EXPERIMENTAL TESTING OF THE EXTENDED
HIGH-GAIN OBSERVER AS A DISTURBANCE
ESTIMATOR
By
Rachel E. Bou Serhal

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Electrical Engineering

2011

ABSTRACT
EXPERIMENTAL TESTING OF THE EXTENDED
HIGH-GAIN OBSERVER AS A DISTURBANCE
ESTIMATOR
By
Rachel E. Bou Serhal

In recent years, the Extended High-Gain Observer (EHGO) has proven valuable in
its use with fully actuated mechanical systems. In this thesis we investigate the performance of the EHGO experimentally. First, we explore its use with underactuated
mechanical systems. Using EHGO to estimate and cancel the disturbance in one link
of an underactuated system results in adding that disturbance to other links. However,
we show that the EHGO can be used to reduce the effect of disturbances in a rotary
pendulum system. We also compare the performance of the EHGO as a disturbance estimator in a fully actuated mechanical system with the performance of the sliding-mode
observer. We use a DC motor to demonstrate the simplicity of applying the EHGO
compared to the complexity of the sliding-mode observer. We then highlight the advantages and drawbacks of using each observer. Overall, our experimental results provide
a check, using the EHGO, for control engineers working with underactuated mechanical system that may simplify controller design. These results also underline the ease in
implementation of the EHGO and its state of the art performance.

To my parents, Lina and Elias.
To my siblings, Chadi, Sara and Georges.

iii

ACKNOWLEDGMENTS

I would like to thank my family for being there through the bad times and the good
times. In particular I would like to recognize my mother and father for all the support
they have given me throughout this journey. I would also like to thank my uncle, Elias
Sayah, for always helping me out when I needed him. I also thank my advisor Professor
Hassan Khalil for guiding me through my work and showing me how to become a good
researcher. I appreciate my friends who have shared this journey with me. Finally, I
thank my students for helping me become a good educator.

iv

You are rewarding a teacher poorly if you remain always a pupil .
- Friedrich Nietzsche

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Background . . . . . . . . . . . . . . . . .
2.1 Observers . . . . . . . . . . . . . . . .
2.1.1 Extended High-Gain Observer .
2.1.2 Sliding Mode Observer . . . . .
2.2 Mechanical Systems . . . . . . . . . .
2.2.1 Rotary Pendulum . . . . . . . .
2.2.2 DC Motor . . . . . . . . . . . .

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4
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7
9
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14

3

Disturbance Cancellation in the Rotary Pendulum Using
Gain Observer . . . . . . . . . . . . . . . . . . . . . . .
3.1 Controller Design . . . . . . . . . . . . . . . . . . . .
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Comparison of Extended High-Gain Observer and Sliding-Mode Observer
using the DCMCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Implementation and Results . . . . . . . . . . . . . . . . . . . . . . .
4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33
34
37
51

Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . .

59

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5

vi

Extended High. . . . . . . .
18
. . . . . . . . . 18
. . . . . . . . . 22

LIST OF TABLES

2.1

List of symbols and values for ROTPEN . . . . . . . . . . . . . . . . . 12

2.2

List of symbols for the DCMCT . . . . . . . . . . . . . . . . . . . . . 16

2.3

List of symbols and values for the DCMCT . . . . . . . . . . . . . . . 16

vii

LIST OF FIGURES

2.1

National Instruments’ ELVIS I Station. For interpretation of the references to color in this and all other figures, the reader is referred to the
electronic version of this thesis. . . . . . . . . . . . . . . . . . . . . . . 10

2.2

ROTPEN trainer developed by Quanser. . . . . . . . . . . . . . . . . . 11

2.3

Free body diagram of the ROTPEN. . . . . . . . . . . . . . . . . . . . 11

2.4

DCMCT trainer developed by Quanser. . . . . . . . . . . . . . . . . . 15

3.1

Estimate of the disturbance in the arm, d2 . . . . . . . . . . . . . . . . 24

3.2

Estimate of the disturbance in the pendulum, d4 . . . . . . . . . . . . . 25

3.3

Estimate of the new disturbance found in the arm angle . . . . . . . . . 25

3.4

θ with no disturbance estimation used . . . . . . . . . . . . . . . . . . 27

3.5

α with no disturbance estimation used . . . . . . . . . . . . . . . . . . 28

3.6

Comparison between ’theta’, the measurement of the arm angle, and
’theta hat’, its estimate using (3.17) . . . . . . . . . . . . . . . . . . . 28

3.7

Comparison between ’theta dot’, the estimated speed of the arm using
(3.2), and ’theta dot hat’, the estimate found using (3.17) . . . . . . . . 29

3.8

Comparison between ’alpha’, the measurement of the arm angle, and
’alpha hat’, its estimate using (3.17) . . . . . . . . . . . . . . . . . . . 29

3.9

Comparison between ’alpha dot’, the estimated speed of the arm using
(3.2), and ’alpha dot hat’, the estimate found using (3.17) . . . . . . . . 30

3.10 x1 with disturbance cancelation. . . . . . . . . . . . . . . . . . . . . . 31
3.11 x3 with disturbance cancelation. . . . . . . . . . . . . . . . . . . . . . 31
4.1

Response of x1 with only PID control. . . . . . . . . . . . . . . . . . . 38

4.2

Close up of transient response of x1 with only PID control. . . . . . . . 39
viii

4.3

Control with only PID control. . . . . . . . . . . . . . . . . . . . . . . 39

4.4

Response of x1 with de = 5V using only PID control. . . . . . . . . . 40

4.5

Close-up of response of x1 with de = 5V using only PID control. . . . 40

4.6

Control with de = 5V using only PID control. . . . . . . . . . . . . . . 41

4.7

Comparison between x1 and its estimate, x1 using (4.14) . . . . . . . . 42
ˆ

4.8

Comparison between x2 using 4.1 and x2 using (4.14) . . . . . . . . . 43
ˆ

4.9

The response of x1 using EHGO. . . . . . . . . . . . . . . . . . . . . . 43

4.10 A close-up of the transient response of x1 using EHGO. . . . . . . . . 44
4.11 Control using EHGO. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.12 Estimate of the inherent disturbance using EHGO. . . . . . . . . . . . . 45
4.13 Response of x1 with de = 5V using EHGO. . . . . . . . . . . . . . . . 46
4.14 Close-up of transient response of x1 with de = 5V using EHGO. . . . . 47
4.15 Estimate of overall disturbance with de = 5V using EHGO. . . . . . . 48
4.16 Comparison between x1 its estimate, x1 using SMO with constant gains 49
ˆ
4.17 Comparison between x2 using 4.1 and z1 using SMO with constant gains 50
4.18 Response using SMO with constant gains. . . . . . . . . . . . . . . . . 50
4.19 Close-up of response using SMO with constant gains. . . . . . . . . . . 51
4.20 Control using SMO with constant gains. . . . . . . . . . . . . . . . . . 51
4.21 Estimate of inherit disturbance using SMO with constant gains. . . . . . 52
4.22 Comparison between x2 using 4.1 and z1 using SMO with time-varying
gains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.23 Response using SMO with time-varying gains. . . . . . . . . . . . . . . 54
4.24 Close-up of transient response using SMO with time-varying gains. . . 55
ix

4.25 Control using SMO with time-varying gains. . . . . . . . . . . . . . . . 55
4.26 Estimate of inherit disturbance using SMO with time-varying gains. . . 56
4.27 Response with de = 5V using SMO with time-varying gains. . . . . . . 56
4.28 Response of x1 with de = 5V using SMO with time-varying gains and
second-order low-pass filters. . . . . . . . . . . . . . . . . . . . . . . . 57
4.29 Close-up of transient response of x1 with de = 5V using SMO with
time-varying gains and second-order low-pass filters. . . . . . . . . . . 57
4.30 Estimate of overall disturbance with de = 5V using SMO with timevarying gains and second-order low-pass filters. . . . . . . . . . . . . . 58

x

CHAPTER 1
Introduction
Disturbance estimation techniques such as the sliding mode observer (SMO) [12], the
disturbance observer (DOB) [7], and the extended high gain observer (EHGO) [6] have
been used in fully actuated mechanical systems to reduce the effect of model errors and
unknown disturbances. A survey of disturbance observers in [10] gives a a good record
of their use in control design. In this thesis we focus on two observers, EHGO and
SMO. We use experimental results to achieve two goals :
• Investigate the potential use of EHGO with an underactuated mechanical system
• Compare the EHGO and the SMO
Although disturbance estimation techniques have shown some promising results in
fully actuated systems, their use in underactuated systems has not yet been thoroughly
explored. Often in these techniques of disturbance estimation and specifically the
method presented by Freidovich and Khalil in [6], the cancellation of disturbance occurs in the control. However, in underactuated mechanical systems, the same control
may appear in more than one link. Thus, cancellation of disturbance in one link adds
disturbance to the other. The work done in [11] showed experimental results of distur1

bance estimation and cancellation in an underactuated mechanical system. However,
their work differs from the work done in this thesis. They estimate and cancel the
disturbance in one of the links, but to deal with added disturbance in the other links,
they design a robust yet complicated control law that overcomes them. Their situation
presents a worst case scenario. We explore the situation where the added disturbance to
the other links actually reduces the already present disturbance. In Chapter 3 we show
a case where this occurs in a rotary pendulum system and use a simple controller to
achieve the desired response. An interesting control problem in underactuated mechanical systems is gantry control (minimization of pendulum oscillations while the arm
tracks a certain reference). Quanser [8] developed a rotary pendulum platform (ROTPEN) for educational purposes. It is an underactuated mechanical system that offers a
good exposure to many control problems, including gantry control. Quanser uses linear state feedback to achieve the desired response. However, this technique does not
consider any unknown disturbances in the system that may affect the system’s performance. For fully actuated mechanical systems, [6] presented a technique utilizing the
EHGO to estimate the disturbance. Their work offered a way to estimate and cancel the
disturbance in a system, while promising transient performance recovery and integral
action. We investigate gantry control and try to minimize the effect of disturbance by
utilizing a variation of the disturbance estimation method presented in [6].
The SMO and the EHGO have both proven to be valuable in disturbance compensation problems [14] [6]. Each observer has a set of design parameters that affect its
performance. The choice of these parameters is dependent on and restricted by the experiment performed. The presence of measurement noise affects the performance of the
EHGO [13]. The effect of measurement noise could be avoided by a better understand-

2

ing of the sensors used in each experiment [1]. With the SMO we deal with chattering
problems from discontinuity [14]. The choice of the parameters involved in designing
the SMO require some knowledge of the unknown states and the bound on the disturbance. This is also true for the EHGO but we show in Chapter 4 that the performance
of the SMO is more sensitive to a better understanding of these bounds. We use the
DC Motor Control Trainer (DCMCT), a fully actuated mechanical system developed
by Quanser [9] to compare the performance of each observer in the same experimental environment.We implemented each experiment using National Instrument’s ELVIS
station and LabView 7.1 to interface with the ROTPEN and the DCMCT.
In this thesis we present a review of each observer and introduce the model for the
ROTPEN and the DCMCT in Chapter 2. We then apply the EHGO to the ROTPEN
and discuss its use with an underactuated mechanical system in Chapter 3. In Chapter
4 we apply both the SMO and EHGO to the DCMCT and discuss the results of each. In
Chapter 5 we summarize the contributions of our work and discuss the future work.

3

CHAPTER 2
Background
This chapter provides the background that is required to understand the contributions of
this thesis. It is divided into two sections. The first section describes the observers designed during our work and the theory behind their use. The second section introduces
the mechanical/electro-mechanical systems used during the experiments.

2.1

Observers

This section discusses the preliminaries needed for designing the observers used in
these experiments. It covers the introduction of the Extended High-Gain Observer and
the Sliding Mode Observer.

2.1.1

Extended High-Gain Observer

Feedback linearization techniques offered a way for designing feedback control for nonlinear systems using linear control theory [6]. Theoretically, these techniques provide
simpler ways of not only achieving stabilization and regulation, but also transient response specifications. However, in practice, model uncertainty and disturbance limit the
4

realization of what feedback linearization can offer. Consequently, the extended highgain observer (EHGO) was developed to combine the idea of disturbance cancellation
with feedback linearization. The design of the EHGO is simple as can be seen in the
following discussion.
Consider a single-input-single-output nonlinear system in the normal form [6]

x = Ax + B[b(x, z, w) + a(x, z, w)u]
Ë™

(2.1)

z = f0 (x, z, w)
Ë™

(2.2)

y = Cx

(2.3)

where A ∈ Rn×n , B ∈ Rn , C ∈ R1×n represent a chain of n integrators, x ∈ Rn
and z ∈ Rm are state variables, u ∈ R is the control input, y ∈ R is the measured
output, and w ∈ Rl is a bounded disturbance input.The functions a(.), b(.), and f0 (.)
are possibly unknown nonlinear functions. Assuming that:
• w(t) belongs to a known compact set W ⊂ Rl .
• a(.), b(.), and f0 (.) are continuously differentiable with locally Lipschitz derivatives.
• There is a radially unbounded function such that
∂(V0 )
(z)f0 (x, z, w) ≤ 0, for z ≥ χ(x, w)
∂z
ensures the internal dynamics (2.3) to be bounded-input-bounded-state stable. With
assumptions of availability of (x, z, w), and knowledge of the functions a(.) and b(.)
feedback linearization
u=

−b(x, z, w) + v
a(x, z, w)

5

(2.4)

could have been used to reduce the model to the target system

x = Ax + Bv, y = Cx.
Ë™

The control v = φ(x) could be chosen as a twice continuously differentiable state feedback control law such thart the closed loop system is locally exponentially stable and
globally asymptotically stable. The state feedback controller v can be chosen using any
linear control design method such as Linear Quadratic Regulator (LQR) or pole placement. However, unavailability of (x, z, w) and uncertainty in the functions a(.) and b(.)
makes this control unrealizable. An EHGO could be used to recover the performance
of the target system.
The extended system is found by augmenting the chain of integrators in (2.1)-(2.9)
with an additional integrator. The EHGO is constructed by designing a high-gain observer to the extended system as
Ë™
x = Aˆ + B[ˆ x) + a(ˆ)(u + σ )] + H(ε)(y − C x)
ˆ
x
b(ˆ
ˆx
ˆ
ˆ
Ë™
σ =
ˆ

αn+1
a(ˆ)εn+1
ˆx

(y − C x)
ˆ

(2.5)
(2.6)

where σ is an estimate of the disturbance, a(.) and ˆ are models of a(.) and b(.)
ˆ
ˆ
b(.)
respectively and
H(ε) =

α1
αn T
, ..., n
.
ε
ε

The choice of α1 , ..., αn , αn+1 must be made such that
sn+1 + α1 sn + ... + αn+1
is Hurwitz. Now, the target system could be reached by choosing the control as
u = −ˆ +
σ

−ˆ x) − φ(ˆ)
b(ˆ
x
a(ˆ)
ˆx

6

= ψ(ˆ, σ ).
x ˆ

However, it is important to saturate the control to protect the system from peaking [6].
The control, thus becomes
ψ(ˆ, σ )
x ˆ
M

u = M sat
where
M > max

−b(x, z, w) + φ(x)
a(x, z, w)

The performance recovery of the target system can be achieved by pushing ε small
enough, such that the error between the trajectories of the actual system and the target
system is minimized. In practice, the choice of ε is bounded from below by the level of
measurement noise in the system [13]. In summary, for mechanical systems, the EHGO
is used to estimate matched disturbances in the system. This estimate is then used in the
controller to cancel the disturbance. The controller will also include an output feedback
control that will recover the target system’s performance.

2.1.2

Sliding Mode Observer

Sliding-mode observers (SMO) are used because they could offer finite-time convergence, and robustness with respect to uncertainties and the possibility of uncertainty
estimation. Unlike high-gain observers, such as the extended high-gain observer, when
using SMO’s we need not worry about peaking [3]. Consider a single-input-singleoutput second-order system, of the form

x1 = x2
Ë™

(2.7)

x2 = a21 x1 + a22 x2 + α(u + d)
Ë™

(2.8)

y = x1

(2.9)

7

where a21 and a22 are known coefficients of x1 and x2 , d is a bounded disturbance
term, α is the control coefficient and u is the control input. The design of the SMO to
estimate the unavailable state, x2 and d is as follows [12]
Ë™
x1 = x2 + M0 sgn(y − x1 )
ˆ
ˆ
ˆ
Ë™
x2 = a21 x1 + a22 x2 + α[u + M1 sgn(z1 )]
ˆ
ˆ
ˆ

(2.10)

τ1 z1 = −z1 + M0 sgn(y − x1 )
Ë™
ˆ
τf zf = −zf + M1 αsgn(z1 )
Ë™

(2.11)

where z1 and zf are estimates of x2 and d respectively. M0 and M1 are constant
gains defined by the bounds on x2 and the d. Their choice will be discussed in more
detail later in the section. The filters in (2.11) are first order low pass filters whose
time constants Ï„1 and Ï„f are chosen according to system parameters. For now, to better
understand the idea behind this method, we investigate the error dynamics. Let
η1 = x1 − x1
ˆ

(2.12)

η 2 = x2 − x2
ˆ
therefore
η1 = η2 − M0 sgn(η1 )
Ë™

(2.13)

η2 = a21 η1 + a22 η2 + d − M1 sgn(z1 )
Ë™
Given that d and x2 are bounded
|x2 | < K1
|d| < K2
then we can guarantee that the estimation error will go to zero in finite time [3] when
we choose
M0 (t) > K1
M1 (t) > K2
8

(2.14)

In sliding-mode the sliding surface is defined as η1 = η2 = 0. The low frequency
behavior yields η1 = η2 = 0 and
Ë™
Ë™
η2 = M0 sgn(η1 )

(2.15)

d = M1 sgn(z1 )
The idea is that the low frequency component of M0 sgn(η1 ) must equal x2 and the the
low frequency component of M1 sgn(z1 ) is an estimate of the disturbance d. Therefore
the low-pass filters (2.11) are designed to appropriately reject the high frequency components and output the estimate of x2 , z1 , and the estimate of d, zf . The value of τ1
is chosen such that the the cut off frequency of the filter is outside of the bandwidth of
x2 . On the other hand the choice of Ï„f must be made close to zero [12]. As can be seen
the design of the SMO is not as intuitive as the EHGO. The SMO also involves design
parameters dependent on some knowledge of the characteristics of the unknown state
and disturbance. The SMO technique does, however, provide good estimation without
the risk of peaking. Experimental results and comparisons between the SMO and the
EHGO will be developed in Chapter 4.

2.2

Mechanical Systems

In this section a description of the mechanical systems used for experimentation will
be presented. Both systems described are developed by Quanser [8] for educational
use in undergraduate laboratories. They are mounted on National Instruments’ ELVIS
I station, shown in Fig.2.1. Labview 7.1 is used as an interface with both of these
platforms.

9

Figure 2.1: National Instruments’ ELVIS I Station. For interpretation of the references
to color in this and all other figures, the reader is referred to the electronic version of
this thesis.

2.2.1

Rotary Pendulum

The Rotary Pendulum (ROTPEN) System, shown in Fig.2.2, is a platform developed
by Quanser [8]. It is used in undergraduate control labs because of the different control
problems it can demonstrate. This system can be used to demonstrate:
• Balance control: balancing the pendulum in the inverted position
• Swing-up contol: bringing the pendulum from the downward position to the inverted position
• Ganty control: minimizing oscillations in the pendulum while the arm tracks a
given reference.
For the purpose of our work the ROTPEN was used to demonstrate gantry control. The
free body diagram of the ROTPEN is shown in Fig.2.3

10

Figure 2.2: ROTPEN trainer developed by Quanser.

Figure 2.3: Free body diagram of the ROTPEN.

Quanser [8] modeled this system by
¨
θ
α
¨

=

1
det(D)

Ë™
d22 f1 − d12 f2 + d22 Km (Vm − Kb θ)/Rm
Ë™
−d12 f1 + d11 f2 − d12 Km (Vm − Kb θ)/Rm

11

Symbol

Numerical Value

Unit

Mass of the pendulum assembly

0.027

Kg

Total length of pendulum

0.191

m

lp

Length of pendulum center of mass
from pivot

0.1524

m

r

Length of arm pivot to pendulum
pivot

0.826

m

g

Gravitational acceleration constant

9.81

Jeq

Equivalent moment of inertia about
motor shaft

1.23 × 10−4

m/s2
Kg.m2

Jc

Pendulum moment of inertia about
its center of mass axis

7.0873 × 10−5

Kg.m2

Jp

Pendulum moment of inertia about
its center of pivot axis

6.9757 × 10−4

Kg.m2

3.3

Ω

Mp
Lp

Description

Rm
Km

Motor armature resistance
Motor torque constant

0.02797

N.m

Kb

Motor back e.m.f. constant

0.02797

V/(rad/s)

Table 2.1: List of symbols and values for ROTPEN
where
d11 = Je q + Mp r2 cos2 θ
d12 = Mp lp r cos θ cos α
d22 = Jp
det(D) = Jeq Jp + Mp r2 Jc cos2 θ sin2 α > 0
Ë™
f1 = Mp r2 θ2 sin θ cos θ + Mp lp rα2 sin α cos θ
Ë™
Ë™
f2 = Mp lp rθ2 sin θ cos α − Mp glp sin α
θ is the arm angle and α is the pendulum angle. Setting all the derivatives to zero
˙ ¨ ˙
θ=θ=α=α=0
¨

12

yields
22
Mp lp rg cos θ cos α = 0
−(Jeq + Mp r2 cos2 θ)Mp glp sin α = 0
The solution is therefore sin α = 0 and the equilibrium points are found at α = αr
and θ = θr , where αr and θr are any given reference. For a gantry control problem we
choose α = 0 and θ = θr . The linearized system is thus found as
¨
θ
α
¨

=

1
∆

22
Ë™
Mp lp rgα + Jp Km (Vm − Kb θ)/Rm
Ë™
−(Jeq + Mp r2 )Mp glp α − Mp lp rKm (Vm − Kb θ)/Rm

where
∆ = Je qJp + Mp r2 Jc .
Using the change of variables
Ë™
x1 = θ − θr , x2 = θ, x3 = α, x4 = α, u = V m
Ë™
the state model is given by
x = Ax + Bu
Ë™
where









0
0 0
1 0


 
 0 a

 b 

22 a23 0 , B =  2 
A=

 
 0
 0 0
0 1
 


0 a42 a43 0
b4

13

(2.16)

2
a22 = Mp lp rg/∆
a23 = −Jp Km Kb /(∆Rm )
a42 = −(Je q + Mp r2 )Mp glp /∆
a43 = Mp lp rKm Kb /(∆Rm )
b2 = Jp Km /(∆Rm )
b4 = −Mp lp rKm /(∆Rm )

As can be seen from (2.16) the ROTPEN is an underactuated system. There is one
control input but two degrees of freedom. Also, notice that the open-loop system is not
stable with a pole at the origin. To achieve the desired response a variety of control
techniques could be used such as pole placement or Linear Quadratic Regulator (LQR).
The exact controllers used in these experiments will be described in Chapter 3.

2.2.2

DC Motor

The DC Motor Control Trainer (DCMCT) shown in Fig.2.4, is another platform developed by Quanser [9]. It demonstrates motor control problems such as speed and position
control using the National Instruments’ ELVIS I station and LabView7.1. Neglecting
the armature inductance and given the relevant parameters shown in Table (2.2.2)(2.2.2), the DCMCT is modeled by [9]
Vm (t) − Rm (t)Im (t) − Eemf (t) = 0

(2.17)

Eemf (t) = kb wm (t) = kwm (t)

(2.18)

where

14

Figure 2.4: DCMCT trainer developed by Quanser.

and
J

dwm
= Tm (t) = kIm (t).
dt

(2.19)

Substituting (2.18) and (2.19) into (2.17) we reach
Vm (t) − Rm (t)

J dwm
− kwm (t) = 0
k dt

(2.20)

Using the change of variables
Ë™
x1 = θm − θr , x2 = θm = wm (t), u = V m
we get
x1 = x2
Ë™
x2 =
Ë™

k2
k
dwm
=−
x2 +
u
dt
Rm J
RmJ

Thus the state model representation of the system is




0
1
0
,B = 

A=
k
k2
0 −R J
Rm J
m
15

(2.21)

(2.22)

Symbol

Description

Unit

θm

Shaft angular position

rad

Vm

Armature voltage

V

Im

Armature current

A

Eemf

Back-electromotive-force(EMF)

V

wm

Shaft angular speed

rad/s

Tm

Torque produced

Nm

Table 2.2: List of symbols for the DCMCT

Symbol
kb
k
J
Rm

Description

Numerical Value

Unit

Back EMF Constant

.0326

V s/rad

Torque constant (k = kb )
Equivalent moment of inertia
of the motor shaft and load

.0326
1.93 × 10−5

N m/A
kg/m2

3.3

Ω

Armature resistance

Table 2.3: List of symbols and values for the DCMCT

16

For the purposes of this thesis we demonstrate position control. To achieve the desired
response many different linear control techniques could be used, such as pole placement
or LQR. The goal is to minimize the tracking error between the position of the shaft and
the reference given to it. The DCMCT is a fully actuated mechanical system with one
control input and one degree of freedom. The exact controllers used for the DCMCT
are presented in Chapter 4.

17

CHAPTER 3
Disturbance Cancellation in the Rotary
Pendulum Using Extended High-Gain
Observer
In this chapter we demonstrate the use of the Extended High-Gain Observer (EHGO)
with the Rotary Pendulum (ROTPEN). Specifically we present a case when disturbance
cancellation could be useful in an underactuated mechanical system. First the control
problem is presented with two different solutions: a controller that utilizes disturbance
cancellation and one that does not. Next, we present the experiments performed. This
chapter concludes with a comparison of the responses while highlighting the benefits of
disturbance cancellation.

3.1

Controller Design

In this section we present the control problem at hand and two different controllers to
achieve the desired response. Recall the linearized state model of the ROTPEN is given
18

by


0

0

1

0





0




 

 b 
 0 a
a23 0

 

22
A=
 , B =  2
 0
 0 0
0 1

 

b4
0 a42 a43 0

(3.1)

2
a22 = Mp lp rg/∆
a23 = −Jp Km Kb /(∆Rm )
a42 = −(Je q + Mp r2 )Mp glp /∆
a43 = Mp lp rKm Kb /(∆Rm )
b2 = Jp Km /(∆Rm )
b4 = −Mp lp rKm /(∆Rm )

The task here is to achieve Gantry control; a controller where the arm angle, θ, can track
a given reference, while minimizing movements in the pendulum angle, α. Specifically
the desired specifications are as follows:
• θ(t) tracks a given reference θr (t) with tp < 1.2s and ts < 2.3s
• |α(t)| < 7.5◦ and ts < 6s
• u = Vm < 5V
There are many control laws that can achieve this goal. Quanser suggests a linear
state feedback controller. With only the positions of the arm and pendulum available,
their speeds are found by passing the measured signals through the first order transfer
function
50s
.
s + 50
19

(3.2)

The pole of (3.2) is chosen far enough to the left such that it does not affect the system
response. The control
Ë™
u = −K1 (θ − θr ) − K2 θ − K3 α − K4 α
Ë™

(3.3)

where
K = [K1 K2 K3 K4 ]
is found using LQR. Using only a linear controller is satisfactory, but is deficient in
compensation of any unknown disturbances in the links. We propose using a fifth order
EHGO to overcome this deficiency.
Here, we present an adjustment to the method mentioned in section 2.1.1 of Chapter
2. The need for an adjustment arises from the fact that the system is underactuated. For
fully actuated systems method [6] suggests estimating the disturbance and then using
the estimate in the control to cancel the existing disturbance. However, in underactuated
mechanical systems such as the ROTPEN, disturbance may exist in either or both the
links. We could use the EHGO to estimate the disturbance and cancel it from one of the
two links. This raises a problem. Although cancellation of disturbance occurs in one of
the links, it is being added to the other because the same control appears in both links.
Let us reconsider the ROTPEN,
x1 = x2
Ë™
x2 = a22 x2 + a23 x3 + b2 u + d2
Ë™
x3 = x4
Ë™
x4 = a42 x2 + a43 x3 + b4 u + d4
Ë™
where d2 and d4 are unknown disturbances in the arm and pendulum angle equations
respectively. If the disturbance d4 and the coefficients a42 , a43 and b4 had been known
20

exactly, the control
u=−

1
(a x + a43 x3 + d4 ) + v
b4 42 2

(3.4)

could have been chosen to reduce the system to
x1 = x2
Ë™
x2 =
Ë™

a22 −

b2 a42
b4

x2 + a23 −

b2 a43
b4

x3 + b2 v + d2 −

b2
d4
b4

(3.5)

x3 = x4
Ë™
x4 = b4 v
Ë™

where v = φ(x) is a linear state feedback controller designed for the system (3.5). This
control could be beneficial if the new disturbance in the arm
b
d2 − 2 d4 < d2 .
b4

(3.6)

However equation (3.4) is unrealizable because the system parameters and the disturbances are unknown exactly. Next we design a fifth order observer, to estimate the
disturbance d4 . To do so we replicate (3.1) and augment it with the disturbance estimator term. However, since a22 , a33 , a42 and a43 are not known exactly, they can be
lumped in as disturbances in the arm and pendulum equations as well. The observer
will then be
α
Ë™
ˆ
x1 = x2 + 1 (x1 − x1 )
ˆ
ε
α
Ë™
x2 = u + 2 (x1 − x1 )
ˆ
ˆ
ε2
α
Ë™
x3 = x4 + 1 (x3 − x3 )
ˆ
ˆ
ε
α
Ë™
x4 = ˆ4 u + σ + 2 (x3 − x3 )
ˆ
b
ˆ
ˆ
ε2

(3.7)

(3.8)

α
Ë™
σ = 3 (x3 − x3 )
ˆ
ˆ
ε3
Here (3.7) is a second order high-gain observer (HGO) used to estimate the arm angle
and its derivative. While (3.8) is a third order EHGO, designed using the method in [6],
21

to estimate the pendulum angle, its derivative and the disturbance. The augmented state
σ is an estimate of the disturbance d4 in the pendulum angle x3 , and any error in the
ˆ
nominal right hand side. The control can now be chosen as
u=−

1
(−ˆ − φ(ˆ)).
σ
x
ˆ
b4

(3.9)

where φ(ˆ) is a linear state feedback controller designed for (3.5). We have now shown
x
two different control schema’s that could be implemented to achieve gantry control. In
the next section we present the results of using (3.9) and (3.11) and compare them.

3.2

Results

Here, we discuss the results of using each controller. The Q and R weighing matrices
are chosen as





11 0 0 0


 0 0.8 0 0 


Q=
,R = 1
 0 0 0 0


0 0 0 12

(3.10)

The choice of (3.10) produces controller gains


2.24


 1.03 


K=

 13.56


0.75
and guarantees that the desired specs are met. The control is thus
Ë™
u = −2.24(θ − θr ) − 1.03θ − 13.56α − 0.75α
Ë™

22

(3.11)

This choice places the eigenvalues of the system at
−48.02
−3.58

(3.12)

−1.49 ± j4.52
The results of this controller will be presented after the other control technique is presented.
Before the disturbance cancelation technique is applied, condition (3.6) must be satisfied. If this condition is not satisfied, the added disturbance would increase the arm’s
inherit disturbance and cancellation must not be used. To estimate the disturbance in
each link we use a third order observer designed using the method in [6],
α
Ë™
ˆ
x1 = x2 + 1 (x1 − x1 )
ˆ
ε
α
Ë™
x2 = b2 u + 2 (x1 − x1 )
ˆ
ˆ
ε2
α
Ë™
σa = 3 (x1 − x1 )
ˆ
ˆ
ε3

(3.13)

α
Ë™
x3 = x4 + 1 (x3 − x3 )
ˆ
ˆ
ε
α
Ë™
ˆ
x4 = b4 u + 2 (x3 − x3 )
ˆ
ε2
α
Ë™
σp = 3 (x3 − x3 )
ˆ
ˆ
ε3

(3.14)

We design α1 = 6, α2 = 11, and α3 = 6. This choice assigns the poles of each EHGO
at −3/ε, −2/ε, −1/ε. The parameter ε is taken as ε = .02. This is a good choice
of ε as it is small enough to closely estimate the the position and speed but also large
enough so that there is little to no effect from measurement noise [13]. The state space
representation of each observer is thus




−300
0 0
300
0




ˆ
ˆ
Aθ =  −27500 0 1 Bθ =  27500 38.58




−750000 0 0
750000
0
23

(3.15)

Figure 3.1: Estimate of the disturbance in the arm, d2





300
0
−300
0 0




ˆα =  −27500 0 1 Bα =  27500 9.9
ˆ
A




−750000 0 0
750000 0

(3.16)

We use (3.15) and (3.16) to estimate the disturbance in the arm and the pendulum
respectively. The estimate of d2 and any error in the right-hand side is shown in Fig.3.1.
The estimate of the disturbance in the pendulum, d4 and any error in the right-hand
side is shown in Fig. 3.2. By inspection we can see that (3.6) will be satisfied since
d2 < 0 and d4 < 0. However, to show the reduction in the inherit disturbance, the new
disturbance in the arm is shown in Fig.3.3. Now that it is verified that condition (3.6)
is verified we proceed to implement the fifth-order observer proposed in (3.7) and (3.8).
We use the same values for the parameters as (3.15) and (3.16). This choice assigns the
√
poles of (3.8) at −3/ε, −2/ε, −1/ε and the poles of (3.7) at −3± 2 . The state model
ε

24

Figure 3.2: Estimate of the disturbance in the pendulum, d4

Figure 3.3: Estimate of the new disturbance found in the arm angle

25

of the observer is thus

−300
1

 −27500 0

ˆ 
A =  −300
0

 −27500 0

−750000 0





1 0
300
0
0 0 0




 0 1
 27500 38.58
0 0 0




ˆ 
 ˆ 
0 C =  0 0
0 1 0 , B =  300




 0
 27500
9.9 
0 0 1




0 0
750000
0
0 0 0

0 0 0




0 0 0


1 0 0

0 1 0

0 0 1
(3.17)

Now that we have estimates of the speed of the arm and the pendulum we can design
an output feedback controller for the system in (3.5). We design the poles of the target
system to be close to the dominant poles of the closed loop system designed by Quanser
in (3.12) Using pole placement and Matlab we design φ(ˆ) from (3.9) to be
x
φ(ˆ) = 3.3166x1 + 22.4640ˆ2 + 1.1224x3 + .003ˆ4 .
x
x
x

(3.18)

From (3.9) the final control to be implemented is thus
u=−

1
(−ˆ − φ(ˆ))
σ
x
9.9

(3.19)

which places the eigenvalues of the closed loop system at
−34.0764
−6.5251

(3.20)

−1.3603 ± j3.9599
Recall, that it is important to saturate the control to protect the system from peaking.
However, in practice the output states of the observer can be saturated before entering
the control. This guarantees that the peaking phenomenon does not occur and allows
for a less conservative bound on the control. In this application in particular a bound on
the speed of the arm and pendulum was found from observing the output of (3.2). The
saturation on σ was done through tuning. We now compare the results of using each
ˆ
controller.
26

Comaprison
The reference to the arm angle is a square signal of amplitude 60â—¦ and frequency of
0.1Hz. To reduce any sharp changes in the reference, it is smoothed using the first
order low pass filter
10
.
s + 10
The responses using only state feedback without any disturbance reduction are shown in
Fig. 3.4 and Fig. 3.5. Fig. 3.4 shows the response of the arm angle θ given the reference
and Fig. 3.5 displays the response of the pendulum angle α. When using the EHGO
for disturbance estimation and cancellation to make sure the choice of ε is appropriate
we compare the estimate of the speeds to those found by Quanser. Fig. 3.6 compares
the measured position x1 to the estimated position x1 . Fig. 3.7 displays a comparison
ˆ
between the speed of the arm found using the first order filter in equation (3.2), and the
estimate of the speed found using the EHGO. The comparison between the estimated
speeds of the pendulum is shown in Fig.3.9. It can be seen from Fig. 3.6-3.9 that the
observer is working well.

Figure 3.4: θ with no disturbance estimation used

27

Figure 3.5: α with no disturbance estimation used

Figure 3.6: Comparison between ’theta’, the measurement of the arm angle, and
’theta hat’, its estimate using (3.17)
Now that we know the observer’s estimates are comparable to those found by
Quanser, let us look at the response when disturbance cancellation is utilized. Looking
at Fig.3.5 and Fig.3.11 we see that the maximum motion of alpha was reduced by a half

28

Figure 3.7: Comparison between ’theta dot’, the estimated speed of the arm using (3.2),
and ’theta dot hat’, the estimate found using (3.17)

Figure 3.8: Comparison between ’alpha’, the measurement of the arm angle, and ’alpha hat’, its estimate using (3.17)

29

Figure 3.9: Comparison between ’alpha dot’, the estimated speed of the arm using
(3.2), and ’alpha dot hat’, the estimate found using (3.17)
when we canceled the disturbance in the pendulum and thus achieving better gantry
control. Fig. 3.10 shows the response of the arm angle when the disturbance is canceled. By comparing Fig. 3.4 to Fig. 3.10, we can see that x1 has a smoother response.
The rise time and settling time are also reduced.

30

Figure 3.10: x1 with disturbance cancelation.

Figure 3.11: x3 with disturbance cancelation.
Conclusion
In this chapter we discussed the use of the EHGO for disturbance estimation and cancelation in an underactuated mechanical system. We demonstrated the benefits of using
disturbance cancellation in a gantry control problem by applying it on the ROTPEN.

31

Our results showed improvement in system performance and a substantial reduction in
the disturbance present in each link. Our experiment raises other questions of interest. For instance, we ran the same experiment but chose to estimate the disturbance in
the arm angle instead of the pendulum angle measurement of the ROTPEN. The outcomes showed the same type of results we demonstrated in above. The disturbance in
the pendulum angle was reduced as a by product of cancelation of the disturbance in
the arm angle. The responses also showed the same enhanced transient performance
recovery. This chapter offers a check for control engineers dealing with underactuated
mechanical systems in practice. If the condition that the new disturbance is less than
the inherit disturbance does not check, then a different control technique must be used.
However, if this condition checks, then disturbance cancellation could be done using
simpler techniques, such as the EHGO.

32

CHAPTER 4
Comparison of Extended High-Gain
Observer and Sliding-Mode Observer
using the DCMCT
In this chapter the goal is to compare the performance of two disturbance estimators,
the Extended High-Gain Observer (EHGO) and the Sliding-Mode Observer (SMO).
The application of disturbance estimators on mechanical systems is important in understanding the limitations and dynamics of each observer design. Measurement noise and
system specifications limit the realization of some theoretical requirements. However,
observers can be tuned to achieve the best response possible for the specific conditions.
We apply the EHGO and the SMO to the DC Motor Control Trainer (DCMCT) and tune
their parameters to achieve the best response observed. The design of each observer will
be demonstrated, then the results will be presented for each experiment. This chapter
concludes with a comparison of the two observers.

33

4.1

Observer Design

In this section we design EHGO and SMO for the DCMCT. Recall from Chapter 2 that
the state model form of the DCMCT is given by

x1 = x2
Ë™
x2 = −
Ë™

k2
k
x2 +
(u + d)
Rm J
RmJ

where d is the matched inherent disturbance due to uncertainty and unmodeled dynamics. The task here is to achieve position control. We would like x1 to follow a
specific reference with the fastest possible rise-time and lowest steady-state error. First
we present the Proportional Integral Derivative (PID) controller designed by Quanser
to achieve the desired response. Since x2 is not available Quanser estimates the speed
using the first-order transfer function
250s
.
s + 250

(4.1)

The pole of (4.1) is chosen far enough to the left such that it does not affect the system
response. For a given reference xr the PID control is
u = −ki

(x1 − xr ) − kp (x1 − xr ) − kd x2

(4.2)

where the gains ki , kp , and kd are found by tuning. The inherent disturbance is not very
large in the DCMCT and its effect on the transient response may not be so detrimental
to the system’s performance. We introduce a matched external disturbance de = 5V to
perturb the system’s performance. The system equations thus become

x1 = x2
Ë™
x2 = −
Ë™

k2
k
x2 +
(u + d + de )
Rm J
RmJ
34

(4.3)

As will be seen in Section 4.2, using only PID control is satisfactory but the response
could be enhanced when disturbance estimation and cancellation is utilized. Next, we
describe the design of the EHGO to use with the DCMCT.
Extended High-Gain Observer Design
To design an EHGO to estimate the speed, x2 , and any inherent disturbance in the
system, we replicate the state equations adding the error terms and augmenting the
system with the disturbance term as follows:
α
Ë™
ˆ
x1 = x2 + 1 (x1 − x1 )
ˆ
ˆ
ε
α
k
k2
Ë™
x2 +
ˆ
(u + σ ) + 2 (x1 − x1 )
ˆ
ˆ
x2 = −
ˆ
Rm J
RmJ
ε2
α3
Ë™
σ =
ˆ
(x1 − x1 )
ˆ
k
ε3 RmJ

(4.4)

thus, the feedback linearization control is
u = −ˆ −
σ

1
k2
(
x − υ)
ˆ
k
Rm J 2
Rm J

(4.5)

and the closed loop system is reduced to
Ë™
x1 = x2
ˆ
ˆ

(4.6)

Ë™
x2 = Ï…
ˆ
where Ï… = K x is a linear output feedback controller designed for (4.6). Next, we
ˆ
design the SMO for the DCMCT.
Sliding-Mode Observer Design
The design of the SMO is similar to the EHGO in that it is a replication of the plant
equations with the added error terms. The sliding-mode observer design and its filters

35

are
Ë™
x1 = x2 + M0 sgn(y − x1 )
ˆ
ˆ
ˆ
k
k2
Ë™
x2 +
ˆ
[u + M1 sgn(z1 )]
x2 =
ˆ
Rm J
RmJ
τ1 z1 = −z1 + M0 sgn(y − x1 )
Ë™
ˆ

(4.7)

k
τf zf = −zf +
Ë™
M sgn(z1 )
RmJ 1

(4.8)

Similar to the technique in the EHGO the control is chosen as
u = −zf −

1
k2
(
z − υ)
k
Rm J 1
Rm J

(4.9)

where Ï… = K x is a linear output feedback controller designed for (4.6). To ensure that
ˆ
the estimation error will go to zero we choose
M0 > |x2 |

(4.10)

M1 > |d|
The filter time constants, τ1 and τf must be chosen appropriately to capture the desired
estimates. The theory behind the choice of the constants M0 and M1 is not very well
defined. The best choice was made through tuning the parameters online during the
experiments. In section 4.2 we show that using constant gains for M0 and M1 does not
produce the desired results. We suggest using the estimates from the EHGO to drive
the terms M0 and M1 . To do so , we implement a third-order EHGO, the same one
described in (4.4), to estimate the speed and the disturbance. The choice of M0 and M1
becomes
M0 = |ˆ2 | + c1
x

(4.11)

M1 = |ˆ | + c2
σ
where x2 is the estimate of x2 and σ is the estimate of the disturbance found using
ˆ
ˆ
the EHGO. The constants c1 and c2 are chosen to ensure (4.10) is satisfied. Equations
36

(4.7) and (4.8) represent first-order low-pass filters, whose outputs are the estimates of
x2 and d respectively. As will be seen in Section 4.2, using first-order low-pass filters
is inadequate when we introduce an external disturbance. For a faster roll-off slope we
choose to implement second-order low-pass filters to estimate x2 and d. The secondorder low-pass filters have the equations
z1 = z2
Ë™
Ë™
2
2Ë™
Ë™
ˆ
τ1 z2 = −z1 − √ τ1 z2 + M0 sgn(y − x1 )
2

(4.12)

zf = zf 2
Ë™
Ë™
2
k
2Ë™
τf zf 2 = −zf − √ τf zf 2 +
M sgn(z1 )
Ë™
RmJ 1
2
The exact choice for these parameters will be discussed in the next section.

4.2

Implementation and Results

In this section we show the exact controllers for each experiment. We discuss the reasoning behind the chosen parameters in each case. Then we present the results of each
experiment. The state-space representation of DCMCT is

A=

0

0

−16.684 1

B=

0
511.8543

(4.13)

With no disturbance cancellation, the gains of the PID controller presented in (4.2) were
found through tuning to be

ki = 2, kd = .025, kp = 2
placing the eigenvalues of the closed-loop system at
−1.0295, −14.2275 ± j28.1421.

37

The response of using only PID control is shown in Fig.4.1. For a better view of the
transient response a close-up is shown in Fig.4.2. The control is shown in Fig.4.2. The
response using only PID is satisfactory has a slow rise-time, no overshoot and steadystate error. We now look at the reponse using only PID when an external matched
disturbance de = 5V is introduced.
The response of (4.3) using only PID control is shown in Fig.4.4 and a close-up of the

Figure 4.1: Response of x1 with only PID control.

transient response is shown in Fig. 4.5 respectively. The control is shown in Fig. 4.6.
It can be seen that the control effort is increased while the response suffers from a slow
rise time and a large steady-state error. Next, we will use the EHGO and the SMO to
estimate the inherent disturbance and cancel it, followed by estimation and cancellation
of the external disturbance, de .

38

Figure 4.2: Close up of transient response of x1 with only PID control.

Figure 4.3: Control with only PID control.

39

Figure 4.4: Response of x1 with de = 5V using only PID control.

Figure 4.5: Close-up of response of x1 with de = 5V using only PID control.

Extended High-Gain Observer
We now implement EHGO to the DCMCT to estimate the inherent disturbance and
cancel it in the control. We design α1 = 6, α2 = 11, and α3 = 6. This choice assigns
the poles of the EHGO at −3/ε, −2/ε, −1/ε. Here, the parameter ε is taken as ε = .01.

40

Figure 4.6: Control with de = 5V using only PID control.

This is a good choice of ε as it is small enough to closely estimate the speed but also
large enough so that there is little to no effect from measurement noise [13]. The third
order EHGO is




−600
0
0
600
0




ˆ
ˆ
A =  −11000 −16.684 511.8543 B =  110000 511.8543




−11722.08
0
0
11722.085
0

(4.14)

The estimates of x1 and x2 are shown in Fig.4.7 and Fig.4.8 respectively. It can be seen
from these figures that the observer produces good estimates of the states.
Now that it can be seen that the estimates from the observer are good we design
the control described in (4.15) for the DCMCT. Substituting for the values of each
parameter we get
u = −ˆ −
σ

1
(16.684ˆ2 − υ)
x
511.8543

(4.15)

where
υ = 994.3995x1 + 28.544ˆ2
x

41

(4.16)

Figure 4.7: Comparison between x1 and its estimate, x1 using (4.14)
ˆ

was designed using pole placement to assign the eigenvalues of the closed loop system
at
−14.2275 ± j28.1421.
The response of x1 when using an EHGO to estimate and cancel the disturbance is
shown in Fig. 4.9. For a clearer view, a close-up of the transient response is shown in
Fig.4.10. The control and the estimate of the disturbance, σ , are shown in Fig.4.11 and
ˆ
Fig.4.12 respectively. We can see from Fig.4.1 and Fig.4.9 that disturbance cancellation
using the EHGO is beneficial to the response of the DCMCT. We see a faster rise time
and a smaller steady-state error than when only using PID control with no disturbance
rejection.
We are now interested in the performance of the EHGO with a larger disturbance. We
introduce the external matched disturbance, de = 5V , to the system and investigate the

42

Figure 4.8: Comparison between x2 using 4.1 and x2 using (4.14)
ˆ

Figure 4.9: The response of x1 using EHGO.

43

Figure 4.10: A close-up of the transient response of x1 using EHGO.

Figure 4.11: Control using EHGO.

44

Figure 4.12: Estimate of the inherent disturbance using EHGO.

results. Recall, with the introduction of de the system equations have now become
x1 = x2
Ë™
x2 = −
Ë™

k2
k
x2 +
(u + d + de )
Rm J
RmJ

We will use the same EHGO described in (4.14) and the control in (4.15). The only
difference is in the estimated disturbance. The disturbance estimator σ is an estimate of
ˆ
d + de . The response using an EHGO with an external matched disturbance is shown in
Fig.4.13 and a close-up is shown in Fig.4.14. It can be seen that the transient response
is better with disturbance cancellation used. We see a lower steady-state error and a
faster rise time. The estimate of the disturbance is shown in Fig.4.15, it can be seen that
its value is close to the sum on the inherent disturbance,d, estimated in Fig.4.12 and the
external matched disturbance de = 5V . Next, we design the SMO for the DCMCT and
present the results of its use.
Sliding-Mode Observer
We now implement the SMO to the DCMCT and observe the response. Recall from
45

Figure 4.13: Response of x1 with de = 5V using EHGO.

Section 4.1 the SMO observer equations are given as
Ë™
x1 = x2 + M0 sgn(y − x1 )
ˆ
ˆ
ˆ
k2
k
x2 =
Ë™
x2 +
ˆ
[u + M1 sgn(z1 )]
Rm J
RmJ
τ1 z1 = −z1 + M0 sgn(y − x1 )
Ë™
ˆ
τf zf = −zf +
Ë™

k
M sgn(z1 )
RmJ 1

A good choice for M0 and M1 was hard to find using tuning. To get a better idea of
how to choose these gains we use the estimates of the speed and the disturbance found
using the EHGO to find a lower bound on M0 and M1 . Using Fig. 4.8 we choose
M0 = 60. From Fig. 4.12 we choose M1 = 5. The choice of the filter time constants
τ1 and τf was done through tuning. The best choice we could find for the first order
filters (4.7) and (4.8) were τ1 = .09 and τf = .008. The estimates of the observer with
Ï„1 = .09 and Ï„f = .008 and gains M0 = 60 and M1 = 5 are shown in Fig. 4.16 and
46

Figure 4.14: Close-up of transient response of x1 with de = 5V using EHGO.

Fig. 4.17. Though the SMO yields a good estimate of x1 , it can be seen from Fig.4.17
that z1 is not a good estimate of the x2 .The response when using constant gains is
shown in Fig.4.18. We see a large overshoot and steady-state error. A close of the
transient response and the steady state error when the reference goes low is shown in
Fig.4.19. The control is shown in Fig.4.20 and the estimated disturbance is shown in
Fig. 4.21.

Comparing Fig.4.21 and Fig.4.12 we see that the choice of a constant M1 affects the
estimate of the inherent disturbance d. The value of d is not constant and the choice
of a constant M1 cannot account for the variations in d. Choosing a constant gain is
not appropriate when the disturbance is time-varying. We investigate the use of timevarying gains as discussed in Section 4.1.

47

Figure 4.15: Estimate of overall disturbance with de = 5V using EHGO.

We now look at the performance of the SMO using time-varying gains for M0 and M1 .
The estimate of x2 when using the time-varying gains in (4.11) is shown in Fig.4.22.
It can be seen that a better estimate is found using (4.11) for M0 and M1 . The response and a close-up of the transient are shown in Fig.4.23 and Fig.4.24 respectively.
We can see that the response using the SMO with time-varying gains had a faster rise
time and less steady-state error than when only using a PID controller as Quanser suggests. The control is shown in Fig. 4.25 and the estimate of the inherent disturbance
is shown in Fig.4.26. From Fig. 4.25 and Fig.4.26 we can see that the estimate of the
disturbance and the control account for the varying values of d.

We now investi-

gate the performance of the SMO when an external matched disturbance de = 5V is
introduced. Note that zf will be an estimate of the sum of the inherent disturbance
and de . As discussed in Section 4.1 the use of first-order low-pass filters is inadequate

48

Figure 4.16: Comparison between x1 its estimate, x1 using SMO with constant gains
ˆ

when de is introduced. The response using first-order filters is shown in Fig.4.27. The
effects of the slow response of a first-order low-pass filter could be seen in Fig.4.27.
The oscillations in the response are caused by the chattering nature of the SMO. To fix
this issue we use the second-order low-pass filters presented in (4.12) to estimate the
overall disturbance and reach a desirable response. Using the same filter time constants
τ1 = .09 and τf = .008 we implement the second-order filters. The response is shown
in Fig.4.28 and a close-up of the transient is shown in Fig.4.29. The estimate of the
overall disturbance is shown in Fig.4.30. It can be seen that the response has a faster
rise time with a little bit of overshoot, and we achieve better steady-state error than with
no disturbance cancellation, shown in Fig.4.4. As expected we’ve shown that using the
EHGO and the SMO for disturbance cancellation enhances the response of the system.
Our main interest, however, is how well each observer performers. In the next section

49

Figure 4.17: Comparison between x2 using 4.1 and z1 using SMO with constant gains

Figure 4.18: Response using SMO with constant gains.

50

Figure 4.19: Close-up of response using SMO with constant gains.

Figure 4.20: Control using SMO with constant gains.

we compare the performances of the SMO and the EHGO in each scenario tested.

4.3

Comparison

We start the discussion by comparing the response of the system in each scenario given
each observer. When estimating and canceling the inherent disturbance we achieve

51

Figure 4.21: Estimate of inherit disturbance using SMO with constant gains.

a better response using the SMO. Comparing Fig.4.10 and Fig.4.24 we see a better
steady-state error when using SMO at a cost of a slight overshoot. One of the reasons
a larger steady-state error is observed when using the EHGO, is the fact that the parameter ε could not be pushed smaller without seeing the effects of measurement noise.
Theoretically we expect the steady-state error to go to zero as ε goes to zero, which is
not realizable in application. The rise time and settling time of each response is very
similar.
In the presence of a larger matched disturbance the response when using the EHGO
was more desirable. Comparing Fig.4.14 and Fig.4.29 we see about the same amount
of overshoot but a faster settling time when using the EHGO than the SMO. thought the
SMO has a smaller steady-state error the settling time is too long.
In the presence of only the inherent disturbance and in the presence of an external

52

Figure 4.22: Comparison between x2 using 4.1 and z1 using SMO with time-varying
gains.

disturbance the estimate of the unknown state and the disturbance was better done using
the EHGO. Looking at Fig. 4.8 and Fig.4.22 we can see that the EHGO yields a very
good estimate of the unknown state x2 when we compare it to the estimate found by
Quanser. The low-pass filtering done to get the estimate of x2 does not account for any
fast changes in x2 . If the cut-off frequency for the low-pass filter is increased we allow
for more chattering in the response. For the SMO, that is the best estimate we could get
in our experimental conditions.
Finally and most importantly, it must be noted that the responses achieved using the
SMO were only as good as they were because we used the estimates found using EHGO.
Without a good understanding of the disturbance, for example knowledge of upper and
lower bounds or whether or the the disturbance is time-varying, a good estimate cannot

53

Figure 4.23: Response using SMO with time-varying gains.

be found using SMO. Even when the EHGO was utilized to drive the terms of the SMO
the response was still pretty comparable to that using only the EHGO. It is superfluous
to use the EHGO and the SMO together to produce results that are comparable to using
only the EHGO. However, we did achieve lower steady state error in Fig.4.24 from utilizing both observers. If having a low steady-state error is necessary the two observers
could be implemented together to produce the desired results.

54

Figure 4.24: Close-up of transient response using SMO with time-varying gains.

Figure 4.25: Control using SMO with time-varying gains.

55

Figure 4.26: Estimate of inherit disturbance using SMO with time-varying gains.

Figure 4.27: Response with de = 5V using SMO with time-varying gains.

56

Figure 4.28: Response of x1 with de = 5V using SMO with time-varying gains and
second-order low-pass filters.

Figure 4.29: Close-up of transient response of x1 with de = 5V using SMO with
time-varying gains and second-order low-pass filters.

57

Figure 4.30: Estimate of overall disturbance with de = 5V using SMO with timevarying gains and second-order low-pass filters.

58

CHAPTER 5
Conclusion and Future Work
In our work we investigated the use of the Extended High-Gain Observer (EHGO)
with the Rotary Pendulum (ROTPEN), an underactuated mechanical system. The issue with disturbance rejection in underactuated mechanical systems is that cancelling
disturbance from one link adds it to the other. We have shown a case where the added
disturbance to the other links actually reduces the inherent disturbance. We use EHGO
to estimate the disturbance in the pendulum of the ROTPEN and cancel it in the control.
The added disturbance from the pendulum actually reduced the inherent disturbance in
the arm. We were thus able to implement a simple controller to achieve better results
than when no disturbance rejection was utilized. We have presented a check for control
engineers working with underactuated mechanical systems. We can use the EHGO to
estimate the disturbance in each link and test the possibility of the added disturbance
being a reduction in the inherent disturbance. When this possibility checks, the use of
a simple controller could be implemented. This idea could be applied to other underactuated mechanical systems and simple control schemes could be applied rather than
complicated robust techniques.

59

We have also applied the Sliding-Mode Observer (SMO) and the EHGO to the DC
Motor Control Trainer (DCMCT) and tested their performance in different scenarios.
We have shown that the SMO is sensitive and needs a lot of tuning for good results.
The EHGO, however, was simple with only two tuning parameters. Its implementation
is simple and the responses it yielded were very comparable to those when using SMO.
Both of these observers show improved results in system performance but the ease of
design and implementation of the EHGO makes it a more desirable choice.
Finally, our work could be used as a pedagogical tool in undergraduate teaching laboratories. The idea of inherent disturbances, those from unmodeled dynamics and model
uncertainties is a very abstract idea. Moreover, the effect of these disturbances may be
undesirable to system performance and there presence must be addressed in a teaching
laboratory. We used Quanser’s [9] [8] educational platforms to estimate the disturbance
and cancel it. Part of our work could be developed into a laboratory project where
the students design disturbance observers, either SMO or EHGO, and implement them.
The students will then have to tune their designs to achieve the best response possible. This will provide the students with a tangible idea of disturbance as they will be
able to see its estimate. The ability to see the estimate will help the students better
understand the concept of inherent disturbance because it has been shown that a visual
concrete representation will help the students learn [2]. Since the students will have to
tune their parameters online to get the best response they will have to use their problem solving skills to better understand the tuning process. In a problem-based learning
environment such as this we expect students to have better retention of information [4].
Implementing this in an undergraduate teaching laboratory also provides opportunity to
fulfill multiple ABET accreditation outcomes [5] such as :

60

• outcome a: ”an ability to apply knowledge of mathematics, science, and engineering”
• outcome b: ”an ability to design and conduct experiments, as well as to analyze
and interpret data”
• outcome k: ”an ability to use the techniques, skills, and modern engineering tools
necessary for engineering practice”
Outcome a is satisfied through the design of the observer. Outcome b is realized in the
implementation of the experiment and the tuning process of the observers. The use of
LabView and the trainers developed by Quanser satisfies outcome k.
Our work calls for more exploration of the use of disturbance estimators with underactuated mechanical system. It also invokes the comparison of other disturbance
estimators so that conclusions on different observers and their use in different circumstances could be drawn.

61

BIBLIOGRAPHY

62

BIBLIOGRAPHY

[1] J.H. Ahrens and H.K. Khalil. High-gain observers in the presence of measurement
noise: A switched-gain approach. Automatica, 45(4):936–943, 2009.
[2] J. Bransford. How people learn: Brain, mind, experience, and school. National
Academies Press, 2000.
[3] J. Davila, L. Fridman, and A. Levant. Second-order sliding-mode observer for
mechanical systems. Automatic Control, IEEE Transactions on, 50(11):1785–
1789, 2005.
[4] C. Dym, A. Agogino, O. Eris, D. Frey, and L. Leifer. Engineering design thinking,
teaching, and learning. 2005.
[5] R.M. Felder and R. Brent. Designing and teaching courses to satisfy the abet engineering criteria. JOURNAL OF ENGINEERING EDUCATION-WASHINGTON-,
92(1):7–26, 2003.
[6] L.B. Freidovich and H.K. Khalil. Performance recovery of feedback-linearizationbased designs. Automatic Control, IEEE Transactions on, 53(10):2324–2334,
2008.
[7] B.K. Kim and W.K. Chung. Advanced disturbance observer design for mechanical
positioning systems. Industrial Electronics, IEEE Transactions on, 50(6):1207–
1216, 2003.
[8] Quanser. Qnet gantry laboratory manual. 2004.
[9] Quanser. Qnet position control laboratory manual. 2004.
[10] A. Radke and Z. Gao. A survey of state and disturbance observers for practitioners.
In American Control Conference, 2006, pages 6–pp. IEEE.
[11] K. Shibayama, V. Kroumov, and A. Inoue. Robust control of underactuated inverted pendulum system in presence of unknown disturbances. In Modelling,
Identification and Control (ICMIC), The 2010 International Conference on, pages
349–353. IEEE.
[12] V.I. Utkin, J. Guldner, and J. Shi. Sliding mode control in electromechanical
systems, volume 9. CRC, 1999.

63

[13] L.K. Vasiljevic and H.K. Khalil. Error bounds in differentiation of noisy signals
by high-gain observers. Systems & Control Letters, 57(10):856–862, 2008.
[14] K.D. Young, V.I. Utkin, and U. Ozguner. A control engineer’s guide to sliding
mode control. Control Systems Technology, IEEE Transactions on, 7(3):328–342,
1999.

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