Move Suppression Calculations for Well-Conditioned MPC

Abstract

Several popular tuning strategies applicable to Model Predictive Control (MPC) schemes such as GPC and DMC have previously been developed. Many of these tuning strategies require an approximate model of the controlled process to be obtained, typically of the First Order Plus Dead Time type. One popular method uses such a model to analytically calculate an approximate value of the move suppression coefficient to achieve a desired condition number for the regularized system dynamic matrix; however it is not always accurate and tends to under-estimate the required value. In this paper an off-line method is presented to exactly calculate the move suppression coefficient required to achieve a desired condition number directly from the unregularized system dynamic matrix. This method involves an Eigen decomposition of the system dynamic matrix -which may be too unwieldy in some cases -and a simpler analytical expression is also derived. This analytical expression provides a guaranteed tight upper bound on the required move suppression coefficient yielding a tuning formula which is easy to apply, even in on-line situations. Both methods do not require the use of approximate or reduced order process models for their application. Simulation examples and perturbation studies illustrate the effectiveness of the methods in both off-line and on-line MPC configurations. It is shown that accurate conditioning and improved closed loop robustness can be achieved.

A Microcontroller-Based Adaptive Model Predictive Control Platform for Process Control Applications

“…The move suppression parameter λ applies a weight on the magnitude of the two projected control moves in the objective function (Equation (9)), and in effect regularizes the control matrix C −1 = (G T G + λI) −1 and allows the condition number with respect to inversion to be manipulated [7]. There is a strong link between this condition number and the closed-loop performance and robustness of the resulting controller; typically, for process control, a condition number of c = 500 can be employed [7].…”

“…There is a strong link between this condition number and the closed-loop performance and robustness of the resulting controller; typically, for process control, a condition number of c = 500 can be employed [7]. In this context, the robustness of an MPC controller refers to the relative sensitivity of the applied controls to errors in the plant mathematical model or state measurement [7]; it is therefore of high importance in the context of adaptive MPC control, where estimates of plant parameters may transiently lag behind actual process parameters until estimator convergence. In this paper, the following method is applied to regulate the control matrix and exactly achieve a specified condition number c. Denote µ max and µ min as the largest and smallest of the eigenvalues of the matrix (G T G).…”

“…The effect of move suppression is to add an identity matrix scaled by λ, and for λ ≥ 0 the resulting effect upon the eigenvalues is that of a uniform additive shifting [7], such that the condition number is modified as follows:…”

“…The major drawback of such a scheme is the large computational burden that results especially in constrained and/or adaptive situations. In addition, a typical MPC controller has many tunable parameters: aside from considerations regarding the process parameterization, the principal ones of interest are the choice of sampling time T, the length of the prediction horizon P and the control horizon M, and the value of the move suppression coefficient  [7,8]. The latter parameter applies a weight on the magnitude of the projected control moves in the objective function.…”

“…The latter parameter applies a weight on the magnitude of the projected control moves in the objective function. Due to the complex relationships between these tunable parameters and the closed loop system properties, many previous authors have suggested 'tuning rules' that allow a user to configure an MPC instance to achieve a desired level of closed-loop performance [7,8]. Taken together, these complications have typically limited the applicability of MPC to situations in which either the required high-bandwidth real-time computing facilities can be justified (or the sampling rate is made appropriately slow), and expert operators are available to help determine an accurate process model and to configure and tune the MPC controller [1][2][3].…”