This paper develops computationally tractable analysis and synthesis methods for nonlinear systems with structured gain-bounded uncertainty . The methods can take account of the size and shape of valid region in the state-space. In contrast with the case of linear systems, the stability of nonlinear system is not always globally achieved. In stabilizing controller design, the region of attraction should be designed to cover the range where the nonlinear system operates. It is known that the analysis and synthesis of nonlinear systems with structured uncertainty are can be characterized by Hamilton-Jacobi inequalities. Since the partial differential inequalities are not always solvable or too difficult to solve, the analysis and synthesis method using the Hamilton-Jacobi inequalities are not practical.

There are some studies about nonlinear robust control which focus on the state-space region of nonlinear system. It has been shown that state-dependent scaling matrices are effective for the analysis and synthesis of nonlinear system . The state-dependent scaling changes the valid region which the Hamilton-Jacobi inequality guarantees. But, the study does not tell us how to find suitable state-dependent scaling matrixes detecting valid region. Another paper has shown that the robustness analysis is characterized by linear matrix inequalities. A computationally tractable approach to maximizing the region of attraction is also proposed. But, it remains to show to what extent the approach has advantage in robustness analysis and synthesis of nonlinear systems.

In this paper, it is shown that the analysis using state-dependent scaling matrices are very useful to estimate the region where the system is robust stable. The analysis using state-dependent scaling matrices can detect larger region than the analysis using constant scaling. The author proposes a computationally tractable analysis method of maximizing region of attraction. This paper clarifies that the most effective state-dependent scaling matrices are found automatically when we try to detect the largest valid region. The state-dependent scaling reduces the conservatism in robust stability analysis. The author also proposes a synthesis method based on the idea of analysis.