# Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System
## 1 Introduction
To find the invariant set of the respective closed-loop control system, Lyapunov-like functions are employed, which transform the stability properties of dynamical systems described by a set of ordinary differential equations into inequalities. Those inequalities can be evaluated using interval computation and finally lead to the desired enclosures of the invariant sets to be determined. Note that the proposed technique does not focus on determining invariant sets by using the La Salle invariance theorem. Instead, only temporal derivatives of the aforementioned Lyapunov-like functions are required during the solution procedure. To depict the examined problem and to distinguish it from a state-of-the-art sliding mode control design [34; 35; 36; 37; 38; 39], consider a single-input system:  
\[\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t),u(\mathbf{x}(t))), \tag{1}\]  
where \(\mathbf{x}\in\mathbb{R}^{n}\) is the state vector and \(\mathbf{f}:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}\) is the evolution function. For simplicity, assume that the input in (1) is given by the state dependent expression \(u(\mathbf{x}(t))\in\mathbb{R}\). The time dependencies of the state vector \(\mathbf{x}:=\mathbf{x}(t)\in\mathbb{R}^{n}\) and of the control vector \(u:=u(t)\) are furthermore not explicitly denoted. Additionally, consider a sliding surface:  
\[s(\mathbf{x})=0\,\quad\text{where}\quad s:\mathbb{R}^{n}\mapsto\mathbb{R}. \tag{2}\]  
For the system model defined in Equation (1), the control input \(u\) should be found in such a way that the function \(\frac{1}{2}s^{2}(\mathbf{x})\), which can be interpreted as a Lyapunov function candidate, decreases monotonically up to the point where perfect tracking of the predefined path (respectively, trajectory) occurs due to \(s(\mathbf{x})=0\). As a consequence, the state trajectories of the controlled system will converge to an invariant set in which the sliding surface should be contained. The paper is structured in the following way. Firstly, the general concept of sliding mode approaches for the control design of nonlinear systems is described in Section 2 with the aim of distinguishing the investigated problem from classical techniques in this field. Then, the proposed novel method for a sliding mode-type controller design is explained in Section 3 in combination with the use of a Grobner basis to solve Bezout's identity. Section 4 deals with the conditions to be satisfied for an invariant set, which are evaluated numerically by using interval analysis as described in Section 5. In Section 6, several applications of the developed method are presented. These applicationsinclude cases with sliding surfaces represented by closed curves in the phase space. Moreover, straight line segments are considered as well. In addition, a concluding example is investigated where Bezout's identity is not solvable. However, the interval based method for enclosing the resulting invariant set of the controlled system can still be employed successfully if a bounded control input for the considered nonlinear system of dimension three is analyzed. Finally, conclusions and an outlook toward future work together with an explanation of how the presented interval technique can be applied to other nonlinear control techniques are given in Section 7.