Appendix D
Derivation of the Optimal Sampling Time for 1D

n_{x} = (\frac{x_{d} - u_{x} T_{O S P}}{2 \sqrt{D_{x} T_{O S P}}}),

(D.1)

Squaring both sides

n_{x}^{2} = (\frac{x_{d}^{2} - 2 x_{d} u_{x} T_{O S P} + u_{x}^{2} T_{O S P}}{4 D_{x} T_{O S P}}),

(D.2)

Multiplying both sides with $4 D_{x} T_{O S P}$ ,

4 D_{x} T_{O S P} n_{x}^{2} = x_{d}^{2} - 2 x_{d} u_{x} T_{O S P} + u_{x}^{2} T_{O S P} .

(D.3)

Sides of the equation are switched.

x_{d}^{2} - 2 x_{d} u_{x} T_{O S P} + u_{x}^{2} T_{O S P} = 4 D_{x} T_{O S P} n_{x}^{2},

(D.4)

Substracting $4 D_{x} T_{O S P} n_{x}^{2}$ from both sides,

x_{d}^{2} - 2 x_{d} u_{x} T_{O S P} + u_{x}^{2} T_{O S P} - 4 D_{x} T_{O S P} n_{x}^{2} = 4 D_{x} T_{O S P} n_{x}^{2} - 4 D_{x} T_{O S P} n_{x}^{2},

(D.5)

Simplifying the equation,

u_{x}^{2} T_{O S P} - (2 x_{d} u_{x} + 4 D_{x} n_{x}^{2}) T_{O S P} + x_{d}^{2} = 0,

(D.6)

The above equation is in the form of $a x^{2} + b x + c = 0$ and therefore the solution to these form of equation are;

T_{{1 D}_{m a x, m i n}} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a},

(D.7)

For $a = u_{x}^{2}$ , $b = - 2 x_{d} u_{x} - 4 D_{x} n_{x}^{2}$ and $c = x_{d}^{2}$ the solution becomes;

T_{{1 D}_{m a x, m i n}} = \frac{- (- 2 x_{d} u_{x} - 4 D_{x} n_{x}^{2}) \pm \sqrt{{(- 2 x_{d} u_{x} - 4 D_{x} n_{x}^{2})}^{2} - 4 u_{x}^{2} x_{d}^{2}}}{2 u_{x}^{2}},

(D.8)

The following steps are simplifications for the aforementioned equation.

T_{{1 D}_{m a x, m i n}} = \frac{2 x_{d} u_{x} + 4 D_{x} n_{x}^{2} \pm \sqrt{{(- 2 x_{d} u_{x} - 4 D_{x} n_{x}^{2})}^{2} - 4 u_{x}^{2} x_{d}^{2}}}{2 u_{x}^{2}},

(D.9)

The part ${(- 2 x_{d} u_{x} - 4 D_{x} n_{x}^{2})}^{2}$ is opened;

T_{{1 D}_{m a x, m i n}} = \frac{2 x_{d} u_{x} + 4 D_{x} n_{x}^{2} \pm \sqrt{(4 x_{d}^{2} u_{x}^{2} + 16 x_{d} u_{x} D_{x} n_{x}^{2} + 16 D_{x}^{2} n_{x}^{4}) - 4 u_{x}^{2} x_{d}^{2}}}{2 u_{x}^{2}},

(D.10)

The equation is simplified;

T_{{1 D}_{m a x, m i n}} = \frac{2 x_{d} u_{x} + 4 D_{x} n_{x}^{2} \pm \sqrt{16 x_{d} u_{x} D_{x} n_{x}^{2} + 16 D_{x}^{2} n_{x}^{4}}}{2 u_{x}^{2}},

(D.11)

$16 n_{x}^{2}$ is taken out of the squareroot.

T_{{1 D}_{m a x, m i n}} = \frac{2 x_{d} u_{x} + 4 D_{x} n_{x}^{2} \pm 4 n_{x} \sqrt{x_{d} u_{x} D_{x} + D_{x}^{2} n_{x}^{2}}}{2 u_{x}^{2}},

(D.12)

T_{{1 D}_{m a x, m i n}} = \frac{x_{d} u_{x} + 2 D_{x} n_{x}^{2} \pm 2 n_{x} \sqrt{x_{d} u_{x} D_{x} + D_{x}^{2} n_{x}^{2}}}{u_{x}^{2}},

(D.13)

T_{{1 D}_{m a x, m i n}} = \frac{x_{d} u_{x} + 2 D_{x} n_{x}^{2} \pm 2 n_{x} \sqrt{D_{x} (D_{x} n_{x}^{2} + x_{d} u_{x})}}{u_{x}^{2}},

(D.14)

Appendix DDerivation of the Optimal Sampling Time for 1D