Stability And Stabilization Of Nonlinear Systems With Multiple Critical Modes And Application To Active Control Of Rotating Stall In Axial Compressors

When an equilibrium of smooth nonlinear dynamical system contains both stable manifold and central manifold,the stability of the system cannot be determined by the linear part,which depends on the higher-order terms.In this thesis,we study the stability conditions and stabilization conditions of systems in this critical case.The main results are as follows:(1)By using the central manifold theorem and the Poincaré normal form transformation,this type of nonlinear systems can be transformed into a normal form: Kolmogrov equations.By analyzing the stability conditions of the normal form,we prove that when the partition matrix satisfying certain conditions,the stability of the corresponding system is equivalent to the stability of a class of lower-order systems.The relationship between stability conditions and diagonal stability over the first quadrant is derived.Several system stability criterias are derived basing on these analyses.(2)When the system has a particular nonlinear control input,by using the results of stability analysis,necessary and sufficient conditions for the stabilization of the second-order and third-order systems are derived.(3)The theoretical results are applied to the axial compressor rotating stall(MooreGreitzer)model.By using the central manifold theorem and Poincaré normal form transformation,the MG model is transformed into an ODE model with n pairs of stall modes.The matrix corresponding to the quadratic term is a circulant matrix,and the relationship between the stability of the system and the shape near the peak of the compressor characteristic curve is further analyzed.(4)Active control is performed on the MG model with n-th stall mode and we construct the stabilizing controller with second order homogeneous functions when the system is non-degenerate.