CS545_Lecture_6 - CS545Contents VI Control Theory II Linear Stability Analysis Linearization of Nonlinear Systems Discretization See

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CS545—Contents VI Control Theory II Linear Stability Analysis Linearization of Nonlinear Systems Discretization Reading Assignment for Next Class See http://www-clmc.usc.edu/~cs545
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Stability Analysis Given the control system How can we the show that a particular choice of a controller generates a stable control system? In order to get started, consider whether the generic dynamical system is stable: ˙ x = f x , u ( ) or ˙ x = Ax + Bu ˙ x = f x ( ) or ˙ x =
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Equilibrium Points and Stability Definition of an Equilibrium Point A state x is an equilibrium state (or equilibrium point) of the system if once x(t) is equal to x, it remains equal to x for all future time. Mathematically, this means: Definition of Stability An equilibrium state x is said to be stable, if, for any R>0, there exists r>0, such that if ||x(0)||<r,then ||x(t)||<R for all t 0. Otherwise, the equilibrium point is unstable. r R Equilibrium
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Linear Stability Analysis (Local Stability Analysis) What is needed at the outset: The system model (linear or nonlinear) An equilibrium point The linearization of the system about the equilibrium point Then, we have the linear(ized) system: This system is stable if and only if: The complete stability definitions are: ˜ ˙ x = A x = x * ˜ x where ˜ x = x - x * ( ) REAL eig ( A ) ( )< 0 continuous system discrete system
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CS545_Lecture_6 - CS545Contents VI Control Theory II Linear Stability Analysis Linearization of Nonlinear Systems Discretization See

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