In this paper, we investigate the problem of system identification for autonomous Markov jump linear systems (MJS) with complete state observations. We propose switched least squares method for identification of MJS, show that this method is strongly consistent, and derive data-dependent and data-independent rates of convergence. In particular, our data-independent rate of convergence shows that, almost surely, the system identification error is $\mathcal{O}\big(\sqrt{\log(T)/T} \big)$ where $T$ is the time horizon. These results show that switched least squares method for MJS has the same rate of convergence as least squares method for autonomous linear systems. We derive our results by imposing a general stability assumption on the model called stability in the average sense. We show that stability in the average sense is a weaker form of stability compared to the stability assumptions commonly imposed in the literature. We present numerical examples to illustrate the performance of the proposed method.
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In this work, we study the concentration behavior of a stochastic approximation (SA) algorithm under a contractive operator with respect to an arbitrary norm. We consider two settings where the iterates are potentially unbounded: (1) bounded multiplicative noise, and (2) additive sub-Gaussian noise. We obtain maximal concentration inequalities on the convergence errors, and show that these errors have sub-Gaussian tails in the additive noise setting, and super-polynomial tails (faster than polynomial decay) in the multiplicative noise setting. In addition, we provide an impossibility result showing that it is in general not possible to achieve sub-exponential tails for SA with multiplicative noise. To establish these results, we develop a novel bootstrapping argument that involves bounding the moment generating function of the generalized Moreau envelope of the error and the construction of an exponential supermartingale to enable using Ville's maximal inequality. To demonstrate the applicability of our theoretical results, we use them to provide maximal concentration bounds for a large class of reinforcement learning algorithms, including but not limited to on-policy TD-learning with linear function approximation, off-policy TD-learning with generalized importance sampling factors, and $Q$-learning. To the best of our knowledge, super-polynomial concentration bounds for off-policy TD-learning have not been established in the literature due to the challenge of handling the combination of unbounded iterates and multiplicative noise.
We present DeepDECS, a new method for the synthesis of correct-by-construction discrete-event controllers for autonomous systems that use deep neural network (DNN) classifiers for the perception step of their decision-making processes. Despite major advances in deep learning in recent years, providing safety guarantees for these systems remains very challenging. Our controller synthesis method addresses this challenge by integrating DNN verification with the synthesis of verified Markov models. The synthesised models correspond to discrete-event controllers guaranteed to satisfy the safety, dependability and performance requirements of the autonomous system, and to be Pareto optimal with respect to a set of optimisation objectives. We use the method in simulation to synthesise controllers for mobile-robot collision mitigation and for maintaining driver attentiveness in shared-control autonomous driving.
We consider the problem of identification of linear dynamical systems from a single trajectory. Recent results have predominantly focused on the setup where no structural assumption is made on the system matrix $A^* \in \mathbb{R}^{n \times n}$, and have consequently analyzed the ordinary least squares (OLS) estimator in detail. We assume prior structural information on $A^*$ is available, which can be captured in the form of a convex set $\mathcal{K}$ containing $A^*$. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm which depend on the local size of the tangent cone of $\mathcal{K}$ at $A^*$. To illustrate the usefulness of this result, we instantiate it for the settings where, (i) $\mathcal{K}$ is a $d$ dimensional subspace of $\mathbb{R}^{n \times n}$, or (ii) $A^*$ is $k$-sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball. In the regimes where $d, k \ll n^2$, our bounds improve upon those obtained from the OLS estimator.
In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. In a first step, we approximate the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical results and demonstrate that the approach can be applied to other nonlocal problems.
We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.
This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schm\"udgen and Putinar Positivstellens\"atze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.
In multi-point contact systems, precise force control is crucial for achieving stable and safe interactions between robots and their environment. Thus, we demonstrate an admittance controller with auto-tuning that can be applied for these systems. The controller's objective is to track the target wrench profiles of each contact point while considering the additional torque due to rotational friction. Our admittance controller is adaptive during online operation by using an auto-tuning method that tunes the gains of the controller while following user-specified training objectives. These objectives include facilitating controller stability, such as tracking the wrench profiles as closely as possible, ensuring control outputs are within force limits that minimize slippage, and avoiding configurations that induce kinematic singularity. We demonstrate the robustness of our controller on hardware for both manipulation and locomotion tasks using a multi-limbed climbing robot.
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.
Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals. We also prove its convergence for any sampling. In addition, we study SPDHG for parallel Magnetic Resonance Imaging reconstruction, where data from different coils are randomly selected at each iteration. We apply SPDHG using a wide range of random sampling methods and compare its performance across a range of settings, including mini-batch size and step size parameters. We show that the sampling can significantly affect the convergence speed of SPDHG and for many cases an optimal sampling can be identified.
When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.
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