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Optimal control

In many problem instances we want to control the system in an optimal way. A goal can be to minimize energy consumption, maximize profit, minimize tracking error, minimize settling time, etc. This natural desire in achieving the best possible system performance is usually formulated as an optimization problem, i.e. mathematical program of the following form

where J is the objective function and u are the system control variables.

The system dynamics can be used to predict the future system evolution over a prediction horizon based on the current system state. In discrete-time case the objective is then formulated as

where xk represents the predicted system state k steps in the future, uk is the concurrent control input, N the prediction horizon and l is usually a weighted norm. Such control formulations fall in the category of model predictive control (MPC).

At each sampling instant an optimal control sequence U* is obtained, but only the first planned control move of the optimal sequence is applied to the system.

The whole procedure is repeated at the next time instance over a shifted horizon (receding horizon control).

Receding horizon control

Thanks to the variety of model types that can be used for the prediction in the MPC framework, and due to the natural incorporation of the system constraints, the MPC quickly became a very popular control strategy in the process industries. However, one should point out the real problem is not solved simply by using the MPC strategy - it is just ''moved'' to the underlying optimization problem. Precisely for that reason the MPC was applied mostly to the relatively slow processes for which one could afford (time- and money-wise) the usage of a significant computational power.

Luckily the things have changed in the last decade. We are now equipped with the techniques - the so-called multi-parametric mathematical programming - that allow the off-line pre-computation of the optimal control law. In this way, instead of solving a difficult optimization problem on-line, all we need to do is evaluate a look-up table like function.

Explicit MPC

 

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