: discrete (no continuous states)

• Fixed-step size: T (Simulink will pick the value of T from MATLAB workspace)

• Stop time: 60 (seconds)

• Tasking mode for periodic sample time: “SingleTasking”

Define the sample time variable T in the MATLAB workspace as T = 0.01.

This Simulink model is now configured to simulate a discrete system, using difference equations

and discrete transfer functions.

We will first test the performance of the digital controller using the digital equivalent of the plant.

Build a discrete feedback loop as shown Model 4 below, using the transfer functions found above.

Note the use of discrete transfer functions P_zoh and Dd.

Model 4. Digital feedback system with compensator and simulated plant

Q1.15: Use your report to compare the time history of y_s of the simulations of the digital

feedback control loop (Model 4) with the analogue (continuous-time) implementation

(Model 2). How does the performance compare in terms of overshooting and settle time?

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1.3.4 EFFECT OF THE SAMPLE TIME

We will now increase the sample time, and re-assess the performance of the controller.

Set the sample time to T = 0.04 in the MATLAB workspace.

Q1.16: Re-compute the discrete equivalent of the plant P_zoh, the lead compensator Dd and

its gain K_d for the new sample time.

1.3.4.1 SIMULATED PLANT

Run the digital feedback system (Model 4) again, with the new sample time of 0.04 s.

Q1.17: Use your report to plot again the time history of y_s. Is it still stable?

Control Laboratory Page 20

2 PART 2: EVALUATION WITH REALISTIC MODEL

2.1 OPEN-LOOP SYSTEM

You will now evaluate how well the linear model corresponds with a realistic model which

approximates the true servo system.

2.1.1 COMPARISON OF SIMULATION MODEL AND REAL SYSTEM

Create an open-loop model, such as the one made before, but this time the sys transfer function

block is replaced with a realistic simulation of the system. Download the Simulink library

qube2servo.slx from the lab section on the Moodle page and open it. You will find two blocks in this

model:

Copy the block named Quanser Qube-2 Servo (which provides a realistic simulation of the servo

system) to your Simulink model. You could also run, in parallel and simultaneously, the model of

the plant developed before, to compare the results, such as in Model 5 below. Save this model

under meaningful name (e.g. as model_5)

Model 5. Open-loop system comparing the simulated plant and the actual servo-motor.

Make sure you set up the Scope block as explained on page 9!

Note that we have used, as a signal source, a Signal Generator block, which allows to generate

various types of signals, and not only a sinusoid. For now, we are interested in sinusoidal inputs, so

make sure that the “wave form” is set to “sine” in its properties, with unit amplitude.

Control Laborator

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