# Global Stabilization of a Reaction Wheel Pendulum: A Discrete-Inverse Optimal Formulation Approach via A Control Lyapunov Function

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## Abstract

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## 1. Introduction

- ✓
- Global stabilization of the discrete version of the RWP dynamic model system via control Lyapunov functions with global exponential convergence and optimality properties.
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- Robustness performance against parametric uncertainties while preserving asymptotic convergence properties.
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- The comparison of the proposed discrete-inverse optimal controllers via control Lyapunov functions with the discrete versions of the passivity-based and Lyapunov-based controllers (from the continuous domain) demonstrating superior performances regarding stabilization of the RWP system with minimum settling times.

## 2. Dynamical Modeling of the Reaction Wheel Pendulum

#### 2.1. Dynamical Model in the Continuous Domain

**Remark**

**1.**

**Remark**

**2.**

#### 2.2. Dynamical Model in the Discrete Domain

## 3. Discrete-Inverse Optimal Formulation via CLF

**Definition**

**1.**

- (i)
- it allows achieving the global exponential stability of the equilibrium point ${x}_{k}=0$ for the discrete system (5), and
- (ii)
- it minimizes a cost functional defined as (8)$$\begin{array}{c}\hfill V\left(\right)open="("\; close=")">{x}_{k}=\sum _{n=k}^{\infty}l\left(\right)open="("\; close=")">{x}_{n}+{u}_{n}^{T}R{u}_{n}& ,\end{array}$$with $l\left(\right)open="("\; close=")">{x}_{k}$, being$$\begin{array}{c}\hfill \mathcal{V}:=V\left(\right)open="("\; close=")">{x}_{k+1}-V\left(\right)open="("\; close=")">{x}_{k}& +{u}_{k}^{\star T}R{u}_{k}^{\star}.\end{array}$$

**Remark**

**3.**

#### 3.1. Global Stability Test

#### 3.2. Optimality Test

**Remark**

**5.**

## 4. Numerical Validations

#### 4.1. Evaluation of the Controller for Different Values of the R Gain

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- The gain R highly influences the behavior of the angular position of the RWP system depicted in Figure 2a, since small values of this allows reaching the desired operational point about in 375 samples, (i.e., 375 ms) (see curves for $R=1$ and $R=10$); however, when this parameter increases then the system exhibits oscillations around the desired point and the settling time is between 400 ms and 700 ms.
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- The angular speed of the pendulum bar depicted in Figure 2b presents a negative acceleration between the interval from 0 ms to 300 ms. However, when the concave form of the angular position changes to a convex one, this velocity is reduced from its negative maximum to zero. In addition, the figure shows the overpass in the angular position due to the effects produced on this by the R gain.
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- For small values of the R gain, it is possible to observe two main saturations of the control gain about 10 and $-10$ (see Figure 2). These saturations imply that the motor, coupled with the reaction wheel, is accelerated to its maximum values to reach the desired operative point in minimum settling times. When the R gain increases, it is possible to observe that the second saturation disappears which enlarge the settling times of the state variables, due to the acceleration of the motor is not at its maximums.

#### 4.2. Performance under Parametric Variations

#### 4.3. Comparison with Nonlinear Controllers

## 5. Conclusions and Future Works

- The feedback control law ${u}_{k}^{\star}$ guarantees an exponential an asymptotically stable behavior with capabilities of working under parametric uncertainties without compromising the convergence to the equilibrium point in settling times lower than 700 ms.
- The control function is ${u}_{k}^{\star}$ is indeed an optimal signal since it minimizes the cost function and make it to reach the global optimum value of the cost functional at ${V}^{\star}\left({x}_{k}\right)=\frac{1}{2}{x}_{0}^{T}P{x}_{0}$ with settling times between 370 ms and 700 ms.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Isidori, A. Nonlinear Control Systems; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Iqbal, J.; Ullah, M.; Khan, S.G.; Khelifa, B.; Ćuković, S. Nonlinear control systems-A brief overview of historical and recent advances. Nonlinear Eng.
**2017**, 6, 301–312. [Google Scholar] [CrossRef] - Lu, Q.; Sun, Y.; Mei, S. Nonlinear Control Systems and Power System Dynamics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 10. [Google Scholar]
- Montoya, O.D.; Gil-González, W. Nonlinear analysis and control of a reaction wheel pendulum: Lyapunov-based approach. Eng. Sci. Technol. Int. J.
**2020**, 23, 21–29. [Google Scholar] [CrossRef] - Montoya, O.D.; Garrido, V.M.; Gil-González, W.; Orozco-Henao, C. Passivity-Based Control Applied of a Reaction Wheel Pendulum: An IDA-PBC Approach. In Proceedings of the 2019 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC), Ixtapa, Mexico, 13–15 November 2019; pp. 1–6. [Google Scholar]
- Olivares, M.; Albertos, P. Linear control of the flywheel inverted pendulum. ISA Trans.
**2014**, 53, 1396–1403. [Google Scholar] [CrossRef] [PubMed] - Correa-Ramírez, V.D.; Giraldo-Buitrago, D.; Escobar-Mejía, A. Fuzzy control of an inverted pendulum Driven by a reaction wheel using a trajectory tracking scheme. TecnoLogicas
**2017**, 20, 57–69. [Google Scholar] - Spong, M.W.; Corke, P.; Lozano, R. Nonlinear control of the Reaction Wheel Pendulum. Automatica
**2001**, 37, 1845–1851. [Google Scholar] [CrossRef] - Baimukashev, D.; Sandibay, N.; Rakhim, B.; Varol, H.A.; Rubagotti, M. Deep Learning-Based Approximate Optimal Control of a Reaction-Wheel-Actuated Spherical Inverted Pendulum. In Proceedings of the 2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), Boston, MA, USA, 6–9 July 2020; pp. 1322–1328. [Google Scholar]
- Montoya, O.D.; Gil-González, W.; Ramírez-Vanegas, C. Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach. Symmetry
**2020**, 12, 1359. [Google Scholar] [CrossRef] - Sanchez, E.N.; Ornelas-Tellez, F. Discrete-Time Inverse Optimal Control for Nonlinear Systems; CRC Press Taylor and Francis Group: Boca Raton, FL, USA, 2017. [Google Scholar]
- Ornelas, F.; Sanchez, E.N.; Loukianov, A.G. Discrete-time inverse optimal control for nonlinear systems trajectory tracking. In Proceedings of the 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA, 15–17 December 2010. [Google Scholar] [CrossRef]
- Montoya, O.D.; Gil-González, W.; Serra, F.M. Discrete-time inverse optimal control for a reaction wheel pendulum: A passivity-based control approach. Rev. UIS Ing.
**2020**, 19, 123–132. [Google Scholar] [CrossRef] - Ohsawa, T.; Bloch, A.M.; Leok, M. Discrete Hamilton-Jacobi Theory. SIAM J. Control Optim.
**2011**, 49, 1829–1856. [Google Scholar] [CrossRef][Green Version] - Block, D.J.; Åström, K.J.; Spong, M.W. The reaction wheel pendulum. Synth. Lect. Control Mechatron.
**2007**, 1, 1–105. [Google Scholar] [CrossRef] - Atkinson, C.; Osseiran, A. Discrete-space time-fractional processes. Fract. Calc. Appl. Anal.
**2011**, 14. [Google Scholar] [CrossRef] - Owolabi, K.M.; Atangana, A. Finite Difference Approximations. In Numerical Methods for Fractional Differentiation; Springer: Singapore, 2019; pp. 83–137. [Google Scholar] [CrossRef]
- Sun, J.; Cheng, X.L. Iterative methods for a forward-backward heat equation in two-dimension. Appl. Math.-A J. Chin. Univ.
**2010**, 25, 101–111. [Google Scholar] [CrossRef] - Keadnarmol, P.; Rojsiraphisal, T. Globally exponential stability of a certain neutral differential equation with time-varying delays. Adv. Differ. Equ.
**2014**, 2014. [Google Scholar] [CrossRef][Green Version] - Teel, A.R.; Forni, F.; Zaccarian, L. Lyapunov-Based Sufficient Conditions for Exponential Stability in Hybrid Systems. IEEE Trans. Autom. Control
**2013**, 58, 1591–1596. [Google Scholar] [CrossRef][Green Version] - Valenzuela, J.G.; Montoya, O.D.; Giraldo-Buitrago, D. Local Control of Reaction Wheel Pendulum Using Fuzzy Logic. Sci. Tech.
**2013**, 18, 623–632. [Google Scholar] - Sanfelice, R.G. On the Existence of Control Lyapunov Functions and State-Feedback Laws for Hybrid Systems. IEEE Trans. Autom. Control
**2013**, 58, 3242–3248. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of a reaction wheel pendulum (taken from [4]).

**Figure 2.**Dynamical behavior of the reaction wheel pendulum (RWP) for different values of the gain R: (

**a**) angular position of the pendulum bar, (

**b**) angular speed of the pendulum bar, and (

**c**) control input.

**Figure 3.**Phase-portrait and Lyapunov function behaviors of the RWP for different values of the gain R: (

**a**) Phase portrait and (

**b**) Lyapunov function.

**Figure 5.**Behavior of the angle of the pendulum bar when compared the proposed inverse optimal control with the Lyapunov-based and the passivity-based approaches.

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**MDPI and ACS Style**

Montoya, O.D.; Gil-González, W.; Dominguez-Jimenez, J.A.; Molina-Cabrera, A.; Giral-Ramírez, D.A.
Global Stabilization of a Reaction Wheel Pendulum: A Discrete-Inverse Optimal Formulation Approach via A Control Lyapunov Function. *Symmetry* **2020**, *12*, 1771.
https://doi.org/10.3390/sym12111771

**AMA Style**

Montoya OD, Gil-González W, Dominguez-Jimenez JA, Molina-Cabrera A, Giral-Ramírez DA.
Global Stabilization of a Reaction Wheel Pendulum: A Discrete-Inverse Optimal Formulation Approach via A Control Lyapunov Function. *Symmetry*. 2020; 12(11):1771.
https://doi.org/10.3390/sym12111771

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Walter Gil-González, Juan A. Dominguez-Jimenez, Alexander Molina-Cabrera, and Diego A. Giral-Ramírez.
2020. "Global Stabilization of a Reaction Wheel Pendulum: A Discrete-Inverse Optimal Formulation Approach via A Control Lyapunov Function" *Symmetry* 12, no. 11: 1771.
https://doi.org/10.3390/sym12111771