Using a two dimensional compressible flow representation of axial compressor dynamics, a control-theoretic input-output model is derived which is of general utility in rotating stall/surge active control studies. The derivation presented here begins with a review of the fluid dynamic model, which is a 2D stage stacking technique that accounts for blade row pressure rise, loss and deviation as well as blade row and inter-blade row compressible flow. This model is extended to include the effects of the upstream and downstream geometry and boundary conditions, and then manipulated into a transfer function form that dynamically relates actuator motion to sensor measurements. Key relationships in this input-output form are then approximated using rational polynomials. Further manipulation yields an approximate model which is in standard form for studying active control of rotating stall and surge. As an example of high current relevance, the transfer function from an array of jet actuators to an array of static pressure sensors is derived. Numerical examples are also presented, including a demonstration of the importance of proper choice of sensor and actuator locations, as well as a comparison between sensor types. Under a variety of conditions, it was found that sensor locations near the front of the compressor or in the downstream gap are consistently the best choices, based on a quadratic optimization criterion and a specific 3-stage compressor model. The modeling and evaluation procedures presented here are a first step toward a rigorous approach to the design of active control systems for high speed axial compressors.
Multidimensional Wave Field Signal Theory: Transfer Function Relationships
