Complexity of Implementation and Synthesis in Linear Parameter-Varying Control

In this paper an analysis of the complexity involved in the implementation and synthesis of linear parameter-varying (LPV) controllers is presented. Its purpose is to provide guidance in the selection of a synthesis approach for practical LPV control problems and reveal directions for further research with respect to complexity issues in LPV control. Standard methods are classified into polytopic, linear fractional transformation and gridding-based techniques with an emphasis on output-feedback synthesis. Carried out as a convex optimization problem via finitely many linear matrix inequalities (LMIs) for both parameter-independent and parameter-dependent Lyapunov functions (PiDLF/PDLF), the complexity of LPV controller existence conditions is assessed in terms of the number of decision variables and total size of the LMI. The implementation complexity is assessed in terms of the number of arithmetic operations required to compute the parameter-varying state space matrices of the controller, as well as the memory requirements to store associated variables. The results are applied to the LPV controller synthesis for a three-degrees-of-freedom robotic manipulator and the charge control of a spark-ignited engine, for which multiple models, as well as associated synthesis results have been reported in the literature.