A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

doi:10.1016/j.automatica.2008.01.006

Automatica

Volume 44, Issue 9, September 2008, Pages 2352-2357

https://doi.org/10.1016/j.automatica.2008.01.006 Get rights and content

Abstract

Immersion and Invariance (I&I) is the method to design asymptotically stabilizing control laws for nonlinear systems that was proposed in [Astolfi, A., & Ortega, R. (2003). Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems. IEEE Transactions on Automatic Control, 48, 590–606]. The key steps of I&I are (i) the definition of a target dynamics, whose order is strictly smaller than the order of the system to be controlled; (ii) the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics; (iii) the design of a control law that renders the manifold attractive and ensures that all signals are bounded. The second step requires the solution of a partial differential equation (PDE) that may be difficult to obtain. In this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps, and invoking physical considerations, it is possible to obviate the solution of the PDE. To underscore the generality of the proposed variation of I&I, we show that it is also applicable to a class of $n$ -dimensional systems that contain, as a particular case, the cart and pendulum system.

Section snippets

Introduction and preliminaries on immersion and invariance

The method of I&I for stabilization of nonlinear systems originated in Astolfi and Ortega (2003) and was further developed in a series of publications that have been recently summarized in Astolfi, Karagiannis, and Ortega (2007), see also Karagiannis, Ortega, and Astolfi (2004). The major result of Astolfi and Ortega (2003), that constitutes the basis of the present note, is the following theorem.

Theorem 1

Consider the system¹ $\dot{x} = f (x$

Model

We consider the classical cart-pendulum system depicted in Fig. 1 (left), and assume that a partial feedback linearization stage has been applied. ² After normalization this yields $Σ : {\begin{cases} {\dot{x}}_{1} = x_{2}, \\ {\dot{x}}_{2} = a sin x_{1} - u b cos x_{1}, \\ {\dot{x}}_{3} = u, \end{cases}$ where $(x_{1}, x_{2}) \in S^{1} \times R$ are the pendulum angle with respect to the upright vertical and its velocity, respectively, and $x_{3} \in R$ is the velocity of the cart, $u \in R$ is the input, and $a > 0$ and $b > 0$ are physical

Simulations

Extensive simulations have been carried out for the four selections of $π_{3}$ (the three on Table 1 and the one on Eq. (19)) described above with different values of $k_{1}, k_{2}$ and $γ$ . The largest domain of attraction among these controllers is achieved by the one calculated from the third line of Table 1. This gives the control law $u = - \frac{1}{1 - k_{2} b} (γ k_{1} + k_{1} x_{2} + γ x_{3} + \frac{γ k_{2}}{cos x_{1}} x_{2} + k_{2} tan x_{1} (\frac{x_{2}^{2}}{cos x_{1}} + a sin x_{1})),$ with $k_{1} > 0, k_{2} > \frac{1}{b}$ and $γ > 0$ . Notice that $V$ has an isolated global minimum at zero and $Δ$ is a constant. The controller

An extension to a class of underactuated mechanical systems

The variation of the I&I procedure presented above can be directly extended to the following systems⁶ ${\dot{x}}_{1} = x_{2},$ ${\dot{x}}_{2} = f_{2} (x_{1}) + g_{2} (x_{1}) u,$ ${\dot{x}}_{3} = f_{3} (x_{1}) + u,$ where $x_{1}, x_{2} \in R^{p}$ and $x_{3}, u \in R^{m}$ . The objective is to stabilize the origin and we assume

Assumption B.1

The matrix $g_{2} (0)$ is not identically equal to zero.

The target dynamics is selected as

Concluding remarks

The main stumbling block for application of the I&I methodology of Astolfi and Ortega (2003) is the need to solve the PDE of the immersion condition—i.e., the computation of the function $π$ that defines the manifold $M$ . ⁷ To overcome this problem we have proposed in this paper to transform this PDE into

Acknowledgments

The work of J.Á. Acosta was supported by MEC-FEDER grants DPI2006-07338 and by The Consejería de Innovación, Ciencia y Empresa of The Junta de Andalucía. This work was partially supported by HYCON. I. Sarras acknowledges the financial support of the Greek Scholarships Foundation (IKY).

References (7)

J.Á. Acosta et al.
Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one
IEEE Transactions on Automatic Control
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A. Astolfi et al.
Nonlinear and adaptive control design and applications
(2007)
A. Astolfi et al.
Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems
IEEE Transactions on Automatic Control
(2003)

There are more references available in the full text version of this article.

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In [1] the I&I method was applied to the cart and pendulum system, rendering the upward position of the pendulum, with the cart stopped, asymptotically stable, by using a target dynamics equivalent to a simple pendulum with viscous friction. In [25] a constructive approach was proposed, inspired by [1], for certain classes of underactuated mechanical systems, where the target dynamics has the form of a mechanical system described in the Hamiltonian framework. However, the class of systems addressed in those works is defined by a change of coordinates on the momenta variable or a pre–feedback controller, such that the Coriolis and centripetal terms (quadratic terms on the velocity) are canceled.
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J.Á. Acosta was born in Huelva, Spain. He obtained both Servo-Electrical and Mechanical Engineering degrees from the University of Huelva, Spain, and a third degree in Electrical Engineering from the University of Seville, Spain. In 2003, he joined the Centre Nacional de la Recherche Scientifique (CNRS, France) at the Laboratoire des Signaux et Systemes, (Supélec, Paris) as a CTS (Control Training Site) fellow. In 2004, he obtained the Ph.D. degree at the University of Sevilla, with a thesis on underactuated nonlinear control systems. In 2005, he was awarded the Ph.D. European Award. In 2006, he was a finalist of the George S. Axelby Outstanding Paper Award in the IEEE Transactions on Automatic Control journal. In 1999 he joined as Research Assistant in the Department of Systems Engineering and Automatic Control at the University of Seville, where he is currently Researcher, Associate Professor and member of the Automatic and Robotic Institute. He has been Visitor in the Laboratoire des Signaux et Systemes (CNRS, France) repeatedly since 2005. His research interests are focused on mathematical control theory, nonlinear control and dynamical systems with emphasis on Underactuated Electro-mechanical and Robotic Systems. He is author of over 50 research publications in scientific journals, internationals conferences and book chapters.

R. Ortega was born in Mexico. He obtained his BSc in Electrical and Mechanical Engineering from the National University of Mexico, Master of Engineering from Polytechnical Institute of Leningrad, USSR, and the Docteur D‘Etat from the Politechnical Institute of Grenoble, France in 1974, 1978 and 1984 respectively.

He then joined the National University of Mexico, where he worked until 1989. He was a Visiting Professor at the University of Illinois in 1987–88 and at the McGill University in 1991–1992, and a Fellow of the Japan Society for Promotion of Science in 1990–1991. He has been a member of the French National Researcher Council (CNRS) since June 1992. Currently he is in the Laboratoire de Signaux et Systemes (SUPELEC) in Paris. His research interests are in the fields of nonlinear and adaptive control, with special emphasis on applications.

Dr Ortega is a Fellow Member of the IEEE. He was the Chairman of the IEEE Working Group on Adaptive Control and Systems Identification, of the IFAC Technical Committee on Adaptive Control and Tuning and of the Automatica Paper Prize Award Committee. He was a member of the IFAC Technical Board and chairman of the IFAC Coordinating Committee on Systems and Signals. Currently he is an Editor at Large of IEEE Transactions on Automatic Control, and an Associate Editor of Systems and Control Letters and International Journal of Adaptive Control and Signal Processing.

A. Astolfi was born in Rome, Italy, in 1967. He graduated in electrical engineering, cum laude, at the University of Rome in 1991. In 1992 he joined ETH-Zurich where he obtained a M.Sc. in Information Theory in 1995 and the Ph.D. degree with Medal of Honour in 1995 with a thesis on discontinuous stabilization of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome “La Sapienza” for his work on nonlinear robust control. Since 1996 he has been with the Electrical and Electronic Engineering Department of Imperial College, London (UK), where he is currently a Professor in Non-linear Control Theory. From 1998 to 2003 he was also an Associate Professor at the Dept. of Electronics and Information of the Politecnico of Milano. Since 2005 he is also Professor at Dipartimento di Informatica, Sistemi e Produzione, University of Rome Tor Vergata. He has been a visiting lecturer in “Nonlinear Control” in several universities, including ETH-Zurich (1995–1996); Terza University of Rome (1996); Rice University, Houston (1999); Kepler University, Linz (2000); SUPELEC, Paris (2001). His research interests are focused on mathematical control theory and control applications, with special emphasis on the problems of discontinuous stabilization, robust stabilization, robust control and adaptive control. He is the author of more than 70 journal papers, of 20 book chapters and of over 160 papers refereed conference proceedings. He is a co-editor of a book on “Modeling and Control of Mechanical Systems” and author (with D. Karagiannis and R. Ortega) of the monograph “Nonlinear and Adaptive Control with Applications” (Springer Verlag). He is Associate Editor of Systems and Control Letters, Automatica, IEEE Trans. Automatic Control, the International Journal of Control, the European Journal of Control, the Journal of the Franklin Institute, and the International Journal of Adaptive Control and Signal Processing. He has also served in the IPC of various international conferences.

I. Sarras was born in Athens, Greece, in 1982. He graduated from the Automation Engineering Department of the Technological Education Institute (T.E.I.) of Piraeus in 2004 and received the Master of Research (M2R) in Automatic Control from the Université Paul Sabatier (Toulouse III), France, in 2006. Since October 2006, he is a PhD candidate at the Laboratoire des Signaux et Systémes (Université Paris-Sud 11), France, under the supervision of Dr. Romeo Ortega. His research interests are in the fields of nonlinear control with emphasis on mechanical systems.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil.

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Brief paperA constructive solution for stabilization via immersion and invariance: The cart and pendulum system☆

Abstract

Section snippets

Introduction and preliminaries on immersion and invariance

Model

Simulations

An extension to a class of underactuated mechanical systems

Concluding remarks

Acknowledgments

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

IEEE Transactions on Automatic Control

Nonlinear and adaptive control design and applications

Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems

IEEE Transactions on Automatic Control

Brief paper
A constructive solution for stabilization via immersion and invariance: The cart and pendulum system☆