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Register a new user$mathcal{L}_1$-$mathcal{GP}$: $mathcal{L}_1$ Adaptive Control with Bayesian Learning
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We present $mathcal{L}_1$-$mathcal{GP}$, an architecture based on $mathcal{L}_1$ adaptive control and Gaussian Process Regression (GPR) for safe simultaneous control and learning. On one hand, the $mathcal{L}_1$ adaptive control provides stability and transient performance guarantees, which allows for GPR to efficiently and safely learn the uncertain dynamics. On the other hand, the learned dynamics can be conveniently incorporated into the $mathcal{L}_1$ control architecture without sacrificing robustness and tracking performance. Subsequently, the learned dynamics can lead to less conservative designs for performance/robustness tradeoff. We illustrate the efficacy of the proposed architecture via numerical simulations.
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This paper introduces an $mathcal{L}_1$ adaptive control augmentation for geometric tracking control of quadrotors. In the proposed design, the $mathcal{L}_1$ augmentation handles nonlinear (time- and state-dependent) uncertainties in the quadrotor dynamics without assuming/enforcing parametric structures, while the baseline geometric controller achieves stabilization of the known nonlinear model of the system dynamics. The $mathcal{L}_1$ augmentation applies to both the rotational and the translational dynamics. Experimental results demonstrate that the augmented geometric controller shows consistent and (on average five times) smaller trajectory tracking errors compared with the geometric controller alone when tested for different trajectories and under various types of uncertainties/disturbances.
The metrization of the space of neural responses is an ongoing research program seeking to find natural ways to describe, in geometrical terms, the sets of possible activities in the brain. One component of this program are the {em spike metrics}, notions of distance between two spike trains recorded from a neuron. Alignment spike metrics work by identifying equivalent spikes in one train and the other. We present an alignment spike metric having $mathcal{L}_p$ underlying geometrical structure; the $mathcal{L}_2$ version is Euclidean and is suitable for further embedding in Euclidean spaces by Multidimensional Scaling methods or related procedures. We show how to implement a fast algorithm for the computation of this metric based on bipartite graph matching theory.
Let $L$ be a finite extension of $mathbb{Q}_p$, and $rho_L$ be an $n$-dimensional semi-stable non crystalline $p$-adic representation of $mathrm{Gal}_L$ with full monodromy rank. Via a study of Breuils (simple) $mathcal{L}$-invariants, we attach to $rho_L$ a locally $mathbb{Q}_p$-analytic representation $Pi(rho_L)$ of $mathrm{GL}_n(L)$, which carries the exact information of the Fontaine-Mazur simple $mathcal{L}$-invariants of $rho_L$. When $rho_L$ comes from an automorphic representation of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real filed $F^+$ which is compact at infinite places and $mathrm{GL}_n$ at $p$-adic places), we prove under mild hypothesis that $Pi(rho_L)$ is a subrerpresentation of the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+})$. In other words, we prove the equality of Breuils simple $mathcal{L}$-invariants and Fontaine-Mazur simple $mathcal{L}$-invariants.
This article considers the $mathcal{H}_infty$ static output-feedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the $mathcal{L}_2$-induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. In addition, some analysis is given for the condition under which the robustly stabilized expanded system implies the robust stability of the original system. A numerical example illustrates the effectiveness of the proposed approach.
We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a emph{single} $mathcal{L}_{omega_1,omega}$-sentence $psi$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $psi$ is consistently equal to $[aleph_0,aleph_{omega_1}]$ and also consistently equal to $[aleph_0,2^{aleph_1})$, where $2^{aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $psi$ is consistently equal to $[aleph_1,aleph_{omega_1}]$ and $[aleph_1,2^{aleph_1})$, where again $2^{aleph_1}$ is weakly inaccessible. This is the first example of an $mathcal{L}_{omega_1,omega}$-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, $psi$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{aleph_0}<aleph_{omega_1}<2^{aleph_1}$ and there exists an $mathcal{L}_{omega_1,omega}$-sentence with models in $aleph_{omega_1}$, but no models in $2^{aleph_1}$. This relates to a conjecture by Shelah that if $aleph_{omega_1}<2^{aleph_0}$, then any $mathcal{L}_{omega_1,omega}$-sentence with a model of size $aleph_{omega_1}$ also has a model of size $2^{aleph_0}$. Our result proves that $2^{aleph_0}$ can not be replaced by $2^{aleph_1}$, even if $2^{aleph_0}<aleph_{omega_1}$.
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