This repository is only for linear quadratic regulator problems with convex constraints on states/control/bounded disturbance, c.f. [1]. Besides, if the dimension of the system is larger than 3, the calculation of mRPI set becomes quite difficult.
This repository includes examples for the tube model predictive control (tube-MPC)[1] as well as the generic model predictive control (MPC) written in MATLAB.
- optimization_toolbox (matlab)
- control_toolbox (matlab)
- Multi-Parametric Toolbox 3 (open-source and freely available at http://people.ee.ethz.ch/~mpt/3/)
If you find this package helpful, giving a "star" to this repositry will be a happy feedback for me! If you find a bug, or have more broader kind of quession about tube MPC,please post that in the issue page. I will try hard to respond to questions via e-mail but, I strongly recommend do it in the issue page. It's much easier for me to keep myself on track.
See example/example_tubeMPC.m and example/example_MPC.m for the tube-MPC and generic MPC, respectively. Note that every inequality constraint here is expressed as a convex set. For example, the constraints on the state Xc is specified as a rectangular, which is constructed with 4 vertexes. When considering a 1-dim input Uc, Uc will be specified by min and max value (i.e. u∊[u_min, u_max]), so it will be constructed by 2 vertexes. For more detail, please see the example codes.
After running example/example_tubeMPC.m, you will get the following figure sequence.
Now that you can see that the green nominal trajectory starting from the bottom left of the figure and surrounding a "tube". At each time step, the nominal trajectory (green line) is computed online.
Let me give some important details. The red region Xc that contains the pink region Xc-Z is the state constraint that we give first. However, considering the uncertainty, the tube-MPC designs the nominal trajectory to be located inside Xc-Z, which enables to put "tube" around the nominal trajectory such that the tube is also contained in Xc-Z. Of course, the input sequence associated with the nominal trajectory is inside of Uc-KZ.
I think one may get stuck at computation of what paper [1] called "disturbance invariant set". The disturbance invariant set is an infinite Minkowski addition Z = W ⨁ Ak*W ⨁ Ak^2*W..., where ⨁ denotes Minkowski addition. Because it's an infinite sum of Minkowski addition, computing Z analytically is intractable. In [2], Racovic proposed a method to efficiently compute an outer approximiation of Z, which seems to be heavily used in MPC community. In this repository, computation of Z takes place in the constructor of DisturbanceLinearSystem class. To understand how Z guarantee the robustness, running example/example_dist_inv_set.m may help you.
I used the maximal positively invariant (MPI) set Xmpi as the terminal constraint set. (Terminal constraint is usually denoted as Xf in literature). Book [3] explains the concept of the MPI and algorithm well in section 2.4. Xmpi is computed in the constructor of OptimalControler.m. Note that the MPI set is computed with Xc and Uc in the normal MPC setting, but in the tube-MPC the MPI set is computed with Xc⊖Zand Uc⊖Z instead.
[1] Mayne, David Q., María M. Seron, and S. V. Raković. "Robust model predictive control of constrained linear systems with bounded disturbances." Automatica 41.2 (2005): 219-224. [2] Rakovic, Sasa V., et al. "Invariant approximations of the minimal robust positively invariant set." IEEE Transactions on Automatic Control 50.3 (2005): 406-410. [3] Kouvaritakis, Basil, and Mark Cannon. "Model predictive control." Switzerland: Springer International Publishing (2016).