Input-to-state stability of infinite-dimensional control systems

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.     

A Lyapunov-based small-gain theorem for interconnected switched systems

Stability of an interconnected system consisting of two switched systems is investigated in the scenario where in both switched systems there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signals neither switch too frequently nor activate non-ISS subsystems for too long, a small-gain theorem can be used to conclude global asymptotic stability (GAS) of the interconnected system. For each switched system, with the constraints on the switching signal being modeled by an auxiliary timer, a correspondent hybrid system is defined to enable the construction of a hybrid ISS Lyapunov function. Apart from justifying the ISS property of their corresponding switched systems, these hybrid ISS Lyapunov functions are then combined to establish a Lyapunov-type small-gain condition which guarantees that the interconnected system is globally asymptotically stable.     

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems.Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.     

A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability of interconnected systems

In this paper, a novel approach to constructing flexible Lyapunov inequalities is developed for establishing Input-to-State Stability (ISS) of interconnection of nonlinear time-varying systems. It aims at a useful tool for using nonlinear small-gain conditions by allowing some flexibility in Lyapunov inequalities each subsystem is to satisfy. In the application of the ISS small-gain “theorem”, achieving a Lyapunov inequality conforming to a nonlinear small-gain “condition” is not a straightforward task. The proposed technique provides us with many Lyapunov inequalities with which a single trade-off condition between subsystems gains can establish the ISS property of the interconnected system. Proofs are based on explicit construction of Lyapunov functions.     

Input-to-state stability analysis for interconnected difference equations with delay

Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.     

Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability

We consider nonlinear systems with input-to-output stable (IOS) unmodeled dynamics which are in the range of the input. Assuming the nominal system is globally asymptotically stabilizable and a nonlinear small-gain condition is satisfied, we propose a first control law such that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin. The design of this controller is based on a gain assignment result which allows us to prove our statement via a Small-Gain Theorem [JTP, Theorem 2.1]. However, this control law exhibits a high-gain feature for all values. Since this may be undesirable, in a second stage we propose another controller with different characteristics in this respect. This controller requires morea priori knowledge on the unmodeled dynamics, as it is dynamic and incorporates a signal bounding the unmodeled effects. However, this is only possible by restraining the IOS property into the exp-IOS property. Nevertheless, we show that, in the case of input-to-state stability (ISS)—the output is the state itself-ISS and exp—ISS are in fact equivalent properties.Yuan Wang was supported in part by NSF Grant DMS-9403924 and by a scholarship from Université Lyon I, France.     

A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals

This paper considers interconnected retarded nonlinear systems. Integral input-to-state stable subsystems and the construction of Lyapunov-Krasovskii functionals for their interconnections are focused on. Both discrete and distributed time-delays in the subsystems and the communication channels are covered. This paper provides a sufficient small-gain type condition for the stability of the interconnected systems with respect to external inputs in the framework of Lyapunov-Krasovskii functionals. Global asymptotic stability is addressed as a special case which deals with time-varying delays in communication channels effectively.     

Input-to-state stability for a class of Lurie systems

We analyze input-to-state stability (ISS) for the feedback interconnection of a linear block and a nonlinear element. This study is of importance for establishing robustness against actuator nonlinearities and disturbances. In the absolute stability framework, we prove ISS from a positive real property of the linear block, by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity. When this growth condition is violated, examples show that the ISS property is lost. The result is used to give a simple proof of boundedness for negative resistance oscillators, such as the van der Pol oscillator. In a separate application, we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear.     

Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.     

An ISS-modular approach for adaptive neural control of pure-feedback systems

Controlling non-affine non-linear systems is a challenging problem in control theory. In this paper, we consider adaptive neural control of a completely non-affine pure-feedback system using radial basis function (RBF) neural networks (NN). An ISS-modular approach is presented by combining adaptive neural design with the backstepping method, input-to-state stability (ISS) analysis and the small-gain theorem. The difficulty in controlling the non-affine pure-feedback system is overcome by achieving the so-called “ISS-modularity” of the controller-estimator. Specifically, a neural controller is designed to achieve ISS for the state error subsystem with respect to the neural weight estimation errors, and a neural weight estimator is designed to achieve ISS for the weight estimation subsystem with respect to the system state errors. The stability of the entire closed-loop system is guaranteed by the small-gain theorem. The ISS-modular approach provides an effective way for controlling non-affine non-linear systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.     

Quasi-ISS/ISDS observers for interconnected systems and applications

This paper considers interconnected nonlinear dynamical systems and studies observers for such systems. For single systems the notion of quasi-input-to-state dynamical stability (quasi-ISDS) for reduced-order observers is introduced and observers are investigated using error Lyapunov functions. It combines the main advantage of ISDS over input-to-state stability (ISS), namely the memory fading effect, with reduced-order observers to obtain quantitative information about the state estimate error. Considering interconnections quasi-ISS/ISDS reduced-order observers for each subsystem are derived, where suitable error Lyapunov functions for the subsystems are used. Furthermore, a quasi-ISS/ISDS reduced-order observer for the whole system is designed under a small-gain condition, where the observers for the subsystems are used. As an application, we prove that quantized output feedback stabilization for each subsystem and the overall system is achievable, when the systems possess a quasi-ISS/ISDS reduced-order observer and a state feedback law that yields ISS/ISDS for each subsystem and therefor the overall system with respect to measurement errors. Using dynamic quantizers it is shown that under the mentioned conditions asymptotic stability can be achieved for each subsystem and for the whole system.     

Analysis of input-to-state stability for discrete time nonlinear systems via dynamic programming

The input-to-state stability (ISS) property for systems with disturbances has received considerable attention over the past decade or so, with many applications and characterizations reported in the literature. The main purpose of this paper is to present analysis results for ISS that utilize dynamic programming techniques to characterize minimal ISS gains and transient bounds. These characterizations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimization problems in nonlinear dissipative systems theory (including L₂-gain analysis and nonlinear H_∞ theory). As such, the results presented address an obvious gap in the literature.     

Construction of Lyapunov–Krasovskii functionals for networks of iISS retarded systems in small-gain formulation

This paper tackles a problem of verifying stability of retarded dynamical networks in a dissipative formulation. Subsystems are assumed to be integral input-to-state stable (iISS). Time-delays are allowed to reside in both subsystems and interconnection channels, and may be both discrete and distributed. No assumption is made on the interconnection topology. A small-gain methodology is developed for constructing a Lyapunov–Krasovskii functional to establish iISS of such a network.     

Lyapunov conditions for input-to-state stability of impulsive systems

This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.     

Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems

We study the stability properties of a class of time-varying non-linear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.     

Input-to-state stability of networked control systems

A new class of Lyapunov uniformly globally asymptotically stable (UGAS) protocols in networked control systems (NCS) is considered. It is shown that if the controller is designed without taking into account the network so that it yields input-to-state stability (ISS) with respect to external disturbances (not necessarily with respect to the error that will come from the network implementation), then the same controller will achieve semi-global practical ISS for the NCS when implemented via the network with a Lyapunov UGAS protocol. Moreover, the ISS gain is preserved. The adjustable parameter with respect to which semi-global practical ISS is achieved is the maximal allowable transfer interval (MATI) between transmission times.     

Strong iISS is preserved under cascade interconnection

In a recent paper, we have introduced the notion of Strong iISS as a compromise between the strength of input-to-state stability (ISS) and the generality of integral ISS (iISS). In this note, we continue the investigations around this property by studying its behavior in an interconnection context. In particular, we show that the cascade of Strongly iISS systems is itself Strongly iISS and we recall some useful tools to study Strongly iISS systems in feedback interconnection.     

A geometrical formulation to unify construction of Lyapunov functions for interconnected iISS systems

In recent years, the ability to accommodate various nonlinearities has become even more important to support systems design and analysis in a broad area of engineering and science. In this line of research, this paper discusses usefulness of the notion of integral input-to-state stability (iISS) in assessing and establishing system properties through interconnection of component systems. The focus is to construct Lyapunov functions which explain mechanism and provide estimate of stability and robustness of interconnected systems. Unique issues arising in dealing with iISS systems are reviewed in comparison with interconnections of input-to-state stable (ISS) systems. The max-separable Lyapunov function and the sum-separable Lyapunov function which are popular for ISS and iISS, respectively, are revisited. The max-separable function cannot be qualified as a Lyapunov function when component systems are not ISS. Level sets of the max-separable function are rectangles, and the rectangles cannot be expanded to encompass the entire state space in the presence of non-ISS components. The sum-separable function covers iISS components which are not ISS. However, it has practical limitations when stability margins are small. To overcome the limitations, this paper brings in a new idea emerged recently in the literature, and proposes a new type of construction looking at level sets of a Lyapunov function. It is shown how an implicit function allows us to draw chamfered rectangles based on fictitious gain functions of component systems so that they provide reasonable estimates of forward invariant sets producing a Lyapunov function applicable to both iISS and ISS systems equally.     

Input-to-state stability of switched systems and switching adaptive control

In this paper we prove that a switched nonlinear system has several useful input-to-state stable (ISS)-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.