New interesting facts about minimum-energy perfect control for LTI nonsquare state-space systems

A study on a new criterion for minimum-energy perfect control in the state-space framework

“…Note, that the right inverse can be any from the wide range of unique and nonunique ones described in detail, for example, as given by Hunek 19 and Levine. 23 The issues concerning the inverses are discussed later.…”

“…The result of the above conjecture can also be obtained through an application of the singular value decomposition (SVD) procedure. The detail information concerning the perfect control energy nature can be found in Hunek’s 19 study.…”

“…17 It shall be emphasized that the classical Moore–Penrose inverses, renamed the minimum-norm and least-squares ones dedicated to right- and left-invertible plants, respectively, usually do not meet the requirements of control robustness. 15,19 Therefore, new inverses in forms of τ , σ , S , H and HM structures have been introduced in order to ensure both the minimum-energy and stability behaviors of perfect control algorithm. 13,23,27 In special cases, those inverses reduce to the Moore–Penrose ones.…”

“…17,18 For such plants, the use of so-called degrees of freedom of nonunique right inverses gives quantifiable minimum-energy benefits, in opposite to the unique T - inverse , for which the said control algorithm often does not exist. 17,19 Therefore, the various inverses have been introduced over the past years, with particular reference to the authors’ τ , σ , S , H and HM inverses involved in the design of robust stable-pole or pole-free control structures. 15 Therefore, the desired tool, next to the aforementioned inverse approaches, seems to be the different performance indices allowing to estimate the energy of the perfect control runs in a objective way.…”