ترغب بنشر مسار تعليمي؟ اضغط هنا

Learning Sparse Dynamical Systems from a Single Sample Trajectory

370   0   0.0 ( 0 )
 نشر من قبل Salar Fattahi
 تاريخ النشر 2019
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

This paper addresses the problem of identifying sparse linear time-invariant (LTI) systems from a single sample trajectory generated by the system dynamics. We introduce a Lasso-like estimator for the parameters of the system, taking into account their sparse nature. Assuming that the system is stable, or that it is equipped with an initial stabilizing controller, we provide sharp finite-time guarantees on the accurate recovery of both the sparsity structure and the parameter values of the system. In particular, we show that the proposed estimator can correctly identify the sparsity pattern of the system matrices with high probability, provided that the length of the sample trajectory exceeds a threshold. Furthermore, we show that this threshold scales polynomially in the number of nonzero elements in the system matrices, but logarithmically in the system dimensions --- this improves on existing sample complexity bounds for the sparse system identification problem. We further extend these results to obtain sharp bounds on the $ell_{infty}$-norm of the estimation error and show how different properties of the system---such as its stability level and textit{mutual incoherency}---affect this bound. Finally, an extensive case study on power systems is presented to illustrate the performance of the proposed estimation method.



قيم البحث

اقرأ أيضاً

When training the parameters of a linear dynamical model, the gradient descent algorithm is likely to fail to converge if the squared-error loss is used as the training loss function. Restricting the parameter space to a smaller subset and running th e gradient descent algorithm within this subset can allow learning stable dynamical systems, but this strategy does not work for unstable systems. In this work, we look into the dynamics of the gradient descent algorithm and pinpoint what causes the difficulty of learning unstable systems. We show that observations taken at different times from the system to be learned influence the dynamics of the gradient descent algorithm in substantially different degrees. We introduce a time-weighted logarithmic loss function to fix this imbalance and demonstrate its effectiveness in learning unstable systems.
A new algorithm is presented for reconstructing stochastic nonlinear dynamical models from noisy time-series data. The approach is analytical; consequently, the resulting algorithm does not require an extensive global search for the model parameters, provides optimal compensation for the effects of dynamical noise, and is robust for a broad range of dynamical models. The strengths of the algorithm are illustrated by inferring the parameters of the stochastic Lorenz system and comparing the results with those of earlier research. The efficiency and accuracy of the algorithm are further demonstrated by inferring a model for a system of five globally- and locally-coupled noisy oscillators.
In this paper, we study the system identification problem for sparse linear time-invariant systems. We propose a sparsity promoting block-regularized estimator to identify the dynamics of the system with only a limited number of input-state data samp les. We characterize the properties of this estimator under high-dimensional scaling, where the growth rate of the system dimension is comparable to or even faster than that of the number of available sample trajectories. In particular, using contemporary results on high-dimensional statistics, we show that the proposed estimator results in a small element-wise error, provided that the number of sample trajectories is above a threshold. This threshold depends polynomially on the size of each block and the number of nonzero elements at different rows of input and state matrices, but only logarithmically on the system dimension. A by-product of this result is that the number of sample trajectories required for sparse system identification is significantly smaller than the dimension of the system. Furthermore, we show that, unlike the recently celebrated least-squares estimators for system identification problems, the method developed in this work is capable of textit{exact recovery} of the underlying sparsity structure of the system with the aforementioned number of data samples. Extensive case studies on synthetically generated systems, physical mass-spring networks, and multi-agent systems are offered to demonstrate the effectiveness of the proposed method.
We present a sample-based Learning Model Predictive Controller (LMPC) for constrained uncertain linear systems subject to bounded additive disturbances. The proposed controller builds on earlier work on LMPC for deterministic systems. First, we intro duce the design of the safe set and value function used to guarantee safety and performance improvement. Afterwards, we show how these quantities can be approximated using noisy historical data. The effectiveness of the proposed approach is demonstrated on a numerical example. We show that the proposed LMPC is able to safely explore the state space and to iteratively improve the worst-case closed-loop performance, while robustly satisfying state and input constraints.
196 - Jiahui Cheng , Sui Tang 2021
In this paper, we study the nonlinear inverse problem of estimating the spectrum of a system matrix, that drives a finite-dimensional affine dynamical system, from partial observations of a single trajectory data. In the noiseless case, we prove an a nnihilating polynomial of the system matrix, whose roots are a subset of the spectrum, can be uniquely determined from data. We then study which eigenvalues of the system matrix can be recovered and derive various sufficient and necessary conditions to characterize the relationship between the recoverability of each eigenvalue and the observation locations. We propose various reconstruction algorithms, with theoretical guarantees, generalizing the classical Prony method, ESPIRIT, and matrix pencil method. We test the algorithms over a variety of examples with applications to graph signal processing, disease modeling and a real-human motion dataset. The numerical results validate our theoretical results and demonstrate the effectiveness of the proposed algorithms, even when the data did not follow an exact linear dynamical system.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا