In the paper, a methodology for the guaranteed cost estimation and control for nonlinear system discretetime systems is proposed. To solve such a challenging problem, the article starts with a general description of the system and assumptions regarding its nonlinearities. The subsequent part of the paper describes the design methodology of the robust observer and controller for the predefined cost function using linear matrix inequalities. The final part of the paper presents an illustrative example oriented towards a practical application to the multiple tank system, which illustrates the performance of the proposed approach.