Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

Abstract

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that - in case of provable asymptotic stability - a certain domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions.

Proving the Stability of the Rolling Navigation

“…To determine the ellipsoid E x , we have to solve the Lyapunov equation: (c) The green ellipse B is not positive invariant which is linear in P. If the system is nonlinear, we apply the Lyapunov method on the linearized system and check the positive invariance using interval analysis [28].…”

“…Boxes. To find a box which is positive invariant, we may use the centered form method [8,28]. For this, check if…”

“…We now want to show that all initial state inside X 0 will converge to 0. We use the centered form [22] for stability [8,28]. For this, we follow the procedure given by relation (7).…”

“…In the context of dynamical systems and stability analysis, Tucker [33] has used interval analysis to prove that the Lorenz attractor exists and efficient solvers (such as CAPD) have been proposed for integrating differential equations [14,35] in a rigorous way. The corresponding methods can then be used for stability analysis of nonlinear systems [5,17,28]. In the context of hybrid systems, even if guaranteed integration has been used for characterizing reachability sets [24,25], to our knowledge, it has never been used to check the stability of dynamical systems where jumps could occur.…”