Model Predictive Control

14.2 Piecewise Affine Systems 285
X1
X7
X3
X2
xc(k)
xc(3)
xc(0)
xc(1)
xc(2)
Figure 14.2 Piecewise affine (PWA) systems. Mode switches are only triggered
by linear threshold events
and the inequalities in (14.1c) should be interpreted component-wise. Each linear
inequality in (14.1c) defines a half-space in R n and a corresponding hyperplane,
that will be also referred to as guardline. Each vector inequality (14.1c) defines a
polyhedron C i = {[ x u] ∈ R n+m : H i x + J i u ≤ K i } in the state+input space R n+m
that will be referred to as cell, and the union of such polyhedral cells as partition.
We assume that C i are full dimensional sets of R n+m , for all i = 1, . . .,s.
A PWA system is called well-posed if it satisfies the following property [34]:
Definition 14.1 Let P be a PWA system of the form (14.1) and let C = ∪ s i=1 Ci ⊆
R n+m be the polyhedral partition associated with it. System P is called well-posed
if for all pairs (x(t), u(t)) ∈ C there exists only one index i(t) satisfying (14.1).
Definition 14.1 implies that x(t+1), y(t) are single-valued functions of x(t) and u(t),
and therefore that state and output trajectories are uniquely determined by the
initial state and input trajectory. A relaxation of definition 14.1 is to let polyhedral
cells C i sharing one or more hyperplanes. In this case the index i(t) is not uniquely
defined, and therefore the PWA system is not well-posed. However, if the mappings
(x(t), u(t)) → x(t + 1) and (x(t), u(t)) → y(t) are continuous across the guardlines
that are facets of two or more cells (and, therefore, they are continuous on their
domain of definition), such mappings are still single valued.
14.2.1 Modeling Discontinuities
Discontinuous dynamical behaviors can be modeled by disconnecting the domain.
For instance, the state-update equation
x(t + 1) =
1
2
X6
X4
X5
x(t) + 1 if x(t) ≤ 0
0 if x(t) > 0
(14.2a)