# A Control Method Based on a Simple Dynamic Optimizer: An Application to Micromachines with Friction

## Abstract

**:**

## 1. Introduction

## 2. The Benchmark Dynamic Optimizer

**Theorem**

**1**

**Proof.**

- There is only one maximum point of the function.
- The function is concave.

## 3. The Recent Control Scheme

**Theorem**

**2**

- There is a real constant value ${k}_{u}$ such that ${\alpha}_{1}-\beta {k}_{u}=-{\u03f5}_{1}<0$, with ${\u03f5}_{1}\in {R}^{+}$,
- There is a real constant value ${k}_{d}$ such that ${\alpha}_{2}-\beta {k}_{d}=-{\u03f5}_{2}<0$, with ${\u03f5}_{2}\in {R}^{+}$,
- There is a real constant value k such that $\beta k=-{\u03f5}_{0}<0$, with ${\u03f5}_{0}\in {R}^{+}$.

- The parameter ${\u03f5}_{2}$ was sufficiently large with respect to ${\u03f5}_{1}$ (${\u03f5}_{2}>>{\u03f5}_{1}$),
- The relation $\frac{{\u03f5}_{0}}{{\u03f5}_{1}}$ was sufficiently small ($\frac{{\u03f5}_{0}}{{\u03f5}_{1}}<<1$).

**Proof.**

- ${\alpha}_{1}-\beta {k}_{u}<0\to {\alpha}_{1}-\beta {k}_{u}=-{\u03f5}_{1}$; ${\u03f5}_{1}\in {R}^{+}$,
- ${\alpha}_{2}-\beta {k}_{d}<0\to {\alpha}_{2}-\beta {k}_{d}=-{\u03f5}_{2}$; ${\u03f5}_{2}\in {R}^{+}$,

## 4. Application to a Micromachine Device

## 5. Future Work

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**A numerical example using $Xcos$ from $Scilab$ application, where the numerical integration method used was $RK54$. The concave function is $f\left(x\right)=-{(x-1)}^{2}$. Note that the notation for the signum function $sgn(\xb7)$ in the $Xcos$ environment is $sign(\xb7)$.

**Figure 3.**A recent control scheme using a dynamic optimizer. The parameter $\tau $ in the derivative estimator block was assumed to be small. This is a well-known derivative block (see, for instance, reference [25], p. 495).

**Figure 4.**Numerical example of the oscillator dynamic (20). (

**Top**): the Xcos block diagram of the system. (

**Bottom**): the system response $z\left(t\right)$ versus time. We use $\dot{z}\left(0\right)=1$ and $z\left(0\right)=-1$.

**Figure 5.**Numerical example: A pulse reference signal case. The green line is the reference signal $a\left(t\right)$, and the black line is the system response $y\left(t\right)$.

**Figure 6.**Numerical example: A sinusoidal reference signal case. The green line is the reference signal $a\left(t\right)$, and the black line is the system response $y\left(t\right)$.

**Figure 7.**An enlarged view of the Figure 5.

**Figure 10.**Results of numerical experiments. Top left: the green line is the reference signal $a\left(t\right)$ and the black line is the position response of the micromachine $y\left(t\right)$. Top right: control signal $u\left(t\right)$. At the bottom: Error signal between system output and reference command $e\left(t\right)$.

**Figure 11.**Results of numerical experiments. Error signal between system output and reference command $e\left(t\right)$. The final part of a long-term simulation.

Challenge | Strategy | Evidence |
---|---|---|

Incorporate a dynamic optimizer in a closed-loop system | Search for a structure where the control signal is integrated to reduce the vibration that the optimizer and the plant could produce | Through numerical experiments |

Closed-loop stability test | Invoke the theory of stability in the sense of Lyapunov | Verifying that the conditions of Lyapunov’s theory are met |

To test control performance in a frictional micromachine using an experimentally validated system model | Use of numerical experiments | From numerical data, observe acceptable performance |

**Table 2.**Effects of the controller parameters on the dynamics of the closed-loop system (data obtained from numerical experimentation by independently increasing each control parameter).

Control Parameter | Steady-State Error | Transient Time Duration |
---|---|---|

k | – | ↓ |

${k}_{u}$ | ↑ | ↓ |

${k}_{d}$ | ↑ | – |

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**MDPI and ACS Style**

Acho, L.
A Control Method Based on a Simple Dynamic Optimizer: An Application to Micromachines with Friction. *Micromachines* **2023**, *14*, 387.
https://doi.org/10.3390/mi14020387

**AMA Style**

Acho L.
A Control Method Based on a Simple Dynamic Optimizer: An Application to Micromachines with Friction. *Micromachines*. 2023; 14(2):387.
https://doi.org/10.3390/mi14020387

**Chicago/Turabian Style**

Acho, Leonardo.
2023. "A Control Method Based on a Simple Dynamic Optimizer: An Application to Micromachines with Friction" *Micromachines* 14, no. 2: 387.
https://doi.org/10.3390/mi14020387