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Research ArticleNonlinear/Linear SwitchedControl of
InvertedPendulumSystem:Stability Analysis and Real-Time
Implementation
Amira Tiga , Chekib Ghorbel , and Naceur Benhadj Braiek
Laboratory of Advanced Systems, Polytechnic High School of
Tunisia, University of Carthage, BP 743,2078 La Marsa, Tunis,
Tunisia
Correspondence should be addressed to Amira Tiga;
[email protected]
Received 8 April 2019; Accepted 15 July 2019; Published 7 August
2019
Academic Editor: Ines Tejado Balsera
Copyright © 2019 Amira Tiga et al. (is is an open access article
distributed under the Creative Commons AttributionLicense, which
permits unrestricted use, distribution, and reproduction in any
medium, provided the original work isproperly cited.
(is paper treats the problems of stability analysis and control
synthesis of the switched inverted pendulum system
withnonlinear/linear controllers. (e proposed control strategy
consists of switching between backstepping and linear state
feedbackcontrollers on swing-up and stabilization zones,
respectively. First, the backstepping controller is implemented to
guarantee therapid convergence of the pendulum to the desired rod
angle from the vertical position. Next, the state feedback is
employed tostabilize and maintain the system on the upright
position inherently unstable. Based on the quadratic Lyapunov
approach, theswitching between the two zones is analyzed in order
to determine a sufficient domain in which the stability of the
desiredequilibrium point is justified. A real-time experimentation
shows a reduction of 84% of the samples below the classical
schemewhen using only the backstepping control in the entire
operating region. Furthermore, the reduction percentage has become
92%in comparison with the composite linear/linear controller.
1. Introduction
For several years, stability analysis and control synthesis
ofswitched systems have received an increasing interest. (isclass
of systems consists of a finite number of subsystemsand a logical
rule that orchestrates switching between them.Mathematically, these
subsystems are usually described bydifferential equations. Examples
of such switched systemscan be found in chemical processes [1],
power converters [2],automotive industry, electrical circuits [3],
and many otherfields.
(e most control structure of switched systems used inliterature
is to design the same number of linear subsystemsand linear
subcontrollers.(erefore, necessary and sufficientstability
conditions are in general based on switched qua-dratic Lyapunov
functions [4]. However, the main disad-vantage of this structure
consists of the complexity indesigning several controllers and that
does not easily ac-count for all interactions between them. Lately,
some re-searchers suggested another switching structure by
applying
only one controller [5]. (us, a common quadratic Lya-punov
function is used for all subsystems which ensures theasymptotic
quadratic stability [5]. Despite the simplicity ofthis structure,
there are many control and performanceproblems that could profit
from it, for instance, the existenceof systems that cannot be
asymptotically stabilized by asingle control law.
Among the large variety of analysis and controlproblems of
switched systems is the guarantee of a suf-ficient stability domain
of the desired equilibrium point.In fact, the knowledge if a given
initial state lies withinsuch a stable region is a question of
practical importance inmany engineering applications as the
synthesis of thepreferment nonlinear controller [6]. (erefore, the
ma-jority of studies concerned with this object are setting inthe
Lyapunov method which is essentially applied tocomplex and
nonlinear systems. Nevertheless, stabilityanalysis of switched
systems used in general is the Popovcriterion [7], the circle
criterion [8], and the positive-reallemma.
HindawiMathematical Problems in EngineeringVolume 2019, Article
ID 2391587, 10 pageshttps://doi.org/10.1155/2019/2391587
mailto:[email protected]://orcid.org/0000-0003-4422-608Xhttps://orcid.org/0000-0003-1279-1500https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/2391587
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In this paper, we have focused on an alternative switchedcontrol
structure which includes both backstepping andlinear feedback
control laws. Indeed, this strategy ensuresthe rapidity of the
system behavior in the closed loop and theminimization of
subsystems and subcontroller numbers.(ereafter, the switch between
these models has been an-alyzed using the quadratic Lyapunov
approach and Syl-vester’s criterion.
One of the most important mechanical systems whichtreats
extensively the issues cited above is the invertedpendulum. (is
system, inherently unstable with highlynonlinear dynamics, belongs
to the class of underactuatedsystems having fewer control inputs
than degrees of free-dom.(is renders the control task more
challenging makingthis process a classical benchmark for testing
and comparingdifferent control techniques.
(e evolution space of the considered system can bedecomposed
into two operating zones: swing-up and sta-bilization zones in
which nonlinear and linear feedbackcontrollers are applied,
respectively.
(is paper is organized as follows: Section 2 depicts theproposed
alternative switched control strategy. Section 3outlines the
mathematical model and the design of non-linear/linear composite
controller of the inverted pendulumsystem. Section 4 deals with
determination of a sufficientdomain in which the stability of the
equilibrium point isjustified.
2. Problem Formulation
A switched system consists of a finite number of
dynamicalsubsystems and a switching unit that orchestrates
betweenthem. Indeed, the most control structure used, as depicted
inFigure 1, contains the same number of linear models Mi, fori � 1,
2, . . . , r, and linear subcontrollers Ci, for i � 1, 2, . . . ,
r.At each switching time instant, only one model and
itssubcontroller are active.
(erefore, some important results of necessary andsufficient
stability conditions have been obtained in [9, 10].However, the
major disadvantage of this structure is in thedesign of several
subcontrollers and that does not easilyaccount for all interactions
between them.
Some other researchers suggested another switchedcontrol
structure, shown in Figure 2, which consists of thedesign of a
linear centralized controller C for all linearmodels Mi, i � 1, 2,
. . . , r. (is technique is simple to im-plement since it requires
the design of only one controller.
On the contrary, there are many control and perfor-mance
problems that could profit from this approach; forinstance, various
objectives cannot be met by a singlecontroller.
In this paper, we have focused on an alternative
switchedstructure, as can be seen in Figure 3. (erefore, we
havedesigned linear and nonlinear controllers in swing-up
andstabilization zones, respectively. Indeed, this method
ensuresthe rapidity of the closed-loop system behavior and
theminimization of subsystems and subcontroller numbers.(ereafter,
the switch between the two zones has beenanalyzed in order to
determine a sufficient stability domain
using the quadratic Lyapunov approach and
Sylvester’scriterion.
(is alternative switch structure can be perfectly appliedto the
inverted pendulum system, the mechanical model ofwhich is
illustrated in Figure 4 [11, 12].
(is process consists of a rod and a moving cart in thehorizontal
direction. Its parameters are defined in Table 1.
(e educational kit of the considered process, shown inFigure 5,
consists of a track of 1m length with limit switchesbetween −0.4
and 0.4m, PC Pentium 4 with PCI-1711 card,cable adaptor, optical
encoders with HCTL2016 ICs, digitalpendulum controller, cart, DC
motor, pendulum withweight, Advantech PCI-1711 device driver, and
adjustablefeet with belt tension adjustment.
In the next section, the proposed control structure will
beapplied to the inverted pendulum system in order to show
itsefficiency.
–+
Subcontrollers System
Switchingunit
M1
M2
Mr
C1
. . .
. . .
C2
Cr
Reference Output
Figure 1: Switched system control structure containing
linearmulticontrollers.
System
Switchingunit
M1
M2
Mr
Controller
C–+Reference Output
Figure 2: Switched system control structure containing
linearcentralized controllers.
Linearcontroller
Nonlinearcontroller
Non
linea
rm
odel
System
OutputSwitchingunit
Controller
–+Reference
Figure 3: Proposed switched system control structure.
2 Mathematical Problems in Engineering
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3. Switched Control Design
Our work aims to design a nonlinear (NL) control law, in
theswing-up zone, to guarantee the rapid convergence of thependulum
from down position into the stabilization zone.(en, the switch to
the linear (L) controller permits tostabilize it at the desired
open-loop unstable equilibriumpoint.
3.1.MathematicalModeling. (emathematical model of theinverted
pendulum is depicted by the following nonlinearequations:
(m + M)€x + b _x + ml€θ cos(θ)−ml _θ2 sin(θ) � F,
ml €x cos(θ) + J + ml2( €θ−mgl sin(θ) + d _θ � 0,
⎧⎨
⎩ (1)
where x (m) is the position of the cart, _x (m/s) is the
velocity,θ (rad) is the rod angle from the vertical position, _θ
(rad/s) isthe angular velocity, and F (N) is the force.
For
X �
x1 � x
x2 � _x
x3 � θ
x4 �_θ
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (2)
Equation (1) can be rewritten as follows:_x1 � x2,
_x2 � f1(X) + g1(X)F,
_x3 � x4,
_x4 � f2(X) + g2(X)F,
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(3)
withf1(X) �
ρ1(X)σ(X)
,
f2(X) �ρ2(X)σ(X)
,
g1(X) �J + ml2
σ(X),
g2(X) �−ml cos x3(
σ(X),
ρ1(X) � J + ml2
mlx24 sin x3( − J + ml
2 bx2
−m2l2g cos x3( sin x3( + ml dx4 cos x3( ,
ρ2(X) � −m2l2x24 sin x3( cos x3( −d(m + M)x4
+(m + M)mgl sin x3( + mlbx2 cos x3( ,
σ(X) � (m + M) J + ml2 − ml cos x3( ( 2.
(4)
3.2. Composite NL/L Controller. (e proposed controlstrategy is
based on switching between backstepping andlinear state feedback on
swing-up and stabilization zones,respectively. (erefore, the
switching rule of the globalcontrol law F is distributed into Fswp
and Fstb in swing-upand stabilization zones, respectively, as
depicted in Figure 6.
3.2.1. Linear Feedback Control in the Stabilization Zone.In this
zone, system (3) can be linearized using the
small-angleapproximations which are as follows: sin(θ)≃ θ, cos(θ)≃
1,and _θ
2≃ 0. (en, it can be described by the following linear
state equation:_X � AX + BFstb, (5)
Table 1: Inverted pendulum parameters.
Symbol Parameter Valuesg Gravity 9.81m/s2l Pole length 0.36 to
0.4mM Mass of cart 2.4 kgm Mass of pendulum 0.23 kgJ Moment of
inertia of the pole 0.009 kg·m2b Cart friction coefficient
0.05Ns/md Pendulum damping coefficient 0.005Nms/rad
θd
x
b
F M
(J, m, l,)
Figure 4: Inverted pendulum system.
Figure 5: Inverted pendulum experiment.
Mathematical Problems in Engineering 3
-
with
A �
0 1 0 0
0 a2 a3 a4
0 0 0 1
0 a5 a6 a7
,
B �
0
b1
0
b2
,
a2 �− J +ml2( )b
(m +M) J +ml2( )−(ml)2,
a3 �−(ml)2g
(m +M) J +ml2( )−(ml)2,
a4 �mld
(m +M) J +ml2( )−(ml)2,
a5 �mlb
(m +M) J +ml2( )−(ml)2,
a6 �(m +M)mgl
(m +M) J +ml2( )−(ml)2,
a7 �−(m +M)d
(m +M) J +ml2( )−(ml)2,
b1 �J +ml2
(m +M) J +ml2( )−(ml)2,
b2 �−ml
(m +M) J +ml2( )−(ml)2.
(6)
Based on the pole placement technique, we have appliedthe linear
state feedback controller:
Fstb � −KstbX, (7)
whereKstb is the state feedback gain. From equations (5) and(7),
we have in closed loop:
_X � ABFX, (8)
where ABF � A−BKstb.
3.2.2. Backstepping Controller in Swing-Up Zone.Backstepping
control is known as a construction approachin the sense that it has
a systematic way of constructing theLyapunov function along with
the control input design. Toensure the negativeness of the
derivative of the every stepLyapunov function, it usually requires
the cancelation of theindenite cross-coupling terms [13].
is NL controller is used to move the cart x until θattains
θlimit. e design of Fswp in the swing-up zone isillustrated as
below:
Step 1. A new control variable ε1 is dened as
ε1 � x1 −xref , (9)
where xref is the desired setpoint. en, the derivative of]1 �
(1/2)ε21 is obtained as
_]1 � ε1 _ε1 � ε1 x2 − _xref( ). (10)
e function of stabilization x2d is presented as follows:
x2d � _xref − c1ε1, (11)
where c1 is a positive constant.
Step 2. A second error ε2 is given by the following
equation:
ε2 � x2 −x2d. (12)
e derivative of ]2 � ]1 + (1/2)ε22 is described asfollows:
_]2 � ε1 x2 − _xref( ) + ε2 f1(X) + g1(X)F− _x2d( ). (13)
ereafter, the backstepping control law Fswp is depictedas
Fswp � g1(X)( )−1 _x2d −f1(X)− ε1 − c2ε2( ), (14)
where c2 is a positive constant.
4. System Analysis in the Stabilization Zone:Estimation of a
Stability Domain
It is necessary to guarantee the switched NL/L controlledsystem
stability, when it is provided by the linear feedbackcontrol law
(7). erefore, we have to determine the su-cient stability
domain:
θ = π
Stabilization zone
θlimit
Swing-up zone
θ = 0θ
xmin = −0.4m xmax = 0.4mx
Figure 6: Swing-up and stabilization zones.
4 Mathematical Problems in Engineering
-
ξ(P, c) � X ∈R4
V(X) ≤ c, _V(X)< 0 , (15)
where c is a real-positive constant and V(X) is the
positivedefinite quadratic Lyapunov function expressed
asfollows:
V(X) � XTPX. (16)
(e symmetric positive definite matrix P is determinedby solving
the Lyapunov equation:
ATBFP + PABF � −Q, (17)
where Q � QT > 0. Asymptotic stability of the equilibriumX �
0R4 of the nonlinear model (3) provided of a feedbacklinear control
law (7) is guaranteed when the derivative ofV(X):
_V(X) � _XTPX + X
TP _X, (18)
is negative definite.Equation (18) can be rewritten as
_V(X) � XTξ(X)X, (19)
where
ξ(X) �
a11(·) a12(·) a13(·) a14(·)
0 a22(·) a23(·) a24(·)
0 0 a33(·) a34(·)
0 0 0 a44(·)
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
, (20)
and the parameters aij(·) are detailed in Appendix.As also, the
matrix ξ(X) can be transformed in the
following symmetric form:
ψ(X) �12
ξ(X) + ξT(X) . (21)
By applying Sylvester’s criterion, the closed-loopsystem is
asymptotically stable if ψ(X)< 0. Accordingly,the elementary
determinants Δi, for i � 1, 2, 3, 4, shouldsatisfy
Δ1 � a11(·) ε,
Δ3 �
2a11(·) a12(·) a13(·)
a12(·) 2a22(·) a23(·)
a13(·) a23(·) 2a33(·)
ε,
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(22)
where ε> 0.Subsequently, by solving the inequality system
(22), the
condition _V(X)< 0 leads to ||x3||≤ θmax which can be
re-written as
‖LX‖≤ θmax, (23)
where L � 0 0 1 0( . (us, we state the following result.
Theorem 1. 8e equilibrium point X � 0R4 of the nonlinearsystem
(3) provided of a feedback linear control law (7) isasymptotically
stable in the attraction domain:
ξ(P, c) � X ∈R4
XTP X≤
θmaxLS−1‖ ‖
2⎧⎨
⎩
⎫⎬
⎭, (24)
if there exist a positive definite matrix P, a solution
ofLyapunov equation (17), and a not unitary orthogonal matrixS such
that P � STS.
Proof 1. (e proof of the above theorem is based on thequadratic
Lyapunov function V(X) described by equation(16) and its time
derivative _V(X) given by equation (19).(erefore, based on the
eigen decomposition, the matrix Pcan be reduced in the form:
P � GTDG, (25)
where G is an orthogonal matrix and D is a diagonal matrix.(en,
by writing D � WTW, we obtain
P � STS, (26)
where S � WG.Consequently, the quadratic Lyapunov function
V(X)
can be expressed by the following equation:
V(X) � ‖Z‖2, (27)
where Z � SX. (en, the condition _V(X)< 0 leads to
in-equality (23) which becomes
LS−1
Z����
����≤ θmax. (28)
On the contrary, we have
LS−1
Z����
����≤ LS−1����
����‖Z‖. (29)
From conditions (28) and (29), we consider that
LS−1
‖Z‖≤ θmax, (30)
or otherwise,
‖Z‖≤θmaxLS−1‖ ‖
. (31)
(erefore, the asymptotic stability domain ξ(P, c) givenin (24)
is justified. □
5. Results and Discussion
(is section deals with the results obtained using the com-posite
NL/L controller in real time. Indeed, we have set thebackstepping
controller parameters in the swing-up zone:c1 � 3.7 and c2 � 2.6.
In the stabilization zone, by selectingthe desired closed-loop
poles: Pdes � −1.5 −2.8 −4.5 −6( ,
Mathematical Problems in Engineering 5
-
we have determined the following state feedback controllergain:
Kstb � −12 −51 −351 −116( ).
Furthermore, by applying the proposed theorem, wehave
numerically obtained the following:
P �
5.93 2.07 0.19 0.63
2.07 8.27 5.73 4.86
0.19 5.73 23.48 9.77
0.63 4.86 9.77 155.81
,
Q �
48 15 −27 84
15 66 −21 51
−27 −21 168 −36
84 51 −36 558
,
G �
0.74 −0.64 0.18 0.006
−0.66 −0.70 0.26 0.08
0.04 0.31 0.94 −0.08
0.004 0.03 0.07 0.99
,
S �
1.53 −1.31 0.37 0.01
−1.88 −1.98 0.74 0.02
0.21 1.55 4.68 −0.41
0.06 0.44 0.93 12.48
,
θmax � 0.58,
c � 8.25.
(32)
Hence, the evolution of the four elementary de-terminants Δ1 ∼
Δ4 is depicted in Figures 7–10, respectively.
Based on the proposed strategy, we have carried out thereal-time
control experiment of the inverted pendulurmsystem. erefore, the
composite backstepping/linear statecontroller feedback has been
implemented to swing rapidlythe pendulum from its initial position
into the desired angleand, then, to guarantee its
stabilization.
e experimental results of position x (m), angle θ (rad),and
force F (N) are shown in Figure 11.
To valid the above experimental study, we have veried, atthe
switch time t1 � 2.92 s corresponding to θlimit � (π/8) rad,that
V(Xlimit) � XTlimitPXlimit � 7.62< c where the real-timevalues
of the considered state variables are given byxlimit � −0.19m,
_xlimit � −0.04m/s, θlimit � (π/8) rad, and_θlimit � 0.14
rad/s.
In conclusion, real-time results show that the NL/Lcontroller
provides excellent performance. In fact, this typeof control
guarantees more rapidity with comparison to theresponse of the
backstepping controller in the entire op-erating region,
illustrated in Figure 12, and on the contrarywith the response of
the composite L/L controller, depictedin Figure 13. Indeed, the
swing-up phase has taken 15.4 and42 seconds to stabilize the
inverted pendulum when usingthe NL controller in the entire
operating zone and thecomposite L/L controller, respectively.
e reader can nd a video that shows the performanceof the
real-time composite backstepping/linear state feedbackcontroller
implemented for the inverted pendulum
system:https://www.youtube.com/watch?reload�9v�m2PsGHmlH04feature�youtu.be.
6. Conclusion
In this paper, we have applied an alternative switchedcontrol
structure consisting of backstepping and linear statefeedback
controllers in swing-up and stabilization zones,respectively.
Indeed, the nonlinear controller was imple-mented to guarantee the
rapid convergence of the pendulumfrom down position into the
stabilization zone. en, theswitch to the linear controller
stabilizes it at the equilibriumpoint.
Based on the quadratic Lyapunov approach and Syl-vester’s
criterion, we have determined a sucient stabilitydomain in which
the equilibrium point of the consideredclosed-loop system is
justied.
A real-time experimentation shows the eectiveness of theproposed
control strategy and proves a reduction of 84% of
θ6420 –2–4–6
a 11
–12
–11.5
–11
–10.5
–10
–9.5
–9
–8.5
–8
–7.5
Figure 7: Evolution of Δ1(θ).
6 Mathematical Problems in Engineering
https://www.youtube.com/watch?reload=9v=m2PsGHmlH04feature=youtu.behttps://www.youtube.com/watch?reload=9v=m2PsGHmlH04feature=youtu.be
-
θ–4 –3 –2 –1 0 1 2 3 4
det (ψ 3
)
–0.58 0.58–2
0
2
4
6
8
10
12 ×106
Figure 9: Evolution of Δ3(θ).
θ–4 –3 –2 –1.34 –1 0 1 1.34 2 3 4
det (ψ 2
)
–2000
–1500
–1000
–500
0
500
1000
Figure 8: Evolution of Δ2(θ).
×1011
5
4
3
2
1
0
–16
42
0–2
–4–4 –3
–2 –10 1
2 34
–6θ· θ
det (ψ)
x: –1.243y: –6.283z: 3.71e + 08
x: –1.247y: –6.283z: 2.25e + 08
Figure 10: Evolution of Δ4(θ, _θ).
Mathematical Problems in Engineering 7
-
t (sec)706050403020100
Posit
ion
(m)
–0.20
0.2
Composite NL/L controller of the inverted pendulum system in
real time
(a)
t (sec)706050403020100 t1 = 2.92
Ang
le (r
ad)
0246
(b)
t (sec)706050403020100
Forc
e (N
)
–50
0
50
(c)
Figure 11: Real-time trajectories of (a) x (m), (b) θ (rad), and
(c) F (N) with the composite NL/L controller.
t (sec)706050403020100
Posit
ion
(m)
–0.2
0
0.2Backstepping control on the entire horizon for the inverted
pendulum system
(a)
t (sec)706050403020100
Ang
le (r
ad)
0246
(b)
t (sec)706050403020100
Forc
e (N
)
–50
0
50
(c)
Figure 12: Real-time trajectories of (a) x (m), (b) θ (rad), and
(c) F (N) with only backstepping control in the entire operating
region.
t (sec)706050403020100
Posit
ion
(m)
–0.3–0.2–0.1
00.1
Composite linear/linear controller of the inverted pendulum
system in real time
(a)
Figure 13: Continued.
8 Mathematical Problems in Engineering
-
the samples below the conventional scheme when applyingonly the
backstepping controller in the entire operating re-gion.
Furthermore, the reduction percentage has become 92%in comparison
with the composite linear/linear controller.
Appendix
(e aij parameters of the triangular matrix ξ(X), given
inequation (20), are as follows:
a11(·) � −p2g1(X)k1 −p4g2(X)k1,
a12(·) � p1 − k1g1(X)p5 − k2g1(X)p2 − k1g2(X)p7 − k2g2(X)p4
−1σ
p2 J + ml2
b +1σ
p4mlb cos x3( ,
a13(·) � p3 − k1g1(X)p6 − k3g1(X)p2 − k3g2(X)p4 − k1g2(X)p9
−12σ
p2(ml)2g sin 2x3( −
1x3σ
p4(m + M)mgl sin x3( ,
a14(·) � −k1g1(X)p7 − k4g1(X)p2 − k4g2(X)p4 − k1g2(X)p10,
a22(·) � p2 − k2g1(X)p5 − k2g2(X)p7 −1σ
p5 J + ml2
b +1σ
p4mlb cos x3( ,
a23(·) � p6 + p3 − k1g1(X)p6 − k3g1(X)p5 − k3g2(X)p7 − k2g2(X)p9
−12σ
p5(ml)2g sin 2x3(
−1
x3σp7(m + M)mgl sin x3( −
1σ
p6 I + ml2
b−1σ
p9mlb cos x3( ,
a24(·) � p4 − k2g1(X)p7 − k4g1(X)p5 − k2g2(X)p10 − k4g2(X)p7
−1σ
p7 J + ml2
b +1σ
p10mlb cos x3( ,
a33(·) � p8 − k3g1(X)p6 − k3g2(X)p9 −12σ
p6(ml)2g sin 2x3( +
1x3σ
p9(m + M)mgl sin x3( ,
a34(·) � p9 − k3g1(X)p7 − k4g1(X)p6 − k3g2(X)p10 − k4g2(X)p9
−12σ
p6(ml)2g sin 2x3(
+1
x3σp9(m + M)mgl sin x3( ,
a44(·) � −k4g1(X)p7 − k4g2(X)p10 +1σ
(ml)2p9x3 sin 2x3(
+1σ
(ml)2p9x4 sin 2x3( +
1σ
J + ml2
mlp6x3 sin x3( +1σ
J + ml2
mlp7x4 sin x3( .
(A.1)
t (sec)706050403020100
Ang
le (r
ad)
024
6
(b)
t (sec)706050403020100
Forc
e (N
)
–50
0
50
(c)
Figure 13: Real-time trajectories of (a) x (m), (b) θ (rad), and
(c) F (N) with the composite L/L controller.
Mathematical Problems in Engineering 9
-
Data Availability
No data were used to support this study.
Conflicts of Interest
(e authors declare that they have no conflicts of interest.
Supplementary Materials
(e video that shows the performance of the real-timecomposite
backstepping/linear state feedback controllerimplemented for the
inverted pendulum system. (Supple-mentary Materials)
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10 Mathematical Problems in Engineering
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