Input-to-State Stability, Integral Input-to-State Stability, and <inline-formula> <tex-math notation="LaTeX">${\cal L}_{2} $</tex-math></inline-formula>-Gain Properties: Qualitative Equivalences and Interconnected Systems

Abstract

Input-to-state stability (ISS) and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain are wellknown robust stability properties that continue to find wide application in the analysis and control of nonlinear dynamical systems and their interconnections. We investigate the relationship between ISS-type and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain properties, demonstrating several qualitative equivalences between these two properties via coordinate transformations. In particular, we investigate and clearly specify the relationship between ISS and linear L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain, and between integral ISS and nonlinear L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain. This latter relationship raises an interesting question as it is known that cascade and feedback interconnections of integral ISS systems are not generally well-behaved, whereas we demonstrate that such interconnections for systems with nonlinear L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain are well-behaved. We subsequently present several new sufficient conditions for the stability of interconnected systems derived by exploiting the derived qualitative equivalences.