lec13_6243_2003

# lec13_6243_2003 - Massachusetts Institute of Technology...

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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 13: Feedback Linearization 1 Using control authority to transform nonlinear models into linear ones is one of the most commonly used ideas of practical nonlinear control design. Generally, the trick helps one to recognize “simple” nonlinear feedback design tasks. 13.1 Motivation and objectives In this section, we give a motivating example and state technical objectives of theory of feedback linearization. 13.1.1 Example: fully actuated mechanical systems Equations of rather general mechanical systems can be written in the form M ( q ( t ))¨ q ( t ) + F ( q ( t ) ,q ˙( t )) = u ( t ) , (13.1) where q ( t ) ∀ R k is the position vector, u ( t ) is the vector of actuation forces and torques, F : R k × R k ≤ R k is a given vector-valued function, and M : R k ≤ R k × k is a given function taking positive definite symmetric matrix values (the inertia matrix). When u = u ( t ) is fixed (for example, when u ( t ) = u cos( t ) is a harmonic excitation), analysis of (13.1) is usually an extremely diﬃcult task. However, when u ( t ) is an unrestricted control effort to be chosen, a simple change of control variable u ( t ) = M ( q ( t ))( v ( t ) + F ( q ( t ) ,q ˙( t ))) (13.2) transforms (13.1) into a linear double integrator model q ¨( t ) = v ( t ) . (13.3) 1 Version of October 29, 2003 2 The transformation from (13.1) to (13.3) is a typical example of feedback linearization , which uses a strong control authority to simplify system equations. For example, when (13.1) is an underactuated model, i.e. when u ( t ) is restricted to a given subspace in R k , the transformation in (13.2) is not valid. Similarly, if u ( t ) must satisfy an a-priori bound, conversion from v to u according to (13.2) is not always possible. In addition, feedback linearization relies on access to accurate information , in the current example – precise knowledge of functions M, F and precise measurement of coor- dinates q ( t ) and velocities ˙ q ( t ). While in some cases (including the setup of (13.1)) one can extend the benefits of feedback linearization to approximately known and imperfectly observed models, information ﬂow constraints remain a serious...
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lec13_6243_2003 - Massachusetts Institute of Technology...

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