constructive nonlinear smc
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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
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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.
I. Introduction
Feedback Linearization
Sliding Mode Control
Control Lyapunov Function
II. Background : Sliding Mode Control
The sliding condition:
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 .
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.
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.
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:
IV. Main Result
IV. Main Result
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:
V. Applications : Simulations of Pendulum on a Cart
V. Applications : Simulations of Inertia Wheel Pendulum
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