Defining the companion model

Let us linearize equation (24.7). Provided that X does not differ greatly from x and a is small, we may approximate the right-hand side by a first-order Taylor series about a point defined by a = a p,  

It is natural to define the outputs, of the linearized system to correspond to those of the original system  

Equations (24.22) and (24.24) define a linear system by which the state differences are driven by parameter variations about their optimal value, and we name this the companion model . The properties of this companion model will clearly depend on Jx and J,. A controllability test will show whether all the elements of the 0 vector can be influenced by variations in the a vector. Provided that the system is controllable, then it should be possible to drive 0 to any transient desired, in the absence of limits on the amplitude or frequency of variation of a. 

The outputs, if, of the companion model may be related to the error between the distorted model and the plant by the following steps  

Thus when a = 0, then if = 0 and hence e(t) = emin(f)-However driving the companion model with model-matching parameter variations, a (t), will cause Y(r) = z(t), and hence e(t) = z(t) - Y(t) = 0. Accordingly, 

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