Robust Model Predictive Control Using Tube

This repository includes examples for the tube model predictive control (tube-MPC)[1, 2] as well as the generic model predictive control (MPC) written in MATLAB.

Requirement

1. optimization_toolbox (matlab)
   
2. control_toolbox (matlab)
   
3. Multi-Parametric Toolbox 3 (open-source and freely available at http://people.ee.ethz.ch/~mpt/3/)

Usage

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 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.

Short introduction to the tube MPC

After running example/example_tubeMPC.m, you will get the following figure.

Now that you can see that the green nominal trajectory starting from the bottom left of the figure and surrounding a "tube". The blue plot means the real trajectory affected by the disturbance. You can see that this real trajectory never stick out from the "tube", and is robustly guided into the region `Xf-Z`.

Protting it in a 3-dim fashion with a minus-time axis in the z-direction, you can easily get why we call that "tube".

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. If there is no disturbance, putting a terminal constraint Xf (set it to Maximum Positively Invariant Set, here) guarantees the stability and feasibility. However, considering the disturbance we have to put the terminal constraint Xf-Z instead. Once the terminal constraint Xf-Z is satisfied, in spite of the disturbance, the following trajectory will be stable around the origin by LQR feedback without violating any constraints.

Disturbance invariant set

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. Obtaining this analytically is impossible, and then, what comes ones' mind first may be an approximation of that by truncation: Z_approx = W ⨁ Ak*W...⨁ Ak^n*W. This approximation, however, leads to Z_approx ⊂ Z, which means Z_approx is not disturbance invariant. So, we must multiply it by some parameter alpha and get Z_approx = alpha*(W ⨁ Ak*W...⨁ Ak^n*W) so that Z ⊂ Z_approx is guaranteed. (see src/LinearSystem.m for this implementation). The order of truncation and alpha are tuning parameters, and I chose values that are large enough. But, if you want to choose them in a more sophisticated way, paper [3] will be useful. Also, example/example_dist_inv_set.m may help you understand how disturbance the invariant set works.

Maximum positively invariant set

I used the maximum positively invariant (MPI) set Xmpi as the terminal constraint set. (Terminal constraint is usually denoted as Xf in literature). Xmpi is computed in the constructor of OptimalControler.m. Instead in the tube-MPC, the terminal constraint is set to Xmpi⊖Z rather than just Xmpi, where ⊖ denotes the pontryagin difference. This operation of ⊖ robustifies the terminal constraint.

Reference

[1] Langson, Wilbur, et al. "Robust model predictive control using tubes." Automatica 40.1 (2004): 125-133. [2] 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. [3] Rakovic, Sasa V., et al. "Invariant approximations of the minimal robust positively invariant set." IEEE Transactions on Automatic Control 50.3 (2005): 406-410.