An official website of the United States government
Linear temporal logic (LTL) with the knowledge operator, denoted as KLTL, is a variant of LTL that incorporates what an agent knows or learns at run-time into its specification. Therefore it is an appropriate logical formalism to specify tasks for systems with unknown components that are learned or estimated at run-time. In this paper, we consider a linear system whose system matrices are unknown but come from an a priori known finite set. We introduce a form of KLTL that can be interpreted over the trajectories of such systems. Finally, we show how controllers that guarantee satisfaction of specifications given in fragments of this form of KLTL can be synthesized using optimization techniques. Our results are demonstrated in simulation and on hardware in a drone scenario where the task of the drone is conditioned on its health status, which is unknown a priori and discovered at run-time.
more »
« less
Brownian bridges for contained random walks
Abstract Using linear operator techniques, we demonstrate an efficient method for investigating rare events in stochastic processes. Specifically, we examine contained trajectories, which are continuous random walks that only leave a specified region of phase space after a set period of time . We show that such trajectories can be efficiently generated through the use of a Brownian Bridge, derived via the solution to the Backward Fokker–Planck (BFP) equation. Using linear operator techniques, we place the BFP operator in self‐adjoint form and show that in the asymptotic limit , the set of paths contained in a specified region is equivalent to paths on a modified potential energy landscape that is related to the dominant eigenfunction of the self‐adjoint BFP operator. We demonstrate this idea on several example problems, one of which is the Graetz problem, where one is interested in the survival time of a particle diffusing in tube flow.
more »
« less
- Award ID(s):
- 2126230
- PAR ID:
- 10641583
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- AIChE Journal
- Volume:
- 71
- Issue:
- 5
- ISSN:
- 0001-1541
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Matni, N; Morari, M; Pappas, G.J. (Ed.)One of the long-term objectives of Machine Learning is to endow machines with the capacity of structuring and interpreting the world as we do. This is particularly challenging in scenes involving time series, such as video sequences, since seemingly different data can correspond to the same underlying dynamics. Recent approaches seek to decompose video sequences into their composing objects, attributes and dynamics in a self-supervised fashion, thus simplifying the task of learning suitable features that can be used to analyze each component. While existing methods can successfully disentangle dynamics from other components, there have been relatively few efforts in learning parsimonious representations of these underlying dynamics. In this paper, motivated by recent advances in non-linear identification, we propose a method to decompose a video into moving objects, their attributes and the dynamic modes of their trajectories. We model video dynamics as the output of a Koopman operator to be learned from the available data. In this context, the dynamic information contained in the scene is encapsulated in the eigenvalues and eigenvectors of the Koopman operator, providing an interpretable and parsimonious representation. We show that such decomposition can be used for instance to perform video analytics, predict future frames or generate synthetic video. We test our framework in a variety of datasets that encompass different dynamic scenarios, while illustrating the novel features that emerge from our dynamic modes decomposition: Video dynamics interpretation and user manipulation at test-time. We successfully forecast challenging object trajectories from pixels, achieving competitive performance while drawing useful insights.more » « less
-
A linear sixth-order partial differential equation (PDE) of “parabolic” type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green’s function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances “feel” the finite boundaries, and show that the derived Green’s function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio.more » « less
-
Abstract In a recent paper, we showed that a subspace of a real ‐triple is an ‐summand if and only if it is a ‐closed triple ideal. As a consequence, ‐ideals of real ‐triples, including real ‐algebras, real ‐algebras and real TROs, correspond to norm‐closed triple ideals. In this paper, we extend this result by identifying the ‐ideals in (possibly non‐self‐adjoint) real operator algebras and Jordan operator algebras. The argument for this is necessarily different. We also give simple characterizations of one‐sided ‐ideals in real operator algebras, and give some applications to that theory.more » « less
-
We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schrödinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the weighted setting are the same as in the regular case after subtracting a finite rank operator corresponding to the threshold resonances. Such matrix Schrödinger operators naturally arise from linearizing a focusing nonlinear Schrödinger equation around a solitary wave. It is known that the linearized operator for the 1D focusing cubic NLS equation exhibits a threshold resonance. We also include an observation of a favorable structure in the quadratic nonlinearity of the evolution equation for perturbations of solitary waves of the 1D focusing cubic NLS equation.more » « less
