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MIT16_30F10_lec22_slides

# MIT16_30F10_lec22_slides - 16.30/31 Feedback Control...

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16.30/31 Feedback Control Systems Overview of Nonlinear Control Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology December 3, 2010 E. Frazzoli (MIT) Nonlinear Control December 3, 2010 1 / 14

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An overview of nonlinear control design methods Extend applicability of linear design methods: Gain scheduling Integrator anti-windup schemes Geometric control Feedback linearization Dynamics inversion Differential ﬂatness Adaptive control Neural network augmentation Lyapunov-based methods/Contraction theory Control Lyapunov Functions Sliding mode control Backstepping Computational/logic approaches Hybrid systems Model Predictive Control E. Frazzoli (MIT) Nonlinear Control December 3, 2010 2 / 14
Gain scheduling Nonlinear system: x ˙ = f ( x , u ). Choose n equilibrium points, i.e., ( x i , u i ), such that f ( x i , u i ) = 0, i = 1 , . . . n . For each of these equilibria, linearize the system and design a “local” control law u i ( x ) = u i K ( x x i ) for the linearization. A global control law consists of: Choose the right control law, as a function of the state: i = σ ( x ) Use that control law: u ( x ) = u σ ( x ) ( x ) E. Frazzoli (MIT) Nonlinear Control December 3, 2010 3 / 14

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Control Lyapunov functions Nonlinear system: x ˙ = f ( x ) + g ( x ) u , with equilibrium at x = 0 A function V : x �→ V ( x ) is a Control Lyapunov Function if It is positive definite V (0) = 0. It is always possible to find u such that V ˙ = V f ( x ) + V g ( x ) u 0 . x x If V is a CLF, it is always possible to design a control law ensuring V ˙ 0! E. Frazzoli (MIT) Nonlinear Control December 3, 2010 4 / 14
Differential Flatness A dynamical system d x dt = f ( x , u ) , z = h ( x , u ) .

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MIT16_30F10_lec22_slides - 16.30/31 Feedback Control...

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