Input-to-state Stability Lyapunov Research Articles

Stability and Performance of Asynchronous Sampled-Data Interconnected Systems: An Integral Approach

In this paper, we focus on the stability of asynchronous sampled-data interconnected systems and the evaluation of how asynchronous sampling and time delays degrade the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula> gains from exogenous inputs to controlled outputs. To judge the stability and evaluate the performance of systems, an integral approach is proposed and it combines a revised form of ISS (input-to-state stability) Lyapunov conditions with an integral description of bounded measurement errors. Practical procedures for the application of the approach are further presented. In the case with shared sampling periods, a time-translation approach is introduced to investigate the relationship between asynchronous sampling and time delays.

The Stochastic Robustness of Nominal and Stochastic Model Predictive Control

In this work, we establish and compare the stochastic and deterministic robustness properties achieved by nominal model predictive control (MPC), stochastic MPC (SMPC), and a proposed constraint-tightened MPC (CMPC) formulation, which represents an idealized version of tube-based MPC. We consider three definitions of robustness for nonlinear systems and bounded disturbances: robust asymptotic stability (RAS), robust asymptotic stability in expectation (RASiE), and RASiE w.r.t. the stage cost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell (\cdot)$</tex-math></inline-formula> used in these MPC formulations ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> -RASiE). Via input-to-state stability (ISS) and stochastic ISS (SISS) Lyapunov functions, we establish that MPC, subject to sufficiently small disturbances, and CMPC ensure all three definitions of robustness without a stochastic objective function. While SMPC also ensures RASiE and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> -RASiE, SMPC does not guarantee RAS for nonlinear systems. Through a few simple examples, we illustrate the implications of these results and demonstrate that, depending on the definition of robustness considered, SMPC is not necessarily more robust than nominal MPC even if the disturbance model is exact.

Control Lyapunov function method for robust stabilization of multistable affine nonlinear systems

AbstractIn this paper, we study the problem of robust stabilization of affine nonlinear multistable systems in the presence of exogenous disturbances. The results are based on the theory of input‐to‐state stability (ISS) and integral input‐to‐state stability (iISS) for systems with multiple invariant sets. The notions of ISS and iISS control Lyapunov functions (CLFs) and the small control property are extended within the multistability framework. Such properties are also complemented by the concept of a weak iISS CLF and corresponding small control property. It is verified that the universal control formula can be applied to yield the ISS (iISS) property for the closed‐loop system. The efficiency of the extended CLF framework in the multistable sense is illustrated for a Duffing system and in application to a noise‐induced transition in a semiconductor‐gas‐discharge gap system.

A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

The exponential input-to-state stability property: characterisations and feedback connections

The exponential input-to-state stability (ISS) property is considered for systems of controlled nonlinear ordinary differential equations. A characterisation of this property is provided, including in terms of a so-called exponential ISS Lyapunov function and a natural concept of linear state/input-to-state L^2-gain. Further, the feedback connection of two exponentially ISS systems is shown to be exponentially ISS provided a suitable small-gain condition is satisfied.

On Input-to-State Stability of Linear Difference Equations and Its Characterization with a Lyapunov Functional

This paper investigates necessary and sufficient Lyapunov conditions for Input-to-State Stability (ISS) of Linear Difference Equations with pointwise delays and an additive exogenous signal. Grounding on recent works in the literature on necessary conditions for the exponential stability of such Difference Equations, we propose a quadratic Lyapunov functional involving the derivative of the so-called delay Lyapunov matrix of the corresponding homogeneous Difference Equation. We prove that the ISS of Linear Difference Equations is equivalent to the existence of an ISS Lyapunov functional. We apply this result to the stability and ISS analysis of hyperbolic Partial Differential Equations of conservation laws.

Data-Driven Stability Certificate of Interconnected Homogeneous Networks via ISS Properties

This letter is concerned with a compositional data-driven approach for stability certificate of interconnected homogeneous networks with (partially) unknown dynamics while providing 100% correctness guarantees (as opposed to probabilistic confidence). The proposed framework enjoys input-to-state stability (ISS) properties of subsystems described by ISS Lyapunov functions. In our data-driven scheme, we first reformulate the corresponding conditions of ISS Lyapunov functions as a robust optimization program (ROP). Due to appearing unknown dynamics of subsystems in the constraint of ROP, we propose a scenario optimization program (SOP) by collecting data from trajectories of each unknown subsystem. We solve SOP and construct an ISS Lyapunov function for each subsystem with unknown dynamics. We accordingly leverage a compositional technique based on -type small-gain reasoning and construct a Lyapunov function for an unknown interconnected network based on ISS Lyapunov functions of individual subsystems. We demonstrate the efficacy of our data-driven approach over a room temperature network containing 1000 rooms with unknown dynamics. Given collected data from each unknown room, we verify that the unknown interconnected network is globally asymptotically stable (GAS) with 100% correctness guarantee.

Input-to-state stability for discrete-time switched systems by using Lyapunov functions with relaxed constraints

&lt;abstract&gt;&lt;p&gt;In this paper, input-to-state stability (ISS) is investigated for discrete-time time-varying switched systems. For a switched system with a given switching signal, the less conservative assumptions for ISS are obtained by using the defined weak multiple ISS Lyapunov functions (WMISSLFs). The considered switched system may contain some or all subsystems which do not possess ISS. Besides, for an ISS subsystem the introduced Lyapunov function could be increasing along the trajectory of the subsystem without input at some moments. Then for a switched system under any switching signal, the relaxed sufficient constraints for ISS are attained by using the defined weak common ISS Lyapunov functions. For this case, each subsystem of the considered system must be ISS. The proposed function may be increasing along the trajectory of each ISS subsystem of the considered system without input at some instants. The relationship between WMISSLFs for a switched system and the defined weak multiple Lyapunov functions for this switched system without input is set up. Three numerical examples are investigated to display the usefulness of the principal outcomes. According to the main conclusions, an intermittent controller is applied to ensure ISS for a discrete-time disturbed Chua's chaotic system.&lt;/p&gt;&lt;/abstract&gt;

LaSalle–Yoshizawa Theorem for nonlinear systems with external inputs: A counter-example

LaSalle–Yoshizawa Theorem is an important tool in guaranteeing convergence of a nonlinear time-varying adaptive system. It claims boundedness and convergence when the derivative of a Lyapunov function is negative semidefinite. For a nonlinear system with an external input, the input-to-state stability (ISS) Lyapunov theorem reveals boundedness of system solutions when the derivative of an ISS Lyapunov function is negative definite with an input term. It is interesting to seek a LaSalle–Yoshizawa like criterion for a nonlinear system with an external input. An intuitive question is whether a certain boundedness property is guaranteed when the derivative of an ISS Lyapunov function is negative semidefinite with an input term. This technical communique gives a negative answer with a counter-example.

Lyapunov characterizations on input-to-state stability of nonlinear systems with infinite delays

This paper addresses Lyapunov characterizations on input-to-state stability (ISS) of time-varying nonlinear systems with infinite delays. With novel ISS definitions in the case of nonlinear systems with infinite delays, we present several results on their ISS Lyapunov characterizations in the form of both ISS Lyapunov theorems and converse ISS Lyapunov theorems. It is shown that an infinite-delayed system is (locally) ISS if it has a (local) ISS Lyapunov functional, and conversely, there exists a (local) ISS Lyapunov functional if it is (locally) ISS. To prove the converse ISS Lyapunov theorems, we establish a key technical lemma bridging ISS/LISS and robust asymptotic stability of systems with infinite delays and two converse Lyapunov theorems concerning robust asymptotic stability of systems with infinite delays. Two distinctive advantages of this work are that a large class of infinite dimensional spaces are allowed and the results are established based on a more general Lipschitz condition, i.e., the right hand side Lipschitz (RS-L) condition. An example is provided for illustration of the obtained results.

Damping-Robustness of Various 1-D Wave PDEs under Boundary Feedback Control

This paper is devoted to the study of the robustness properties of the 1-D wave equation for an elastic vibrating string under four different damping mechanisms that are usually neglected in the study of the wave equation: (i) friction with the surrounding medium of the string (or viscous damping), (ii) thermoelastic phenomena (or thermal damping), (iii) internal friction of the string (or Kelvin-Voigt damping), and (iv) friction at the free end of the string (the so-called passive damper). The passive damper is also the simplest boundary feedback law that guarantees exponential stability for the string. We study robustness with respect to distributed inputs and boundary disturbances in the context of Input-to-State Stability (ISS). By constructing appropriate ISS Lyapunov functionals, we prove the ISS property expressed in various spatial norms.

ISS small-gain criteria for infinite networks with linear gain functions

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional ℓ ∞ -type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a consequence, the small-gain condition can be formulated in terms of a generalized spectral radius of the gain operator. By an example, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some sort of symmetry. While our main result provides an ISS Lyapunov function in implication form for the overall network, an ISS Lyapunov function in a dissipative form is constructed under mild extra assumptions.

Input-to-State Stability and Stabilization of Nonlinear Impulsive Positive Systems

This paper focuses on the problems of input-to-state stability (ISS) and stabilization for nonlinear impulsive positive systems (NIPS). Using the max-separable ISS Lyapunov function method, a sufficient condition on ISS is given for general NIPS. On that basis, the ISS criteria for linear impulsive positive systems (LIPS) and affine nonlinear impulsive positive systems (ANIPS) are given. Through them, ISS properties can be directly judged from the algebraic and differential characteristics of the systems. Then, utilizing the ISS criteria, state-feedback and impulsive controllers are designed for LIPS and ANIPS, respectively, which make the systems input-to-state stabilizable. Lastly, some numerical examples are given to verify the effectiveness of our results.

Input-to-state stability and integral input-to-state stability of non-autonomous infinite-dimensional systems

In this paper, we provide Lyapunov-based tools to establish input-to-state stability (ISS) and integral input-to-state stability (iISS) for non-autonomous infinite-dimensional systems. We prove that for a class of admissible inputs the existence of an ISS Lyapunov function implies the ISS of a system in Banach spaces. Furthermore, it is shown that uniform global asymptotic stability is equivalent to their integral input-to-state stability for non-autonomous generalised bilinear systems over Banach spaces. The Lyapunov method is provided to be very useful for both linear and nonlinear tools including partial differential equations (PDEs). In addition, we present a method for construction of iISS Lyapunov function in Hilbert spaces. Finally, two examples are given to verify the effectiveness of the proposed scheme.

ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.

Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

Lyapunov functions enable analyzing the stability of dynamic systems described by ordinary differential equations without finding the solution of such equations. For nonlinear systems, devising a Lyapunov function is not an easy task to solve in general. In this paper, we present an approach to the construction of Lyapunov funtions to prove stability in estimation problems. To this end, we motivate the adoption of input-to-state stability (ISS) to deal with the estimation error involved by state observers in performing state estimation for nonlinear continuous-time systems. Such stability properties are ensured by means of ISS Lyapunov functions that satisfy Hamilton–Jacobi inequalities. Based on this general framework, we focus on observers for polynomial nonlinear systems and the sum-of-squares paradigm to find such Lyapunov functions.

INPUT-TO-STATE STABILITY FINITE-TIME LYAPUNOV FUNCTIONS FOR CONTINUOUS-TIME SYSTEMS

In this paper we propose an input-to-state stability (ISS) criterion for continuous–time systems based on a finite–time decrease condition for a positive definite function of the norm of the state. This yields a so–called ISS finite–time Lyapunov function, which allows for easier choice of candidate functions compared to standard ISS Lyapunov functions. An alternative converse ISS theorem in terms of ISS finite– time Lyapunov functions is also provided. Moreover, we prove that ISS finite–time Lyapunov functions are equivalent with standard ISS Lyapunov functions using a Massera–type construction. The developed ISS framework can be utilized in combination with Sontag’s “universal” stabilisation formula to develop input–to–state stabilizing control laws for continuous–time nonlinear systems that are affine in the control and disturbance inputs, respectively. MSC: 93C10, 93D09, 93D30, 93D15

Stability analysis of a class of non-simultaneous interconnected impulsive systems

This paper provides sufficient conditions for input-to-state stability of two interconnected nonlinear impulsive systems whose jump instants are not necessarily identical. Unlike prior results, each subsystem is allowed to possess stabilizing or destabilizing flows. In this regard, a candidate exponential input-to-state stable (ISS) Lyapunov function is constructed for the overall system. Then, by bounding the trajectory, for each possible combination of impulsive subsystems, sufficient conditions are presented which ensure input-to-state stability of the interconnected system. Furthermore, to render the newly-derived conditions less conservative, the coefficients of the candidate exponential ISS Lyapunov function of each subsystem are considered to be time-varying. The applicability of the theoretical outcomes is verified through some numerical examples.

On Input-to-State Stability of Discrete-Time Switched Nonlinear Time-Varying Systems

In this paper, input-to-state stability (ISS) for discrete-time switched nonlinear time-varying (SNTV) systems is investigated. Starting with discrete-time nonlinear time-varying (NTV) systems, some improved sufficient conditions are proposed to verify the ISS of systems by using the weak implication-form ISS (WI-ISS) Lyapunov function, weak dissipative-form ISS (WD-ISS) Lyapunov function, and interval descent technique. Then, the results obtained are extended to study the ISS of discrete-time SNTV systems, several relaxed conditions are given by using piecewise WI-ISS and WD-ISS Lyapunov functions, minimum dwell time, and infinite switching methods, respectively. Comparing with the existing results, the obtained conditions release the requirement on negative definiteness of the differences of (piecewise) Lyapunov functions, moreover, all subsystems are allowed to be unstable in the case of infinite switching. Finally, a numerical example is given to illustrate the theoretical results.

Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.