Studies on the adaptive flight control systems in the presence of destructive limit-cycle oscillation (LCO) or pilot induced oscillation has received recent interest in order to construct a highly reliable control laws. Potential adverse LCO is inherent in control systems with piecewise nonlinearities such as actuator and/or rate saturation, especially when we need to apply a sufficiently high gain to the system for high performance. This paper provides a theoretical foundation to analyze the LCO and to optimize the loop gain for the control systems with piecewise nonlinearities. By establishing the piecewise nonlinearities as equivalent analytic functions we can apply Floquet theory to identify LCO and check its stability. In addition, we determine the least upper bound of the loop gain based on the framework provided by the Floquet theory and piecewise linear system analysis. We take an example of a short period mode flight control system with rate saturated servo mechanism to demonstrate the proposed framework of analysis.