ABSTRACT
The Bounded Input (BI) stability concept is essentially different from Lyapunov’s (LY) stability concept for its following characteristics:
–The BI stability concept concerns the system behavior under actions of input variables, while the original LY stability concept treats the system behavior in the free regime considered in the framework of deviations, i.e., in the total nominal regime in terms of the total coordinates 93.
–The original BI stability concept in the framework of the linear continuous-time dynamical systems demands boundedness of the system behaviors for bounded input variables and for all zero initial conditions [117, p. 311], which was broaden to the boundedness of the system behaviors for bounded input variables and any bounded initial conditions [68], [69], while the LY stability concept demands Lyapunov’s ɛ - δ closeness of the system behaviors to the zero equilibrium vector in the framework of deviations under zero input deviations. This means that various new BI stability properties were introduced and defined, and complex domain criteria were discovered for them. E. D. Sontag et al. 118, 120, 121 established the theory of “input to state stability”that concerns both the bounded input vector and arbitrary initial state conditions. It is valid in the general setting of nonlinear systems. Their results, which hold in time domain, exploit the Lyapunov method in combination with the comparison functions of W. Hahn 83. 344
–The BI stability concept characterizes various stability properties of a dynamical system, while the LY stability concept determines various stability properties of a desired (nominal) system behavior in terms of total coordinates, i.e., of an equilibrium vector in terms of derivations.
–The BI stability concept does not demand asymptotic convergence ( a s t → ∞ ) $ (as t \to \infty ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781138039629/a28624dc-7a42-40c0-abd8-1d07ecbd20fe/content/inline-math14_1.tif"/> of the system behaviors to the equilibrium vector, while LY stability properties – attraction, asymptotic and exponential stability of the equilibrium vector −do.
