# Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System
## 1 Introduction  
During the last few decades, interval analysis has been shown to provide efficient numerical approaches for solving various tasks in scientific computing, as well as in application-oriented disciplines in which numerical techniques with a result verification are helpful or required. In such a way, they help to solve problems in control engineering, robotics, including rigorous methods for localization and/or path planning, as well as guaranteed obstacle avoidance, computational (bio-)mechanics, civil engineering, or bio-engineering, just to mention a few of the large number of different areas [1; 2; 3; 4; 5; 6; 7]. In these applications, not only the possibility of interval analysis to treat the effects of round-off and truncation errors of numerical solution techniques (such as truncated temporal Taylor series expansions for the solution of initial value problems to ordinary differential equations) is exploited [8; 9; 10]. In addition, these applications also benefit from a further fundamental property of interval analysis, namely that it provides guaranteed outer enclosures of all possible solutions to a mathematical problem at hand in the presence of ambiguities in the solutions or in the presence of bounded uncertainty in selected parameters. Here, a selection of useful techniques ranges from the extension of fundamental arithmetic operations to interval valued expressions [1; 2; 3], which were recently standardized in the IEEE standard 1788 [11], over the development of set valued counterparts for zero finding techniques for sets of algebraic equations such as the Krawczyk operator [12; 13; 14], interval based techniques for reachability analysis [15], the verified global optimization [16; 17; 18], to the solution of identification tasks bymeans of set inversion techniques via interval analysis (e.g., the algorithms set inversion via interval analysis (SIVIA) and problem specific generalizations) [2; 19]. If control applications are concerned, interval analysis can be applied both to an offline design and verification of control procedures under consideration of feasibility and safety requirements such as input and state constraints and to an online interval evaluation in terms of real-time capable robust control strategies generalizing the ideas of variable-structure control techniques and backstepping [20; 21; 22; 23; 24; 25]. Further, approaches for the control of systems with input saturations that do not rely on interval analysis were handled, for example, in [26] and the references therein. In addition to the implementation of robust controllers, also the dual task of interval based state estimation and observer design can be solved; see, for example, [27; 28; 29; 30; 31; 32]. This paper addresses the following two issues: First, a method for finding a nonlinear state-feedback controller is derived for the exact tracking of predefined paths in the phase-space. These paths are assumed to be described in terms of a sliding surface for a nonlinear dynamic system. Second, this control design procedure is interfaced with techniques from the field of interval analysis that are employed to determine enclosures of the invariant sets of the resulting controlled system in a reliable and computationally efficient way. For a continuous-time system model with the state vector \(\mathbf{x}\in\mathbb{R}^{n}\), a subset \(S\subset\mathbb{R}^{n}\) is called positively invariant if the relation \(\mathbf{x}(t_{1})\in S\) is satisfied for all possible \(\mathbf{x}(t_{0})\in S\) if \(t_{1}\geq t_{0}\) holds [33]. Due to the fact that we are only interested in a forward-in-time propagation of state trajectories, these sets are briefly called invariant throughout this paper. For the proposed control design, an appropriate Grobner basis is used to check if the occurring Bezout identity has a solution, such that a controller can be designed that does not contain any structural singularities.