Abstract— This paper addresses the problem of risk-aware fixed-time stabilization of a class of uncertain, output-feedback nonlinear systems modeled via stochastic differential equations. First, novel classes of certificate functions, namely riskaware fixed-time- and risk-aware path-integral-control Lyapunov functions, are introduced. Then, it is shown how the use of either for control design certifies that a system is both stable in probability and probabilistically fixed-time convergent (for a given probability) to a goal set. That is, the system trajectories probabilistically reach the set within a finite time, independent of the initial condition, despite the additional presence of measurement noise. These methods represent an improvement over the state-of-the-art in stochastic fixed-time stabilization, which presently offers bounds on the settling-time function in expectation only. The theoretical results are verified by an empirical study on an illustrative, stochastic, nonlinear system and the proposed controllers are evaluated against an existing method. Finally, the methods are demonstrated via a simulated fixed-wing aerial robot on a reach-avoid scenario to highlight their ability to certify the probability that a system safely reaches its goal.
B. Risk-Aware Path Integral CLF A potential drawback to using the RA-FxT-CLF given by Definition 4 for controller design and/or verification is that pg g, which, for large V or may produce g 1. This may make (11) difficult to satisfy in practice, especially in the presence of other hard system constraints (like safety). The following notion of the RA-PI-CLF, which is inspired by the risk-aware control barrier functions introduced by and allows for arbitrary pg [0.5, 1) (thus opening up the interval [0.5, g)), helps mitigate this issue. Definition 5. Suppose that Assumptions 1 and 2 hold, and consider a set Sg defined by (4) for a twice continuously differentiable function V : X (cid:55) R satisfying (5) and (6). The function V is a risk-aware path integral control Lyapunov function (RA-PI-CLF) for the interconnected system ((1), (3)) w.r.t. Sg if there exists pg [0.5, 1) such that on every sample path , the following holds t T ,
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LxV (xt(), u) c1W (t, )1 c2W (t, )2 , (14) inf uU where W (t, ) = I V (t, )+ 2T erf 1(pg)+V +T LV , (15) with I V defined according to (10) and = (V, Og) for defined by (8). Theorem 2. If V is a RA-PI-CLF for the interconnected system ((1), (3)) w.r.t. the set Sg, then the set Sg is rendered g-FxTS with probability p p Proof. It will be shown that if V is a RA-PI-CLF then Sg is 1) stable in probability, and 2) locally finite-time attractive with probability p g = pg(1 ). g and uniformly bounded settling time. The first components of this proof mirror the proof of Theorem 1, and thus we skip to comparing V and V as follows: c 0, pv(c) P (cid:26) sup tT V (x) c (cid:27) P (cid:26) sup tT V (x) c (cid:27) pv(c). let Bc = {x X | V (x) c} and R(c) = Now, supxBc x, and note that R(c) < for c < due to V satisfying (5). Observe that for a given sample path pv(c) = P (cid:26) sup tT (cid:2)V (x0) + IV (t, ) + wt (cid:3) c (cid:27) , P
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= P (cid:26) (cid:26) V + sup tT sup tT wt < (cid:27) IV (t, ) + sup tT c V suptT IV (t, ) wt c , (cid:27) . Since P{suptT |IV (t, ) I V (t, )| tLV } 1 from Lemma 3, it is true that IV (t, ) I V (t, )+tLV I V (t, ) + T LV with probability of at least 1 . Thus, pv(c) P (cid:40) sup tT wt < c V suptT I V (t, ) T LV (cid:41) (1 ). With W (t, ) 0 whenever I V (t, ) 2T erf 1(pg) V T LV , it follows by (14) that when V is a RA-PI-CLF, LxV 0 whenever I V (t, ) 2T erf 1(pg) V I V (t, ) = 0. Then, by Lemma 1, T LV , and thus suptT (cid:19) (cid:18) c V T LV pv(c) erf (1 ). 2T Taking the above with equality and setting (c) = 1 pv(c), it holds that, R(c) 0, P (cid:26) sup tT x(t; x0) R(c) (cid:27) 1 (c), which by Definition 2 implies that Sg is stable in probability. Now, by similar arguments, it follows that pg P (cid:26) inf tT V (x) 0 (cid:27) P (cid:26) inf tT V (x) 0 (cid:27)
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As such, following the same steps as above (omitting ), pg erf (cid:18) V inf 0tT I V (t) T LV 2T (cid:19) (1 ), which implies that pg p for which the following holds, t [, T ], g = pg(1) provided that T 2T erf 1(cid:0)pg (cid:1) V T LV . I V (t) Therefore, the above is satisfied when the function W given W = by (15) satisfies W (t) 0, t [, T ]. Note that LxV (x, u), and so (14) is equivalent to W c1W 1 c2W 2 , which by [5, Lemma 1] renders W fixed-time stable to the origin, i.e., W 0 as t T (x0) Tg given by (7) and W (t) = 0 for all t [Tg, T ]. Therefore, Sg is p g-FxTS, i.e., finite-time attractive with probability p g = pg(1) and uniformly bounded settling time, T (x0) Tg.
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