# Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach

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## Abstract

**:**

## 1. Introduction

- ✔
- The application of the discrete-inverse optimal control to regulate all the state variables of the ball and beam system guaranteeing passivity, stability, and optimality properties.
- ✔
- The numerical validation via simulations by working the discrete equivalent nonlinear model of the system without any special assumption on the open- or closed-loop dynamics.
- ✔
- The robustness and effectiveness of the discrete-inverse optimal control design when parametric variations affect the discrete dynamical model.

## 2. Dynamical Model and Discretization

## 3. Inverse Optimal Control Design

**Definition**

**1.**

**Definition**

**2.**

**P**such that the following inequality is held.

#### 3.1. Passivity

**Theorem**

**1.**

**Proof.**

#### 3.2. Stability

**Theorem**

**2.**

**Proof.**

#### 3.3. Optimality

**Theorem**

**3.**

**Proof.**

#### 3.4. General Commentaries

- ✔
- To stabilize a nonlinear discrete dynamical system with the form defined in (4) it is used the optimal control law $\left({u}_{k}=\beta \left({x}_{k}\right)\right)$ guaranteeing passivity, stability, and optimallity properties.
- ✔
- The application of the inverse optimal control design is subject to the fact that the dynamical system be zero detectable, which can be expressed as presented in Definition 3.
**Definition****3.**A system (4) is locally zero-state observable (locally zero-state detectable) if there is a neighborhood $\mathcal{Z}$ of ${x}_{k}=0\in {\mathbb{R}}^{n}$ such that for all ${x}_{0}\in \mathcal{Z}$$$\begin{array}{c}\hfill {y}_{k}{|}_{{u}_{k}=0}=h\left(\varphi \left(k,{x}_{0},0\right)\right)=0\forall k\to {x}_{k}=0,\end{array}$$

## 4. Numerical Validation

#### 4.1. Regulation of the State Variables

- ✔
- All the state variables are regulated when have passed 4000 samples, i.e., ~4 s, which implies that the discrete-inverse optimal control fulfill the control objective when the control input (28) is applied to the discrete equivalent system (3).
- ✔
- The ball position exhibits a smooth dynamical behavior from the initial position (like a second-order system), i.e., ${x}_{{1}_{0}}=15$ cm, to the origin with a minimum overpass, which implies that the selection of the control gains was appropriate. Nevertheless, this behavior can also be improved (smoothing) if an optimization procedure over gains in $\mathbf{P}$ is made as recommended in [13].
- ✔
- The control input ${u}_{k}$ presented in Figure 2c reaches the zero value when all the state variables are in the origin of coordinates, which is a natural behavior as it is a nonlinear function of all these variables working as a proportional controller that reduces its amplitude when the regulated variables are near to the origin of coordinates.

#### 4.2. Dynamical Performance for Different Control Gains

#### 4.3. Effect of the Parameter Variations

## 5. Conclusions and Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Time-domain behavior of the state variables and control input in the ball and beam system: (

**a**) position and and speed of the ball, (

**b**) angular position and angular speed of the beam, and (

**c**) control input.

**Figure 3.**MATLAB/OCTAVE implementation of the inverse optimal controller for the ball and beam system.

**Figure 5.**Dynamical behavior of the ball position when parametric variations are experimented in h and ${k}_{bb}$ parameters.

Parameter | Value | Unity |
---|---|---|

${L}_{\mathrm{beam}}$ | 42.55 | cm |

${r}_{\mathrm{arm}}$ | 2.54 | cm |

R | 1.27 | cm |

m | 64 | mg |

g | 9.81 | m/s^{2} |

${J}_{b}$ | 4.1290$\times {10}^{6}$ | kgm^{2} |

${k}_{1}$ | 1.76 | rad/sv |

$\tau $ | 28.5 | ms |

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**MDPI and ACS Style**

Danilo Montoya, O.; 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.
https://doi.org/10.3390/sym12081359

**AMA Style**

Danilo Montoya O, 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(8):1359.
https://doi.org/10.3390/sym12081359

**Chicago/Turabian Style**

Danilo Montoya, Oscar, Walter Gil-González, and Carlos Ramírez-Vanegas.
2020. "Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach" *Symmetry* 12, no. 8: 1359.
https://doi.org/10.3390/sym12081359