Decision Making Problem

1 2 2 ₌ d c , 4 1 = b a < 2 2 d c .

Substituting the data into the formula (6.14) gives y_max = 4 and y_min = c₁u_min(1) + d₁u(_min2) = 2, according to the formula (6.12). The set D_y is then determined by the inequality 2 ≤y ≤ 4.

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6.3 Decision Making Problem

It is an inverse problem to the analysis problem formulated in Sect. 6.2, and for the plant described by the function y = Φ(u), it consists in deter- mining such a decision u = u* that the respective output y = Φ(u*) is equal to the given required value y*. This problem for the functional plant has been considered in Sect. 3.1. In the relational plant it is not possible to satisfy the requirement y = y* but it has a sense to formulate the require- ment in the form y∈D_y for the fixed set D_y, and to find a decision u for which this requirement is satisfied. Solving the problem consists in deter- mining the set D_u of all possible (or acceptable) decisions, i.e. in deter- mining all values u for which the property y∈D_y will be fulfilled.

Decision making (control) problem: For the given relation R(u, y) and the set D_y⊂ Y determining a user’s requirement one should find the larg- est set D_u⊂ U such that the implication

u∈D_u→y∈D_y (6.15)

is satisfied.

The general form of the problem solution is as follows:

Du = {u∈U: Dy(u) ⊆Dy} (6.16)

where D_y(u) is defined by the formula (6.6). Then, D_u is the set of all such values u for which the set of possible values y belongs to the given set D_y. A remark on difficulties connected with the determination of a fi- nal solution for concrete forms of R(u, y) and D_y, and on a universal al- gorithm in the case of logical operations is now analogous to that for the analysis problem in Sect. 6.2. Similarly as in Sect. 6.2 it is worth noting

that the decision problem for the relational plant may be considered as a generalization of the respective problem for the functional plant where the input property u∈D_u and the output property y∈D_y are reduced to the forms u = u* and y = y*, respectively.

The solution of the decision problem under consideration may not exist, i.e. D_u may be an empty set. Such a case is illustrated in Fig. 6.4: For the given interval D_y, the interval D_u for which the implication (6.15) could be satisfied does not exist. It means that the requirement is too strong, i.e. that the interval D_y is too small. The requirement may be satisfied for the greater interval D_y (see Fig. 6.2). If D_u = ∅ (empty set), we can say that the plant R(u, y) is non-controllable for the requirement y∈D_y. For exam- ple, let D_y = [y₁, y₂] in the example illustrated by Fig. 6.1, i.e. the property y₁ ≤ y ≤ y₂ is required by a user. It is easy to note that the solution for

0 1> y is as follows: D_u =

[

]

and the controllability condition has the form

1 1 c y ≤ 2 2 c y .

If external disturbances z act on the plant and as a result of measurement it is known that z∈D_z then the decision problem is formulated as follows: For the given relation R(u, y, z) and the given sets D_z and D_y one should find the largest set D_u for which the implication

(u∈D_u) ∧ (z∈D_z) →y∈D_y is satisfied.

The general form of the decision problem solution is now the following: Du = {u∈U: z D z∈ [Dy(u, z) ⊆ Dy]} (6.17) where D_y(u, z) = { y∈Y: (u, y, z)∈R(u, y, z)}. (6.18)

6.3 Decision Making Problem 129

y

u y

D

Fig. 6.4. Illustration of the case when solution does not exist

It is then the set of all decisions u such that for every z belonging to D_z the set of all possible values y belongs to the given set D_y. One may say that the solution D_u is robust with respect to z, which means that it is not sensitive to z, i.e. it gives a satisfying solution y for every value of the dis- turbance z from the fixed D_z. For the fixed z, the set of possible decisions is defined by the formula (6.16) in which D_y(u) should be determined ac- cording to the formula (6.6) for the given relation R(u, y, z), i.e. the rela- tion R(u, y; z)⊂U×Y with the parameter z. Consequently

D_u(z) = { u∈U: D_y(u, z) ⊆D_y} ∆= R(z,u) (6.19) where D_y(u, z) is defined by the formula (6.18). The formula (6.19) de- fines a relation between z and u which has been denoted by R(z,u). This relation may be considered as a description of a relational decision (con- trol) algorithm in the open-loop system (Fig. 6.5) or the relational repre- sentation of a knowledge on the control (i.e. the knowledge representation of the controller in the open-loop system). For the functional plant the sys- tem in Fig. 6.5 is reduced to the system presented in Fig. 3.1. It is interest- ing and important to note that for an uncertain plant one obtains an uncer- tain control algorithm determined by using a knowledge representation of the plant. In the case of the relational description of uncertainty considered in this chapter, it is the relational plant and the corresponding relational control algorithm.

R(z,u) R(u,y,z)

z

u y

Fig. 6.5. Open-loop control system with relational plant and relational control al- gorithm

Example 6.2. Consider the plant with two scalar inputs u, z and the single output y, described by the inequality

cu + z≤y≤ 2cu + z, c > 0. (6.20) Determine the set D_u for the given sets D_z = [z₁, z₂] and D_y = [y₁, y₂]. In other words, we want to obtain the set of all control decisions satisfying the requirement y₁ ≤y≤y₂ for every z from the interval [z₁, z₂]. The set (6.18) is now defined directly by the inequality (6.20). According to (6.17) the set D_u is then defined by the inequality

c z y₁− ₁ ≤u≤ c z y 2 2 2 − _.

The solution exists if 2( y1 – z1) ≤ ( y2 – z2). For the given value z the set

D_u(z) is determined by the inequality

c z y₁− ≤u≤ c z y 2 2− _.

It is R(z,u) or the relational control algorithm in the open-loop system.

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