# Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System
## 2 Classical Sliding Mode Control  
The main concept in sliding mode control is to replace an \(n\)th order tracking problem by a stabilization problem of the order \((n-1)\)[40]. Looking at the dynamic system:  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},u), \tag{3}\]  
the guaranteed stabilizing, variable structure input signal \(u\) is chosen according to:  
\[u= \begin{cases}u^{+},&\text{if }s(\mathbf{x},t)>0\\ u_{\text{eq}},&\text{if }s(\mathbf{x},t)=0\\ u^{-},&\text{if }s(\mathbf{x},t)<0.\end{cases}  \tag{4}\]  
For that purpose, a switching function \(s(\mathbf{x},t)\) needs to be defined. In classical sliding mode procedures, which are applied to solve trajectory tracking tasks, this switching function is usually composed of the output tracking error and a weighted linear combination of the temporal derivatives of this error signal up to the order \(\delta-1\), where \(\delta\) denotes the system's relative degree [24]. In the remainder of this section, it is assumed that the relative degree \(\delta\) equals the system order according to \(n=\delta\). Then, a stability proof of the closed-loop dynamics can be performed in terms of a square Lyapunov function candidate as was mentioned in the Introduction of this paper. In general, the switching function \(s(\mathbf{x},t)\) determines under which conditions the input signal of the system changes structurally, so that \(u\) becomes discontinuous across \(s(\mathbf{x},t)=0\)[40; 41]. In such a way, the sliding surface, which is defined by \(s(\mathbf{x},t)=0\), divides the state space into two mutually disjoint domains where either \(s(\mathbf{x},t)>0\) or \(s(\mathbf{x},t)<0\) holds. In those domains, different dynamics of the system occur due to different definitions of the system's input \(u\). The sliding surface is included in neither of these subdomains as a trajectory, whereas each point of the sliding surface \(s(\mathbf{x},t)=0\) has to be accessible by a trajectory from either side [41]. To make the system converge to the sliding surface, the input has to be chosen in such a way that it always makes the state trajectories approach the sliding surface, i.e., they have to be \"pushed\" towards \(s(\mathbf{x},t)=0\). This phase is called the transient or reaching phase. The possibilities for the design of controllers for this reaching phase can be found, for example, in [42; 43] and the references therein. Note that the design of the corresponding control for this phase needs to account for input constraints in terms of the limited range of the admissible system inputs \(u\). If scenarios occur, in which the system input reaches its lower or upper saturation value in such a phase, a careful stability proof is inevitable, cf. [26]. The next phase, following the completion of the reaching phase, is called sliding mode. The system's dynamics are then represented by the switching function itself. As mentioned before, this function typically has a lower order than the open-loop system [38; 39]. Thus, in this state, the system's dynamic behavior only depends on the parameters included in the switching function, which means that this form of control is not only robust against uncertainties in plant parameters, but also against disturbances satisfying suitable matching conditions (for example, disturbances entering the state Equations (3) via the same channels as the system inputs \(u\)). From an intuitive point of view, this results from the fact that even if the system leaves the sliding surface slightly, it will immediately return to it, because of the switching-type behavior of the input \(u\) and the proven stability (i.e., attraction property) of the sliding surface. However, the switching between structurally different inputs as defined in (4) may lead to a phenomenon called chattering, which implies high control activity and thus leads to a rapid wear of the actuator. In addition, the associated high frequencies may also excite dynamics that were not considered in the model [40].