## 3 Concept of the Robustness \(r(x)\) of the Control Value \(x\) to an Uncertain Disturbance  
First, the concept of the control robustness will be explained with a simple real-life example. Let us assume that we have bought an apartment and we want to insure it against certain possible risks. The set of considered risks is as follows: \(r_{1}\)--fooding with water, \(r_{2}\)--apartment fire, \(r_{3}\)--apartment theft. We do not know the probability of individual threats occurring in the new block of flats where we have bought the apartment. We only know that such threats are possible. We can insure an apartment for only one selected risk, for 2 selected risks, or for all 3 risks. Which decision will be the most robust against possible threats? Obviously, the decision to have 2-risk insurance is more robust than 1-risk insurance. Covering 3 risks is a more robust decision for a set of possible risks than covering 2 risks or 1 risk. As we will see, in complex systems there will be multi-dimensional disturbance states defined by a triple \((a_{2},a_{1},a_{0})\), which are independent from us. It will be necessary to determine such a control value \(x\) (making a decision about the control value \(x\)) that will ensure the achievement of the control goal with as many system disturbance states as possible. If the number of all possible states of a discrete system is \(n\), and the number of states at which a given control \(x\) ensures satisfying the control requirements is \(m\), the robustness \(r(x)\) of the control value \(x\) to the system uncertainty is \(r(x)=m/n\). The robustness \(r(x)\) can also be called the degree of adjustment of the \(x\)-control value to uncertain states of the system. In the following, the concept of robustness \(r(x)\) will be used in determining the optimal control \(x\) in the problem of plant fertilization, which is described by the full quadratic equation \([\underline{a}_{2},\overline{a}_{2}]x^{2}+[\underline{a}_{1},\overline{a}_ {1}]x+[\underline{a}_{0},\overline{a}_{0}]=[\underline{y}_{T}\cdot\overline{ y}_{T}]\). However, the sense of robustness will be explained here only for the simplified version (13) and (14) of the full quadratic equation. \[[\underline{a},\overline{a}]x^{2} =[\underline{y}_{T},\overline{y}_{T}] \tag{13}\] \[[2,4]x^{2} =[4,20] \tag{14}\]  
The simplified version (13) enables the visualization of the problem in 2D space \(A\times X\), while the full version of the QIE exists in 4D space. Step 1: We transfer the conventional form of Equation (14) into the MIA form containing the variables RDM \(\gamma_{a}\) and \(\gamma_{y}\) and solve the equation for \(x\). \[a\in[2,4]:\;a=2+2\gamma_{a},\;\;\gamma_{a}\in[0,1]\] \[y_{T}\in[4,20]:\;y_{T}=4+16\gamma_{y},\;\;\gamma_{y}\in[0,1]\] \[x_{1}(\gamma_{a},\gamma_{y})=-\sqrt{\frac{4+16\gamma_{y}}{2+2 \gamma_{a}}},\;\;\;x_{2}(\gamma_{a},\gamma_{y})=+\sqrt{\frac{4+16\gamma_{y}}{ 2+2\gamma_{a}}} \tag{15}\]  
Note that the values of the roots \(x_{1}(\gamma_{a},\gamma_{y})\) and \(x_{2}(\gamma_{a},\gamma_{y})\) are not independent. They are conjugated by the RDM variables \(\gamma_{a}\),\(\gamma_{y}\). One pair \((\gamma_{a},\gamma_{y})\) corresponds to one pair of roots. Step 2: Determine the spans \(SP_{x1}\) and \(SP_{x2}\) of both roots (16). Figure 1 shows the relationship between the roots' intervals \([\underline{x}_{1},\overline{x}_{1}]\) and \([\underline{x}_{2},\overline{x}_{2}]\) and values of the disturbance \(a\). \[SP_{x1} =[\min_{\gamma_{a},\gamma_{y}}x_{1}(\gamma_{a},\gamma_{y}),\max_{ \gamma_{a},\gamma_{y}}x_{1}(\gamma_{a},\gamma_{y})]=[-\sqrt{10},-1]\approx[-3.16,-1]\] \[SP_{x2} =[\min_{\gamma_{a},\gamma_{y}}x_{2}(\gamma_{a},\gamma_{y}),\max_{ \gamma_{a},\gamma_{y}}x_{2}(\gamma_{a},\gamma_{y})]=[1,\sqrt{10}]\approx[1,3.16] \tag{16}\]  
Figure 1: Illustration of the relationship between the possible values of the roots \(x_{1}\), \(x_{2}\) and possible values of the uncertain variable \(a\) in equation \([2,4]x^{2}=[4,20]\).