Some Fundamental Control Theory I: Controllability, Observability, and Duality

A new aspect of Hankel matrices via Krylov matrix

“…It also plays an important role in determining controllability and observability for a linear time-invariant control system (see [15]). …”

Some Fundamental Control Theory II: Feedback Linearization of Single Input Nonlinear Systems

“…By the use of state feedback, u = Kx with K a 1 x n matrix, such systems may be expressed in the particularly simple form, z?1) = v, where v is a new reference input that is available for control purposes. For convenience we summarize the main result of Part I in Theorem 1; see [12] for definitions of the relevant concepts. Theorem 1.…”

“…In Part I we showed that, conversely, if one of the matrices on the left is nonsingular (for some b, respectively c), then the other matrix is also nonsingular (for some c, respectively b). Nonsingularity on the right in (5) implies an observability condition and a controllability condition [12]. The geometric interpretation of the zeros on the right hand side is that the null space of the linear functional y = cTx is the (i1 -1) dimensional space, span {b,Ab,..., A"-2b}.…”

“…If such a transformation is possible, one can exploit the special control-theoretic properties discussed in [12] for that special form. In such a case, a nonlinear system can be controlled using linear control methods.…”

“…The idea is that feedback by it = a(x) simplifies the system equations by cancelling nonlinearities when possible, and then subsequent controls v are available to control the dynamics in a desired manner. The next definition recalls the behavior of the special outputs that yield observability and a companion form in the linear case of [12]. The definition is motivated by the form of the terms that appear when a function is differentiated repeatedly along system (6a).…”