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 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(a(t))q(t)+F(a(t), it))=u(t), (13.1) where q(t)E R is the position vector, u(t)is the vector of actuation forces and torques. F:R×R→ R is a given vector- valued function,andM:R←→ RXK is a given function taking positive definite symmetric matrix values(the inertia matrix). When u= u(t)is fixed(for example, when u(t)=uo cos(t)is a harmonic excitation (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))(u(t)+F(q(t),i(t)) (13.2) transforms(13. 1)into a linear double integrator model Version of october 29. 2003Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 13: Feedback Linearization1 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) ∀ Rk is the position vector, u(t) is the vector of actuation forces and torques, F : Rk × Rk ≤� Rk is a given vector-valued function, and M : Rk ≤� Rk×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) = u0 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) 1Version of October 29, 2003