# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 1 Introduction
Some simplifications of the cost function of the identification task (with respect to unknown model parameters), which are claimed to be more suited for practical applications, have been proposed in [21]. A systematic heuristic procedure to build an input signal for the identification of the nonlinear multi-input multi-output (MIMO) dynamic model of the closed-loop rotor/radial active magnetic bearing (rotor/RAMB) system has been proposed in [22]. For the synthesis of the input signal which can reduce a potential correlation problem caused by perturbation signals, an optimal parallel amplitude-modulated pseudo-random binary sequence (PAPRBS) has been proposed. By applying the (PAPRBS) generator, the accuracy of the identified MIMO dynamics could be guaranteed to some extent. Identification of Frequency Response Functions (FRFs) is essential in different control issues of mechatronic systems--from system design and validation to controller design and diagnostics. In the last years, optimal design of experiments for FRF identification of multivariable motion systems focused the interest of researchers. In [23], the method of FRF identification subject to element-wise power constraints has been proposed. Developed computational techniques for multivariable excitation design frameworks are fast and suitable for application to large dimensional realistic systems. For the synthesis of control algorithms for industrial robots, both the modeling and the identification problems are challenging (since the required accuracy is very high, the system complex and the real-time requirements are sometimes critical). In [24], it has been proved that by conducting experiments only in the selected robot configurations, the experiment time can be reduced and the accuracy of the parameter estimate can be increased (both in terms of bias and standard deviation). The study also indicates that conducting more experiments in the optimal configurations can be equivalent to experiments in additional configurations. The methodology presented in this paper is classified as an optimal average performance input design for control [25]. The \(\mathcal{H}_{\infty}\) norm of the weighted relative error and the positive real Kalman-Yakubovich-Popov (KYP) lemma are used to appraise the stability and performance indices of the closed-loop system [26; 27]. The minimum FIR filter out put power and the corresponding spectrum selection problem have been specified as the least-costly experiment design [28]. In the manuscript, the context of real-time closed-loop system identification is considered. The industrial processes are mostly run as closed-loop control systems, and it is often not possible to open the loop. Generally, optimal experiment design problems correspond to non-convex programming tasks, and their applicability to the industry has been restricted. The contribution of this work is to propose a framework for converting the input design problem into a convex optimization under linear matrix inequality constraints. The solution of the input spectrum design, applying minimization of the worst-case quality measure (over all models in an ellipsoidal region), has been presented in [10]. We propose a different approach where the input spectrum power minimization is performed, subject to the fixed bound on the ellipsoidal quality constraint. The results of the closed-loop system identification considering the optimization problem performed jointly, concerning the fixed compensator and the optimal spectrum of the input signal, are presented. Another objective has been to check the numerical stability issues of the methods used for evaluating the ellipsoidal quality constraints, which have a great impact on real-time system identification. The frequency grid and KYP methods have been compared for the worst-case minimization between the identified plant and models in the uncertainty region.