sgn (FUN) ¶ FUNCTION sgn : INT This function will return the result of the signum function applied to input value \(x\) : \[\begin{split}\text{sgn}(x) = \begin{cases} -1 &\mbox{if } x < 0 \\0 &\mbox{if } x = 0 \\1 &\mbox{if } x > 0 \end{cases}\end{split}\] InOut: Scope Name Type Comment Return sgn INT Input lr LREAL input value \(x\)

analytical functions ¶ CalcRootLin (FunctionBlock) CalcRootParable (FunctionBlock) Derivative (FunctionBlock) Integral (FunctionBlock) PolynomialValue (Function)

CalcRootLin (FB) ¶ FUNCTION_BLOCK CalcRootLin By use of this function block the root of a linear function \(y = a \cdot x + b\) with \(a\mbox{, }b\mbox{, }x \in \mathbb{R}\) , if there is one, will be calculated. InOut: Scope Name Type Comment Input lrParam1 LREAL gradient of straight line (corresponds to \(a\) ) lrParam0 LREAL height at which line cuts the vertical y-axis (corresponds to \(b\) ) Output byRoots BYTE number of roots 0: If line is parallel, but not identical to horizontal x-axis | 255: If line is identical to x-axis 1: Else lrRoot LREAL root

CalcRootParable (FB) ¶ FUNCTION_BLOCK CalcRootParable By use of this function block the root of a linear function \(y = a \cdot x^{2} + b \cdot x + c\) with \(a\mbox{, }b\mbox{, }c\mbox{, }x \in \mathbb{R}\) , if there are some, are calculated. InOut: Scope Name Type Comment Input lrParam2 LREAL multiplier of \(x^{2}\) (corresponds to \(a\) ) lrParam1 LREAL multiplier of \(x\) (corresponds to \(b\) ) lrParam0 LREAL multiplier of \(1\) (corresponds to \(c\) ) Output byRoots BYTE number of roots (0, 1, 2 or 255 in case of infinite roots) lrRoot0 LREAL first root lrRoot1 LREAL second root

Derivative (FB) ¶ FUNCTION_BLOCK Derivative This function block will approximate the first derivative \(f\prime\) of a function \(f = f(t)\) at the actual time \(t_{n}\) with the respect to the values of the last three function calls according to the BDF method: \[f\prime (t_{n}) \doteq \frac{3 (f(t_{n}) - f(t_{n-3})) + f(t_{n-1}) - f(t_{n-2})}{3(t_{n-2}-t_{n-3})+4(t_{n-1}-t_{n-2})+3(t_{n}-t_{n-1})}\] InOut: Scope Name Type Comment Input xEnable BOOL reset lrInputValue LREAL actual function value udiTM UDINT length of time interval \([t_{n-1}, t_{n}]\) (equals time passed since last call to function) Output lrDerivative LREAL approxmated value of first derivative xValid BOOL Validity of result FALSE : If the number of calls neccessary for approximating has not been executed yet.

Integral (FB) ¶ FUNCTION_BLOCK Integral This function block will approximate the integral function of the fuction \(f = f(t)\) over the time interval between the first function call \(t_{0}\) and the actual time \(t_{n}\) : \(\int_{t_{0}}^{t_{n}}f(t)\mbox{d}t\) . The size of the time intervals \([t_{i+1}, t_{i}]\) are integers and measured in micro seconds. The approximation is carried out by use of the explicit ( \(x = f(t_{n-1})\) ) resp. implicit ( \(x = f(t_{n})\) ) Euler method: \[\int_{t_{0}}^{t_{n}}f(t)\mbox{d}t \doteq \int_{t_{0}}^{t_{n-1}}f(t)\mbox{d}t + (t_{n} - t_{n-1}) \cdot x\] InOut: Scope Name Type Initial Comment Input xEnable BOOL reset lrInputValue LREAL function value (corresponds to :math`x`) udiTM UDINT size of time interval \([t_{n-1}, t_{n}]\) (equals time passed since last call to function) Output lrIntegral LREAL approximated value of integral xOverflow BOOL FALSE error flag TRUE : If an overflow has occured

PolynomialValue (FUN) ¶ FUNCTION PolynomialValue : LREAL This function block evaluates a given polynomial of arbitrary degree, \(p(x) = c_{n}x^{n} + c_{n-1}x^{n-1} + \dots + c_{1}x + c_{0}\) with \(c_{i} \in \mathbb{R}, n \in \mathbb{N_{0}}\) at a point \(x \in \mathbb{R}\) by use of the Horner scheme. InOut: Scope Name Type Comment Return PolynomialValue LREAL Input siDegree SINT degree of polynomial plr POINTER TO LREAL pointer to Array of polynomial coefficients; at first there is the coefficient of the monomial of highest degree (corresponds to \(c_{n}\) in the formula) lrValue LREAL point \(x\) , where polynomial has to be evaluated

function manipulators ¶ CharCurve_DINT (FunctionBlock) CharCurve_LREAL (FunctionBlock)

CharCurve_DINT (FB) ¶ FUNCTION_BLOCK CharCurve_DINT This function block will evaluate a piecewise linear function (the characteristic curve) at an integral point \(x \in \mathbb{Z}\) . The characteristic curve is specified by a defined number of integral sampling points \((x_{1}, y_{1}), \dots , (x_{n}, y_{n}) \in \mathbb{Z^{2}}\) InOut: Scope Name Type Comment Input diInputValue DINT interpolation point \(x \in \mathbb{Z}\) usiNoPoints USINT number \(N\) of sampling points defining the characteristic curve ( \(2 \leq N \leq 11\) ) Inout ap2diPoints ARRAY [0..10] OF POINT2_DINT array of \(N\) two dimensional sampling points \((x_{i}, y_{i})\) with \(1 \leq i \leq N\) Output diOutputValue DINT interpolated value at point \(x \in \mathbb{Z}\) xError BOOL error flag wErrorID WORD information on error 0: No error 1: error within array of sampling points (i.e. the sampling points aren’t arranged in ascending order) 2: interpolation point diInputValue is outside of area covered by sampling points ( \(x \notin [x_{1}, x_{n}]\) ) 4: invalid number of sampling points

CharCurve_LREAL (FB) ¶ FUNCTION_BLOCK CharCurve_LREAL This function block will evaluate a piecewise linear function (the characteristic curve) at an integral point \(x \in \mathbb{R}\) . The characteristic curve is specified by a defined number of integral sampling points \((x_{1}, y_{1}), \dots , (x_{n}, y_{n}) \in \mathbb{R^{2}}\) InOut: Scope Name Type Comment Input lrInputValue LREAL interpolation point \(x \in \mathbb{R}\) usiNoPoints USINT number \(N\) of sampling points defining the characteristic curve ( \(2 \leq N \leq 11\) ) Inout ap2lrPoints ARRAY [0..10] OF POINT2_LREAL array of \(N\) two dimensional sampling points \((x_{i}, y_{i})\) with \(1 \leq i \leq N\) Output lrOutputValue LREAL interpolated value at point \(x \in \mathbb{R}\) xError BOOL error flag wErrorID WORD information on error 0: No error 1: error within array of sampling points (i.e. the sampling points aren’t arranged in ascending order) 2: interpolation point diInputValue is outside of area covered by sampling points ( \(x \notin [x_{1}, x_{n}]\) ) 4: invalid number of sampling points