System structure and singular control

Equivalence of AR‐representations in the light of the impulsive‐smooth behaviour

“…Furthermore, ' imp can be decomposed to a direct sum of two subalgebras ' pÀimp the purely impulsive distributions (linear combinations of d and its distributional derivatives) and ' sm the smooth distributions. f 2 ' imp can be written uniquely as f ¼ f 1 þ f 2 where its impulsive part is f 1 and its smooth part is f 2 : Then f ð0þÞ :¼ lim t#0 f 2 ðtÞ ¼ f 2 ð0þÞ: If f 2 ' sm ; then the distributional derivative of f ; f * d ð1Þ ; equals f ð1Þ þ f ð0þÞ (with f ð0þÞ ¼ f ð0þÞd), where f ðiÞ denotes the distribution that corresponds to the ordinary ith derivative of f on R þ : For more on the properties of ' imp see [20,21]. Consider a non-regular linear time-invariant system described by the AR-representation S : Að@ÞxðtÞ ¼ 0; t 2 ½0; þ1Þ ð1Þ…”

Modelling, Well-Posedness, and Stability of Switched Electrical Networks

“…For the moment, we restrict ourselves to so-called Bohl functions (sines, cosines, polynomials, exponentials and their sums and products) as inputs w, which leads to Bohl functions as solutions to (11) (see [10]). More precisely, a function f is called a Bohl function (or Bohl type) if f (t) = He F t G for some matrices F , G, and H of appropriate sizes.…”

The role of the generalised continuous algebraic Riccati equation in impulse-free continuous-time singular LQ optimal control

“…[1], [11]), whereas when R is positive semidefinite, the problem is called singular. The singular cases have been treated within the framework of geometric control theory, see for example [9], [18], [15], [13] and the references cited therein. In particular, in [9] and [18] it was proved that an optimal solution of the singular LQ problem exists for all initial conditions if the class of allowable controls is extended to include distributions.…”