General Overview

This report is the result of the use of the Python 3.4 package Sympy (for symbolic mathematics), as means to translate published models to a common language. It was created by Verónika Ceballos-Núñez (Orcid ID: 0000-0002-0046-1160) on 29/7/2015, and was last modified on lm.

About the model

The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Hilbert & Reynolds (1991).

Abstract

A model is developed that considers the allocation of carbon and nitrogen substrates to a protein compartment in the shoots, shoot structural components, and root biomass. Inclusion of a shoot-protein compartment allows variation in shoot-specific activity to be modelled as a function of leaf nitrogen concentration. Allocation to the biomass compartments is controlled by two partitioning variables that are defined by explicitly using the balanced activity hypothesis. The model produces balanced activity where the shoot-specific activity, as well as root and shoot biomass, vary in response to the above-ground (light and CO\(_2\)) and below-ground (nitrogen) environments. The predicted patterns of both root: shoot ratio and leaf nitrogen concentration in response to environmental resource availability are qualitatively consistent with general trends observed in plants.

Space Scale

global

Available parameter values

State Variables

The following table contains the available information regarding this section:

Additional Variables

The following table contains the available information regarding this section:

Allocation Coefficients

The following table contains the available information regarding this section:

Auxiliary Variables

The following table contains the available information regarding this section:

Components

The following table contains the available information regarding this section:

Pool model representation

Flux description

Figure 1: Pool model representation

Input fluxes \(W_{p}: \frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\) \(W_{s}: \frac{W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\) \(W_{r}: \frac{W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\) \(W_{C}: \frac{I\cdot W_{s}\cdot h_{max}\cdot\sigma_{c}}{\rho\cdot\left(I + h_{half}\right)}\) \(W_{N}: W_{r}\cdot\sigma_{r}\) Internal fluxes \(W_{C} > W_{p}: \frac{B\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(B\cdot I\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r} + B\cdot f_{C}\cdot f_{N}\cdot\rho\cdot\left(I + h_{half}\right) + f_{N}^{2}\cdot\rho\cdot\left(I + h_{half}\right)\right)}\) \(W_{C} > W_{s}: \frac{W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}^{2}\cdot f_{cs}\cdot\kappa\cdot\rho\cdot\left(I + h_{half}\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(B\cdot I\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r} + B\cdot f_{C}\cdot f_{N}\cdot\rho\cdot\left(I + h_{half}\right) + f_{N}^{2}\cdot\rho\cdot\left(I + h_{half}\right)\right)}\) \(W_{C} > W_{r}: \frac{B\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{C}\cdot f_{N}\cdot f_{cr}\cdot\kappa\cdot\rho\cdot\left(I + h_{half}\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(B\cdot I\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r} + B\cdot f_{C}\cdot f_{N}\cdot\rho\cdot\left(I + h_{half}\right) + f_{N}^{2}\cdot\rho\cdot\left(I + h_{half}\right)\right)}\) \(W_{N} > W_{p}: \frac{B\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(B\cdot I\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r} + B\cdot f_{C}\cdot f_{N}\cdot\rho\cdot\left(I + h_{half}\right) + f_{N}^{2}\cdot\rho\cdot\left(I + h_{half}\right)\right)}\) \(W_{N} > W_{s}: \frac{W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}^{2}\cdot f_{ns}\cdot\kappa\cdot\rho\cdot\left(I + h_{half}\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(B\cdot I\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r} + B\cdot f_{C}\cdot f_{N}\cdot\rho\cdot\left(I + h_{half}\right) + f_{N}^{2}\cdot\rho\cdot\left(I + h_{half}\right)\right)}\) \(W_{N} > W_{r}: \frac{B\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{C}\cdot f_{N}\cdot f_{nr}\cdot\kappa\cdot\rho\cdot\left(I + h_{half}\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(B\cdot I\cdot W_{r}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r} + B\cdot f_{C}\cdot f_{N}\cdot\rho\cdot\left(I + h_{half}\right) + f_{N}^{2}\cdot\rho\cdot\left(I + h_{half}\right)\right)}\)

The right hand side of the ODE

\(\left[\begin{matrix}\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\\\frac{W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\\\frac{W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\\\frac{I\cdot W_{s}\cdot h_{max}\cdot\sigma_{c}}{\rho\cdot\left(I + h_{half}\right)} -\frac{W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right)\\-\frac{W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) + W_{r}\cdot\sigma_{r}\end{matrix}\right]\)

The Jacobian (derivative of the ODE w.r.t. state variables)

\(\left[\begin{matrix}-\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} & -\frac{I^{2}\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}^{2}\cdot W_{s}\cdot f_{C}^{2}\cdot f_{cp}\cdot h_{max}^{2}\cdot\kappa\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I^{2}\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}^{2}\cdot W_{s}\cdot f_{C}^{2}\cdot f_{np}\cdot h_{max}^{2}\cdot\kappa\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I^{2}\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}^{2}\cdot W_{s}\cdot f_{C}^{2}\cdot h_{max}^{2}\cdot\kappa\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} & -\frac{I^{2}\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}^{2}\cdot f_{C}^{2}\cdot f_{cp}\cdot h_{max}^{2}\cdot\kappa\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I^{2}\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}^{2}\cdot f_{C}^{2}\cdot f_{np}\cdot h_{max}^{2}\cdot\kappa\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I^{2}\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}^{2}\cdot f_{C}^{2}\cdot h_{max}^{2}\cdot\kappa\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{p}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} &\frac{I\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{N}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} &\frac{I\cdot W_{C}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{I\cdot W_{C}\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cp} - f_{np} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\\-\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} & -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{cs}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{ns}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} & -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{s}^{2}\cdot f_{cs}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{s}^{2}\cdot f_{ns}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{s}^{2}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} &\frac{W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{N}\cdot W_{s}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{N}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} &\frac{W_{C}\cdot W_{s}\cdot f_{N}\cdot f_{cs}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{s}\cdot f_{N}\cdot f_{ns}\cdot\kappa}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{s}\cdot f_{N}\cdot\kappa\cdot\left(- f_{cs} - f_{ns} + 1\right)}{B\cdot f_{C}\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\\-\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} & -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}^{2}\cdot f_{C}\cdot f_{cr}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}^{2}\cdot f_{C}\cdot f_{nr}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}^{2}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cr} - f_{nr} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} & -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cr}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{nr}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}\cdot\left(- f_{cr} - f_{nr} + 1\right)}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{2\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{3}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{N}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} &\frac{W_{N}\cdot W_{r}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{N}\cdot W_{r}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{N}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} &\frac{W_{C}\cdot W_{r}\cdot f_{cr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{r}\cdot f_{nr}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{C}\cdot W_{r}\cdot\kappa\cdot\left(- f_{cr} - f_{nr} + 1\right)}{\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\\-\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{2\cdot W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) &\frac{I\cdot h_{max}\cdot\sigma_{c}}{\rho\cdot\left(I + h_{half}\right)} -\frac{W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(-\frac{I^{2}\cdot W_{p}\cdot W_{r}^{2}\cdot W_{s}\cdot f_{C}^{2}\cdot f_{cp}\cdot h_{max}^{2}\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{I\cdot W_{p}\cdot W_{r}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{I\cdot W_{r}^{2}\cdot f_{C}\cdot f_{cr}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{cs}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) +\frac{2\cdot W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) & -\frac{W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(-\frac{I^{2}\cdot W_{p}\cdot W_{r}\cdot W_{s}^{2}\cdot f_{C}^{2}\cdot f_{cp}\cdot h_{max}^{2}\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{I\cdot W_{p}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cr}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} -\frac{I\cdot W_{s}^{2}\cdot f_{cs}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}}\right) +\frac{2\cdot W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) & -\frac{W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) & -\frac{W_{C}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{cp}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{cr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{cs}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right)\\-\frac{I\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\kappa\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{2\cdot W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) & -\frac{W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(-\frac{I^{2}\cdot W_{p}\cdot W_{r}^{2}\cdot W_{s}\cdot f_{C}^{2}\cdot f_{np}\cdot h_{max}^{2}\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{I\cdot W_{p}\cdot W_{r}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{I\cdot W_{r}^{2}\cdot f_{C}\cdot f_{nr}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} -\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{ns}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) +\frac{2\cdot W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) & -\frac{W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(-\frac{I^{2}\cdot W_{p}\cdot W_{r}\cdot W_{s}^{2}\cdot f_{C}^{2}\cdot f_{np}\cdot h_{max}^{2}\cdot\sigma_{c}^{2}\cdot\sigma_{r}^{2}}{f_{N}^{2}\cdot\rho^{2}\cdot\left(I + h_{half}\right)^{2}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{I\cdot W_{p}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} -\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{nr}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}} +\frac{f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} -\frac{I\cdot W_{s}^{2}\cdot f_{ns}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{B\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)^{2}}\right) +\frac{2\cdot W_{C}\cdot W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{3}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) +\sigma_{r} & -\frac{W_{N}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right) & -\frac{W_{C}\cdot\kappa}{\left(W_{p} + W_{r} + W_{s}\right)^{2}}\cdot\left(\frac{I\cdot W_{p}\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot f_{np}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)} +\frac{W_{r}\cdot f_{nr}}{\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}} +\frac{W_{s}\cdot f_{N}\cdot f_{ns}}{B\cdot f_{C}\cdot\left(\frac{I\cdot W_{r}\cdot W_{s}\cdot f_{C}\cdot h_{max}\cdot\sigma_{c}\cdot\sigma_{r}}{f_{N}\cdot\rho\cdot\left(I + h_{half}\right)} + 1 +\frac{f_{N}}{B\cdot f_{C}}\right)}\right)\end{matrix}\right]\)

References