# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 3 Spectrum Representation  
The only factor affecting the covariance matrix \(P\) is the spectrum of the excitation signal. Proper parameterization of the input spectrum leads to the formulation of the finite-dimensional convex problem [33]. Consider that the closed-loop system identification is performed using the spectral matrix \(\Phi_{\kappa}\) defined in (7). This matrix can be expressed as  
\[\Phi_{\kappa}(\omega)=\sum_{k=-\infty}^{\infty}\widetilde{c}_{k}\Re_{k}\Big{(} e^{j\omega}\Big{)}, \tag{19}\]  
where the scalar basis functions \(\big{\{}\Re_{k}\big{(}e^{j\omega}\big{)}\big{\}}_{k=0}^{\infty}\) are proper, stable, and rational with \(\widetilde{c}_{k}=\widetilde{c}_{-k}^{T}\). Using proper parameterization of the signal spectrum, it is possible to solve a finite-dimensional convex optimization problem with matrix coefficients \(\widetilde{c}_{k}\). We assume that \(\Re_{k}\big{(}e^{-j\omega}\big{)}=\Re_{k}^{*}\big{(}e^{j\omega}\big{)}\) so that \(\widetilde{c}_{k}\in\mathbb{R}\). Therefore, the factors \(\widetilde{c}_{k}\) must satisfy the below constraint:  
\[\Phi_{\kappa}(\omega)\succeq 0,\ \ \forall\omega. \tag{20}\]  
The most common choice of the basis function is \(\Re_{k}\big{(}e^{j\omega}\big{)}=e^{-j\omega k}\). A finite-dimensional parameterization is then acquired by substituting the spectrum \(\Phi_{\kappa}(\omega)\) in (19) with the following approximation:  
\[\Phi_{\kappa}(\omega)=\Psi\Big{(}e^{j\omega}\Big{)}+\Psi^{*}\Big{(}e^{j\omega }\Big{)}, \tag{21}\]  
where \(\Psi(e^{j\omega})\) is a finite-dimensional estimation of the positive real part of \(\Phi_{\kappa}(\omega)\) with \(\Psi^{*}(e^{j\omega})=\Psi^{T}(e^{-j\omega})\). Finally, \(m\) initial components of (19) are used to obtain the input spectrum:  
\[\Psi\Big{(}e^{j\omega}\Big{)}=\sum_{k=0}^{m-1}c_{k}\Re_{k}\Big{(}e^{j\omega} \Big{)}, \tag{22}\]  
where \(c_{-k}^{T}=c_{k}\) and \(m\) is chosen large enough. Finite-dimensional parameterization can be achieved using various methods. One of them is the positive real lemma that arises from the KYP technique [27]:  
**Lemma 1.**_If \(A\), \(B\), \(C\), \(D\) is a controllable state-space implementation of the positive real part of the input spectrum \(\Psi\big{(}e^{i\omega}\big{)}\), then a \(Q=Q^{T}\) like the following exists:_  
\[\mathrm{K}(Q,\{A,B,C,D\})=\begin{bmatrix}Q-A^{T}QA&-A^{T}QB\\ -B^{T}QA&-B^{T}QB\end{bmatrix}+\begin{bmatrix}0&C^{T}\\ C&D+D^{T}\end{bmatrix}\succeq 0, \tag{23}\]  
_if and only if \(\Phi_{\kappa}(\omega)=\sum\limits_{k=0}^{m-1}c_{k}\big{[}\Re_{k}\big{(}e^{i \omega}\big{)}+\Re_{k}^{*}(e^{i\omega})\big{]}\geq 0\), \(\forall\omega\), and \(\mathrm{K}\) is the Kautz function._  
Thus, the necessary condition considering (20) to hold for the shortened sequence is the inequality (23) if a \(Q\) matrix exists. The formulation becomes an LMI in \(\mathrm{c}_{k}\) and \(Q\) if only \(C\) and \(D\) matrices are linearly dependent on the \(c_{k}\) coefficients [27]. Since the finite-dimensional spectrum parameterization can be approximated using the m-order FIR filter, the positive real KYP lemma is verified in the paper.