The Inverted Pendulum Benchmark in Nonlinear Control Theory: A Survey

Abstract

Abstract For at least fifty years, the inverted pendulum has been the most popular benchmark, among others, in nonlinear control theory. The fundamental focus of this work is to enhance the wealth of this robotic benchmark and provide an overall picture of historical and current trend developments in nonlinear control theory, based on its simple structure and its rich nonlinear model. In this review, we will try to explain the high popularity of such a robotic benchmark, which is frequently used to realize experimental models, validate the efficiency of emerging control techniques and verify their implementation. We also attempt to provide details on how many standard techniques in control theory fail when tested on such a benchmark. More than 100 references in the open literature, dating back to 1960, are compiled to provide a survey of emerging ideas and challenging problems in nonlinear control theory accomplished and verified using this robotic system. Possible future trends that we can envision based on the review of this area are also presented.

Keywords

Inverted Pendulum Robotic Benchmarks Nonlinear Control Theory

1. Introduction

Control theory is a field dealing with disciplines and methods that lead to an automatic decision process in order to improve the performance of a control system. The evolution of control theory is related to research advances in technology, theoretical controller design methods and their real-time implementation [1–3]. Control theory progress is also associated with education and control systems engineering [4–6].

Concerning engineering systems, many robotic benchmarks of high interest exist in the literature. They are frequently used to realize experimental models, validate the efficiency of emerging control techniques and verify their implementation. The most common robotic benchmarks are Acrobot [7], Pendubot [8], the Furuta Pendulum [9], the inverted pendulum [10], the Reaction Wheel Pendulum [11], the bicycle [12], the VTOL aircraft [13], the Beam-and-Ball system [14] and TORA [15].

In spite of its simple structure, the inverted pendulum is considered, among the last examples, the most fundamental benchmark. For this system, different versions exist, offering a variety of interesting control challenges.

The most familiar types of the inverted pendulum are the rotational single-arm pendulum [16], the cart inverted pendulum [17] and the double inverted pendulum [18]. The less common versions are the rotational two-link pendulum [19], the parallel type dual inverted pendulum [20], the triple inverted pendulum [21], the quadruple inverted pendulum [22] and the 3D or spherical pendulum [23].

This paper proposes to enhance the wealth of the pendulum benchmark and attempt to provide an overall picture of historical, current and trend developments in nonlinear control theory based on its simple structure and its rich nonlinear model.

The paper is organized as follows. The wealth of the inverted pendulum model will be explained in the next section. In Section 3, different control design techniques will be surveyed through applications of this benchmark. Trends and challenges will be addressed in the last section.

2. Why the inverted pendulum is the most popular benchmark?

The inverted pendulum benchmark (see Figure 1) can be considered as the simplest robotic system, with only one rigid body and only one rotational joint. For this robotic system, let us denote by θ the angle between vertical and the pendulum, which is positive in the clock-wise direction and by m, J and ℓ, the mass of the pendulum, the moment of inertia with respect to the pivot point and the distance from the pivot to the centre of mass, respectively. The acceleration of gravity is g and the acceleration of the pivot is u.

Figure 1.

The inverted Pendulum

Using the Newton Euler approach and under some predefined assumptions, the equations of the motion of the pendulum can described by [16]:

J \overset{..}{θ} = m g ℓ s i n θ - m u ℓ c o s θ

(1)

The energy of the system can be deduced as:

E = \frac{J_{P} {\dot{θ}}^{2}}{2} + m g ℓ (\cos θ - 1)

(2)

where:

J_{P} = m ℓ^{2} + J

(3)

Let ω₀ = mgℓ/J, u = vg and $x = [\begin{matrix} x_{1} & x_{2}] = [\begin{matrix} θ & \dot{θ} \end{matrix}] \end{matrix}$ . The normalized state space system is then described by:

{\begin{cases} {\dot{x}}_{1} = \dot{θ} \\ {\dot{x}}_{2} = (\sin θ - v \cos θ) ω_{0}^{​} \end{cases}

(4)

As can be deduced from system (4), the inverted pendulum system is a typical nonlinear dynamic system including a stable equilibrium point when the pendulum is in a pending position and an unstable equilibrium point when the pendulum is in an upright position. When the system is moved up from the pending position to the upright position, the model is strongly nonlinear with the pendulum angle.

The structure of the inverted pendulum seems to be quite simple, which explains the development of many virtual models [24–25], real devices [26–28] and web based remote control laboratories [29–32].

However, though the simplicity of its structure, many standard techniques in control theory are ineffective when tested on the inverted pendulum benchmark. The study of the system is much more difficult than it appears to be for many reasons, e.g., many geometric properties of the system are lost when the pendulum moves through horizontal positions [33]. Indeed, the output-zeroing manifold does not contain any equilibrium points, the relative degree of the system is not constant and the controllability distribution of the system does not have a constant rank. The system is therefore not linearizable [34].

The control problem to be solved is generally composed of two specific tasks: the first one is the upswing control of the pendulum and the second one is the stabilization of the pendulum around the unstable equilibrium point [35–36]. Finding a continuous feedback that can render the vertical upward position of the pendulum globally asymptotically stable has been, until recently, considered a major problem.

3. Major accomplishments in control theory

Since the 1950s, the inverted pendulum benchmark has been used for teaching linear feedback control theory [37] to stabilize open-loop unstable systems. The first solution to this problem was described in 1960 by Roberge [38] and then by Schaefer and Cannon in 1966 [39]. This benchmark was considered in [40] as a typical model for root-locus analysis and was used in [41] to solve the linear optimal control problem.

The inverted pendulum benchmark has in addition maintained its usefulness in nonlinear control theory.

Table 1 illustrates this idea. Illustrative academic books and survey papers on nonlinear control theory are reported in the first and the second columns, respectively. Research papers where the same emerging control techniques are illustrated and verified using the inverted pendulum benchmark are given in the third column.

Table 1.

Nonlinear Control techniques illustrated by the inverted pendulum

Control design	Academic Books	Survey papers	Research papers with illustration
Bang-Bang control	-	[42]	[43, 44]
Fuzzy logic control	[45]	[46, 47]	[36,48]
Neural Network control	[49–50]	[51]	[52]
PID Adaptive control	[53]	[54– 57]	[58]
Energy based control	[59–60]	[61]	[16, 34] [62–67]
Hybrid control	[68–70]	[71–75]	[33,76–77]
Feed-forward control	-	[78]	[79–80]
Sliding mode control	[81– 83]	[84–85]	[86–88]
Time optimal control	[89–91]	[92]	[93–96]
Predictive control	[97–100]	[101–104]	[105–108]

The inverted pendulum is effectively a fundamental benchmark in control theory. It was used, as shown by Table 1, to illustrate all emerging ideas in nonlinear control theory.

A comparative study of control approaches verified on this benchmark leads to the following statements: The Bang-Bang control strategy requires complex calculation and control variables are usually characterized by abruptly switching between two states. Fuzzy Logic and Neural Network controllers have simple structures and usually avoid unnecessary and lengthy computations. For their simplicity, they are currently considered as the most popular control techniques. However, stability conditions are often not specified when such approaches are applied. PID adaptive controllers can give satisfactory results in performances but should be frequently tuned using evolutionary techniques or convex optimization tools such as Linear Matrix Inequalities. The most rigorous control approaches applied to the inverted pendulum benchmark are the so-called energy-based methods, energy-shaping techniques and the hybrid control approaches, for which Lyapunov tools are usually used and global stability conditions of the overall system are mathematically well proven. When robustness performances are required, a sliding mode control approach is applied. In such cases, stability conditions are well proven and robustness performances are often guaranteed. Time optimal control and predictive control techniques can give very satisfactory results by solving optimization problems. However, the predictive control approach suffers in many cases of a lack of stability condition proofs.

4. Future trends and challenges

In this section, possible future trends that we can envision based on the review of this area are presented. One of the more complex problems is the global asymptotic stabilization of the origin by a single and smooth continuous feedback for hybrid systems. For the inverted pendulum case, the problem can be approached by designing a single controller to realize both the upswing and the stabilization control. A complete solution to this crucial problem was recently proposed in [65, 109–110]. For non linear systems, handling delays, instable internal dynamics and actuator saturation are also considered as challenging problems. For the rotational single-arm pendulum, the output regulation problem under uncertain conditions in the presence of nonlinear backlash effects is solved in [111] using hybrid architecture, which combines Type-1 or Type-2 fuzzy logic systems and genetic algorithms. A solution to the same crucial problem for the cart inverted pendulum is proposed in [109] using an input–output linearization, energy control and singular perturbation theory.

Alternatively, friction is a very complicated phenomenon that occurs in all control systems. When friction effects are not modelled, feedback controllers may fail. Indeed, frictions are usually associated with oscillatory behaviours or in limiting cycles. In this framework, very few results exist, see for example [112] and references therein.

Regarding the question of the lack of a mathematical method proving the stability and robustness of the fuzzy control approach, a solution to this problem was recently proposed in [113], where type-1 and type-2 fuzzy logic controllers are based on a Lyapunov theory. First, the control problem is solved for the rotational single-arm pendulum without a gravity effect to prove stability conditions and then robustness conditions are proved for the nonlinear system with a gravity effect. An extension of this approach for non-smooth mechanical systems is found in [114].

On one hand, based on the inverted pendulum stabilization principle, many robotic control strategies are emerged. Recent major accomplishments include control design of under actuated robotic systems [115–117], mobile wheeled inverted pendulums [118–121] and gait pattern generation for bipedal and humanoid robots [122–132]. In most cases, the non linear control problem is based on finding a suitable reference trajectory obeying a certain criteria formulated as a constrained optimization problem (see [133] and [134] and references therein). For such complex control problems, new hybrid control approaches, such as fuzzy-neural control approaches [135, 136, 137], are emerging. The use of neural networks has increased in recent years in control theory, since they do not involve any mathematical theories. They reduce the development cost of controllers for complex systems, particularly nonlinear ones [138]. For tuning controller parameters and solving constrained control optimization problems of robotic systems, bio-inspired methods, such as genetic algorithms [111], particle swarm optimization [139, 140] and ant colony optimization [141], are promising optimization algorithms. The recent concise review in [142] explains and discusses, for example, the design of type-2 fuzzy systems using such optimization methods. It is expected that these approaches and similar ones could emerge in the near future in the area of nonlinear control theory and especially for robotic applications.

On the other hand, chaos theory is an open research area extensively investigated nowadays in nonlinear control theory including very promising research fields such as chaos analysis, chaos control and chaos synchronization. In this framework, the damped driven pendulum can be considered as a basic example. Indeed, it is recently shown, that if the pendulum is placed in certain spots, the corresponding motion will become chaotic [143, 144]. The coexistence of uncountable non-periodic motions and countable periodic motions of the pendulum is proved. New results on chaos synchronization of the pendulum motion with other systems can be also found in [145].

5. Conclusion

In this paper, it has been shown that the inverted pendulum system is a fundamental benchmark in nonlinear control theory. The particular interest in this application lies in its simple structure and the wealth of its nonlinear model. The simple structure allows real and virtual experiments to be carried out. The richness of the model has shown its usefulness in illustrating all emerging ideas in nonlinear control theory.

At the moment, energy based control, energy-shaping techniques, hybrid control approaches and variable structure control approaches can be considered as the most mathematical rigorous approaches to solve the upswing control and the stabilization of the pendulum around the unstable equilibrium point since global stability is based on Lyapunov theory. However, new hybrid control approaches, such as fuzzy-neural control approaches and bio-inspired methods like genetic algorithms, particle swarm optimization and ant colony optimization, have attracted more attention because they reduce the complexity of controllers.

Finally, based on this review, we can confirm that handling for nonlinear systems, delays, instable internal dynamics, uncertainty conditions, actuator saturation and chaos dynamics are the future trends and the most challenging problems to be solved in control theory.

References

Murray

R. M.

Åström

K. J.

Boyd

S. P.

Brockett

R.W.

Stein

, “Future directions in control in an information-rich world” IEEE Control Systems Magazine, vol. 23, n°2, pp. 20–33, 2003.

Bars

Colaneri

de Souza

C. E.

Dugard

Allgöwer

Kleimenov

. “Theory, algorithms and technology in the design of control systems” Annual Reviews in Control, vol. 30, n°1, pp. 19–30, 2006.

Bars

Colaneri

Dugard

Allgower

Kleimenov

Scherer

C. W.

, “Trends in theory of control system design - status report by the IFAC coordinating committee” Proceedings of the 17th IFAC World Congress, Seoul, vol. 17, n°1, pp. 93–114, 2008.

Horác□ek

, “Laboratory experiments for control theory courses: A survey” Annual Reviews in Control, vol. 24, pp. 151–162, 2000.

Kheir

N. A.

Åström

K. J.

Auslander

Cheok

K. C.

Franklin

G. F.

Masten

, “Control systems engineering education” Automatica, vol. 32, n°2, 147–166, 1996.

Bencomo

S. D.

, “Control learning: Present and future” Annual Reviews in Control vol. 28, n°1, pp. 115–136, 2004.

Spong

M. W.

, “The swing up control problem for the acrobat” IEEE Control Systems Magazine, vol. 15, pp. 72–79, 1995.

Fantoni

Lozano

Spong

M. W.

, “Energy based control of the Pendubot” IEEE Transactions on Automatic Control, vol. 45, n°4, pp. 725–729, 2000.

Acosta

J. Á.

, “Furuta's pendulum: A conservative nonlinear model for theory and practice” Mathematical Problems in Engineering, vol. 2010, 2010.

10.

Furuta

Yamakita

Kobayashi

, “Swing-up control of inverted pendulum using pseudo-state feedback” Journal of Systems and Control Engineering” vol. 14, pp. 263–269, 1992.

11.

Block

D. J.

Astrom

K. J.

Spong

M.W.

, The Reaction Wheel Pendulum, San Rafael: Morgan and Claypool, 2007.

12.

Åström

K. J.

Klein

R. E.

Lennartsson

, “Bicycle dynamics and control” IEEE Control Systems Magazine, vol. 25, n°4, pp. 26–47, 2005.

13.

Martin

Devasia

Paden

, “A different look at output tracking: Control of a VTOL aircraft” Automatica, vol. 32, n°1, pp. 101–107, 1996.

14.

Andreev

Auckly

Gosavi

Kapitanski

Kelkar

White

, “Matching, linear systems, and the ball and beam” Automatica, vol. 38, n°12, pp. 2147–2152, 2002.

15.

Jankovic

Fontaine

Kokotović

P. V.

, “TORA example: Cascade- and passivity-based control designs” IEEE Transactions on Control Systems Technology, vol. 4, n°3, pp. 292–297, 1996.

16.

Åström

K. J.

Furuta

“Swinging up a pendulum by energy control”

Automatica, vol. 36, n°2, pp. 287–295, 2000.

17.

Yoshida

, “Swing-up control of an inverted pendulum by energy-based methods” in Proceedings of the American Control Conference, San Diego, pp. 4045–4047, 1999.

18.

Furuta

Okutani

Sone

, “Computer control of a double inverted pendulum” Computer and Electrical Engineering, vol. 5, pp. 67–84, 1978.

19.

Bortoff

S. A.

, “Robust swing up control for a rotational double pendulum”, in Proceedings of the 13th World Congress of IFAC, San Francisco, pp. 413–419, 1996.

20.

Tsai

Shen

B. H.

, “Synchronization control of parallel dual inverted pendulums driven by linear servomotors” IET Control Theory and Applications, vol. 1, n°1, pp.20–327, 2007.

21.

Furuta

Ochiai

Ono

, “Attitude control of a triple inverted pendulum” International Journal of Control, vol. 39, n°6, pp. 1351–1365, 1984.

22.

Miao

Wang

, “Variable universe adaptive fuzzy control on the quadruple inverted pendulum” Science in China, Series E: Technological Sciences, vol. 45, n°2, pp. 213–224, 2002.

23.

Shen

Sanyal

A. K.

Chaturvedi

N. A.

Bernstein

McClamroch

, “Dynamics and control of a 3D pendulum” in Proceedings of the 43rd IEEE Conference on Decision and Control, Nassau, vol. 1, pp. 323–328, 2004.

24.

Wollherr

Buss

, “Cost-oriented virtual reality and real-time control system architecture” Robotica, vol. 21, n°3, pp. 289–294, 2003.

25.

Sánchez

Morilla

Dormido

Aranda

Ruipérez

, “Virtual control lab using Java and Matlab: A qualitative approach” IEEE Control Systems Magazine, vol. 22, n°2, pp. 8–20, 2002.

26.

H. L.

Chang

B. C.

Jagadish

Kwatny

H. G.

, “A DSP microprocessor hybrid control of an inverted pendulum” in Proceedings of the 2009 IEEE International Conference on Control and Automation, Christchurch, pp. 2317–2322, 2009.

27.

Koga

Toriumi

Sampei

, “An integrated software environment for the design and real-time implementation of control systems” Control Engineering Practice, vol. 6, n° 10, pp. 1287–1293, 1998.

28.

Benrejeb

Boubaker

, “FPGA modeling and real-time embedded control design via LabVIEW Software: Application for swinging-Up a pendulum” International Journal on Smart Sensing and Intelligent Systems, vol. 5, n°3, pp. 576–591, 2012.

29.

Sánchez

Morilla

Dormido

, “Teleoperation on an inverted pendulum through the World Wide Web” in Proceedings of the IFAC Workshop on Internet Based Control Education, Madrid, pp. 37–42, 2001.

30.

Gillet

Salzmann

Huguenin

, “A distributed architecture for teleoperation over the internet with application to the remote control of an inverted pendulum” Lecture Notes in Control and Information Sciences, vol. 258, n° 399–407, 2000.

31.

Sánchez

Morilla

Pastor

Dormido

, “A Java/Matlab-based environment for remote control system laboratories: Illustrated with an inverted pendulum” IEEE Transactions on Education, vol. 47, n°. 3, pp. 321–329, 2004.

32.

Yao

Dai

Tian

Zhang

, “MATALB & internet based remote control laboratory” in Proceedings of the Chinese Control and Decision Conference, Guilin, pp. 1262–1268, 2009.

33.

Zhao

Spong

M.W.

, “Hybrid control for global stabilization of the cart-pendulum system” Automatica, vol. 37, n°12, pp. 1941–1951, 2001.

34.

Chung

C. C.

Hauser

, “Nonlinear control of a swinging pendulum” Automatica, vol. 31, n°6, pp. 851–862, 1995.

35.

Muskinja

Tovornik

, “Swinging up and stabilization of a real inverted pendulum” IEEE Transaction on Industrial Electronics, vol. 53, n°2, pp. 631–639, 2006.

36.

Yubazaki

Hirota

, “Upswing and stabilization control of inverted pendulum system based on the SIRMs dynamically connected fuzzy inference model” Fuzzy Sets and Systems, vol. 122, n° 1, pp. 139–152, 2001.

37.

Kailath

, Linear Systems, Englewood Cliffs: Prentice-Hall, 1980.

38.

Roberge

J. K.

, The Mechanical Seal, Massachusetts Institute of Technology: Bachelor's thesis, 1960.

39.

Schaefer

J. F.

Cannon

R. H.

, “On the control of unstable mechanical systems” Proceedings of the third Automatic and Remote Control, London, vol. 3, n°1, pp. 12–24, 1966.

40.

Siebert

W. McC.

, Circuits, Signals, and Systems, Cambridge: MIT Press, 1998.

41.

Kwakernaak

Sivan

, Linear Optimal Control Systems, New York: Wiley, 1972.

42.

Sonneborn

Van Vleck

, “The bang-bang principle for linear control systems” SIAM Journal of Control, vol. 2, pp. 151–159, 1965.

43.

Mori

Nishihara

Furuta

, “Control of unstable mechanical system- control of pendulum” International Journal of Control, vol. 23, n°5, pp. 673–692, 1976.

44.

Furuta

Yamakita

Kobayashi

, “Swing up control of inverted pendulum” Proceedings of the 1991 International Conference on Industrial Electronics, Control and Instrumentation, Kobe, pp. 2193–2198, 1991.

45.

Wang

L. X.

, Adaptive fuzzy systems and control - Design and stability analysis, Englewood Cliffs: Prentice Hall, 1994.

46.

Lee

C.C.

, “Fuzzy logic in control systems: Fuzzy logic controller - part I” IEEE Transactions on Systems, Man and Cybernetics, vol. 20, n°2, pp. 404–418, 1990.

47.

Lee

C.C.

, “Fuzzy logic in control systems: Fuzzy logic controller - part II” IEEE Transactions on Systems, Man and Cybernetics, vol. 20, n° 2, pp. 419–435, 1990.

48.

Wang

H. O.

Tanaka

Griffin

M. F.

, “An approach to fuzzy control of nonlinear systems: Stability and design issues” IEEE Transactions on Fuzzy Systems, vol. 4, n°1, pp. 14–23, 1996.

49.

Lewis

F. L.

Yesilidrek

Jagannathan

, Neural network control of robot manipulators and nonlinear systems, Philadelphia: Taylor & Francis, 1999.

50.

S. S.

Lee

T. H.

Harris

C. J.

, Adaptive neural network control of robotic manipulators, Singapore: World Scientific, 1998.

51.

Hunt

K. J.

Sbarbaro

Zbikowski

Gawthrop

P. J.

, “Neural networks for control systems - A survey” Automatica, vol. 28, n°6, pp. 1083–1112, 1992.

52.

Anderson

C. W.

, “Learning to control an inverted pendulum using neural networks” IEEE Control Systems Magazine, 1989, vol. 9, n°3, pp. 31–37, 1989.

53.

Astrom

K.J.

Wittenmark

, Adaptive Control. New York: Addison-Wesley, 1995.

54.

Landau

I. D.

, “A survey of model reference adaptive techniques-theory and applications” Automatica, vol. 10, n°4, pp. 353–379, 1974.

55.

Åström

K. J.

, “Theory and applications of adaptive control-A survey” Automatica, vol. 19, n°5, pp. 471–486, 1983.

56.

Ortega

Tang

, “Robustness of adaptive controllers-A survey” Automatica, vol. 25, n°5, pp. 651–677, 1989.

57.

Åström

K. J.

Hägglund

Hang

C. C.

W. K.

, “Automatic tuning and adaptation for PID controllers - a survey” Control Engineering Practice, vol. 1, n°4, pp. 699–714, 1993.

58.

Chang

W. D.

Hwang

R. C.

Hsieh

J. G.

, “A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach” Journal of Process Control, vol. 12, n°2, pp. 233–242, 2002.

59.

Khalil

H. K.

, Nonlinear systems, New Jersey: Prentice Hall, 2002.

60.

Marino

Tomei

, Nonlinear control design, Englewood Cliffs: Prentice Hall, 1995.

61.

Kokotović

Arcak

, “Constructive nonlinear control: A historical perspective” Automatica, vol. 37, n° 5, pp. 637–662, 2001.

62.

Lozano

Fantoni

Block

D. J.

, “Stabilization of the inverted pendulum around its homoclinic orbit” Systems and Control Letters, vol. 40, n°3, pp. 197–204, 2000.

63.

Angeli

, “Almost global stabilization of the inverted pendulum via continuous state feedback” Automatica, vol. 37, n°7, pp. 1103–1108, 2001.

64.

Chatterjee

Patra

Joglekar

H. K.

, “Swing-up and stabilization of a cart-pendulum system under restricted cart track length” Systems and Control Letters, vol. 47, n°4, pp. 355–364, 2002.

65.

Aström

K. J.

Aracil

Gordillo

, “A family of smooth controllers for swinging up a pendulum” Automatica, vol. 44, n°7, pp.1841–1848, 2008.

66.

Yang

J. H.

Shim

S. Y.

Seo

J. H.

Lee

Y. S.

, “Swing-up control for an inverted pendulum with restricted cart rail length” International Journal of Control Automation and Systems, vol. 7, n°4, pp. 674–680, 2009.

67.

Siuka

Schoberl

, “Applications of energy based control methods for the inverted pendulum on a cart” Robotics and Autonomous Systems, vol. 57, pp. 1012–1017, 2009.

68.

Mariton

, Jump linear systems in automatic control, New York: Marcel Dekker, 1990.

69.

Savkin

A. V.

Evans

R. J.

, Hybrid dynamical systems: controller and sensor switching problems, Boston: Birkhauser, 2002.

70.

Liberzon

, Switching in systems and control, Boston: Birkhauser, 2003.

71.

Liberzon

Morse

A. S.

, “Basic problems in stability and design of switched systems” IEEE Control Systems Magazine, vol. 19, n°5, pp. 59–70, 1999.

72.

Labinaz

Bayoumi

M. M.

Rudie

, “A survey of modeling and control of hybrid systems” Annual Reviews in Control, vol. 21, pp. 79–92, 1997.

73.

Decarlo

R. A.

Branicky

M. S.

Pettersson

Lennartson

, “Perspectives and results on the stability and stabilizability of hybrid systems” Proceedings of the IEEE, vol. 88, n°7, pp. 1069–1082, 2000.

74.

Rugh

W. J.

Shamma

J. S.

, “Research on gain scheduling” Automatica, vol. 36, n°10, pp. 1401–1425, 2000.

75.

Sun

S. S.

, “Analysis and synthesis of switched linear control systems” Automatica, vol. 41, n°2, pp. 181–195, 2005.

76.

Åström

K. J.

, “Hybrid Control of Inverted Pendulums” In Yamamoto

Hara

(Eds.), Learning, control & hybrid systems, Berlin: Springer, 1999.

77.

Aracil

Gordillo

, “The inverted pendulum: A benchmark in nonlinear control” Proceedings of the Sixth Biannual World Automation Congress-Intelligent Automation and Control Trends, Principles, and Applications, Seville, 2004, pp. 468–482.

78.

Mazenc

Praly

, “Adding integrations, saturated controls, and stabilization for feed-forward systems” IEEE Transactions on Automatic Control, vol. 41, n°11, pp. 1559–1578, 1996.

79.

Mazenc

Praly

, “Asymptotic tracking of a reference state for systems with a feed-forward structure” Automatica, vol. 36, n°2, pp. 179–187, 2000.

80.

Mazenc

Bowong

, “Tracking trajectories of the cart-pendulum system” Automatica, vol. 39, n°4, pp. 677–684, 2003.

81.

Utkin

V. I.

, Sliding Modes in Control Optimization, Berlin: Springer Verlag, 1992.

82.

Utkin

Guldner

Shi

, Sliding mode control in electromechanical systems, New York: Taylor & Francis, 1999.

83.

Edwards

Spurgeon

S. K.

, Sliding mode control: Theory and applications, Taylor and Francis, 1998.

84.

Utkin

V. I.

, “Variable structure systems with sliding modes” IEEE Transactions on Automatic Control, vol. 22, n°2, pp. 212–222, 1977.

85.

Hung

J. Y.

Gao

Hung

J.C.

, “Variable structure control. A survey” IEEE Transactions on Industrial Electronics, vol. 40, n°1, pp. 2–22. 1993.

86.

Riachy

Orlov

Floquet

Santiesteban

Richard

J. P.

, “Second-order sliding mode control of underactuated mechanical systems I: Local stabilization with application to an inverted pendulum” International Journal of Robust and Nonlinear Control, vol. 18, n°4–5, pp. 529–543, 2008.

87.

Riachy

Orlov

Floquet

Santiesteban

Richard

J. P.

, “Second-order sliding mode control of underactuated mechanical systems II: Orbital stabilization of an inverted pendulum with application to swing up/balancing control” International Journal of Robust and Nonlinear Control, vol. 18, n°4–5, pp. 544–556, 2008.

88.

Park

M. S.

Chwa

, “Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method” IEEE Transactions on Industrial Electronics, vol. 56, n°9, pp. 3541–3555, 2009.

89.

Lewis

F.L.

Syrmos

V.L.

, Optimal control, New York: John Wiley & Sons, 1995.

90.

Vinter

R. B.

, Optimal control, Boston: Birkhäuser, 2000.

91.

Boscain

Piccoli

, Optimal syntheses for control systems on 2-D Manifolds, Berlin: Springer-Verlag, 2004.

92.

Mayne

D. Q.

, “Optimal control-A survey of computational methods” IEEE Conference Publication, vol. 96, pp. 236–244, 1973.

93.

Iwase

Furuta

, “Time optimal swing-up control of single pendulum” Journal of Dynamic Systems, Measurement and Control, vol. 123, n°3, pp. 518–527, 2001.

94.

Holzhüter

, “Optimal regulator for the inverted pendulum via Euler-Lagrange backward integration” Automatica, vol. 40, n°9, pp. 1613–1620, 2004.

95.

Chernousko

F. L.

Reshmin

S. A

, “Time-optimal swing-up feedback control of a pendulum” Nonlinear Dynamics, vol. 47, n°1–3, pp. 65–73, 2007.

96.

Mason

Broucke

Piccoli

, “Time optimal swing-up of the planar pendulum” IEEE Transactions on Automatic Control, vol. 53, n°8, pp. 1876–1886, 2008.

97.

Allgöwer

Zheng

, Nonlinear model predictive control, Berlin: Birkhauser, 2000.

98.

Maciejowski

J. M.

. Predictive control with constraints. New York: Prentice Hall, 2002.

99.

Findeisen

Allgöwer

Biegler

, Assessment and future directions of nonlinear model predictive control, Berlin: Springer, 2006.

100.

Magni

Raimondo

D. M.

Allgöwer

, Nonlinear predictive control-Towards new challenging applications, Berlin: Springer, 2009.

101.

García

C. E.

Prett

D. M.

Morari

, “Model predictive control: Theory and practice-A survey” Automatica, vol. 25, n°3, pp. 335–348, 1989.

102.

Morari

Lee

H. J.

, “Model predictive control: Past, present and future” Computers and Chemical Engineering, vol. 23, n°4–5, pp. 667–682, 1999.

103.

Mayne

D. Q.

Rawlings

J. B.

Raoand

C.V.

Scokaert

P.O.M.

, “Constrained model predictive control: Stability and optimality” Automatica, vol. 36, n°6, pp. 789–814, 2000.

104.

Magni

Scattolini

, “Stabilizing model predictive control of nonlinear continuous time systems” Annual Reviews in Control, vol. 28, n°1, pp. 1–11, 2004.

105.

Magni

Scattolini

Astro□m

, “Global stabilization of the inverted pendulum using model predictive control” Proceedings of the 15^th IFAC World Congress, Barcelona, 2002.

106.

Ba□lan

Ma□ties□

Stan

, “A solution of the inverse pendulum on a cart problem using predictive control” Proceedings of the IEEE International Symposium on Industrial Electronic, Dubrovnik, pp. 63–68, 2005.

107.

Gawthrop

P. J.

Wang

, “Intermittent predictive control of an inverted pendulum” Control Engineering Practice, vol. 14, n°11, pp. 1347–1356, 2006.

108.

Mills

Wills

Ninness

, “Nonlinear model predictive control of an inverted pendulum” Proceedings of the American Control Conference, St. Louis, pp. 2335–2340, 2009.

109.

Srinivasan

Huguenin

Bonvin

, “Global stabilization of an inverted pendulum-control strategy and experimental verification,” Automatica, vol. 45, n°1, pp. 265–269, 2009.

110.

Gordillo

Aracil

, A new controller for the inverted pendulum on a cart” International Journal of Robust and Nonlinear Control, vol.18, n°17, pp. 1607–1621, 2008.

111.

Cázarez-Castro

N. R.

Aguilar

L. T.

Castillo

Oscar

, “Fuzzy logic control with genetic membership function parameters optimization for the output regulation of a servomechanism with nonlinear backlash” Expert Systems with Applications, vol. 37, n°6, pp. 4368–4378, 2010.

112.

Campbell

S. A.

Crawford

Morris

, “Friction and the inverted pendulum stabilization problem” Journal of Dynamic Systems, Measurement and Control, vol. 130, n°5, 2008.

113.

Castillo

Aguilar

L. T.

Cázarez-Castro

N. R.

Cardenas

, “Systematic design of a stable type-2 fuzzy logic controller” Applied Soft Computing, vol. 8, n°3, pp. 1274–1279, 2008.

114.

Cázarez-Castro

N. R.

Aguilar

L. T.

Castillo

, “Designing type-1 and type-2 fuzzy logic controllers via fuzzy Lyapunov synthesis for non smooth mechanical systems” Engineering Applications of Artificial Intelligence, vol. 25, n°5, pp. 971–979, 2012.

115.

Ortega

Spong

M. W.

Gómez-Estern

Blankenstein

, “Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment” IEEE Transactions on Automatic Control, vol. 47, n°8, pp. 1218–1233, 2002.

116.

Spong

M. W.

, “Underactuated Mechanical Systems” In Siciliana

Valavanis

K. P.

(Eds.), Control problems in robotics and automation, Berlin: Springer, 1998.

117.

Liu

Wane

Yang

, “On tracking control of a pendulum-driven cart-pole underactuated system” International Journal of Modeling, Identification and Control, vol. 4, n°4, pp. 357–372, 2008.

118.

Y. S.

Yuta

, “Trajectory tracking control for navigation of the inverse pendulum type self-contained mobile robot” Robotics and Autonomous Systems, vol. 17, n°1–2, pp. 65–80, 1996.

119.

Grasser

D'Arrigo

Colombi

Rufer

, “Joe: A mobile, inverted pendulum” IEEE Transactions on Industrial Electronics, vol. 49, n°1, pp. 107–114, 2002.

120.

Takei

Imamura

Yuta

, “Baggage transportation and navigation by a wheeled inverted pendulum mobile robot” IEEE Transactions on Industrial Electronics, vol. 56, n°10, pp. 3985–3994, 2009.

121.

Younis

Abdelati

, “Design and implementation of an experimental segway model” Proceedings of AIP Conference Proceedings, vol. 1107, pp. 350–354, 2009.

122.

Kuo

A. D.

, “The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective” Human Movement Science, vol. 26, n°4, pp. 617–656, 2007.

123.

Kajita

Kanehiro

Kaneko

Fujiwara

Yokoi

Hirukawa

, “Biped walking pattern generation by a simple three-dimensional inverted pendulum model” Advanced Robotics, vol. 17, n°2, pp. 131–147, 2003.

124.

Noh

K. K.

Kim

J. G.

Huh

U. Y.

, “Stability experiment of a biped walking robot with inverted pendulum” Proceedings of the 30th Annual Conference of IEEE Industrial Electronics Society, Busan, pp. 2475–2479, 2004.

125.

Tang

M. J.

, “Humanoid 3D gait generation based on inverted pendulum model” Proceedings of the 22^nd IEEE International Symposium on Intelligent Control, Singapore, pp. 339–344, 2007.

126.

Erbatur

Kurt

, “Natural ZMP trajectories for biped robot reference generation” IEEE Transactions on Industrial Electronics, vol. 56, n°3, pp. 835–845, 2009.

127.

Shibuya

Suzuki

Ohnishi

, “Trajectory planning of biped robot using linear pendulum mode for double support phase” Proceedings of IEEE Industrial Electronics Society, Jules Verne, pp. 4094–4099, 2006.

128.

Motoi

Suzuki

Ohnishi

, “A bipedal locomotion planning based on virtual linear inverted pendulum mode” IEEE Transactions on Industrial Electronics, vol. 56, n°1, pp. 54–61, 2009.

129.

Kajita

Morisawa

Miura

Nakaoka

Harada

Kaneko

, “Biped walking stabilization based on linear inverted pendulum tracking” Proceedings of the IEEE/RSJ 2010 International Conference on Intelligent Robots and Systems, Taipei, pp. 4489–4496, 2010.

130.

Lee

S. H.

Goswami

, “The Reaction mass pendulum (RMP) model for humanoid robot gait and balance control” in Choi

(Eds.), Humanoid Robots, Rijeka: InTech, pp. 167–186, 2009.

131.

Ito

Sasaki

, “Motion control of biped lateral stepping based on zero moment point feedback for adaptation to slopes” in Filho

A.C. Pina

(Eds.), Biped Robots, Rijeka: InTech, pp.15–34, 2011.

132.

Aoyama

Sekiyama

Hasegawa

Fukuda

, “Passive dynamic autonomous control for the multi-locomotion robot” in Filho

A.C. Pina

(Eds.), Biped Robots, Rijeka: InTech, pp.115–128, 2011.

133.

Cardenas-Maciela

S. L.

Castillo

Aguilarc

L. T.

, “Generation of walking periodic motions for a biped robot via genetic algorithms” Applied Soft Computing, vol. 11, pp. 5306–5314, 2011.

134.

Aloulou

Boubaker

, “Minimum jerk-based control for a three dimensional bipedal robot” Lecture Notes in Computer Science, vol. 7102, pp. 251–262, 2011.

135.

Rashid

M. K.

, “Simulation of intelligent single wheel mobile robot” International Journal of Advanced Robotic Systems, vol. 4, n°1, pp. 73–80, 2007.

136.

Cardenas-Maciel

S. L.

Castillo

Aguilar

L.T.

Castro

J. R.

, “A T-S fuzzy logic controller for biped robot walking based on adaptive network fuzzy inference system” International Joint Conference on Neural Networks, Barcelona, pp. 1–8, 2010.

137.

Cardenas-Maciel

S. L.

Castillo

Aguilar

L. T.

, “Neuro-fuzzy based output feedback controller design for biped robot walking” Studies in Computational Intelligence, vol. 318, pp. 423–444, In: Soft Computing for Intelligent Control and Mobile Robotics, 2011.

138.

Pezeshki

Badalkhani

Javadi

, “Performance Analysis of a Neuro-PID Controller Applied to a Robot Manipulator” International Journal of Advanced Robotic Systems, vol. 9, pp. 163–173, 2012.

139.

Mehdi

Boubaker

, “Position/force control optimized by particle swarm intelligence for constrained robotic manipulators” Proc. of the 11^th IEEE International Conference on Intelligent Systems Design and Applications(ISDA'2011), Córdoba, Spain, pp. 190–195, 2011.

140.

Mehdi

Boubaker

, “Impedance controller tuned by particle swarm optimization for robotic arms” International Journal of Advanced Robotic Systems, vol.8, n°5, pp. 93–103, 2011.

141.

García

M. A. P.

Montiel

Castillo

Sepúlveda

Melin

, “Path Planning for Autonomous Mobile Robot Navigation with Ant Colony Optimization and Fuzzy Cost Function Evaluation” Applied Soft Computing Journal, vol. 9, pp. 1102–1110, 2009.

142.

Castillo

Melin

, “Optimization of type-2 fuzzy systems based on bio-inspired methods: A concise review” Information Sciences, vol. 205, pp. 1–19, 2012.

143.

, “An alternative approach for chaos: A plane pendulum with oscillating torque” Communications in Nonlinear Science and Numerical Simulation, vol.16, n°12, pp. 4725–4730, 2011.

144.

Sakthivel

Rajasekar

, “Diffusion dynamics near critical bifurcations in a nonlinearly damped pendulum system” Communications in Nonlinear Science and Numerical Simulation, vol.17, n°3, pp.1303–1311, 2012.

145.

Luo

A. C. J.

Min

, “The chaotic synchronization of a controlled pendulum with a periodically forced, damped Duffing oscillator” Communications in Nonlinear Science and Numerical Simulation, vol. 16, n°12, pp. 4704–4717, 2011.