A dynamic analysis of a compressor system is presented to characterize its behavior and establish a mathematical framework for identifying stable and unstable operating regions. The study is grounded in the nonlinear Moore–Greitzer model, which describes compressor dynamics in terms of mass flow and pressure rise as functions of rotor speed. To predict the onset of surge and system instability, advanced nonlinear techniques are employed, including the Jacobian matrix, linear parameter-varying (LPV) modeling, Bendixson’s criterion, and phase plane analysis. These tools enable the identification of both stable and unstable regions, as well as the limit cycle associated with surge phenomena. All of these analyses of the compressor are innovative. Accurate prediction of compressor surge and instability is essential for defining and designing effective control strategies, as surge can damage the compressor, interrupt downstream flow, and inherently represents an unstable operating condition. However, analysis alone is insufficient for practical compressor operation. Therefore, three active control methods are considered: Proportional–Integral–Derivative (PID), Linear Quadratic Regulator (LQR), and Model Predictive Control (MPC). The comparative analysis reveals that insufficient consideration of varying system conditions in LQR design may lead to inferior performance relative to MPC and PID control, particularly under changing disturbances. In contrast, MPC and PID exhibit stronger robustness to disturbance variations and provide effective disturbance rejection. In the proposed approach, MPC simulations are conducted to evaluate controller performance. Due to disturbances in the closed-loop model, the LQR controller demonstrates reduced robustness compared to PID and MPC. Under surge-related disturbances, the minimum input mass flow by both PID and MPC controllers is 0.495 (very close to setpoint), and both controllers exhibit an overshoot of 33% and a rise time of 3 s.
Hosseindokht et al. (Tue,) studied this question.