# DC Drive Adaptive Speed Controller Based on Hyperstability Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Methods

#### 2.1. Disadvantages of Conventional Cascade DC Drive Control System

^{0}) and structurally implemented as a PI-controller:

^{0}) is applied more often. In such a case, the speed PI-controller is defined as:

#### 2.2. Problem Statement

**Assumption**

**1.**

_{c}is matched with the control action signal$u$.

^{2}is the state vector; ${\mathsf{\tau}}_{e}$ is the disturbance, which is caused by the load torque ${M}_{c}$ and matched with the control action signal; $\Delta \left(u\right)$ is a function to describe unmodeled dynamics (the disturbance, which is caused by the armature current loop dynamics and could not be compensated by the control signal $u$), $A$ ∈ R

^{2×2}is the state matrix, and $B$ ∈ R

^{2×1}is the input matrix. The pair $\left(A,B\right)$ is controllable. The following assumption is introduced about the disturbance, $\Delta \left(u\right)$.

**Assumption**

**2.**

**Remark**

**1.**

**Remark**

**2.**

^{2}is the reference model state vector, ${A}_{ref}$ ∈ R

^{2×2}is the reference model state matrix, ${B}_{ref}$ ∈ R

^{2×2}is the reference model input matrix. The disturbance $\Delta \left(u\right)$ is calculated as the difference between the current value of the armature current I and the control action $u$: $\Delta \left(u\right)=I-u$. The values of the parameters ${a}_{ref}^{0}$ and ${a}_{ref}^{1}$ are chosen so as to make the matrix ${A}_{ref}$ be Hurwitz one. If, additionally, ${a}_{ref}^{0}$ and ${a}_{ref}^{1}$ are chosen according to the following expressions (${a}_{\mathsf{\omega}}$ = 4 is the conventional value):

#### 2.3. Main Result

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 3. Results and Discussion

^{−3}s; $c{\Phi}_{e}$ = 0.08; ${J}_{\Sigma}$ = 10.67 μkg·m

^{2}; ${u}_{\mathrm{max}}=-{u}_{\mathrm{min}}$ = 1 A. The armature current controller was calculated using the motor parameters values according to the known Equation (2) [9,14] to make the armature current loop follow the modulus optimum requirements. The output of the armature current controller was bounded by values of ±10 V.

^{−6}s).

#### 3.1. Demonstration of Disadvantages of Conventional Cascade Control

#### 3.2. First Experiment

#### 3.3. Second Experiment

#### 3.4. Third Experiment

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof.**

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**Figure 5.**Comparison of motor speed curves when ${\mathsf{\tau}}_{e}\ne 0$ and ${\widehat{K}}_{ref}\left(0\right)-{K}_{ref}=0$.

**Figure 7.**Comparison of motor speed curves when ${\mathsf{\tau}}_{e}=0$, ${\widehat{K}}_{ref}\left(0\right)-{K}_{ref}\ne 0$ and $\widehat{K}\left(0\right)-K\ne 0$.

**Figure 8.**Comparison of transient curves of normalized errors $\Vert \tilde{K}\Vert \cdot {\Vert \tilde{K}\left(0\right)\Vert}^{-1}$ and $\left|{\tilde{K}}_{ref}\right|\left|{\tilde{K}}_{ref}^{-1}\left(0\right)\right|$.

**Figure 9.**Comparison of transients of motor speed when ${\mathsf{\tau}}_{e}\ne 0$, ${\widehat{K}}_{ref}\left(0\right)-{K}_{ref}\ne 0$ and $\widehat{K}\left(0\right)-K\ne 0$.

Plant | $\mathit{\sigma}$, % | ${\mathit{t}}_{\mathit{s}\mathit{e}\mathit{t}\mathit{t}\mathit{l}\mathit{e}},\mathbf{s}$ | n |
---|---|---|---|

Nominal values | 16.8 | 0.0243 | 0 |

0.5 ${J}_{\Sigma}$/1.5 R/1.5 L | 26.3 | 0.01435 | 3 |

2 ${J}_{\Sigma}$/1.5 R/1.5 L | 14.2 | 0.0464 | 1 |

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**MDPI and ACS Style**

Glushchenko, A.; Lastochkin, K.; Petrov, V.
DC Drive Adaptive Speed Controller Based on Hyperstability Theory. *Computation* **2022**, *10*, 40.
https://doi.org/10.3390/computation10030040

**AMA Style**

Glushchenko A, Lastochkin K, Petrov V.
DC Drive Adaptive Speed Controller Based on Hyperstability Theory. *Computation*. 2022; 10(3):40.
https://doi.org/10.3390/computation10030040

**Chicago/Turabian Style**

Glushchenko, Anton, Konstantin Lastochkin, and Vladislav Petrov.
2022. "DC Drive Adaptive Speed Controller Based on Hyperstability Theory" *Computation* 10, no. 3: 40.
https://doi.org/10.3390/computation10030040