TRAJECTORY PLANNING OF DOUBLE INVERTED PENDULUM: GAIN SCHEDULED GENERALIZED PREDICTIVE CONTROL (GS-GPC) APPROACH USING LAGUERRE FUNCTIONS

AbstraktAbstract

This paper discusses the systematic approach to implement Gain-Scheduled Generalized Predictive Control (GS-GPC) strategy for a Double Inverted Pendulum (DIP) system using orthonormal basis functions approach- the Laguerre Functions. The problem focused herein is a trajectory planning problem thereby incorporating the approximation (prediction) of trajectories, in the context of reference trajectory tracking. The Laguerre model is used to approximate the future control effort which provides basis for the system’s response prediction against reference trajectory change, therefore by ensuring the precise prediction and reference tracking. The contribution posed in this work is that, this approach is a purely nonlinear control approach and is applicable to any nonlinear system, (which most of real-time systems and processes are). We have also presented a dynamic decomposition approach in this paper for the derivation of DIP nonlinear mathematical model with state-dependent matrices. The dynamic decomposition is performed to approximate the system’s model in actuated and under-actuated parts. The point here to investigate is that, GPC can work (and it should!) with “good” prediction of model if real (actual) measurements of system’s behaviour are available. The nonlinear dependencies in the system’s model are treated as parameter variations or as “Gain Scheduling” control based on numerical parameter optimization. Such gain scheduling control formalism of GPC is applicable and valid for each sampling instant and is sensitive to every single reference trajectory variation per sample within the entire permissible ranges (boundary conditions) along entire prediction horizon. This is shown with the help of simulation results.

This paper discusses the systematic approach to implement Gain-Scheduled Generalized Predictive Control (GS-GPC) strategy for a Double Inverted Pendulum (DIP) system using orthonormal basis functions approach- the Laguerre Functions. The problem focused herein is a trajectory planning problem thereby incorporating the approximation (prediction) of trajectories, in the context of reference trajectory tracking. The Laguerre model is used to approximate the future control effort which provides basis for the system’s response prediction against reference trajectory change, therefore by ensuring the precise prediction and reference tracking. The contribution posed in this work is that, this approach is a purely nonlinear control approach and is applicable to any nonlinear system, (which most of real-time systems and processes are). We have also presented a dynamic decomposition approach in this paper for the derivation of DIP nonlinear mathematical model with state-dependent matrices. The dynamic decomposition is performed to approximate the system’s model in actuated and under-actuated parts. The point here to investigate is that, GPC can work (and it should!) with “good” prediction of model if real (actual) measurements of system’s behaviour are available. The nonlinear dependencies in the system’s model are treated as parameter variations or as “Gain Scheduling” control based on numerical parameter optimization. Such gain scheduling control formalism of GPC is applicable and valid for each sampling instant and is sensitive to every single reference trajectory variation per sample within the entire permissible ranges (boundary conditions) along entire prediction horizon. This is shown with the help of simulation results.