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.

A new geometric approach providing the minimum-energy issue for inverse model control-related perfect regulation of linear time-invariant multi-input/single-output plants described in the discrete-time state-space framework is proposed in the paper. Recent results have shown that the minimum-norm T-inverse does not guarantee the minimum-energy perfect control design, which has been confirmed by heuristic studies only. The new proposal, postulated throughout the manuscript, certifies the potential of nonunique σ-inverse regarding the minimum-energy behavior of inverse model control-based structures. After application of the proposed geometric approach dedicated to some class of state-space systems, we can precisely calculate the total energy of the multivariable perfect control runs. Thus, the analytical new methodology allows to obtain the minimum-energy inverse model control schemes, what constitutes the main accomplishment of the paper. Additionally, the aim of future analytical exploration covering the entire class of right-invertible state-space systems is clearly focused.

The application of the switching control framework to the perfect control algorithm is presented in this paper. Employing the nonunique matrix inverses, the different closed-loop properties are obtained and further enhanced with possible switching methodology implementation. Simulation examples performed in the MATLAB/Simulink environment clearly show that the new framework can lead to benefits in terms of the control energy, speed, and robustness of the perfect control law. The possibility of transferring the new obtained results to the symmetrical nonlinear plants seems to be immediate.

In this paper a new geometric-based approach to the maximum-speed state- and output-variable pair under perfect control design is proposed. Up to now, the maximum-speed perfect control has been connected with the pole-free instance having all closed-loop control system poles located at zero. The new results obtained through this paper show that the maximum-speed perfect control algorithm should be associated with the stable-pole case. The analytical results supported by the simulation example confirm the truthness of presented here innovative approach.

In this paper a minimum-energy problem of perfect control algorithm dedicated to LTI MIMO plants described in the discrete-time state-space framework is shown. Following the new facts concerning the above-mentioned issue, a new solution based on the mathematical norms utilizing some generalized inverse formulas is given in this paper. The simulation instance confirms the correctness of a new intriguing method, which certainly leads to the final proof of whole, undiscovered yet, modern control theory.

In this paper, the powerful issue of a non-full rank inverse model control investigation is provided. It is broadly known, that the inverse-based methodology is associated with the full-rank control systems only. However, following the recently obtained authors’ results in this matter, it should be concluded, that for single-delayed non-full rank state-space plants, the inverse model control-related perfect control expression can also be established. It is shown here, that for all right- and left-oriented multivariable LTI non-full rank systems, including the square ones, governed by the discrete-time state-space domain arranging the zero-reference value, the maximum-speed pole-free instance of such inverse model control strategy can be achieved. Thus, the new non-full rank algorithm does not coincide with the z-transfer-function scenario, which additionally sounds the intriguing peculiarity. The innovative content of the manuscript, that has never been seen before, is strongly supported by the numerical examples. Henceforth, the presented methodology covers the entire set of LTI MIMO state-space-oriented plants in the discrete-time domain, which is also a consequence of conducted research investigation in the past.