Model Predictive Control from Scratch: Derivation and Python Implementation-Optimal Control Tutorial

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In this control engineering, system identification, and control theory tutorial, we explain:
1) How to derive a model predictive control algorithm from scratch.
2) How to implement the model predictive control algorithm in Python from scratch.

Starting from a state space model, we formulate an optimization problem and explain how to compute the closed form of the model predictive control solution.

The GitHub page with all the codes is given here:
https://github.com/AleksandarHaber/Mo...

The website tutorial is given here:
https://aleksandarhaber.com/model-pre...

The MPC algorithm implementation in C++:
   • Model Predictive Control (MPC) from S...  

The main challenge was how to make a tutorial that was easy for beginners without going immediately into nonlinear and complex optimization worlds, where students immediately get lost and immediately give up on studying MPC. This comment also applies to modern control theory. A number of control scientists publishing papers in Automatica/IEEE TAC simply do not put effort into explaining things such that everyone can understand the basic concepts. There is a trend to make the control theory as complex as possible and as a pure math discipline. I think that was not the vision of the founders of control theory, who were actually engineers solving real-life problems. In fact, control theory is super applicable and relatively easy. The only issue is that one has to conquer math to be able to apply it. Consequently, I start with the MPC formulation for linear systems and I try to stick as much as possible to basic linear algebra.

We explain how to formulate a least-squares cost function, and how to minimize it. The solution can be expressed in a closed form as a solution of a weighted least-squares problem. The input constraints can be implicitly handled by properly selecting weighting matrices. We then explain how to implement this algorithm in Python from scratch and in a disciplined and clean manner (by using Python classes). We test the algorithm on a classical system consisting of two objects connected by springs and dampers. In the second part of this tutorial series, we will consider constrained linear systems. In the third part, we will consider nonlinear smooth systems (the most general formulation). After we complete the Python model predictive control tutorials, we will start a new tutorial series on how to implement the algorithm in C++ by using the Eigen Library.