Chapter 1

Uncertainty Representation Based on Set Theory

Real systems are often complex due to several factors: the system’s nature (e.g. mechanical, electrical and chemical systems), interactions between its different components (e.g. multivariable systems), and its different behavior in a dynamic environment (e.g. influence of disturbances, noises and uncertainties). All these aspects have to be considered when modeling a given system, sometimes leading to a complicated model. In the context of control systems, a mathematical model is frequently used to describe the system behavior. On the one hand, the accuracy of the mathematical model is important to analyze and design control strategies for the considered system; on the other hand, in the context of industrial applications, it is suitable to use unsophisticated controllers designed using a simple model. In this context, a trade-off must be found: the system model should be simple but precise enough to characterize the dynamical behavior of the original system. Thus, the simple/simplified mathematical model cannot represent the real system exactly due to a lack of knowledge of, or unreliable information about, the system. To validate this model, some uncertainties can be added to the mathematical model. Frequently, perturbations influencing the real system have to be taken into account in the mathematical model in order to ensure a similar behavior of the real system and the mathematical model. The importance of uncertainties in system design has been discussed by many authors (the interested reader can refer to [MAY 79, AUG 06, AYY 06] and the references therein). In the literature, there are two ways to represent uncertainties: the statistical (or stochastic) approach and the deterministic approach. An overview of these two directions is provided in the following.

In the stochastic approach, the uncertainty is modeled by a random process with a known statistical property. This technique is widely used in various domains (e.g. economics [BAT 08], biology [ULL 11] and engineering [MAY 79]), especially when estimates of the probability distribution of the uncertain parameters are available. But in many applications, there are situations when the probability distribution of the uncertain parameters is not known and only bounds of the uncertain domain can be fixed. In this case, the probabilistic assumptions on the uncertainty are no longer valid, making this method unsuitable for modeling the uncertainties.

In the deterministic approach, the uncertainty is assumed to belong to a set: a classical (crisp) set1 or a fuzzy set2. In the literature, different families of classical sets are used depending on their properties. Usually, the accuracy and the complexity of the uncertainties’ representation are inversely proportional, depending on the particular problem related to the choice of a suitable geometric form. In the following sections, some popular families of sets are presented with their advantages and weaknesses. Note that in this book only convex classical sets are considered due to the important role of convexity in the optimization theory [BER 03].

1.1. Basic set definitions: advantages and weaknesses

Before presenting the most well-known families of sets, some basic set definitions and operations used in this book are introduced.

DEFINITION 1.1.– A set S ⊂ is called a convex set if for any x1,x2,…,xk ∈ S and any such that then the element is in S.

DEFINITION 1.2.– A convex hull of a given set S, denoted conv(S), is the smallest convex set containing S.

DEFINITION 1.3.– A set S ⊂ is called a C-set if S is compact, convex and contains the origin. This is a proper set if its interior is not empty.

DEFINITION 1.4.– The inclusion operator between two sets is defined by if and only if then x ∈ Y. This means that X is a subset of Y.

DEFINITION 1.5.– The intersection operator of two sets X and Y is defined by .

DEFINITION 1.6.– The image of a set S under a map (projection) is the set .

DEFINITION 1.7.– The Minkowski sum of two sets X and Y is defined by

DEFINITION 1.8.– The Pontryagin difference of two sets X and Y is defined by .

DEFINITION 1.9.– Let X and Y be two non-empty sets. The distance between these two sets X and Y is defined by .

DEFINITION 1.10.– Let X and Y be two non-empty sets. The Hausdorff distance of these two sets X and Y is defined by with

The Hausdorff distance allows characterizing the quality of the approximation of X by Y [HUN 93]. If X and Y have the same closure3, then the Hausdorff distance is equal to 0. Figure 1.1 illustrates the difference between the “normal” distance (Definition 1.9) which is equal to 0 and the Hausdorff distance dH(X, Y) between the two sets X and Y.

Figure 1.1. Illustration of the Hausdorff distance between two sets

1.1.1. Interval set

A very simple way to define uncertainties is by using the interval notion. This is based on the idea of enclosing numerical errors into an interval. In many cases, obtaining the probability of occurrence of different uncertainties is not possible. Therefore, it can be easier and thus suitable to bound the uncertainties by intervals. Moreover, the interval analysis allows us to simplify most of the standard operations [MOO 66, HAN 65, JAU 01]. This approach is developed in many domains (e.g. identification, diagnosis and estimation), especially when a short computation time is required.

DEFINITION 1.11.– An interval is defined by the set

DEFINITION 1.12.– The center and the radius of an interval are, respectively, defined by and rad(I) =

DEFINITION 1.13.– The unitary interval is denoted by B = [–1, 1].

DEFINITION 1.14.– The set of real compact intervals [a,b], where a,b ∈ and a ≤ b, is denoted by

DEFINITION 1.15.– A box is an interval vector.

DEFINITION 1.16.– A unitary box in , denoted by Bn, is a box composed of n unitary intervals.

Consider the intervals An operation o between these two intervals can be formalized as:

[1.1]

The four basic operations [MOO 66] with intervals are the following:

1)

2)

3)

4)

Despite the simplicity of interval analysis, a drawback of this approach is that the computation results can sometimes be conservative due to the dependency effect4 and the wrapping effect5 [MOO 66, KÜH 98b, JAU 01]. These two effects are further analyzed through two examples.

EXAMPLE 1.1.– DEPENDENCY EFFECT – Consider two functions f1(x, y) = x – y and f2(x) = x – x, with the variables x, y ∈ [–1, 1]. Using the interval analysis, the value domain of f1 and f2 is the same [–1, 1] – [–1, 1] = [–2, 2], even if the real value domain of f2 is 0. This problem, called the “dependency effect”, relies on the fact that the occurrence of the same variable x in the function f2 is independently considered and can lead to an important over-approximation of the result.

EXAMPLE 1.2.– WRAPPING EFFECT – Consider a function , with and . Using the interval analysis leads to , with the value domains and , represented in dark gray in Figure 1.2. The vertices of the exact solution of f (see Figure 1.2) are obtained by computing the product of the matrix A with each vertex of the unitary box containing x and y (e.g. [–0.5, 0]T = A . [–1, 1]T). Comparing these solutions, an important over-approximation of the interval analysis solution can be observed. If this operation is repeated several times, the difference between the exact solution and the solution of the interval analysis is more and more important. This problem is called the “wrapping effect”.

Figure 1.2. Illustration of the wrapping effect in interval analysis

1.1.2. Ellipsoidal set

Another way to represent bounded uncertainties is based on the use of bounded ellipsoids. Moreover, due to its low complexity, the ellipsoidal set is widely used in a large class of applications in automatic control [SCH 68, WIT 68, BER 71, CHE 81, MIL 96, KUR 96, DUR 01, POL 04]. Some basic properties of ellipsoids are discussed in...