We give a brief introduction on boundary modeling via the convex hull in Section 2.1, and some definitions and facts on the convex hull for a set of points in Section 2.2. Refer to [2, 4] for more details regarding the convex hull mathematical representation.
2.1 Boundary modeling in modelbased calibration
The behavior of automotive engines is represented by the state space representation. One of the simplest formulations is as follows:
$$\left\{ \begin{array}{rcl} \frac{d\mathbf{x}}{dt} &=& f(\mathbf{x}, \mathbf{u}), \\ \mathbf{y} &=& g(\mathbf{x}, \mathbf{u}), \end{array} \right. $$
where t is time, x,u and y are vectors which represent the state of the automotive engine, the input signals into the engine and the output signals from the engine, respectively.
Control theory, statistics and optimization are applied to such mathematical models of automotive engines to design more fuelefficient and/or ecofriendly engines. MBC is a systematic approach for aiding such an efficient design of automotive engines and consists of some processes, such as the design of experiments and the response surface methodology.
Boundary modeling is a functionality used in MBC, and is applied to define an AOD for a mathematical engine model. Input signals for automotive engines under development have specific operating ranges and dynamics. In addition, automotive engines may not behave normally when some specific input signals are used, leading to undesirable events such as misfire and knock of the engine. In boundary modeling, one approximates/represents a region of input signals where automotive engines behave normally, e.g., without misfire and knock of the engine.
One of the approximations of the AOD is the convex hull of a set of a finite number of input signals by which the automotive engine behaves normally. This approximation may be too rough, but is a simple way to define an AOD in practice. In fact, it is implemented in some MBC software, such as [10]. Figure 1 displays examples of the approximation of the AOD by the convex hull. In Fig. 1, black circles are input signals by which the automotive engine behaves normally, and red circles indicates input signals by which the automotive engine does not behave normally. The blue region is the approximation of the AOD via the convex hull.
Note that as we mentioned, the approximation of the AOD by the convex hull may be rough. In fact, it does not always represent the region where the automotive engine behave normally. For instance, the approximation at the right of Fig. 1 contains red circles, which means that the automotive engine does not behave normally around the circle.
The approximation of the AOD is used in other processes in MBC as follows:

(P1)
Problem of determining whether a new point is in the approximated AOD or not. This is mainly used in design of experiment of MBC and mathematically formulated as the problem of determining
$$\hat{\mathbf{v}} \in P \, \text{or}\, \hat{\mathbf{v}} \not\in P, $$
where \(\hat {\mathbf {v}}\) is a new point and P is an approximation of the AOD.

(P2)
Optimization of some objective functions over the approximated AOD or a subset of the AOD for more realistic situation in response surface methodology. This is mathematically formulated as
$$\min_{\mathbf{v}\in\mathbb{R}^{n}}\left\{f(\mathbf{v}) : g_{j}(\mathbf{v}) \ge 0 \ (j=1, \ldots, k), \mathbf{v}\in P\right\}, $$
where f(v) is the objective function and g
_{
j
}(v)≥0 is an engine operating constraint.
2.2 Convex hull for a set of points in \(\mathbb {R}^{n}\)
Let V={v
_{1},…,v
_{
m
}} be a finite set of distinct points in \(\mathbb {R}^{n}\). A point
$$\mathbf{x} = \sum_{i=1}^{m} \alpha_{i} \mathbf{v}_{i}, \text{where} \sum_{i=1}^{m}\alpha_{i}=1, \alpha_{i}\ge 0~\text{for}~i=1, \ldots, m, $$
is called a convex combination of v
_{1},…,v
_{
m
}. In particular, the set {α
a+(1−α)b:0≤α≤1} is called the line segment with the endpoints
a
and
b and denoted by [ a,b].
A set \(K\subseteq \mathbb {R}^{n}\) is convex if for every a,b∈K, the line segment [ a,b] is contained in K. We define the empty set ∅ as a convex set. Figure 2 displays an example of convex and nonconvex sets. In fact, for the set at the left of Fig. 2, we see that for every a,b in the set, the line segment [ a,b] is contained in the set, which implies that the set is convex. In contrast, the line segment [ a,b] is not contained in the set at the right of Fig. 2.
Let \(K\subseteq \mathbb {R}^{n}\) be a convex set. A point x∈K is an extreme point or vertex of K if y,z∈K,0<α<1 and x=α
y+(1−α)z imply x=y=z. In other words, an extreme point of K is not a convex combination of other points in K. For instance, at the set of the left in Fig. 2, the black circles at the corners indicate an extreme point of the convex set. We denote the set of extreme points in K by ext(K).
The convex hull conv(A) of a subset \(A\subseteq \mathbb {R}^{n}\) is the set of all convex combination of points from A.
For a set V={v
_{1},…,v
_{
m
}} of distinct points in \(\mathbb {R}^{n}, \text {conv}(V)\) is formulated mathematically as
$$\begin{array}{*{20}l}{} \text{conv}(V) &= \left\{ \mathbf{v}\in\mathbb{R}^{n} : \mathbf{v} = \sum_{i=1}^{m}\alpha_{i} \mathbf{v}_{i}~ \text{for some }\right.\\ &\quad\left.\sum_{i=1}^{m}\alpha_{i} = 1, \alpha_{i}\ge 0 \ (i=1, \ldots, m) \right\}. \end{array} $$
Since some points in V are extreme points of the convex hull, this representation of conv(V) is called the vertex representation (abbr. Vrepresentation). Figure 3 displays an example of the convex hull of V={(0,0),(2,0),(3,2),(1,1),(0,1)}. Since all points except for (1,1) are extreme points, ext(conv(V))={(0,0),(2,0),(3,2),(0,1)}. In fact, (1,1) is not the extreme point of the convex hull because (1,1) can be represented by a convex combination with (2,0),(3,2) and (0,1). In addition, we see conv(V)=conv(ext(V)) in Fig. 3.
A bounded convex set \(K\subseteq \mathbb {R}^{n}\) is a polytope if ext(K) is a finite set. Clearly the convex hull of a set of a finite numbers of points in \(\mathbb {R}^{n}\) is a polytope. A halfspace is a set which is defined as \(\{x\in \mathbb {R}^{n} : \mathbf {a}^{T}\mathbf {x} \le b\}\), with suitable \(\mathbf {a}\in \mathbb {R}^{n}\) and \(b\in \mathbb {R}\). A set P is called polyhedron if P is formed as the intersection of finitely many halfspaces, i.e., there exist \(\mathbf {a}_{1}, \ldots, \mathbf {a}_{k}\in \mathbb {R}^{n}\) and \(b_{1}, \ldots, b_{k}\in \mathbb {R}\) such that \(P =\left \{ \mathbf {x} \in \mathbb {R}^{n} : \mathbf {a}_{i}^{T}\mathbf {x} \le b_{i} \ (i=1, \ldots, k) \right \}\).
MinkowskiWeyl’s theorem ensures that every polytope can be reformulated as a polyhedron. This implies that one can describe the convex hull of a set of points by some halfspaces in addition to the Vrepresentation, which is called the halfspace representation (abbr. Hrepresentation).
Theorem 2.1
(MinkowskiWeyl) Every polytope is polyhedron, i.e., for a given polytope P, there exist \(\mathbf {a}_{1}, \ldots, \mathbf {a}_{k}\in \mathbb {R}^{n}\) and \(b_{1}, \ldots, b_{k}\in \mathbb {R}\) such that \(P=\{x\in \mathbb {R}^{n} : \mathbf {a}_{i}^{T}\mathbf {x}\le b_{i} \ (i=1, \ldots, k)\}\). Moreover, every bounded polyhedron is also polytope, i.e., for a given polyhedron P, there exist v
_{1},…,v
_{
m
}∈P such that P=conv(V), where V={v
_{1},…,v
_{
m
}}.
We give two examples of the V and Hrepresentations. We see from these examples that one needs to choose a suitable representation of the convex hull from the viewpoint of computation.
Example 2.2
(ndimensional unit cube) Let \(P = \{\mathbf {x}\in \mathbb {R}^{n} : 0\le x_{i}\le 1 \ (i=1, \ldots, n)\}\). P is called the ndimensional unit cube. Figure 4 displays an example of 3dimensional unit cube. This is already the Hrepresentation. In fact, we define \(\mathbf {a}_{i}\in \mathbb {R}^{n}, b_{i}\in \mathbb {R}\) (i=1,…,2n) as follows:
$${{}{\begin{aligned} \mathbf{a}_{i} = \left\{ \begin{array}{cl} \mathbf{e}_{i} & (i=1, \ldots, n), \\ \mathbf{e}_{i} & (i=n+1, \ldots, 2n), \end{array} \right. \text{and}~b_{i} = \left\{ \begin{array}{cl} 1 & (i=1, \ldots, n), \\ 0 & (i=n+1, \ldots, 2n), \end{array} \right. \end{aligned}}} $$
where e
_{
i
} is the ith ndimensional standard unit vector. Then P can be reformulated by \(\{\mathbf {x} \in \mathbb {R}^{n} : \mathbf {a}_{i}^{T}\mathbf {x}\le b_{i} \ (i=1, \ldots, 2n)\}\). On the other hand, for \(\text {ext}(P) = \{\mathbf {x}\in \mathbb {R}^{n} : x_{i} = 0 \text {or} 1\}\), the Vrepresentation of P is
$$\begin{aligned} P& = \text{conv}(\{(0, 0, \ldots, 0), (1, 0, \ldots, 0),\\&\quad (0, 1, \ldots, 0), \ldots, (1, 1, \ldots, 1)\}). \end{aligned} $$
We remark that the Vrepresentation of P needs 2^{n} extreme points in ext(P), whereas the Hrepresentation needs only 2n halfspaces.
Example 2.3
(Crosspolytope) Let \(P = \{x\in \mathbb {R}^{n} : x_{1}+\cdots +x_{n}\le 1\}\). P is called the ndimensional crosspolytope. Figure 4 displays an example of the 3dimensional crosspolytope. The Hrepresentation of P is
$$ P = \left\{\mathbf{x}\in\mathbb{R}^{n} : \begin{array}{lcl} x_{1}+x_{2}+\cdots+x_{n}&\le& 1\\ x_{1}+x_{2}+\cdots+x_{n}&\le& 1\\ x_{1}x_{2}+\cdots+x_{n}&\le& 1\\ x_{1}x_{2}+\cdots+x_{n}&\le& 1\\ &\vdots&\\ x_{1}x_{2}\cdotsx_{n}&\le& 1\\ \end{array} \right\}. $$
Here the Hrepresentation is the intersection of 2^{n} halfspaces. In contrast, the Vrepresentation of P can be formulated by 2n points in \(\mathbb {R}^{n}\). In fact, since both e
_{
i
} and −e
_{
i
} are extreme points in P, the Vrepresentation of P is P=conv({±e
_{1},…,±e
_{
n
}}).
A more compact representation of the convex hull is often useful from the viewpoint of computation. For instance, the Vrepresentation in Example 2.2 and the Hrepresentation in Example 2.3 require more computer memory even for small n, whereas the Hrepresentation in Example 2.2 and the Vrepresentation in Example 2.3 need less memory even for large n. Hence the Hrepresentation in Example 2.2 and the Vrepresentation in Example 2.3 are more suitable to deal with in actual computers when the dimension n is large.