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.

Due to the constant trend of building wind turbine sets and photovoltaic cells, the so-called renewable energy sources, it is important to develop a mathematical model that will allow us to analyse the economic viability of their operation. By using this model, multivariate calculations should be carried out and developed in the form of universal nomograms that will allow us to estimate this viability almost on an immediate basis. In this paper, such a model for the unit cost of electricity production from renewable energy sources is presented. It is based on a continuous-time Net Present Value notation. The multi-variant calculations performed with it take into account different subsidy levels and operating time of the renewable energy sources. The paper presents values of unit costs of electricity production from renewable energy sources and values of maximum subsidies for this production. It turns out that subsidies are necessary for renewable energy sources to exist on the energy market. An additional disadvantage is the fact that renewable energy sources are not able to provide a stable electricity supply throughout the year.

In this paper, we propose the application of a new geometric procedure in order to calculate a set of transmission zeros of a propagation environment. Since the transmission zeros play a crucial role in modern communication systems, there is a need to apply the efficient solutions characterized by a maximum speed operation. It turns out that the classical method based on the Smith–McMillan factorization is time-consuming, so its contribution to the detection of transmission zeros could be unsatisfactory. Therefore, in order to fill the gap, we present a new algorithm strictly dedicated to the multivariable telecommunications systems described by the transfer-function approach. Consequently, a set of new achievements resulted, particularly in terms of computational efforts. Indeed, the proposed procedure allows us to overcome obstacles derived from technological limitations. The representative simulation examples confirm the great potential of this new method. Finally, it has been pointed out that the newly introduced geometric-originated approach has significantly reduced the computational burden. Indeed, for the randomly selected matrix of the $5\times 5$ dimension describing the sensor-related propagation environment, two representative scenarios were performed in order to manifest the crucial properties. In the first scenario, the sets of multiple transmission zeros were analyzed, ultimately leading to intriguing results. The Smith–McMillan solution took three times longer to discover the mentioned sets. On the other hand, the second instance brought us the same result. Naturally, the discussed difference has increased as a function of the number of matrix elements. For the square matrices involving 100 components, we have observed the respective differences, both over $Q_I=100$ and $Q_{II}=60$. It should be emphasized that the finding derived from the Smith–McMillan factorization corresponds to the geometric-related approach in the context of some mechanisms. This is particularly visible when appointing the greatest common divisors.

A new nonlinear algorithm covering the inverse model control (IMC) methodology is presented in this article. The perfect control procedure is devoted to the multivariable systems defined in the discrete-time state-space domain. Following the recently introduced perfect control-originated approaches employing the linear continuous- and discrete-time plants of different domains, the new nonlinear instance of such control strategy is proposed here. The innovative scenario follows the behaviors characteristic for its predecessors, in particular in terms of maximum-speed and maximum-accuracy properties. It should be emphasized that the new solution is more general than the existing linear ones, providing the complex original peculiarities. The numerical examples clearly confirm the correctness of the proposed methodology and discuss the key control stability requirement.

In the previous research study it has been pointed out that employing the approximation of the induced matrix norm can be useful during design of minimum-energy perfect control law for LTI MIMO discrete-time state-space plants. It has also been shown that this scenario was related to the case involving the closed-loop control matrix norm being less than unity, which was preserved by proper selection of applied nonunique inverses. This paper focuses on the systems which have been excluded from our past considerations. The methodology of minimum-energy approach for plants with norm greater than one is postulated here and some peculiarities in this matter additionally are given. This innovative method allows to predefine in a simple way a minimum-energy behavior of this intriguing control procedure.