# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 4 Finite-Dimensional Parameterization and Ellipsoidal Constraints  
We introduce the ellipsoidal quality constraints, which are required to fulfill the demands for models within the system identification set (12). We consider the robust stability definition (11) for all models inside the system uncertainty set [33]:  
\[\underset{\omega,\theta\in U_{D}}{max}\left|\Delta\right|^{2}\leq\gamma. \tag{24}\]  
Using the quality measure (24), we require that about 95% of the estimated models will fulfill the constraints imposed by the ellipsoidal region. Let \(W_{n}\), \(W_{d}\), \(X_{n}\), and \(X_{d}\) be finite-dimensional transfer functions, and \(\mathrm{Y}_{n}\) is as follows:  
\[Y_{n}=Y_{n}^{*}Y_{n}, \tag{25}\]  
where \(\mathrm{Y}_{n}\) is some stable finite-dimensional transfer function and \(\mathrm{Y}_{d}\), \(\mathrm{K}_{d}\), and \(\mathrm{K}_{n}\) are defined in the same way. Furthermore, \(R\) is a positive definite matrix. A suggested quality measure of the model, including (24), is formulated as  
\[ \begin{array}{l}F(\omega,\eta)\leq\gamma,\;\forall\omega\;\text{and}\; \forall\;\eta\in\Gamma,\\ \Gamma=\Big{\{}\eta:(\eta-\eta_{0})^{T}R(\eta-\eta_{0})\leq 1\Big{\}},\end{array}  \tag{26}\]  
where  
\[F(\omega,\eta)=\frac{[W_{n}G(\eta)+X_{n}]^{*}Y_{n}[W_{n}G(\eta)+X_{n}]+\mathrm{ K}_{n}}{[W_{d}G(\eta)+X_{d}]^{*}Y_{d}[W_{d}G(\eta)+X_{d}]+\mathrm{K}_{d}}. \tag{27}\]  
The transfer function \(G(\omega,\eta)\) includes the parameter vector \(\eta\) which is not affected by the noise, and \(W_{n}\), \(W_{d}\), \(X_{n}\), \(X_{d}\), \(\mathrm{Y}_{n}\), \(X_{d}\), \(\mathrm{K}_{n}\), and \(\mathrm{K}_{d}\) are finite-dimensional stable transfer functions. The Equation (26), is a max-norm bound on \(F\), subject to \(\omega\), and it should be fulfilled for all \(\eta\) which belong to the set \(\Gamma\). The uncertainty set \(\Gamma\) is obtained using the prediction error method. In the considerations that follow it has been assumed that transfer functions are given by \(W_{n}=W_{d}=1\), \(X_{n}=-G_{0}\), \(Y_{n}=T\), \(X_{d}=\mathrm{K}_{n}=\mathrm{K}_{d}=0\). Considering the closed-loop robust stability criterion (11) and the transfer functions defined above, we need to fulfill (12) within the system identification set, using a fixed bound on the ellipsoidal quality constraint. This assumption will be implemented in the experimental part of the work. Another performance criterion that guarantees that input design is convex in \(P^{-1}\) arises from the KYP lemma. If the base functions are rational, the positivity of the inputsignal spectrum is represented by an LMI with the shortened coefficient vector. Then, \(\tau(\omega)\) is defined by  
\[ \begin{array}{l}\tau(\omega)=\Psi\big{(}e^{i\omega}\big{)}+\Psi^{*}\big{(}e^{i \omega}\big{)},\\ \Psi\big{(}e^{i\omega}\big{)}=\sum\limits_{k=0}^{K-1}\tau_{k}\Re_{k}\big{(}e^{i \omega}\big{)},\end{array}  \tag{28}\]  
using some base functions, namely, \(\Re_{k}\), \(k=0\),..., \(K-1\), and there is a set \(\{\Lambda_{k}\}\) expressed by  
\[ \begin{array}{l}\Lambda(\omega)\geq 0,\ \ \forall\omega,\\ \sum\limits_{k=0}^{P}\Lambda_{k}\Big{(}e^{-kj\omega}+e^{kj\omega}\Big{)}\geq 0,\ \ \forall\omega,\end{array}  \tag{29}\]  
where the elements \(\{\tau_{k}\}\) and the ingredients of the matrix \(R\) in (26) are linear in \(\{\Lambda_{k}\}\). Then, there exists a state-space realization \(\{A_{\tau}\), \(B_{\tau}\), \(C_{\tau}\), \(D_{\tau}\}\) of the positive part of \(\tau(\omega)\), where the variables \(\{\tau_{k}\}\) are linear in \(C_{\tau}\) and \(D_{\tau}\). Similarly, the same applies to the state-state representation \(\{A_{\Lambda}\), \(B_{\Lambda}\), \(C_{\Lambda}\), \(D_{\Lambda}\}\). Furthermore, it holds (26) if only \(Q_{\tau}=Q_{\tau}^{T}\) and \(Q_{\Lambda}=Q_{\Lambda}^{T}\) exists. All the theorems and their proofs used to formulate the input design problem (30) considering the KYP lemma are available in [27].