Linear time-variant systems: Lyapunov functions and invariant sets defined by Hölder norms

— In this paper we introduce a new type of Lyapunov functions in a general framework particularly suitable for the analysis of systems with noise and uncertainty. These Lyapunov functions may depend on parameters possibly satisfying differential equations or inequalities. The main differences with respect to classical Lyapunov functions and classical tools for the design of composite Lyapunov functions are discussed through examples. A design tool for the design of composite filtered Lyapunov functions is given and examples show improvements over existing literature.

In this paper, we introduce a new type of Lyapunov functions in a general framework particularly suitable for the analysis and control of systems with noise and uncertainty. These Lyapunov functions may depend on parameters possibly satisfying differential equations. The main differences with respect to classical Lyapunov functions and classical tools for the design of composite Lyapunov functions are discussed through examples. A design tool for the design of composite filtered Lyapunov functions is given, and examples show improvements over existing literature.

This paper develops a new sampling-based method for stability verification of piecewise continuous nonlinear systems via Lyapunov functions. Depending on the nonlinear system dynamics, the candidate Lyapunov function and the set of states of interest, verifying stability requires solving complex, possibly non-convex or infeasible optimization problems. To avoid such problems, the proposed approach firstly distributes the verification of Lyapunov's inequality on a finite sampling of a bounded set of states of interest. Secondly, it extends the validity of Lyapunov's inequality to an infinite, bounded set of states by automatically exploiting local continuity properties. A sampling-based method for estimating the domain of attraction (DOA) via level sets of a validated Lyapunov function is also presented. Efficient state-space exploration is achieved using multi-resolution sampling and hyper-rectangles as basic sampling blocks. The operations that need to be performed for each sampling point in the state-space can be carried out in parallel, which improves scalability. The proposed methodology is illustrated for various benchmark examples from the literature.

This article addresses stability and stabilization of infinite delay systems, which are more general but also more challenging to deal with than bounded delay systems. Based on a general model of infinite delay systems and a newly approved key technical lemma, several new Lyapunov theorems for uniform asymptotical stability and exponential stability, respectively, are established. The stability results are more general than existing stability results, and the corresponding conditions are more easily satisfied than existing ones. Therefore, the new stability results are expected to be more widely applicable. These new Lyapunov theorems are then applied to the problems of stabilizing both time-invariant and time-varying linear systems with distributed infinite input delays, and the corresponding stabilizing controllers are developed. Key to obtaining the stabilizing controllers is to construct appropriate Lyapunov functionals for these systems via introduction of a regulation term into the Lyapunov functional. A distinctive advantage of the Lyapunov-based time-domain method proposed in this article over the existing frequency domain method is that the former can be adopted to deal with more general systems, such as time-varying linear systems or even nonlinear systems. Several examples are provided to illustrate the effectiveness of our results.

We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical systems in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to search for a Lyapunov function from a family of parameterized Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a nonconvex mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates in the next iteration. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.

In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

Our paper proposes a framework that allows the construction of several types of diagonal Lyapunov functions for discrete- and continuous-time interval systems. The Lyapunov function candidates are defined as weighted vector norms, where the weighting matrices are diagonal and positive definite. For the existence of such Lyapunov functions, we provide necessary and sufficient conditions in the case of 1 and ℝ norms, as well as two sufficient conditions in the case of 2-norm. The results offer construction procedures focusing on diagonal Lyapunov functions with the fastest decreasing rate, fact that also ensures the identification of the fastest contractive invariant sets shaped in accordance with the aforementioned norms. The exposition covers the discrete- and continuous-time dynamics, in parallel, aiming to emphasize both similarities and differences between the two types of results. The applicability of the theoretical developments is illustrated by a numerical example.

In this paper, a generalized memristor-based recurrent neural network model with variable delays and impulse effects is considered. By using an impulsive delayed differential inequality and Lyapunov function, the exponential stability of the impulsive delayed memristor-based recurrent neural networks is investigated. Several exponential and uniform stability criteria of this impulsive delayed system are derived, which promotes the study of memristor-based recurrent neural networks. Finally, the effectiveness of obtained results is illustrated by two numerical examples.

In this paper we derive a sufficient condition for the stability of second-order switched systems. The results extend and complete the previously presented 45° criterion. The stability condition is formulated in terms of the eigenvalues of the constituent systems and is therefore coordinate independent and easily applicable. Moreover two types of Lyapunov functions (a quadratic and a piecewise linear type) are employed to establish stability. In case of the existence of the piecewise linear Lyapunov function we show that the set of Lyapunov functions can be easily constructed in terms of the subsystems' parameters.

The design of one qubit quantum gates with Lyapunov-based method is studied in this paper. Firstly, one qubit quantum control system is described in the coherence vector formation, then two types of Lyapunov functions are selected to implement NOT gate. The performances of control laws derived from both Lyapunov functions are compared and analyzed by means of the simulations for the NOT gate implementation.

The main objective of this paper is to give an interpretation of using non-convex and disconnected level sets Lyapunov functions in the stability analysis of discrete time systems obtained by the discretization of a continuous time Lur'e system. For simplicity reasons, Euler discretization scheme is used to illustrate the features of the proposed method. The main result of this paper shows that it is possible to build, for the original continuous time system, a sequence of bounded connected sets that converges to the origin using this type of Lyapunov functions. To this end, sufficient LMI conditions ensuring the stability of the discrete-time model and an upper bound on the error between the sampled state and the continuous trajectory are used to prove the proposed results. An example will be considered to illustrate this questioning.

This paper is concerned with fault estimation design for continuous-time Takagi-Sugeno (T-S) fuzzy systems with time delay by using adaptive fault diagnostic observer. Through constructing an appropriate type of Lyapunov function, criteria is established to reduce the conservatism of the design procedure. Based on appropriate type of Lyapunov function, new sufficient delay-dependent conditions are given to guarantee the stability of error dynamics in terms of Linear Matrix Inequalities (LMI). Finally, the effectiveness of the proposed approach is illustrated through a simulation example.

In this paper, we study the stability properties of a general class of nonautonomous discrete-time switched nonlinear systems. The switched systems consist of stable and unstable subsystems. Based on Lyapunov functions, some sufficient conditions for uniform stability, uniform asymptotic stability and uniform exponential stability are established.

l∞-based stability criteria and its applications on FLC systems

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.