(110f) Computing Robust Controlled Invariant Sets of Nonlinear Dynamical Systems

The notion of controlled invariant set (CIS) plays an important role in the analysis and control of dynamical systems. Given a dynamical system, a subset of the state space is said to be controlled invariant set if the state of the set can be maintained within the set forever with the help of control. When a set is CIS even in the presence of uncertainties, it is said to be a robust controlled invariant set (RCIS). CISs are very useful in the stability analysis of dynamical systems and in characterizing the safety region that can be controlled for system operations. Particularly, invariant sets have been found to be important in addressing stability, constraint satisfaction and feasibility issues in model predictive control strategies. Because of their implicit properties, invariant sets have received considerable attention in the dynamics and control literature especially for (uncertain) linear dynamical systems. It is however, well known that estimating invariant sets is not a trivial task. Moreover, robust controlled invariant sets estimation for uncertain nonlinear dynamical systems with control inputs is a very challenging problem.

In this work, we use the notion of symbolic image proposed by Osipenko (2007) as a tool for the analysis of dynamical systems. A symbolic image is a finite approximation of the dynamics of a system using directed graphs. This allows the use of graph algorithms to analyze dynamical systems. It has been successfully used in the analysis of autonomous dynamical systems and in the determination of controlled invariant sets for non-autonomous dynamical systems. Specifically, in this work, we extend this tool further to the analysis of systems with both control inputs and disturbances. Particularly, we pose the problem as a differential game with two players – where player one seeks to keep the states of the system within the invariant set while player two seeks to drive the states out of the invariant set – and qualitatively determine parts of the set where player one always wins. Furthermore, we describe how the concept of feedback linearization can be leveraged to reduce the computational needs of the algorithm for a class of input affine nonlinear systems with additive disturbances.

References

Osipenko, G. (2007). Dynamical systems, graphs, and algorithms. Springer.