constructive nonlinear smc

14
ve Nonlinear Sliding Mod Class of Underactuated Sy with Parametric Uncertainties M. Lopez-Martinez, J.A. Acosta and J.M. Cano In Proceedings of ….. Conference 2008 ออออออออ อออออออออ - ออออออออออออออ ออออ 6 - อออออออออออออออออ อออออออออออออออออออออออ

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Page 1: Constructive nonlinear smc

Constructive Nonlinear Sliding Mode Surfacesfor a Class of Underactuated Systems

with Parametric Uncertainties

M. Lopez-Martinez, J.A. Acosta and J.M. Cano

In Proceedings of ….. Conference 2008

อิ�ทธิ�เดช มูลมู� งมู�- นิ�สิ�ตปริ�ญญาเอิก เทอิมู 6- อิาจาริย์�ปริะจ�าสิาขาวิ�ทย์าศาสิตริ�และเทคโนิโลย์� มูสิธิ.

Page 2: Constructive nonlinear smc

J. Á. Acosta was born in Huelva, Spain. He obtained both the Servo-Electrical and Mechanical Engineering degrees at the University of Huelva, and the Electrical Engineering degree at the University of Seville, Spain. Ph.D. degree in 2004, in the Department of Systems Engineering and Automatic Control at the University of Seville.

Ph.D. European Award in 2005. Outstanding Paper Award in the IEEE Transactions on Automatic Control journal in 2006. In 1999 he joined as Research Assistant in that Department, where he is currently Professor and Researcher member of the Automatic and Robotics Institute. He also has been Visitor in the Laboratoire des Signaux & Systèmes (CNRS, France) repeatedly since 2005 up to today and, Academic Visitor Researching in the Electrical & Electronic Engineering Department as member of the Control & Power Group at Imperial College London in 2008 and 2009.

His research interests are in the fields of Nonlinear Control of Dynamical Systems with emphasis on Electro-Mechanical and Robotic systems. He is author of a number of research publications on Non-linear Control System Theory in Internationals Conferences and Scientific Journals.

Page 3: Constructive nonlinear smc

I. Introduction

Feedback Linearization

Sliding Mode Control

Control Lyapunov Function

Page 4: Constructive nonlinear smc

II. Background : Sliding Mode Control

The sliding condition:

Page 5: Constructive nonlinear smc

II. Background : Collocated Partial Feedback Linearization

Director of Center for

Autonomous Engineering Systems and Robotics at

the University of Illinois

at Urbana-Champaign

M. W. Spong

References:M. W. Spong, “Energy based control of a class of underactuated mechanical systems”, 1996 IFAC World Congress, July 1996.

Spong has shown that all underactuated systems in the form:

where andcan be globally partially linearized using a change of control

where and .

Page 6: Constructive nonlinear smc

II. Background : Collocated Partial Feedback Linearization

References:M. W. Spong, “Energy based control of a class of underactuated mechanical systems”, 1996 IFAC World Congress, July 1996.

Page 7: Constructive nonlinear smc

Remarks:

fully linearized system (using a change of

control) Impossible partially linearized system (q2 is transformed into a double integrator) Possible the new control u appears in the both nonlinear (internal dynamics) (q1, p1) subsystem and linear (q2, p2) subsystem the system has m-vector relative degree (2,…,2)T with respect to the output q2

this procedure is called collocated partial linearization suitable for energy based controller design unsuitable for any cascade controller design ex. Backstepping, DSC (not being in strict feedback form)

II. Background : Collocated Partial Feedback Linearization

References:M. W. Spong, “Energy based control of a class of underactuated mechanical systems”, 1996 IFAC World Congress, July 1996.

Page 8: Constructive nonlinear smc

III. Preliminaries : Constructive Feedback Linearization

The proposed “constructive” output:

References: J. A. Acosta and M. Lopez-Martinez “Constructive Feedback Linearization of Mechanical Systems with Friction and Underactuation Degree One”, Proceedings of the European Control Conference, 2007.

Defining the free smooth function

(decreasing of storage function of zero dynamics)

Output redesign:(the corresponding output to the passive output for the storage function)

Consider

Let

reduces the input-output map to

Constructive linearizing law:

The external controller:

Page 9: Constructive nonlinear smc

IV. Main Result

Page 10: Constructive nonlinear smc

IV. Main Result

Page 11: Constructive nonlinear smc

V. Applications : Pendulum on a Cart

Dynamic equations:

The partially feedback-linearized system:

Constructive sliding surface:

Find the linearizing law from

with

The total control input:

Page 12: Constructive nonlinear smc

V. Applications : Simulations of Pendulum on a Cart

Page 13: Constructive nonlinear smc

V. Applications : Simulations of Inertia Wheel Pendulum

Page 14: Constructive nonlinear smc

VI. Conclusions

2 dof. Underactuated Mechanical Systems

Collocated Partial Feedback Linearization

Design the Fictitious Output(using the concept of passive systems)

Feedback Linearization Controller + Sliding Mode Controller

Constructive Design of Sliding Surface