Eigenvalue analysis characteristic value

Some of the linear combinations will be well defined and others poorly defined. The latter may be eliminated in a filtering procedure, referred to in the literature under the names characteristic value filtering, eigenvalue filtering, and principal component analysis. If the parameter set is not homogeneous, but includes different types, relative scaling is important. Watkin (1994) recommends that the unit be scaled such that similar shifts in all parameters lead to similar changes in the error function S. 

The above equation is called the characteristic equation of the system. It is the system s most important dynamic feature. The values of s that satisfy Eq. (6.67) are called the roots of the characteristic equation (they are also called the eigenvalues of the system). Their values, as we will shortly show, will dictate if the system is fast or slow, stable or unstable, overdamped or underdamped. Dynamic analysis and controller design consists of finding out the values of the roots of the characteristic equation of the system and changing their values to give the desired response. The rest of this book is devoted to looking at roots of characteristic equations. They are an extremely important concept that you should fully understand. ... 

Next, we review concepts from linear algebra that are used in stability analysis. Suppose that A is an n X n matrix where n is the dimension of the state vector, x. Let X denote an eigenvalue of A. By definition, the eigenvalues are the n values of X that satisfy the equation x = Ax (Strang, 2005). The corresponding values of x are the eigenvectors of A. The eigenvalues are the roots of the characteristic equation. 

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