# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 2 Robust Estimation of Model Parameters
Let us introduce \(\kappa(t)=[u(t)\ e_{0}(t)]^{T}\) where \(e_{0}(t)\) represents the true noise. We consider the information matrix, \(P^{-1}\), as a function of the frequency distribution of the spectrum \(\Phi_{\kappa}\). The spectrum distribution is as follows:  
\[\Phi_{\kappa}= \begin{bmatrix}\Phi_{u}&\Phi_{ue}\\ \Phi_{ue}^{*}&\lambda_{0}\end{bmatrix} , \tag{7}\]  
where \(\Phi_{u}\) is the input signal spectrum and \(\Phi_{ue}^{*}\) denotes the complex conjugate of the cross-spectrum between \(u(t)\) and \(e_{0}(t)\). The information matrix, \(P^{-1}\), is a measure of the errors in the parameters, is related to the spectrum distribution \(\Phi_{\kappa}\), and is described as [30]  
\[P^{-1}=\frac{1}{2\pi\lambda_{0}}\int\limits_{-\pi}^{\pi}{F(q,\theta_{0})\Phi_ {\kappa}F^{*}(q,\theta_{0})d\omega}, \tag{8}\]  
Figure 1: The closed–loop system diagram. where \(\psi(t,\theta_{0})=F(q,\theta_{0})\kappa(t)\) and \(F(q,\theta_{0})=\left[F_{u}(q,\theta_{0})\quad F_{e}(q,\theta_{0})\ \right]\) with  
\[F_{u}(q,\theta)=H^{-1}(q,\theta)\frac{dG(q,\theta)}{d\theta}, \tag{9}\]  
\[F_{e}(q,\theta)=H^{-1}(q,\theta)\frac{dH(q,\theta)}{d\theta}. \tag{10}\]  
The relation between the parameters of the model and the covariance matrix \(P\) is obtained, using [PERSON]'s theorem. It should be noted that the input spectrum \(\Phi_{u}\) and the cross-spectrum \(\Phi_{ue}\) are used to create the information matrix \(P^{-1}\). The other variables in (8) apply to the real system [27]. In control design applications, it is a common practice to have frequency-by-frequency conditions on the function estimate error. One such condition is robust stability, expressed as the weighted relative error in the following form [31]:  
\[\Delta(\omega,\theta)=T(\omega)\frac{G_{0}(\omega,\theta_{0})-G(\omega,\theta )}{G(\omega,\theta)}, \tag{11}\]  
where \(G_{0}(\omega,\theta_{0})\) is the true system and \(G(\omega,\theta)\) is the model of the system. When the weight function \(T(\omega)\) is designed as the sensitivity complementary function, then \(\mathcal{H}_{\infty}\) norm of (11) is used to quantify the stability and performance of the closed-loop system [31]. When the parameter vector of the model \(G(\omega,\theta)\) is estimated during an identification experiment, then it is included in an uncertainty set. The parameter uncertainty region is restricted by the \(\chi\) constant:  
\[U_{\theta}=\Big{\{}\theta:N(\theta-\theta_{0})^{T}P^{-1}(\Phi_{\chi})(\theta- \theta_{0})\leq\chi\Big{\}}, \tag{12}\]  
where \(\chi\) defines the size of the uncertainty region [30]. The goal of such an identification experiment is to guarantee that the weighted relative error \(\left\|\Delta(\omega,\theta)\right\|_{\infty}\leq\gamma\) covers all models in the uncertainty set (12) and \(\gamma\) is the preselected constant value. The complementary sensitivity function is computed as the zero-order-hold discretization of  
\[T(s)=\frac{\omega_{0}^{2}}{s^{2}+2\xi\omega_{0}s+\omega_{0}^{2}}, \tag{13}\]  
where \(\omega_{0}\) is used to change the bandwidth of \(T(s)\) with a damping value \(\xi=0.7\). In the experimental part, it was assumed that the size of the confidence region \(\chi\) is determined like that of \(Pr(\chi^{2}(n)\leq\chi)=0.95\) where \(n\) is the number of parameters of the model \(G(\omega,\theta)\).