## 3 Controller Design  
### TSMC for Matrix Second-Order System  
As a first step, define a sliding surface for the system (1) as  
\[\mathbf{s}(t)=\mathbf{\Lambda}\mathbf{e}(t)-\mathbf{Q}\mathbf{\psi}(\mathbf{e}_{\rm x})\,, \tag{8}\]  
where \(\mathbf{\Lambda}=[\mathbf{P}\ \mathbf{I}_{n}]\), \(\mathbf{P}\in\mathbb{R}^{n\times n}\) represents a gain matrix, \(\mathbf{I}_{n}\) is an identity matrix of size \(n\), \(\mathbf{Q}\in\mathbb{R}^{n\times n}\), \(\mathbf{Q}=\mathbf{Q}^{\rm T}>0\) is a positive-definite matrix, \(\mathbf{\psi}:\mathbb{R}^{n}\to\mathbb{R}^{n}\) is an \(n\times n\) matrix with entries that are non-linear, non-negative functions of \(\mathbf{e}_{\rm x}\), and \(\mathbf{e}(t)=[\mathbf{e}_{\rm x}(t)\ \mathbf{e}_{\rm dx}(t)]^{\rm T}\) is a vector composed of the actual and expected trajectory \(\mathbf{x}_{\rm d}(t)\) and \(\dot{\mathbf{x}}_{\rm d}(t)\); that is  
\[\mathbf{e}_{\rm x}(t)=\mathbf{x}(t)-\mathbf{x}_{\rm d}(t)\,, \tag{9}\]  
\[\mathbf{e}_{\rm dx}(t)=\dot{\mathbf{x}}(t)-\dot{\mathbf{x}}_{\rm d}(t)\,. \tag{10}\]  
As a next step, a control law shall be designed to guarantee the existence of the sliding mode in the closed-loop system. Consider the following formula:  
\[\mathbf{u}(t)= -\mathbf{H}(\mathbf{x})^{-1}\mathbf{\Lambda}\mathbf{W}(\dot{\mathbf{x}},\mathbf{x})+\mathbf{ H}(\mathbf{x})^{-1}\mathbf{Q}\frac{\mathrm{d}\mathbf{\psi}(\mathbf{e}_{\rm x})}{\mathrm{d}t}\] \[+\mathbf{H}(\mathbf{x})^{-1}\mathbf{P}\left(\dot{\mathbf{x}}_{\rm d}(t)+\dot{\mathbf{ x}}_{\rm d}(t)\right)\] \[-\mathbf{H}(\mathbf{x})^{-1}\mathbf{S}(t)\mathbf{z}^{\rm max}-\sigma\mathbf{H}(\mathbf{x} )^{-1}\mathbf{S}(t)\mathbf{s}_{\eta}(t)\] \[-\gamma\mathbf{H}(\mathbf{x})^{-1}\mathbf{s}(t)\,, \tag{11}\]  
where \(\sigma>0\) and \(\gamma>0\) are positive coefficients,  
\[\mathbf{W}(\dot{\mathbf{x}},\mathbf{x})= \begin{bmatrix}\dot{\mathbf{x}}\\ -\mathbf{G}(\mathbf{x})-\mathbf{F}(\dot{\mathbf{x}},\mathbf{x})\dot{\mathbf{x}}\end{bmatrix} \,, \tag{12}\]  
\[\mathbf{S}(t)=\mathrm{diag}\left(\mathrm{sgn}\left(s_{1}(t)\right),\,\mathrm{sgn} \left(s_{2}(t)\right),\,\ldots,\,\mathrm{sgn}\left(s_{n}(t)\right)\right), \tag{13}\]  
\[\mathbf{s}_{\eta}(t)= \begin{bmatrix}|s_{1}(t)|^{\eta}&|s_{2}(t)|^{\eta}&\ldots&|s_ {n}(t)|^{\eta}\end{bmatrix} ^{\rm T}\,, \tag{14}\]  
and \(\eta\) is a ratio of two odd positive integers, such that \(0<\eta<1\). The control input (11) to achieve the required sliding surface dynamics can be considered as a function of several parameters; that is, the matrices \(\mathbf{P}\), \(\mathbf{Q}\), and \(\mathbf{\psi}\) and the scalars \(\gamma\), \(\sigma\), and \(\eta\). Some of these parameters influence the sliding surface (8) itself, as well. We can then define a performance index (objective function) for the closed-loop system for evaluation and comparison purposes. The evaluationand comparison criteria can, for example, include indicators related to tracking accuracy, tracking response, robust performance, control effort, chattering behavior, and so on. Weighting factors can be applied to these indicators to define the level of their importance in the performance index. Considering the criteria that are important from an application-specific point of view and combining them into an objective function gives us an optimization problem, the solution of which leads to the best parameters. The problem of selecting the best values of the parameters, in the sense of the defined objective function, has already shown up in the optimal control design literature [43; 44; 45]. Under the Assumptions 1-3, the trajectory of the system (1), (5), when forced by the control (11), will reach the sliding surface (8), starting from the initial condition (5), within a finite time and then remain on this surface.