**5. Conclusions**

Using variations of mathematical optimization to provide state, rate, and decision/control provides virtual sensing information useful as sensor replacements. In this instance, arbitrary position and rate sensors were modeled as ideal sensors, plus Gaussian random noise and algorithms were presented and compared that provide very smooth (not noisy) signals for position, rate, and acceleration (manifest in the decision/control). There was no acceleration sensor, so the notion of sensor replacement is manifest for acceleration, while the position and rate information was provided by the selected algorithm acting as a vital sensor. Real-time optimal (nonlinear) state estimation using the Moore–Penrose pseudoinverse (implemented in MATLAB using the *pinv* command) was revealed to be the most advised approach with very highly accurate estimates and essentially mathematically optimally low costs of utilization. The real-time optimal inverse calculation becomes poorly conditioned as the end-state is approached due to rank deficiency in the matrix inversion, so switching to the open loop optimal in the very end was implemented when the determinant of the matrix became nearly zero.

**Funding:** This research received no external funding. The APC was funded by the author.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** While the simulation codes used to produce these results are presented in the manuscript and the appendix, data supporting reported results can be obtained by contacting the corresponding author.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Appendix A**

This appendix contains a table of variable and acronym definitions used in the manuscript.

