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Unformatted text preview: Linearization Bob Bitmead, April 19, 2010. Linearized ODEs Consider the linearization of the first-order vector ordinary differential equation ˙ x = f ( x,u ) . We use Taylor’s theorem to linearize the solution about some (¯ x t , ¯ u t ) functions. Often the choice of point (¯ x t , ¯ u t ) about which to linearize is made as a constant equilibrium ( x e ,u e ) or some known function, such as a known solution for a specific initial value. However, as we shall see, the choice of (¯ x t , ¯ u t ) need not be so restricted. Apply Taylor’s theorem at (¯ x t , ¯ u t ) to the write the solution as x t = ¯ x t + ˜ x t , u t = ¯ u t + ˜ u t and expand vector function f accordingly. ˙ x t = ˙ ¯ x t + ˙ ˜ x t = f (¯ x t , ¯ u t ) + ∂f ∂x ¯ x t , ¯ u t ˜ x t + ∂f ∂u ¯ x t , ¯ u t ˜ u t + O ( | ˜ x t | 2 + | ˜ u t | 2 ) . (1) The ODE (1) describes how the solution ( x t ,u t ) varies about nominal value (¯ x t , ¯ u t ) . The error term can be quantified in many different and revealing ways. If we drop the error term and recognize that we are making a local (close to (¯ x t , ¯ u t ) ) approximation, then the linearized ODE is ˙ ˜ x t = ∂f ∂x ¯ x t , ¯ u t ˜ x t + ∂f ∂u ¯ x t , ¯ u t ˜ u t + f (¯ x t , ¯ u t )- ˙ ¯ x t . (2) For linearization about the equilibrium, we have ˙ ¯ x e = 0 and f (¯ x e , ¯ u e ) = 0 , and the final two additive terms are zero. For linearization about a particular solution, we have ˙ ¯ x t = f (¯ x t , ¯ u t ) and the last two terms add to zero. In these two most common applications of linearization, we arrive at ˙ ˜ x t = ∂f ∂x ¯ x t , ¯...
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