## 3 Control Design Based on Identified Model  
Herein, as example, the \(H\)o controller design is performed. To synthesize it, a state space linear mathematical model is needed. Due to the lack of priory information of constructive parameters regarding the axial piston pump, the preferred approach for modelling is to evaluate a \"black box\"-type model [27; 28]. The data acquisition experiment is performed according to scheme presented in Figure 3. The applied input signal is white Gaussian noise in order to obtain sufficient excitation. As a result, the discrete time data for  
Figure 1: System for rapid prototyping of control algorithms for axial piston pumps. Figure 2: Photo of the implemented system. the output pump flow rate, pressure, and input signal (control action) are measured. In Figure 4, part of the measured dataset is depicted. It is supposed that the dynamics of the axial piston pump can be described by a \"black box\" discrete-time state-space model  
\[x((k+1)\ T_{S})=Ax(k\ T_{S})+Bu(k\ T_{S})+Ke(k\ T_{S}), \tag{1}\]  
\[y(k\ T_{S})=C\ x(k\ T_{S})+D\ u(k\ T_{S})+e(k\ T_{S}), \tag{2}\]  
where \(x=(x_{1},x_{2},\cdots,x_{n})^{T}\) and \(y=\left(y_{flow},y_{pres}\right)^{T}\) are state and output vectors, respectively; \(y_{flow}\) is the pump flow rate in liters per minute; \(y_{pres}\) is pump output pressure in bars; \(u\) is the control signal in millivolts. The real matrices \(A\in R^{n\times n}\), \(B\in R^{n\times 1}\), \(C\in R^{2\times n}\), \(D\in R^{2\times 1}\), and \(K\in R^{n\times 2}\) present system dynamics for a given operating point. The \(T_{S}=0.01\ \ \mathrm{s}\) is the sample time. The residual signal vector \(e=\left(e_{1},e_{2}\right)^{T}\) represents the error that occurred between the measurement and model data. It is well-known that if the model is adequate then the residuals are a white Gaussian sequence. The estimation with prediction error method model of the sixth order is adequate, as shown in [29]. Herein, only some results for its validation are presented. In Figure 5, the results from the comparison between outputs \(y=\left(y_{flow},y_{pres}\right)^{T}\) of the sixth order model and measured pump flow rate and pressure are presented. As can be seen, the obtained fit for the pump flow rate is approximately 61% and that for pump pressure is 64%, which shows the performance of the estimated sixth order model. Figure 4: Pump flow rate, pressure, and control signal used for identification experiment. Figure 3: Structure scheme of open-loop identification experiment. The obtained values for the identified model parameters are  
\[ \begin{split}\text{A}=\begin{pmatrix}0&1&0&0&0&0\\ 0&0&1&0&0&0\\ -0.3&0.43&0.84&-0.01&0.04&-0.03\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0.57&-11.17&10.83&0.72&-1.62&1.9\\ \end{pmatrix},\text{ B}=\begin{pmatrix}0.005\\ 0.161\\ 0\\ 0.029\\ 0.046\\ 0.058\end{pmatrix},\text{ K}=\begin{pmatrix}0&0.075\\ 0.082&-0.015\\ 0.031&0.031\\ -0.033&0.259\\ 0.141&-0.029\end{pmatrix},\\ \text{C}=\begin{pmatrix}1&0&0&0&0&0\\ 0&0&0&1&0&0\end{pmatrix}\text{ and D}=\begin{pmatrix}0\\ 0\end{pmatrix}.\end{split}  \tag{3}\]  
The model (1) is used for \(H\infty\) controller design according to the scheme presented in Figure 6, where \(W_{p}\) and \(W_{u}\) are the transfer functions of performance weighting filters. As can be seen, the error performance filter \(W_{p}\) has three inputs that form a control output \(y_{c}\) as:  
\[y_{c}= \begin{bmatrix}e_{flow}&e_{\text{int},flow}&-y_{pres}\end{bmatrix} ^{T}, \tag{4}\]  
where:  
\[e_{flow}=r_{flow}-y_{flow} \tag{5}\]  
Figure 5: Comparison between the model simulated output and the identification data set. Figure 6: Structure scheme of extended plant used for \(H\infty\) controller design.