The mu analysis method is applied to a class of multivariable separable autonomous nonlinear feedback systems. The nonlinear element is replaced by its sinusoidal input describing function (SIDF) to develop a quasilinear nominal model for control analysis and design. Some properties of the (infinity)-norm and structured singular values (SSVs) are shown to hold for the type of systems where the output is a function of control input-signal amplitude and frequency. The model uncertainty induced by the SDIF approximation is calculated and incorporated in the control design as an input multiplicative perturbation to the quasilinear nominal model. The mu analysis technique is then employed to study the stability and performance robustness of the nonlinear control system. Using small gain theorem, the condition for the absence of limit cycle is derived. The information obtained from this analysis is then used to design a bank of controllers using the H-infinity optimization-based mu synthesis technique. These controllers are designed such that the closed loop system is robust over a range of control-input amplitude. A simple scheme is then proposed to select the appropriate controller for the operating range. The simulation results illustrate that the proposed technique may be used to design a high performance robust controller for nonlinear systems and the absence of the limit cycle oscillation may be ensured.