Application of Generalized Inverses in the Minimum-Energy Perfect Control Theory

Application of generalized inverses in solving the inverse model control (IMC)-oriented minimum-energy perfect control design (PCD) problem for linear time-invariant multi-input/multi-output systems governed by the discrete-time
$d$
-state-space structure is presented in this article. For this reason, an appropriate class of polynomial generalized inverses is investigated. Moreover, it can be stated that the nonunique right
$\sigma $
-inverse, based on properly selected so-called degrees of freedom (DOFs), outperforms the well-known unique Moore–Penrose (MP) minimum-norm right
$T$
-inverse in terms of the energy consumption of perfect control (PC) input signals. However, the analytical confirmation of such an intriguing statement has only been established for the special class of the single-delayed plants with a zero reference value. Moreover, because of the complexity of the IMC, the objects with a time delay
$d>1$
having a nonzero setpoint have never been analytically explored in regard to the PC energy context until now. Thus, the newly introduced analytical methods defined in this article allow us to designate the proper forms of
$\sigma $
-inverse-related DOFs that guarantee the minimum-energy PCD for the entire set of LTI multivariable nonsquare systems with the delay
$d \geq 1$
. Moreover, the new original results, supported by numerical examples, strongly contest the well-established control and systems theory canons related to the optimal minimum-energy-originated peculiarity of the MP pseudoinverse.