# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control
## 7 Concluding Remarks  
In this paper, a complete analysis of the monowheel system is carried out, along with different control methods for dynamic stability and self-balancing, followed by the formulation of a reference model-assisted adaptive control structure that helps the system to adapt to sudden disturbances in parameters. A realistic estimate of the monowheel's physical parameters was made by referring to Ryno Motor's patent US20120217072A1. Based on that, all other system parameters were calculated, followed by an analysis of the kinematics and the system's dynamics using the Lagrangian dynamic formulation and representing the system in its state-space form. Both pole placement and LQR control methods were explored using a MATLAB Simulink model to mimic the system. Various modifications were made to allow the system to withstand a wide range of rider masses with weights ranging from 12-180 kg while in motion. However, very smooth responses are available in a range of 30-180 kg, under the assumption that the frame would withstand such masses. More modifications were made to consider the self-balancing property under a wide variation of velocity changes and account for external disturbances introduced as white noise. Next, an LQG controller was designed to account for any additional wind disturbances. The results were compared with that of the regular LQR controller, followed  
Figure 24: Comparing reference input velocity and actual velocity output.  
by implementing a Kalman filter to estimate the position state of the monowheel based on all other available measured parameters. The maximum rider speed is reached at 20 km/h, and all analysis is performed within the range of 0-20 km/h. However, higher speeds are achievable by changing the calculations accordingly.  
Next, we moved on to designing a reference model-assisted adaptive control structure for the system so that the system output responses can adapt to a sudden change in parameters and maintain a favorable outcome. For this structure, nominal and reference models were designed, followed by the formulation of error dynamics and adaptive update and control laws. The Lyapunov stability analysis of this system was carried out. A reference model stabilizer was formulated using the standard LQR control method. Although the self-balancing property was achieved for a wide range of masses using LQR control and the system has been made adaptive to load disturbances, the system still has a few shortcomings:  
Self-balancing and stability have not been analyzed for cases with sudden braking;  
All analysis was conducted under the assumption that the frame's mass is 35 kg and the maximum velocity to be attained is 20 km/h;  
A standard incline gradient ratio of 1:4000 feet is assumed. For inclinations greater than that, there will be a change in the calculations and analysis;  
After introducing a load disturbance, the system takes around 20 s to stabilize at the desired value.;  
The system is adaptively controllable only within the parameter ranges given in Table 4.  
Upgraded designs to account for such shortcomings and attempt to implement these control methods in a physical model will be our primary focus of research in the future.  
Conceptualization, I.S. and S.G.; methodology, I.S., S.G., and D.D.; software, I.S. and S.G.; validation, I.S. and S.G.; formal analysis, I.S., S.G. and D.D.; writing--original draft preparation, I.S. and S.G.; writing--review and editing, D.D. and S.O.; supervision, D.D.; funding acquisition, S.O. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the European Regional Development Fund in \"A Research Platform focused on Industry 4.0 and Robotics in Ostrava Agglomeration\" project, Reg. No. _CZ.02.1_01/0.0/0.0/17_049/0008425 within the Operational Programme Research, Development and Education.  
Not Applicable.  
The authors declare no conflict of interest.