## 5 Discussion and Conclusions  
It should be noted that the system described by the matrix-vector second-order differential equation (1) is not equivalent to the system described by the matrix-vector differential equation of the first-order (7). Indeed, the system (1) can be rewritten equivalently into the first-order form as:  
\[\dot{\bar{\mathbf{x}}}(t)+\tilde{\mathbf{G}}(\bar{\mathbf{x}})=\tilde{\mathbf{H}}(\bar{\mathbf{x}}) \mathbf{u}(t)+\tilde{z}(\dot{\bar{\mathbf{x}}},\bar{\mathbf{x}},t)\,, \tag{39}\]  
where  
\[\bar{\mathbf{x}}= \begin{bmatrix}\mathbf{x}\\ \dot{\bar{\mathbf{x}}}\end{bmatrix} ,\ \ \bar{\mathbf{z}}= \begin{bmatrix}\mathbf{0}\\ \mathbf{z}\end{bmatrix} ,\ \ \tilde{\mathbf{G}}(\bar{\mathbf{x}})= \begin{bmatrix}\dot{\mathbf{x}} \\ \mathbf{F}(\dot{\mathbf{x}},\mathbf{x})\dot{\bar{\mathbf{x}}}+\mathbf{G}(\mathbf{x})\end{bmatrix} ,\ \ \text{ and }\ \tilde{\mathbf{H}}(\bar{\mathbf{x}})= \begin{bmatrix}\mathbf{0}\\ \mathbf{H}(\mathbf{x})\end{bmatrix} \,. \tag{40}\]  
It is easy to show that the matrix \(\tilde{\mathbf{H}}\) is non-singular, and that Assumption 3 cannot hold. This means that both systems cannot be considered to be equivalent and should be discussed separately. This is a natural conclusion as, generally, the dynamic properties of second-order differential equations are different to those of first-order differential equations. However, the second class can be considered as an approximation of the first one, which occurs for slowly-varying systems where the second derivative of the system state can be neglected, when compared to the first derivative. It is well-known that TSMC is very effective in systems subjected to both parameter uncertainties and disturbances. The parameters of the systems considered in this paper had the form of the matrices \(\mathbf{F}\), \(\mathbf{G}\), and \(\mathbf{H}\). The elements of these matrices were, in general, non-linear functions of the system state variables. If the parameter uncertainties can be modeled as continuous functions of the state variables, then it is mathematically proved, by Theorems 1 and 2, that such uncertainties have no influence on the performance of the closed-loop systems, which means that the design control schemes are robust against parameter uncertainties. Such a situation has been demonstrated in the provided examples, where the parameters of the RLC and RC electrical circuits (that is, the capacitances \(C\), \(C_{1}\), \(C_{2}\) and resistance \(R\)) contained such types of uncertainties. The case when the parameters of the systems (that is, the elements of the matrices \(\mathbf{F}\), \(\mathbf{G}\), and \(\mathbf{H}\)) are also functions of time \(t\) has formally neither been covered nor proven. In such a case, we would need to apply an alternative theory more suitable for non-stationary systems. The effectiveness of the proposed control approach has been demonstrated on simple electrical circuits. However, the stabilization scheme can also be applied successfully to mechanical, electromechanical [47], electronic, mechatronic, and even pure software systems. The most important aspect of this paper is the proof that the trajectory of the closed-loop system reaches, and remains on, the sliding surface after a finite amount of time. This finite time can be calculated analytically. Furthermore, the simulation of the dynamical behavior of electrical circuits with non-linear elements showed the effectiveness of the proposed method. The numerical calculations and computer simulations were performed in the MathWorks(tm) MATLAB(r)/Simulink(r) environment (The MathWorks, Inc., Natick, MA, USA). It should be noticed here that in practical applications (and in computer simulations), sliding mode controllers induce high-frequency oscillations, known as chattering in the system output and control. Chattering is a harmful phenomenon, which degrades the performance of the system and can damage its physical components.