# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 7 Conclusions  
The main focus of this work is on real-time optimal input spectrum design for the closed-loop system identification task. The minimization of the input spectrum power has been performed as a standard convex optimization problem, considering LMI constraints. When the compensator is close to the complementary sensitivity function, the \(\mathcal{H}_{\infty}\) norm of weighted relative error (11) has been used to ensure stability and measure the performance indices of the cascaded tanks model of the real system. It has been observed that all estimated models are clustered inside the contour \(\left\|\Delta(\omega,\theta)\right\|_{\infty}=0.1\), as desired. It has also been proven that the identification performed subject to the fixed bound on the ellipsoidal quality constraint yields consistent estimates of the true system. In fact, more than 90% of the models with the obtained parameter values meet the performance constraint.  
The results show the benefits of conducting closed-loop system identification experiments, as discussed in the paper, where the time for acquiring measurement data is shortened and the power of the excitation signal is minimized. It has been found that the input design problem considering the frequency grid is much more numerically stable than the KYP method implemented in Moose2. It is related to linear matrix inequalities that become too large when using the KYP lemma, which is why they are the bottleneck for solving optimal input design problems. The problems encountered using the KYP lemma can probably be solved sequentially using an adaptive frequency grid instead and will be reformulated in future works. For real-time system identification purposes, the frequency grid method is preferred. The multi-objective optimization concerning both the parameters of the compensator and the external reference signal power will also be studied.  
There are still several open problems in application-oriented input design for closed-loop system identification, that should be resolved. The results of the optimal input design could be extended to MIMO systems used in industry. The more general class of controllers with implicit control laws also should be analyzed. It should be investigated whether the lower order of the chosen model has an impact on the quality of closed-loop identification. The application-oriented input design for nonlinear system identification (not linearized) is another topic that should be further explored. The fractional-order models often guarantee a more precise description of system dynamics, because they are represented as the infinite distribution of time constants. Computation time has great importance in real-time applications, which is why truncated time window approximation methods (e.g., the long-memory prediction error method) for fractional-order system identification should be applied.  
Conceptualization, [PERSON]; methodology, W.J.; software, W.J. and M.S.; validation, W.J. and M.S.; formal analysis, W.J. and M.S.; investigation, W.J.; writing--original draft preparation, W.J.; writing--review and editing, W.J. and M.S.; visualization, W.J. All authors have read and agreed to the published version of the manuscript.  
The present study was supported by a grant WZ/WI-III/5/2023 ([PERSON]) and WE/WE-IAEE/1/2020 ([PERSON]) from the Bialystok University of Technology and was funded from the resources for research by the Polish Ministry of Education and Science.  
Not applicable.  
Data Availability Statement: The data presented in this study are available upon request from the first author [PERSON].