Revisiting the iISS small-gain theorem through transient plus ISS small-gain regulation

International audienceRecently, the small-gain theorem for input-to-state stable (ISS) systems has been extended to the class of integral input-to-state stable (iISS) systems. Feedback connections of two iISS systems are robustly stable with respect to disturbance if an extended small-gain condition is satisfied. It has been proved that at least one of the two iISS subsystems needs to be ISS for guaranteeing globally asymptotic stability and iISS of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iISS small gain theorem as transient plus ISS small-gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the iISS small-gain theorem

Interpreting the iISS Small-Gain Theorem as Transient Plus ISS Small-Gain Regulation

International audienceThis paper addresses the problem of establishing stability of interconnections of integral input-to-state stable (iISS) systems. Recently, the small-gain theorem for input-tostate stable (ISS) systems has been extended to the class of iISS systems. It has been also proved that at least one of the two iISS subsystems comprising a feedback interconnection needs to be ISS with respect to the state of the other subsystem for guaranteeing the iISS of the overall system. This paper shows that making use of this necessary condition enables to provide more insight on the iISS small gain theorem by giving an alternative proof of this result from the perspective of transient plus ISS small-gain regulation

Causality of general input–output systems and extended small-gain theorem for their feedback connection

For the small-gain theorem derived by Zames in 1966, the later studies after a few decades elaborated on its derivation through defining system causality, which was not assumed by Zames. In connection with the treatment of causality, however, these studies made some unnecessary assumptions on the subsystems in feedback connection and failed to handle general systems described by an input–output relation rather than mapping (which we call input-intolerant/-output-unsolitary systems). On the other hand, although the treatment by Zames can handle such subsystems, it instead turns out to lead to larger values for the induced norms of subsystems compared with the later treatment. This paper is concerned with developing an extended form of the small-gain theorem through the same induced norms as in the later studies while dealing with general input–output causal subsystems. Since causality of subsystems plays a key role in such development, our research direction strongly motivates us to study how causality should be defined for general input–output systems. Thus, much of the arguments in this paper is devoted to such a study, which provides us with profound and thorough understandings on causality of different restricted classes of general input–output systems. Mutual relationships among adequate causality definitions for different classes are also clarified, which should be important in its own right. After deriving an extended form of the small-gain theorem, an example illustrates the importance of dealing with such general subsystems, as well as usefulness of the extension

Observer Design for Interconnected Systems and Implementation via Differential-Algebraic Equations

A new approach to the design of observers of nonlinear dynamical systems is presented. Generally, linear or nonlinear control systems are expressed as explicit systems of differential equations and solved either analytically or numerically. If numerically, they are implemented using standard ordinary differential equation (ODE) solvers. In this thesis, a system is decomposed and modeled as an interconnection between two observer subsystems, particularly, as canonical DAE observers. In general, control design engineers may be faced with a formidable problem of solving this system analytically or in obtaining closed-form solutions. To attest to the complexity and complications in treating a system of interconnected DAE observer systems, a scaled-down version of a publication on “Small-Gain Theorem” is included in the appendix for the reader’s perusal. (A brief introduction to “Small-Gain Theorem” can be found in Chapter 4). The premise of this thesis is to demonstrate that, where the design of an observer plays a major role involving output feedback, there may be advantages in formulating a control system as a differential-algebraic equation (DAE), especially in the case of interconnected subsystems. An implicit system of interconnected DAE observers is considered and shown implementable using an existing DAE solver, whose resolution allows one the capability of computing input and output bounds. This is based on fixed or variable timesteps within the operating interval of each subsystem to ensure input-output stability (IOS) and the observability property of the interconnected observer system. The observer design method is based on the extended linearization approach. The basic background is provided for the design process of an interconnected observer system using DAE. Note, the application of the new approach has not been considered previously for the case of an interconnected DAE observer system