Analyzing And Computing The Regions Of Stability For Piecewise Affine Systems Based On The Boundary

Piecewise Affine(PWA)systems could model nonlinear characteristics such as switching,saturation,dead zone,gap and hysteresis accurately,which makes it a practical modeling method.As a fundamental property of dynamic systems,stability is critical when analyzing and designing PWA systems.Except global stable equilibrium points and global divergency,the PWA systems may involve other behaviors such as limit cycles,local stability,and multiple steady-state solutions.In these cases,the stability could be analyzed by finding the Region of Stability(RoS).Most existing RoS analyzing methods are based on Lyapunov functions,which are not easy to find.What is worse,it is usually difficult to analyze the RoS of limit cycles by the existing methods,and the results are usually subregions of the whole RoS with restricted shapes.In this thesis,a method of analyzing Ro S based on the RoS boundary is proposed for PWA systems.This method does not rely on Lyapunov function,and is applicable to both equilibrium points and limit cycles.Furthermore,the results are no longer subregions with restricted shapes.An RoS boundary meshing algorithm and an RoS computing algorithm based on the Binary Space Partition(BSP)tree are also proposed in this thesis.Both together,they make it possible to compute the RoS automatically,and extend the application scope of this method.The research consists the following parts:(1)The definitions of PWA systems' model and solutions are introduced.PWA systems' special behaviors,such as tangency,sliding mode,and Zeno phenomenon,are analyzed.Then the conditions for the existence and uniqueness of PWA systems' solution are given.At last,the algorithm for computing numerical solutions are introduced.The algorithm consists two parts,i.e.,for computing the state transition function and the switching time function.(2)The relationship between the stability of PWA systems and the continuity of discrete transition function are made.According to the property of trajectories,points on switching surfaces are classified as passable points and impassable points.According to the continuity of the discrete transition function,continuous points and discontinuous are defined.It is proved that the RoS boundary consists of impassable points and discontinuous points.Based on this theory,the formula for computing RoS bonary are derived by analyzing the continuity of discrete transition function.Theoretical analysis indicates that there are three types of discontinuous points,i.e.,generalized tangent points,infinity-switching-time points,and intersection of switching surfaces.Together with the formulas of computing impassable points,the RoS boundary points are classified into four categories,and four similar formulas are given to compute them.(3)A graphical RoS analyzing method is proposed.Multiple different examples are analyzed.They involve different stability properties,such as global stability,local stability,multiple steady solutions and limit cycles.The RoS solving method for these properties is introduced accordingly,and the results are verified.(4)Based on a further analysis on the RoS boundary,a RoS boundary meshing algorithm is proposed.Firstly,the RoS boundary is parameterized;then a weight transformation is applied;at last,the RoS boundary is meshed via Delaunay triangulation.The poposed weight transformation algorithm reduces the distortion when triangulating.The meshing algorithm converts the RoS boundary to triangles(simplices)in the state space,makes it possible to compute the RoS boundary by computation geometry.(5)An RoS computing algorithm based on the BSP tree is proposed.By modifying the constructing and partitioning algorithms,the BSP tree is made applicable for computing Ro S.Firstly,the state space of PWA systems are partitioned by the meshed RoS boundary,resulting some polyhedrons(polytypes);then the faces(facets)of these polyhedrons are colored;at last,the belonging RoS of these polyhedrons(polytypes)are determined by the flood fill algorithm.The innovatory contributions of this thesis are as follows:(1)An RoS analyzing method based on boundary is proposed.Instead of Lyapunov function,this method is based on the continuity of discrete transition function.The relationship between stability and the continuity of discrete transition function is made.Furthermore,this relationship is applicable to not only PWA systems,but also other hybrid dynamic systems.(2)The formula for computing the Ro S boundary is proposed based on analyzing the continuity of PWA systems' discrete transition function.With the ideas such as general tangency,the Ro S boundary points are classified into four categories,and four formulas with similar properties are given.They makes it possible to compute the RoS boundary.(3)Two algorithms,i.e.,the RoS boundary meshing algorithm and BSP-tree-based RoS computing algorithm are proposed.The restrictions of graphical method are overcome,and the RoS could be computed automatically.With these algorithms,the application range of the proposed method is further extended.