## 4 Model-Assisted Reduced-Order Extended State Observer (ESO) Based Cascade Controller Design  
### Model-Assisted Reduced-Order ESO  
In this section, the reduced-order ESO is designed to estimate the lumped disturbances in \(x_{2}\)-subsystem and \(x_{3}\)-subsystem. Since \(x_{2}\) and \(x_{3}\) are available, based on the method in [33], specified reduced-order ESO is given as follows:  
\[\left\{ \begin{array}{l}\dot{z}_{2}=-\beta_{1}z_{2}-\beta_{1}^{2}x_{2}-\beta _{1}r_{1}x_{3}\\ \dot{d}_{1}=z_{2}+\beta_{1}x_{2},\beta_{1}>0\end{array} \right. \tag{22}\]  
\[\left\{ \begin{array}{l}\dot{z}_{3}=-\beta_{2}z_{3}-\beta_{2}^{2}x_{3}-\beta _{2}r_{2}u\\ \dot{d}_{2}=z_{3}+\beta_{2}x_{3},\beta_{2}>0\end{array} \right. \tag{23}\]  
where \(\dot{d}_{1}\) is the estimate of the lumped uncertainties (the sum term of \(f_{1}(x_{1},x_{2})\) and \(d_{1}\)) in the \(x_{2}\)-subsystem, \(\dot{d}_{2}\) is the estimate of the lumped uncertainties (the sum term of \(f_{2}(x_{1},x_{2},x_{3})\) and \(d_{2}\)) in the \(x_{3}\)-subsystem, \(z_{2}\) and \(z_{3}\) are the auxiliary state of the observer, \(\beta_{1}\) and \(\beta_{2}\) are the observer gain. Note that the internal dynamics \(f_{1}(x_{1},x_{2})\) and \(f_{2}(x_{1},x_{2},x_{3})\) are available, to reduce the estimate burden of the observer, with this given model information, the model-assisted reduced-order ESO is proposed as follows:  
\[\left\{ \begin{array}{l}\dot{z}_{2}=-\beta_{1}z_{2}-\beta_{1}^{2}x_{2}-\beta_{ 1}[f_{1}(x_{1},x_{2})+r_{1}x_{3}]\\ \dot{d}_{1}=z_{2}+\beta_{1}x_{2},\beta_{1}>0\end{array} \right. \tag{24}\]  
\[\left\{ \begin{array}{l}\dot{z}_{3}=-\beta_{2}z_{3}-\beta_{2}^{2}x_{3}-\beta_{ 2}[f_{2}(x_{1},x_{2},x_{3})+r_{2}u]\\ \dot{d}_{2}=z_{3}+\beta_{2}x_{3},\beta_{2}>0\end{array} \right. \tag{25}\]  
where \(\dot{d}_{1}\) is the estimate of the lumped uncertainty \(d_{1}\) in the \(x_{2}\)-subsystem, \(\dot{d}_{2}\) is the estimate of the lumped uncertainty \(d_{2}\) in the \(x_{3}\)-subsystem. Based on (3) and (24), the error dynamics of disturbance estimate for \(x_{2}\)-subsystem can be derived as follows:  
\[ \begin{array}{ll}\dot{e}_{o1}&=\dot{d}_{1}-\dot{d}_{1}=\dot{d}_{1}-(\dot{z} _{2}+\beta_{1}\dot{x}_{2})\\ &=\dot{d}_{1}+\beta_{1}z_{2}+\beta_{1}^{2}x_{2}+\beta_{1}[f_{1}(x_{1},x_{2})+r _{1}x_{3}]-\beta_{1}[f_{1}(x_{1},x_{2})+r_{1}x_{3}+d_{1}]\\ &=\dot{d}_{1}+\beta_{1}(z_{2}+\beta_{1}x_{2})-\beta_{1}d_{1}\\ &=\dot{d}_{1}+\beta_{1}(\dot{d}_{1}-d_{1})\\ &=-\beta_{1}e_{o1}+\dot{d}_{1}\end{array}  \tag{26}\]  
Using the same derivation process, the error dynamics of disturbance estimate for \(x_{3}\)-subsystem can be derived as follows:  
\[\dot{e}_{o2}=-\beta_{2}e_{o2}+\dot{d}_{2} \tag{27}\]  
Together with (26) and (27), the overall error dynamics of disturbance estimate for the LEMA system can be described in the following compact form:  
\[\dot{\mathbf{E}}_{o}=\mathbf{A}_{o}\mathbf{E}_{o}+\dot{\mathbf{d}}_{o} \tag{28}\]  
where  
\[\mathbf{E}_{o}=\left[ \begin{array}{cc}e_{o1}&e_{o2}\end{array} \right]^{\rm T}= \left[ \begin{array}{cc}d_{1}-\dot{d}_{1}&d_{2}-\dot{d}_{2}\end{array} \right]^ {\rm T}\!,\mathbf{d}_{o}=\left[ \begin{array}{cc}\dot{d}_{1}&\dot{d}_{2}\end{array}  \right]^{\rm T}\!,\mathbf{A}_{o}=\text{diag}\Big{\{} \begin{array}{cc}-\beta_{1}& -\beta_{2}\end{array} \Big{\}}\]  
Suppose the eigenvalues of the matrix \(\mathbf{A}_{o}\) have negative real parts, i.e., the matrix \(\mathbf{A}_{o}\) is a Hurwitz matrix, so there exists a symmetric positive definite matrix \(\mathbf{P}\) and \(\mathbf{Q}\), such that  
\[\mathbf{A}_{o}^{\rm T}\mathbf{P}+\mathbf{P}\mathbf{A}_{o}=-\mathbf{Q} \tag{29}\]  
Let \(\min(|\beta_{i}|),i=1,2\) denote the smallest eigenvalue of \(\mathbf{A}_{o}\).