There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper recalls a few past achievements in Model Predictive Control, gives an overview of some current developments and suggests a few avenues for future research.

Model Predictive Control

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

http://pdclab.seas.ucla.edu/Publications/PDChristofides/PDChristofides_RScattolini_DMunozdelaPena_JLiu_CCE_2013_21_DMPC_A_Tutorial_Review_and_Future_Research_Directions.pdf

Distributed model predictive control: A tutorial review and future research directions

Two-person Cooperative Games

http://www.esi2.us.es/~alamo/Archivos/Certificaciones/Sec_8_Articulos_Revista/2008/Limon_Aut_08.pdf

Brief paper: MPC for tracking piecewise constant references for constrained linear systems

Robust tube-based MPC for tracking of constrained linear systems with additive disturbances

A comparative analysis of distributed MPC techniques applied to the HD-MPC four-tank benchmark

In this paper, a novel model predictive control (MPC) formulation has been proposed to solve tracking problems, considering a generalized offset cost function. Sufficient conditions on this function are given to ensure the local optimality property. This novel formulation allows to consider as target operation points, states which may be not equilibrium points of the linear systems. In this case, it is proved in this paper that the proposed control law steers the system to an admissible steady state (different to the target) which is optimal with relation to the offset cost function. Therefore, the proposed controller for tracking achieves an optimal closed-loop performance during the transient as well as an optimal steady state in case of not admissible target. These properties are illustrated in an example.

MPC for tracking with optimal closed-loop performance

This paper presents a novel tracking predictive controller for constrained nonlinear systems capable to deal with sudden and large variations of a piece-wise constant setpoint signal. The uncertain nature of the setpoint may lead to stability and feasibility issues if a regulation predictive controller based on the stabilizing terminal constraint is used. The tracking model predictive controller presented in this paper extends the MPC for tracking for constrained linear systems to the more complex case of constrained nonlinear systems. The key idea is the addition of an artificial reference as a new decision variable. The considered cost function penalizes the deviation of the predicted trajectory with respect to the artificial reference as well as the distance between the artificial reference and the setpoint. Closed-loop stability and recursive feasibility for any setpoint are guaranteed, thanks to an appropriate terminal cost and extended stabilizing terminal constraint. Also, two simplified formulations are shown: the design based on a terminal equality constraint and the design without terminal constraint. The resulting controller ensures recursive feasibility for any changing setpoint. In the case of unreachable setpoints, asymptotic stability of the optimal reachable setpoint is also proved. The properties of the controller have been tested on a constrained continuous stirred tank reactor simulation model and have been experimentally validated on a four-tanks plant.

I. Alvarado

Papers

Brief paper: MPC for tracking piecewise constant references for constrained linear systems

Robust tube-based MPC for tracking of constrained linear systems with additive disturbances

A comparative analysis of distributed MPC techniques applied to the HD-MPC four-tank benchmark

MPC for tracking with optimal closed-loop performance

Nonlinear MPC for Tracking Piece-Wise Constant Reference Signals