Computing Bounds on $L_{\infty}$-induced Norm for Linear Time-Invariant Systems Using Homogeneous Lyapunov Functions
Quadratic Lyapunov function has been widely used in the analysis of linear time invariant (LTI) systems ever since it has shown that the existence of such quadratic Lyapunov function certifies the stability of the LTI system. In this work, the problem of finding upper and lower bounds for the $L_{\infty}$-induced norm of the LTI system is considered. Quadratic Lyapunov functions are used to find the star norm, the best upper on the $L_{\infty}$-induced norm, by bounding the unit peak input reachable sets by inescapable ellipsoids. Instead, a more general class of homogeneous Lyapunov functions is used to get less conservative upper bounds on the $L_{\infty}$-induced norm and better conservative approximations for the reachable sets than those obtained using standard quadratic Lyapunov functions. The homogeneous Lyapunov function for the LTI system is considered to be a quadratic Lyapunov function for a higher-order system obtained by Lifting the LTI system via Kronecker product. Different examples are provided to show the significant improvements on the bounds obtained by using Homogeneous Lyapunov functions.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Data-dependent Generalization Bounds for Multi-class Classification
In this paper, we study data-dependent generalization error bounds exhibiting a mild dependency on the number of classes, making them suitable for multi-class learning with a large number of label classes. The bounds gen…
ClassificationGeneral ClassificationGeneralization BoundsMulti-class ClassificationEfficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation
Lipschitz constants are connected to many properties of neural networks, such as robustness, fairness, and generalization. Existing methods for computing Lipschitz constants either produce relatively loose upper bounds o…
FairnessTwo Regimes of Generalization for Non-Linear Metric Learning
A common approach to metric learning is to seek an embedding of the input data that behaves well with respect to the labels. While generalization bounds for linear embeddings are known, the non-linear case is not well un…
Generalization BoundsMetric LearningVocal Bursts Valence PredictionStochastic approximation with cone-contractive operators: Sharp $\ell_\infty$-bounds for $Q$-learning
Motivated by the study of $Q$-learning algorithms in reinforcement learning, we study a class of stochastic approximation procedures based on operators that satisfy monotonicity and quasi-contractivity conditions with re…
Q-LearningReinforcement LearningRobust Training and Verification of Implicit Neural Networks: A Non-Euclidean Contractive Approach
This paper proposes a theoretical and computational framework for training and robustness verification of implicit neural networks based upon non-Euclidean contraction theory. The basic idea is to cast the robustness ana…