Robust control for generalized nonlinear systems with uncertainty via matrix partial order and nonzero-sum games

In this paper, we propose the use of matrix partial order: Löwner and GL partial orders from algebra to investigate some classes of generalized nonlinear systems with uncertainties. Due to the presence of unmatched and structural uncertainties in control and disturbance channels, the considered class of nonlinear systems often faces challenges such as modeling difficulties, increased complexity in robust control design, and significant obstacles in stability analysis. Then, to address the aforementioned challenges, from the novel theoretical perspective of matrix partial orders, corresponding Löwner and GL partial orders techniques are employed to construct the auxiliary systems and introduce the corresponding performance indices, thereby reformulating the original robust control problems as optimal control problems. Next, the trajectory consistency between the auxiliary systems and the original systems is rigorously established through the theorems concerning global asymptotic stability (GAS). Furthermore, based on the constructed auxiliary systems and the new designed controllers, the long-time passivity (LTP) behavior of the original systems is analyzed. Leveraging GAS and LTP, the analyses of the uniform ultimate boundedness (UUB) are conducted. Finally, several practical control problems are presented to demonstrate the feasibility and effectiveness of the proposed theoretical results.