## 3 Controller Design
### TSMC for Matrix First-Order System  
Define a sliding surface for the system (7) by  
\[\mathbf{s}(t)=\mathbf{x}(t)-\mathbf{x}_{\rm d}(t)\,, \tag{23}\]  
where \(\mathbf{x}_{\rm d}(t)\) stands for the expected trajectory. In order to enable the sliding mode in the system, design the control law as  
\[\mathbf{u}(t)= \mathbf{H}(\mathbf{x})^{-1}\left(\mathbf{G}(\mathbf{x})+\dot{\mathbf{x}}_{\rm d}(t) \right)-\mathbf{H}(\mathbf{x})^{-1}\mathbf{S}(t)\mathbf{z}^{\rm max}\] \[-\sigma\mathbf{H}(\mathbf{x})^{-1}\mathbf{S}(t)\mathbf{s}_{\eta}(t)-\gamma\mathbf{H}( \mathbf{x})^{-1}\mathbf{s}(t)\,, \tag{24}\]  
where \(\sigma>0\) and \(\gamma>0\) are positive coefficients,  
\[\mathbf{S}(t)=\text{diag}\left(\text{sgn}\left(s_{1}(t)\right),\,\text{sgn}\left( s_{2}(t)\right),\,\ldots,\,\text{sgn}\left(s_{n}(t)\right)\right), \tag{25}\]  
\[\mathbf{s}_{\eta}(t)=\left[|s_{1}(t)|^{\eta}\quad|s_{2}(t)|^{\eta}\quad\ldots\quad| s_{n}(t)|^{\eta}\right]^{\rm T}\,, \tag{26}\]  
where \(\eta\) is a ratio of two odd positive integers, such that \(0<\eta<1\).  
The control input (24) can be considered to be a function of three scalar parameters: \(\sigma\), \(\gamma\), and \(\eta\). The optimal values of these parameters can be identified during the optimization procedure by minimizing a certain objective function, which is defined to achieve particular system performance goals. When the closed-loop system reaches the sliding surface (23), then \(\mathbf{s}(t)=\mathbf{0}\), and consequently, \(\mathbf{x}(t)=\mathbf{x}_{\rm d}(t)\) for \(t>t_{\rm r}\). This means that the trajectory of the system \(\mathbf{x}(t)\) reaches the desired trajectory \(\mathbf{x}_{\rm d}(t)\). Thus, the formulated problem is equivalent to the classical stabilization problem.  
**Theorem 2**.: _Under the Assumptions 1-3, the trajectory of the system (7), when forced by the control (24), will reach the sliding surface (23), starting from the initial condition \(\mathbf{x}_{0}\), in finite time and then remain on this surface._  
Proof.: Consider the following Lyapunov function candidate for the closed-loop system (7) and (24):  
\[V(t)=0.5\mathbf{s}(t)^{\rm T}\mathbf{s}(t). \tag{27}\]  
Calculating the time derivative of \(V\) and presenting it along the solutions of the system (7) gives  
\[\dot{V}(t) =\mathbf{s}(t)^{\rm T}\mathbf{\dot{s}}(t)\] \[=\mathbf{s}(t)^{\rm T}\left(\dot{\mathbf{x}}(t)-\dot{\mathbf{x}}_{\rm d}(t)\right)\] \[=\mathbf{s}(t)^{\rm T}\left(-\mathbf{G}(\mathbf{x})+\mathbf{H}(\mathbf{x})\mathbf{u}(t)+ \mathbf{z}(\dot{\mathbf{x}},\mathbf{x},t)-\dot{\mathbf{x}}_{\rm d}(t)\right)\,. \tag{28}\]Substitution of (24) into (28) yields  
\[\dot{V}(t)=-\mathbf{s}(t)^{\mathrm{T}}\mathbf{S}(t)\mathbf{z}^{\mathrm{max}}-\sigma\mathbf{s}(t)^ {\mathrm{T}}\mathbf{S}(t)\mathbf{s}_{\eta}(t)-\gamma\mathbf{s}(t)^{\mathrm{T}}\mathbf{s}(t)+\mathbf{ s}(t)^{\mathrm{T}}\mathbf{z}(\dot{\mathbf{x}},\mathbf{x},t)\,. \tag{29}\]  
By Assumption 2, it follows that  
\[\dot{V}(t)\leq-\gamma\,\|\mathbf{s}(t)\|^{2}-\sigma\,\|\mathbf{s}(t)\|^{\eta+1}=-2 \gamma V(t)-2^{\frac{\eta+1}{2}}\sigma V(t)^{\frac{\eta+1}{2}}\,. \tag{30}\]  
By taking \(\alpha=2\gamma>0\), \(\beta=2^{\frac{\eta+1}{2}}\sigma>0\), \(\kappa=\frac{\eta+1}{2}\), and according to the thesis of Lemma 1, the trajectory of the system (7) will reach the sliding surface in finite time, and after that, remain on this surface.