Linealizacion

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1.1.1

Linearization via Taylor Series

In order to linearize general nonlinear systems, we will use the Taylor Series expansion of functions. Consider a function f (x) of a single variable x,and suppose that x is a point such that f (¯) = 0. In this ¯ x case, the point x is called an equilibrium point of the system x = f (x), since we have x = 0 when x = x ¯ ˙ ˙ ¯ (i.e., the system reachesan equilibrium at x). Recall that the Taylor Series expansion of f (x) around the ¯ point x is given by ¯ f (x) = f (¯) + x This can be written as f (x) = f (¯) + x df dx (x − x) + higher order terms.¯
x=¯ x a

df dx

(x − x) + ¯
x=¯ x

1 d2 f 2 dx2

(x − x)2 + ¯
x=¯ x

1 d3 f 6 dx3

(x − x)3 + · · · . ¯
x=¯ x

For x suﬃciently close to x, these higher order terms will be veryclose to zero, and so we can drop them ¯ to obtain the approximation f (x) ≈ f (¯) + a(x − x) . x ¯ Since f (¯) = 0, the nonlinear diﬀerential equation x = f (x) can be approximated near theequilibrium x ˙ point by x = a(x − x) . ˙ ¯ To complete the linearization, we deﬁne the perturbation state (also known as delta state) δx = x − x, ¯ and using the fact that δx = x, we obtain the linearizedmodel ˙ ˙ δx = aδx . ˙ Note that this linear model is valid only near the equilibrium point (how “near” depends on how nonlinear the function is). Extension To Functions of Multiple States and Inputs Theextension to functions of multiple states and inputs is very similar to the above procedure. Suppose the evolution of state xi is given by xi = fi (x1 , x2 , . . . , xn , u1 , u2 , . . . , um ) , ˙for some general function fi . Suppose that the equilibrium points are given by x1 , x2 , . . . , xn , u1 , u2 , . . . , um , ¯ ¯ ¯ ¯ ¯ ¯ so that fi (¯1 , x2 , . . . , xn , u1 , u2 , . . . , um ) = 0 ∀i∈ {1, 2, . . . , n} . x ¯ ¯ ¯ ¯ ¯ Note that the equilibrium point should make all of the functions fi equal to zero, so that all states in the system stop moving when they reach equilibrium. The...