nonlinear/linearswitchedcontrolofinvertedpendulumsystem: … · 2019. 8. 6. · is paper treats the...

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

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

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.


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( ,


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).


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


θ–4 –3 –2 –1.34 –1 0 1 1.34 2 3 4

det (ψ 2

)

–2000

–1500

–1000

–500

0

500

1000


×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).


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.


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( ,


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( ,


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( ,


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.


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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