# Application-Oriented Input Spectrum Design in Closed-Loop Identification
## 4 Finite-Dimensional Parameterization and Ellipsoidal Constraints
When the input is shaped by the FIR filter sequence with \(m\) frequency points and the coefficients \(\tau_{k}\) in (28) are the auto-covariance sequence \(r_{k}\) of the input \(\Phi_{u}(\omega)\), the input variance constraint can be defined as \(r_{0}\leq\alpha\) (see Lemma 3.3 in [27]). Then, the constraint input design problem based on the KYP lemma is fulfilled for all models inside the ellipsoidal region (12). The minimum required power \(\alpha\) of the input signal can be obtained from  
\[  
When the cross-sectional surface of the tank outlet hole, considering an ideal sharp-edged orifice, is known, the outflow of water of each tank can be established using Torricelli's law and is given by  
\[Q_{out}(t)=a\cdot\sqrt{2 gh(t)}\,, \tag{32}\]  
where \(a\) is the cross-sectional area of the hole, and the gravitational constant value \(g\) is equal to \(9.8\) m/s\({}^{2}\). Substituting expression (32) with expression (31) and assuming that the output of the first tank is interconnected with the input of the second tank, the system of two water tanks can be described by the following nonlinear differential equations:  
\[\left\{ \begin{array}{ll}A_{1}\frac{dh_{1}(t)}{dt}=-a_{1}\cdot\sqrt{2 gh_{1}(t )}+Q_{in}(t)\\ A_{2}\frac{dh_{2}(t)}{dt}=a_{1}\cdot\sqrt{2 gh_{1}(t)}-a_{2}\cdot\sqrt{2 gh_{2}(t )}\end{array} \right. \begin{array}{ll}h_{1}(0)=h_{10},\\ h_{2}(0)=h_{20}.\end{array}  \tag{33}\]  
where \(i=1\), \(2\) represents the corresponding tank. We have performed the following substitutions \(Q_{in}(t)=u(t)\), \(x_{1}(t)=h_{1}(t)\), \(x_{2}(t)=h_{2}(t)\), \(y(t)=h_{1}(t)\) to obtain the ordinary form of the first-order model state-space equations:  
\[\left\{ \begin{array}{ll}\dot{x}_{1}=-\frac{a_{1}}{A_{1}}\cdot\sqrt{2 gx_{1}} +\frac{1}{A_{1}}u&x_{1}(0)=h_{10},\\ \dot{x}_{2}=\frac{a_{1}}{A_{2}}\cdot\sqrt{2 gx_{1}}-\frac{a_{2}}{A_{2}}\cdot \sqrt{2 gx_{2}}&x_{2}(0)=h_{20}.\end{array} \right. \tag{34}\]  
where \(x_{1}=x_{1}(t,a_{1})\), \(x_{2}=x_{2}(t,a_{1},a_{2})\). The tanks' water levels have obvious physical limitations:  
\[h_{i,\,max}\geq x_{i}(t)\;\geq 0\;\;i=1,\,2. \tag{35}\]  
The physical parameters and the limitations of the water tanks are shown in Table 1.  
Figure 2: The interacting water tanks process scheme. The nonlinear model of the system (34) was linearized around the steady-state point \(x_{1}^{0}\) = 1.274 [m], \(x_{2}^{0}\) = 0.885 [m], and then discretized. The linearized model matrices, computed using a first-order Taylor expansion, are  
\[ \begin{array}{c}A_{I}=\left[\begin{array}{cc}-\tau_{1}&0\\ \tau_{3}&-\tau_{4}\end{array}\right],\ \ B_{I}=\left[\begin{array}{cc}\frac{u_{0}}{A_{1}} \end{array}\right],\ C_{I}=\left[\begin{array}{cc}1&0\end{array}\right],\\ \tau_{1}=\frac{a_{1}}{A_{1}}\sqrt{\frac{\delta}{2x_{1}^{0}}},\ \tau_{3}=\frac{a_{1}}{A_{2}} \sqrt{\frac{\delta}{2x_{1}^{0}}},\ \tau_{4}=\frac{a_{2}}{A_{2}}\sqrt{\frac{ \delta}{2x_{2}^{0}}}.\end{array}  \tag{36}\]  
Then, the zero-order hold discretization of (34) can be formulated as  
\[A_{d}=e^{A_{1}},\ \ B_{d}=\int\limits_{0}^{1}e^{A_{1}(1-t)}B_{I}dt,\ \ C_{d}=C_{I}. \tag{37}\]  
The discrete-time state-space water tanks model obtained for the 1 Hz sampling rate is  
\[A_{d}= \begin{bmatrix}0.937&0\\ 0.116&0.828\end{bmatrix} ,\ \ B_{d}= \begin{bmatrix}0.161\\ 0.010\end{bmatrix} . \tag{38}\]  
In the experimental part of this paper, the input-output description of the process plant defined by (1) is considered.