# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 1 Introduction  
Identification methods for control have received great interest during the last decades [1, 2, 3]. Many industrial processes reveal slow response which can cause long and costly identification experiments. It has been reported that three-quarters of the total costs related to real-life control projects are absorbed by modeling [4]. The costs of the identification experiment can be associated with the experiment duration and properties of the input signal. Another complex cost can be caused by the requirement of acceptable control performance when providing the model-based control design. The typical industrial practice for model identification purposes is to use the application-oriented input design [5]. The framework is based on an optimal input design, where the objective is to find a trade-off between minimal departure from nominal working conditions and the accuracy of the model parameters to be estimated [6, 7]. The advantage of this model-based method relies on including the cost of the experiment in the input design framework (IDF) through the objective function to be minimized. Application-oriented input design is the cheapest identification method, ensuring the performance of the plant model [8]. In system identification, the relation between the assumed accuracy of the designated model and the experimental costs should be balanced. The optimal input design methods in the open-loop system have received considerable attention [1]. Typical optimal input design problem corresponds to non-convex solvers and, therefore, the applicability of the input signal design in the industry has been limited. A way to solve the problemof model accuracy has been applied to high-order expressions (i.e., Box-Jenkins models) which endow the identified model with a certain robustness [9]. Unfortunately, these expressions are not convenient for handling frequency-wise constraints used in many control applications. Considering these limitations, particular attention to the optimal input design has been paid to finite-order models [10; 11]. The results indicate reasonable approximation accuracy also for models that do not exceed the real system order. It has been shown that convex optimization methods are efficient solutions to many open-loop input design problems [12]. In many real-world processes, the control performance degrades when the accuracy of the model of the system is getting worse. The feedback control instability often causes unexpected consequences. The industrial processes are mostly run as closed-loop control systems, and it is often not feasible to open the loop. When the prior process data are achievable, it is possible to execute a closed-loop identification experiment using the chosen controller. Such a closed-loop experiment should be stable and have a short duration [13; 14]. Another important thing is to ensure that the identification experiment is plant-friendly and meets industrial demands [15; 16; 17]. The spectrum of the excitation signal affects the model parameters to be estimated during the identification experiment. Therefore, the input design problem could be obtained by the signal spectrum approximation using the finite impulse response (FIR) process, where the optimal filter coefficients are computed by convex optimization [18]. Some interesting approaches to the problem of optimal input signal synthesis in the frequency domain for dedicated fields of applications have been published in recent years, and some of them have been presented in [19]. In the paper [20], a heuristic solution has been suggested to tune a bounded amplitude multisine control signal in a closed-loop setup. The solution is based on the combination of two algorithms--the first one minimizes the Crest Factor of the control signal and the second one shapes (tunes) the spectrum of the control to obtain convergence to the desired reference spectrum.