Input-to-state Stability Lyapunov Functions Research Articles

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.

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.

Safety-critical dynamic event-triggered control of nonlinear systems

This paper is concerned with the problem of safety-critical dynamic event-triggered control (ETC) for nonlinear systems. A novel safety-critical dynamic ETC strategy is proposed to simultaneously guarantee event-triggered safety and stability based on the union of input-to-state safe barrier function (ISSf-BF) and input-to-state stable Lyapunov function (ISS-LF). Also, an appropriate dynamic event-triggered mechanism (DETM) is constructed, and Zeno phenomenon of ETC in the safety-critical control framework is definitely excluded based on the DETM constructed. Compared with the existing safety-critical control designed in the framework of continuous-time feedback control, the proposed strategy constructs safety-critical event-triggered controller and an DETM, which greatly alleviates the unnecessary waste of communication and computation resources. Finally, three examples are presented to demonstrate the effectiveness of the proposed design strategy.

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.

Estimate of the domain of attraction for interconnected systems

In this paper, we estimate the domain of attraction of the origin of two interconnected nonlinear systems. For each subsystem we introduce an auxiliary system and assume that the origin is locally robustly asymptotically stable. Then an integral input-to state stable Lyapunov function for each subsystem on a bounded set is constructed via Zubov’s method and the introduced auxiliary system. A local version of small gain theorem is proposed. It is easier to check if the conditions of the proposed small gain theorem are satisfied. Moreover, an estimate for the domain of attraction for the whole system is obtained based on the proposed small gain theorem and the computed integral input-to state stable Lyapunov functions. We illustrate the effectiveness of the main results by an academic example.

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.

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.

Input-to-state Stability of Nonlinear Positive Systems

In this paper, input-to-state stability (ISS), as a useful tool for robust analysis, is first applied to continuous-time and discrete-time nonlinear positive systems. For continuous-time and discrete-time positive systems, some new definitions of ISS are introduced. Different from the usual ISS definitions for nonlinear systems, our ISS definitions can fully reflect the positiveness requirements of states and inputs of the positive systems. By introducing the max-separable ISS Lyapunov functions, some ISS criterions are given for general nonlinear positive systems. Based on that, the ISS criterions for linear positive systems and affine nonlinear homogeneous systems are given. Through them, the ISS properties can be judged directly from the differential and algebraic characteristics of the systems. Simulation examples verify the validity of our results.

ISS Lyapunov functions for cascade switched systems and sampled-data control

Input-to-state stability (ISS) of switched systems is studied where the individual subsystems are connected in a serial cascade configuration, and the states are allowed to reset at switching times. An ISS Lyapunov function is associated to each of the two blocks connected in cascade, and these functions are used as building blocks for constructing ISS Lyapunov function for the interconnected system. The derivative of individual Lyapunov functions may be bounded by nonlinear decay functions, and the growth in the value of Lyapunov function at switching times may also be a nonlinear function of the value of other Lyapunov functions. The stability of overall hybrid system is analyzed by constructing a newly constructed ISS-Lyapunov function and deriving lower bounds on the average dwell-time. The particular case of linear subsystems and quadratic Lyapunov functions is also studied. The tools are also used for studying the observer-based feedback stabilization of a nonlinear switched system with event-based sampling of the output and control inputs. We design dynamic sampling algorithms based on the proposed Lyapunov functions and analyze the stability of the resulting closed-loop system.

Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces

We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems.

The Role of Convexity in Saddle-Point Dynamics: Lyapunov Function and Robustness

This paper studies the projected saddle-point dynamics associated to a convex–concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity–concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity–concavity of the saddle function. When strong convexity–concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability (ISS) of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.