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Model predictive control for piece-wise affine systems

Project members: B. De Schutter, T.J.J. van den Boom

We have extended our results on model predictive control (MPC) for discrete event systems (see Project 4.8) to a class of hybrid systems that can be described by a continuous piecewise-affine state space model. More specifically, we have considered systems of the form

\begin{eqnarray*}
x(k) & = & \mathcal{P}_x (x(k-1),u(k)) \\
y(k) & = & \mathcal{P}_y (x(k),u(k)) \enspace ,
\end{eqnarray*}

where $x$, $u$ and $y$ are respectively, the state, the input and the output vector of the system, and where the components of $\mathcal{P}_x$ and $\mathcal{P}_y$ are continuous piecewise-affine (PWA) scalar functions, i.e., functions that satisfy the following conditions:
  1. The domain space of is divided into a finite number of polyhedral regions;
  2. In each region can be expressed as an affine function;
  3. $f$ is continuous on any boundary between two regions.

We have shown that continuous PWA systems are equivalent to max-min-plus-scaling systems (i.e., systems that can be modeled using maximization, minimization, addition and scalar multiplication). Next, we have considered MPC for these systems. In general, this leads to nonlinear non-convex optimization problems. However, we have developed a method based on canonical forms for max-min-plus-scaling functions to solve these optimization problems in a more efficient way than by just applying nonlinear optimization as was done in previous research. More specifically, the proposed algorithm consists in solving several linear programming problems [20].


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