# Continuous-Time Perfect Control Algorithm--A State Feedback Approach
## 1 Introduction  
The design of control systems for multivariable plants is a wide research topic that is still subject to considerable worldwide interest. Systems with multiple input and output variables often deliver challenges and opportunities that are not available in the Single-Input Single-Output (SISO) plants. For example, in the Multi-Input Multi-Output (MIMO) approach, the Inverse Model Control (IMC) extends the potential of nonunique solutions for a given problem [1; 2; 3; 4; 5; 6; 7].  
One of the most popular inverse model controls is the Minimum Variance Control (MVC) used for both discrete and continuous plants [8; 9; 10; 11]. On the other hand, the perfect control, being a deterministic equivalent of MVC, has been developed mainly for the discrete-time framework. Therefore, some efforts to define the Continuous-Time Perfect Control (CTPC) have recently been undertaken [12]. The preliminary studies have shown that this topic is yet an unexplored area of control theory. Thus, a new approach to CTPC design is confirmed within this paper.  
The key part of perfect control synthesis for multivariable systems is the proper solution of inverse problem derived from the control law. For decades, the unique Moore-Penrose inverse has been used due to its minimum-norm property [13; 14; 15; 16; 17; 18]. On the other hand, admission of the nonunique inverses has resulted in measurable benefits in terms of control speed, energy minimization, or control stability [19]. Thus, in this paper, a short comparison of two selected inverses, namely, unique (Moore-Penrose) right \(T\)-inverse and nonunique right \(\sigma\)-inverse, is shown. Of course, this comparison will be conducted in the context of CTPC employment.  
Stability of such a control scenario will be analyzed with the application of well-known state-feedback or pole-placement methods [20; 21]. Additionally, in contrast to the preliminary studies [12], here, the state-feedback form is given in a straightforward manner, allowing us to obtain the closed-loop poles immediately for all classes of LTI MIMO state-space systems [22]. Of course, as in the discrete-time equivalent, the stability should be understood in terms of the Bounded-Input Bounded-State (BIBS) or input-to-state approach [23].  
However, both control law analysis and synthesis will be made using continuous-time solvers provided by some incremental time-originated step \(\Delta t\), which can be found,e.g., in the Matlab environment [24; 25]. The influence of the time interval will be taken into account during the entirety of the process of perfect control realization and validation.  
This paper is organized as follows. After a short introduction, the system description is given. In Section 3, a brief reminder considering unique and nonunique inverses of nonsquare matrices is presented. The crucial CTPC law is given in Section 4, together with the full calculation procedure covering its closed-loop behavior. Discussion over stability of such a perfect control plant is made with respect to the solver-related peculiarities, leading to interesting observations shown in Section 5. Simulation study given in the penultimate section shows that the proposed control law can successfully be applied in the Matlab/Simulink environment. Finally, the conclusions and open problems are manifested.