| The sufficient condition of the small-gain theorem has been proved for general nonlinear systems. However the question of whether the small-gain condition is necessary for general nonlinear systems with feedback connection has been open since 1966. In fact, the necessary condition of the small-gain theorem does not hold for general nonlinear systems with feedback connection, which has recently been shown by Freeman's counterexample. This example illustrates the fact that robust stability need not imply performance when the system is nonlinear, namely, it is possible to have robust stability even when the conditions of the small-gain theorem are violated.; Motivated by these results, we introduce the notion of new-conditional gain (and, more generally, conditional dissipativity) and further derive the proof of the necessity of new small-gain theorem for general nonlinear systems with feedback connection, which has faced us more than 30 years. When using the new performance measure, we develop and prove the equivalence law for the linear dissipative systems and linear conditional dissipative systems, such that we finally give the better answer to relationships among finite gain, conditional-gain, passive systems, dissipative systems, and conditional dissipative systems. We also provide a storage function characterization of conditional dissipativity to be used in robust stability analysis. This lets us find a new way to study the relationship between the performance and stability. We extend and prove the general stability theorems for general nonlinear systems with feedback connection, which unify various performances such as passivity, dissipativity, conditional dissipativity, quadratic form, and so on, and develop a simple and efficient law of the stability for designing a general nonlinear system. |
