What is muAO-MPC?¶

$\muaompc$ stands for microcontroller applications online model predictive control. It mainly consists of the muaompc Python package.

The generated C code is fully compatible with the ISO C89/C90 standard, and is platform independent. The code can be directly used in embedded applications, using popular platforms like Arduino, and Raspberry, or any other application on which C/C++ code is accepted, like many current generation programable logic controllers (PLC). Additionally, MATLAB/Simulink interfaces to the generated code are provided.

$\muaompc$ is free software released under the terms of the three-clause BSD License.

The ltidt module¶

This module creates a model predictive control (MPC) controller for a linear time-invariant (lti) discrete-time (dt) system with input and (optionally) state constraints, and a quadratic cost function. The MPC problem is reformulated as a condensed convex quadratic program, which is solved using an augmented Lagrangian method together with Nesterov’s gradient method, as described in [KZF12], [KF11].

The MPC problem description is written in a file we call the system module (see The system module for details). After writing this file, the next step is to actually auto-generate the C code. This is done in two easy steps:

create an muaompc object from to the system module, and
write the C-code based on that object.

The MPC setup¶

The plant to be controlled is described by $x^+ = A_d x + B_d u$ , where $x \in \mathcal{C}_x \subseteq \mathbb{R}^n$ , and $u \in \mathcal{C}_u \subset \mathbb{R}^m$ are the current state and input vector, respectively. The the state at the next sampling time is denoted by $x^+$ . The discrete-time system and input matrices are denoted as $A_d$ and $B_d$ , respectively.

The MPC setup is as follows:

$\underset{u}{\text{minimize}} & \;\; \frac{1}{2} \sum\limits_{j=0}^{N-1} ((x_j - x\_ref_j)^T Q (x_j - x\_ref_j) + \\ & (u_j - u\_ref_j)^T R (u_j - u\_ref_j)) + \\ & \frac{1}{2} (x_{N} - x\_ref_N)^{T} P (x_{N} - x\_ref_N) \\ \text{subject to} & \;\; x_{j+1}=A_d x_j + B_d u_j, \;\; j = 0, \cdots, N-1 \\ & \;\; u\_lb \leq u_j \leq u\_ub, \;\; j = 0, \cdots, N-1 \\ & \;\; e\_lb \leq K_{x} x_j + K_{u} u_j \leq e\_ub, \;\; j = 0, \cdots, N-1 \\ & \;\; f\_lb \leq F x_N \leq f\_ub \\ & \;\; x_0 = x \\$

where the integer $N \geq 2$ is the prediction horizon. The symmetric matrices $Q$ , $R$ , and $P$ are the state, input, and final state weighting matrices, respectively. The inputs are constrained by a box set, defined by the lower and upper bound $u\_lb$ and $u\_ub$ , respectively. Furthermore, the state and input vectors are also constrained (mixed constraints), using $K_{x}$ and $K_{u}$ , and delimited by the lower and upper bounds $e\_lb$ and $e\_ub \in \mathbb{R}^q$ , respectively. Additionally, the terminal state needs to satisfy some constraints defined by $f\_lb$ , $f\_ub \in \mathbb{R}^r$ , and the matrix $F$ .

The sequences $x\_ref$ and $u\_ref$ denote the state and input references, respectively. They are specified online. By default, the reference is the origin.

For other setups, check out version 1.x of muaompc.