We consider the problem of computing the (two-sided) Hausdorff distance between the unit
ℓ
p
1
and
ℓ
p
2
norm balls in finite dimensional Euclidean space for
1
≤
p
1
<
p
2
≤
∞
, and derive a ...closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the
k
1
and
k
2
unit
D
-norm balls, which are certain polyhedral norm balls in
d
dimensions for
1
≤
k
1
<
k
2
≤
d
. When two different
ℓ
p
norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different
ℓ
p
unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.
In this article the problem of ellipsoidal bounding of convex set-valued data, where the convex set is obtained by the <inline-formula><tex-math notation="LaTeX">p</tex-math></inline-formula>-sum of ...finitely many ellipsoids, for any real <inline-formula><tex-math notation="LaTeX">p\geq 1</tex-math></inline-formula> is studied. The notion of <inline-formula><tex-math notation="LaTeX">p</tex-math></inline-formula>-sum appears in the Brunn-Minkowski-Firey theory in convex analysis, and generalizes several well-known set-valued operations, such as the Minkowski sum of the summand convex sets (here, ellipsoids). We derive an outer ellipsoidal parameterization for the <inline-formula><tex-math notation="LaTeX">p</tex-math></inline-formula>-sum of a given set of ellipsoids, and compute the tightest such parameterization for two optimality criteria: minimum trace and minimum volume. For such optimal parameterizations, several known results in the system-control literature are recovered as special cases of our general formula. For the minimum volume criterion, our analysis leads to a fixed point recursion over a scalar that parameterizes the shape matrix of the outer ellipsoid. This recursion is proved to be contractive, and found to converge fast in practice. We apply these results to compute the forward reach sets for a linear control system subject to different convex set-valued uncertainty models for the initial condition and control, generated by varying <inline-formula><tex-math notation="LaTeX">p\in 1,\infty </tex-math></inline-formula>. Our numerical results show that the proposed fixed point algorithm offers more than two orders of magnitude speed-up in computational time for <inline-formula><tex-math notation="LaTeX">p=1</tex-math></inline-formula>, compared to the existing semidefinite programming approach without significant effect on the numerical accuracy. For <inline-formula><tex-math notation="LaTeX">p>1</tex-math></inline-formula>, the reach set computation results reported here are novel. Our results are expected to be useful in real-time safety critical applications, such as decision making, for collision avoidance of autonomous vehicles, where the computational time scale for reach set calculation needs to be much smaller than the vehicular dynamics time scale.
In this article the problem of ellipsoidal bounding of convex set-valued data, where the convex set is obtained by the Formula Omitted-sum of finitely many ellipsoids, for any real Formula Omitted is ...studied. The notion of Formula Omitted-sum appears in the Brunn–Minkowski–Firey theory in convex analysis, and generalizes several well-known set-valued operations, such as the Minkowski sum of the summand convex sets (here, ellipsoids). We derive an outer ellipsoidal parameterization for the Formula Omitted-sum of a given set of ellipsoids, and compute the tightest such parameterization for two optimality criteria: minimum trace and minimum volume. For such optimal parameterizations, several known results in the system-control literature are recovered as special cases of our general formula. For the minimum volume criterion, our analysis leads to a fixed point recursion over a scalar that parameterizes the shape matrix of the outer ellipsoid. This recursion is proved to be contractive, and found to converge fast in practice. We apply these results to compute the forward reach sets for a linear control system subject to different convex set-valued uncertainty models for the initial condition and control, generated by varying Formula Omitted. Our numerical results show that the proposed fixed point algorithm offers more than two orders of magnitude speed-up in computational time for Formula Omitted, compared to the existing semidefinite programming approach without significant effect on the numerical accuracy. For Formula Omitted, the reach set computation results reported here are novel. Our results are expected to be useful in real-time safety critical applications, such as decision making, for collision avoidance of autonomous vehicles, where the computational time scale for reach set calculation needs to be much smaller than the vehicular dynamics time scale.
Introduction: Women undergoing elective Lower Segment Caesarean Section (LSCS) delivery under spinal anaesthesia usually experience anxiety and stress due to unpleasant operative environment. Music ...therapy/music medicine has been found to relieve anxiety and stress besides decreasing pain and stable cardiorespiratory parameters. Aim: To assess the effect of perioperative music therapy/music medicine on pain and cardio respiratory parameters in women undergoing elective LSCS under spinal anaesthesia. Materials and Methods: The present study was case-control study in which 60 consecutive parturient women, planned for elective LSCS delivery under spinal anaesthesia, were randomly divided into music and non music groups of 30 each. Demographic characteristics of women were recorded. The ‘music group’ received preselected music three times for 20 minutes each along with standardised analgesia protocol. The ‘non music group’ received standardised analgesia protocol only. Respiratory and haemodynamic parameters were assessed during perioperative period. Postoperative pain was assessed using Visual Analogue Scale (VAS) scores in addition to comparisons of time for rescue analgesia between study and control groups. Statistical differences were derived using Mann-Whitney U-test. Results: Demographic characteristics of women were statistically insignificant. Significant differences were observed in Heart Rate (HR) at 30, 45 and 60 minutes Postanaesthesia Care Unit (PACU) (p-value 0.0278, 0.0151 and 0.02852, respectively) and Respiratory Rate (RR) at 60 minutes PACU (p-value 0.04884). This study found beneficial effect of music on pain as assessed by VAS in the postoperative period. Beneficial effect of music on pain observed in study group vs. controls at 1 hour (0.64 vs. 1.51), 2 hours (1.88 vs. 2.76) and 3 hours (3.07 vs. 3.47) of PACU stay (p-value 0.0003, 0.00152 and 0.02444, respectively). The study also detected a significant delay (29 minutes) in time for first rescue analgesia in music group (p-value 0.01732). Conclusion: Music therapy/medicine during elective LSCS delivery is beneficial on HR, respiratory rate, VAS and time to rescue analgesia.
This is the first of a two part paper investigating the geometry of the integrator reach sets, and the applications thereof. In this Part I, assuming box-valued input uncertainties, we establish that ...this compact convex reach set is semialgebraic, translated zonoid, and not a spectrahedron. We derive the parametric as well as the implicit representation of the boundary of this reach set. We also deduce the closed form formula for the volume and diameter of this set, and discuss their scaling with state dimension and time. We point out that these results may be utilized in benchmarking the performance of the reach set over-approximation algorithms.
In this article, we study the Schrödinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state ...probability density function (PDF) to another over a finite-time horizon, subject to a controlled stochastic differential evolution of the state vector. As such, it can be seen as a stochastic optimal control problem in continuous time with endpoint density constraints-A topic that originated in the physics literature in 1930s, and in the recent years, has garnered burgeoning interest in the systems-control community. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the "Schrödinger factors"-so named since their product at any time yields the optimal controlled joint state PDF at that time. We show that if the drift is a gradient vector field, or is of mixed conservative-dissipative nature, then it is possible to transform these PDEs into a pair of initial value problems (IVPs) involving the same forward Kolmogorov operator. Combined with a recently proposed fixed point recursion that is contractive in the Hilbert metric, this opens up the possibility to numerically solve the SBPs in these cases by computing the Schrödinger factors via a single IVP solver for the corresponding (uncontrolled) forward Kolmogorov PDE. The flows generated by such forward Kolmogorov PDEs, for the two aforementioned types of drift, in turn, enjoy gradient descent structures on the manifold of joint PDFs with respect to suitable distance functionals. We employ a proximal algorithm developed in our prior work that exploits this geometric viewpoint, to solve these IVPs and compute the Schrödinger factors via weighted scattered point cloud evolution in the state space. We provide the algorithmic details and illustrate the proposed framework of solving the SBPs with nonlinear prior dynamics by numerical examples.
This paper presents a probabilistic model validation methodology for nonlinear systems in time-domain. The proposed formulation is simple, intuitive, and accounts both deterministic and stochastic ...nonlinear systems with parametric and nonparametric uncertainties. Instead of hard invalidation methods available in the literature, a relaxed notion of validation in probability is introduced. To guarantee provably correct inference, algorithm for constructing probabilistically robust validation certificate is given along with computational complexities. Several examples are worked out to illustrate its use.
We propose a stochastic reachability computation framework for occupancy prediction in automated driving by directly solving the underlying transport partial differential equation (PDE) governing the ...advection of the closed-loop joint density functions. The resulting nonparametric gridless computation is based on integration along the characteristic curves, and it allows online computation of the time-varying collision probabilities. We provide numerical simulations for multi-lane highway driving scenarios to highlight the scope of the proposed method.
We provide a variational interpretation of the DeGroot-Friedkin map in opinion dynamics. Specifically, we show that the nonlinear dynamics for the DeGroot-Friedkin map can be viewed as mirror descent ...on the standard simplex with the associated Bregman divergence being equal to the generalized Kullback-Leibler divergence, i.e., an entropic mirror descent. Our results reveal that the DeGroot-Friedkin map elicits an individual's social power to be close to her social influence while minimizing the so called "extropy"-the entropy of the complimentary opinion.
We consider the problem of verifying safety for a pair of identical integrator agents in continuous time with compact set-valued input uncertainties. We encode this verification problem as that of ...certifying or falsifying the intersection of their reach sets. We transcribe the same into a variational problem, namely that of minimizing the support function of the difference of the two reach sets over the unit sphere. We illustrate the computational tractability of the proposed formulation by developing two cases in detail, viz. when the inputs have time-varying norm-bounded and generic hyperrectangular uncertainties. We show that the latter case allows distributed certification via second order cone programming.
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