Estimates of reachable sets of a nonlinear dynamical system with impulsive vector control and uncertainty

Filippova, Tatiana F.

doi:10.1109/stab.2018.8408353

Cited by 4 publications

(9 citation statements)

References 15 publications

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“…From (11) it follows that the set of all k that satisfy |k| Ĝ(x) † + Ĝ(x) † f (x) ≤ 1 is a subset of all k that satisfy (9). In other words, if:…”

Section: Iii-b Ball Underapproximationmentioning

confidence: 99%

“…then k satisfies (9). We note Ĝ(x) † = 0 from the definition of the Moore-Penrose pseudoinverse and because G(x) = 0 from Lemma 1.…”

Section: Iii-b Ball Underapproximationmentioning

confidence: 99%

“…is a subset of all k that satisfy (9). The term G(x) † is the inverse of the smallest nonzero singular value of G(x).…”

Section: Iii-c Advanced Convex Underapproximationmentioning

confidence: 99%

“…is a subset of all k that satisfy (9). Next, we bound Ĝ(x) † d using the product and triangle inequalities for matrices:…”

Section: Iii-c Advanced Convex Underapproximationmentioning

confidence: 99%

“…Apart from [7], work on reachability under uncertainty considered computation of reachable sets with dynamics generated by a finite number of uncertain parameters [9], [10] or having bounded disturbances [11], [12]. Work in adaptive and robust control [8], [13], [14] assumes more knowledge on the magnitude or structure of the system dynamics, and classical data-driven learning methods [2], [15], [16] collect data through repeated system runs.…”

Section: Introductionmentioning

confidence: 99%

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Reachability of Nonlinear Systems with Unknown Dynamics

Shafa¹,

Ornik²

2021

Preprint

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Determining the reachable set for a given nonlinear control system is crucial for system control and planning. However, computing such a set is impossible if the system's dynamics are not fully known. This paper is motivated by a scenario where a system suffers an adverse event mid-operation, resulting in a substantial change to the system's dynamics, rendering them largely unknown. Our objective is to conservatively approximate the system's reachable set solely from its local dynamics at a single point and the bounds on the rate of change of its dynamics. We translate this knowledge about the system dynamics into an ordinary differential inclusion. We then derive a conservative approximation of the velocities available to the system at every system state. An inclusion using this approximation can be interpreted as a control system; the trajectories of the derived control system are guaranteed to be trajectories of the unknown system. To illustrate the practical implementation and consequences of our work, we apply our algorithm to a simplified model of an unmanned aerial vehicle.Notice of Previous Publication. This manuscript substantially improves the work of [1]. Theory has been generalized to include a class of non-invertible matrices and improved to provide a larger set of reachable states. All lemmas, corollaries, and Theorems 1, 2, and 4 are entirely novel. Theorem 3 has been slightly modified from existing theorems in [1] given our new results.

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“…From (11) it follows that the set of all k that satisfy |k| Ĝ(x) † + Ĝ(x) † f (x) ≤ 1 is a subset of all k that satisfy (9). In other words, if:…”

Section: Iii-b Ball Underapproximationmentioning

confidence: 99%

“…then k satisfies (9). We note Ĝ(x) † = 0 from the definition of the Moore-Penrose pseudoinverse and because G(x) = 0 from Lemma 1.…”

Section: Iii-b Ball Underapproximationmentioning

confidence: 99%

“…is a subset of all k that satisfy (9). The term G(x) † is the inverse of the smallest nonzero singular value of G(x).…”

Section: Iii-c Advanced Convex Underapproximationmentioning

confidence: 99%

“…is a subset of all k that satisfy (9). Next, we bound Ĝ(x) † d using the product and triangle inequalities for matrices:…”

Section: Iii-c Advanced Convex Underapproximationmentioning