Final Report Summary - CONNECT (Design of Advanced Controllers for Economic, Robust and Safe Manufacturing Performance)
The objective of the project is the development of multi-parametric predictive controllers based on constrained linear and hybrid models, suitable for practical application in low-level control. Offset-free tracking formulations for MPC controllers featuring efficient disturbance rejection based on the Kalman filter have been applied with explicit (multiparametric) MPC controllers that have been previously used only with specific tracking implementations.
A new technique investigated was proper orthogonal decomposition (POD) or method of empirical eigenfunctions, as it is otherwise called. POD consists of taking appropriate (empirical) samples of system's outputs in time and correlating them in a two-dimensional matrix. Through eigenanalysis of this matrix a number of global basis functions that span the hyperspace of (approximately) all systems outputs can be constructed. The (typically few) of this basis functions that capture almost all of the system's energy (the system's dynamics) are then identified. By projecting the original large-scale system onto these few basis functions a low-dimensional system can be obtained that can accurately represent the original large-scale one and that is amenable for on-line MPC. However, for non-linear systems the resulting reduced models are also non-linear. Nevertheless, in this project we wanted to take advantage of powerful linear MPC techniques including parametric MPC (the software for which was developed in work package (WP) 5). For this purpose we developed technologies to automatically compute piece-wise linear approximations (subject to a prechosen error) of the reduced non-linear models. One of the novelties in this work was to combine POD with the finite element method (FEM). Therefore, local FEM basis functions can be used to consistently compute the needed spatial numerical derivatives and FEM Galerkin projection is exploited to systematically project the (discretised) system onto the POD-derived global basis functions (the PODs). Furthermore, a trajectory piece-wise linearisation (TPWL) method was developed to automatically compute the optimal number of linear segments (subject to a given error) of the reduced (one-dimensional) non-linear model (constraints in the MPC formulation). This way the resulting quadratic problem with piece-wise linear constraints can be directly solved using quadratic programming (QP) or parametric MPC.
A major innovation is that local linear analysis for offset-free tracking formulations for MPC controllers featuring efficient disturbance rejection based on the Kalman filter is provided. Analysis results may be visualised in time domain, frequency domain, or on the complex plane. It is required for tuning of feedback properties of MPC controllers for efficient and robust performance, which is most important in low-level control applications. Analysis may be performed with plant-to-model mismatch, with a set of candidate true models. With online MPC controllers, analysis is performed for the unconstrained region; with explicit (multiparametric) MPC controllers, the performance in constrained regions may also be examined.
Explicit MPC eliminates the need for online optimisation by solving the optimisation problem parametrically off-line (in advance), allowing control implementation using inexpensive microcontrollers or programmable logic controllers.
A new technique investigated was proper orthogonal decomposition (POD) or method of empirical eigenfunctions, as it is otherwise called. POD consists of taking appropriate (empirical) samples of system's outputs in time and correlating them in a two-dimensional matrix. Through eigenanalysis of this matrix a number of global basis functions that span the hyperspace of (approximately) all systems outputs can be constructed. The (typically few) of this basis functions that capture almost all of the system's energy (the system's dynamics) are then identified. By projecting the original large-scale system onto these few basis functions a low-dimensional system can be obtained that can accurately represent the original large-scale one and that is amenable for on-line MPC. However, for non-linear systems the resulting reduced models are also non-linear. Nevertheless, in this project we wanted to take advantage of powerful linear MPC techniques including parametric MPC (the software for which was developed in work package (WP) 5). For this purpose we developed technologies to automatically compute piece-wise linear approximations (subject to a prechosen error) of the reduced non-linear models. One of the novelties in this work was to combine POD with the finite element method (FEM). Therefore, local FEM basis functions can be used to consistently compute the needed spatial numerical derivatives and FEM Galerkin projection is exploited to systematically project the (discretised) system onto the POD-derived global basis functions (the PODs). Furthermore, a trajectory piece-wise linearisation (TPWL) method was developed to automatically compute the optimal number of linear segments (subject to a given error) of the reduced (one-dimensional) non-linear model (constraints in the MPC formulation). This way the resulting quadratic problem with piece-wise linear constraints can be directly solved using quadratic programming (QP) or parametric MPC.
A major innovation is that local linear analysis for offset-free tracking formulations for MPC controllers featuring efficient disturbance rejection based on the Kalman filter is provided. Analysis results may be visualised in time domain, frequency domain, or on the complex plane. It is required for tuning of feedback properties of MPC controllers for efficient and robust performance, which is most important in low-level control applications. Analysis may be performed with plant-to-model mismatch, with a set of candidate true models. With online MPC controllers, analysis is performed for the unconstrained region; with explicit (multiparametric) MPC controllers, the performance in constrained regions may also be examined.
Explicit MPC eliminates the need for online optimisation by solving the optimisation problem parametrically off-line (in advance), allowing control implementation using inexpensive microcontrollers or programmable logic controllers.
