robust-tube-mpc

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robust-tube-mpc

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Example implementation for robust model predictive control using tube

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- example_dist_inv_set.m
- example_MPC.m
- example_optimalcontrol.m
- example_tubeMPC.m
- tube_mpc.gif
- .gitignore
- idea.tex
- - convert_Poly2Mat.m
  - number2string.m
- ConstraintManager.m
- DisturbanceLinearSystem.m
- Graphics.m
- LinearSystem.m
- ModelPredictiveControl.m
- OptimalControler.m
- TubeModelPredictiveControl.m
.gitignore
LICENSE.md
README.md

// repository documentation

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# Robust Model Predictive Control Using Tube
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.

## Requirement

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

## Feedback, bug reports, contributions
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](https://github.com/HiroIshida/robust-tube-mpc/issues). 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.

## 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 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.

## Short introduction to the tube MPC
After running `example/example_tubeMPC.m`, you will get the following figure sequence.
![the gif file](https://raw.githubusercontent.com/HiroIshida/robust-tube-mpc/master/fig/tube_mpc.gif)

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`.

## 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](https://en.wikipedia.org/wiki/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.

## Maximum positively invariant set
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⊖Z`and `Uc⊖Z` instead.

# Reference
[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).

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