Computationally Efficient Nonlinear Model Predictive Control Using the L1 Cost-Function

Abstract

Model Predictive Control (MPC) algorithms typically use the classical L2 cost function, which minimises squared differences of predicted control errors. Such an approach has good numerical properties, but the L1 norm that measures absolute values of the control errors gives better control quality. If a nonlinear model is used for prediction, the L1 norm leads to a difficult, nonlinear, possibly non-differentiable cost function. A computationally efficient alternative is discussed in this work. The solution used consists of two concepts: (a) a neural approximator is used in place of the non-differentiable absolute value function; (b) an advanced trajectory linearisation is performed on-line. As a result, an easy-to-solve quadratic optimisation task is obtained in place of the nonlinear one. Advantages of the presented solution are discussed for a simulated neutralisation benchmark. It is shown that the obtained trajectories are very similar, practically the same, as those possible in the reference scheme with nonlinear optimisation. Furthermore, the L1 norm even gives better performance than the classical L2 one in terms of the classical control performance indicator that measures squared control errors.

Fast Model Predictive Control of PEM Fuel Cell System Using the L1 Norm

“…This paper aims to present a computationally simple MPC-L 1 control strategy and its application to the PEM process. This work extends previous research concerned with computationally efficient nonlinear MPC-L 1 in which a neural approximator of the absolute value function is utilised [30]. Unfortunately, such a control strategy's effectiveness heavily depends on the approximator's accuracy.…”

“…This work uses two analytical approximations of the absolute value function. It is also possible to solve this problem using neural approximators [30].…”

“…In order to solve this problem, an advanced on-line trajectory linearisation is carried out. It may be proved elsewhere [30] that the linear approximation of the predicted trajectory of the control error over the prediction horizon embedded in the nonlinear function α can be written in general form…”

“…In all MPC algorithms cited above, the MPC cost-function used considers the sum of absolute values of the predicted control errors (the L 2 norm). However, in a few publications concerned with other applications, it is suggested that better control is possible when the sum of absolute predicted control errors is utilised as the MPC cost function (the L 1 norm) [26][27][28][29][30]. This paper aims to present a computationally simple MPC-L 1 control strategy and its application to the PEM process.…”

Stabilization of the Cart–Inverted-Pendulum System Using State-Feedback Pole-Independent MPC Controllers

“…There are different types of control methods that have been applied to the inverted pendulum systems [ 1 ], including model predictive control (MPC) and non-MPC methods. With regard to the complicated characteristics of the inverted pendulum plants, the needed controlled system performance, and the limited control input effort resource, time-domain optimization techniques, such as the MPC [ 2 , 3 , 4 , 5 , 6 ], seem to be one of the most convenient ways to tackle the above control problem, especially when state and control input constraints are considered. The key feature of the MPC method is based on the following three successive steps [ 3 ]: (i) the explicit use of a model and system measurements to predict the future behavior of the controlled variables over a specified future time horizon, (ii) the calculation of a control sequence minimizing a cost function, and (iii) the application of the first control signal of the sequence for a given time before returning to step (i).…”