# A Sensor-Aided System for Physical Perfect Control Applications in the Continuous-Time Domain

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Representation

## 3. Preliminaries

#### 3.1. Generalized $\sigma $-Inverse

**Remark**

**1.**

#### 3.2. Continuous-Time Perfect Control

**Remark**

**2.**

#### 3.3. The LQ Regulation

**Q**—positive symmetric semi-definite matrices. The value of the first one is responsible for the input of the system, while the second, for its output. These structures are important in the robustness properties of the control system.

#### 3.4. The LQR with Integrating Action

#### 3.5. Systems under Consideration

#### 3.5.1. The Cascade Multi-Tank System

#### 3.5.2. The Two-Level Thermal Object

**Remark**

**3.**

#### 3.6. Quality Criteria

- ISE—Integral of Squared Error defined by$$ISE={\int}_{{t}_{0}}^{t}{e}^{2}\left(t\right)dt,$$
- MOE—Minimum of energy which is an integral of squared control signal$$J\left(u\right)={\int}_{{t}_{0}}^{t}{u}^{T}\left(t\right)u\left(t\right)dt;$$
- RT—Regulation time which is a time considered from the beginning of the simulation to receiving the tolerance range $\pm 5\%$ of the expected value by the system output.

## 4. The Real Continuous-Time Perfect Control

## 5. Simulation Studies

#### 5.1. The Cascade Multi-Tank System Control

#### 5.2. The Two-Level Thermal Object Control

## 6. Discussion on the Obtained Simulation Results

## 7. The Sensor-Aided System—A Real experiment Setup

## 8. A Real Experiment on a Sensor-Aided Servomechanism

#### 8.1. The First Experiment with the RCTPC Law

#### 8.2. The Second Experiment with the RCTPC Law

#### 8.3. Experiment with PID Regulator

## 9. Discussion on the Obtained Sensor-Aided System Control Results

## 10. Conclusions and Open Problems

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Symbol | Description | Value | Unit |
---|---|---|---|

${H}_{1max}$ | height of the first tank | 0.35 | m |

${H}_{2max}$ | height of the second tank | 0.35 | m |

${H}_{3max}$ | height of the third tank | 0.35 | m |

${C}_{1}$ | cross-section of the first valve | $1.0057\phantom{\rule{-0.166667em}{0ex}}\xb7\phantom{\rule{-0.166667em}{0ex}}{10}^{-4}$ | m^{2} |

${C}_{2}$ | cross-section of the second valve | $1.1963\phantom{\rule{-0.166667em}{0ex}}\xb7\phantom{\rule{-0.166667em}{0ex}}{10}^{-4}$ | m^{2} |

${C}_{3}$ | cross-section of the third valve | $9.8008\phantom{\rule{-0.166667em}{0ex}}\xb7\phantom{\rule{-0.166667em}{0ex}}{10}^{-4}$ | m^{2} |

${H}_{10}$ | initial liquid height of the 1st tank | 0.12 | m |

${H}_{20}$ | initial liquid height of the 2nd tank | 0.8 | m |

${H}_{30}$ | initial liquid height of the 3rd tank | 0.15 | m |

${\alpha}_{1}$ | flow factor of the first tank | 0.5 | - |

${\alpha}_{2}$ | flow factor of the second tank | 0.5 | - |

${\alpha}_{3}$ | flow factor of the third tank | 0.5 | - |

R | radius of the third tank | 0.365 | m |

a | the base of the first tank | 0.25 | m |

b | distance between tanks | 0.348 | m |

c | base of the second tank | 0.1 | m |

w | the width of all tanks | 0.035 | m |

${Q}_{f}$ | set flow through the pump | $9.8008\phantom{\rule{-0.166667em}{0ex}}\xb7\phantom{\rule{-0.166667em}{0ex}}{10}^{-5}$ | (m^{3})/s |

q | liquid inlet to the upper tank | - | (m^{3})/s |

${q}_{0}$ | initial condition of liquid inlet | 0.035 | (m^{3})/s |

Symbol | Description | Value | Unit |
---|---|---|---|

${T}_{int}$ | interior temperature | - | °C |

${T}_{int0}$ | initial interior temperature | 0 | °C |

${T}_{ext}$ | exterior temperature | - | °C |

${T}_{ext0}$ | initial exterior temperature | −20 | °C |

${T}_{att}$ | attic temperature | - | °C |

${T}_{att0}$ | initial attic temperature | 0 | °C |

${Q}_{h}$ | heat power | - | W |

${Q}_{0}$ | initial heat power | 20,000 | W |

${C}_{vin}$ | interior thermal capacity | 25,714.29 | J/K |

${C}_{vat}$ | attic thermal capacity | 5714.29 | J/K |

${K}_{ae}$ | loss coefficient of the roof | 60 | - |

${K}_{ie}$ | loss coefficient of the external walls | 80 | - |

${K}_{ia}$ | loss coefficient of the ceiling | 50 | - |

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**Figure 1.**The arrangement scheme of cascade multi-tank system [46].

**Figure 3.**Geometry of the tank in which the water level was maintained—dimensions given in cm (source: authors).

**Figure 8.**Diagram of DC motor [47].

**Figure 10.**The received angular position of the model and the sensor-aided object—results of the three attempts with 100° reference value (source: authors).

**Figure 11.**The received velocity of the model and the sensor-aided object for 100° reference value (source: authors).

**Figure 12.**The received angular position of the model and the sensor-aided object—results of the three attempts with 300° reference value (source: authors).

**Figure 13.**The received velocity of the model and the sensor-aided object for 300° reference value (source: authors).

**Figure 14.**The received angular position of the the sensor-aided object for 100° reference value (source: authors).

**Figure 15.**The received angular position of the the sensor-aided object for 300° reference value (source: authors).

**Figure 16.**Graphical presentation of Table 4 (source: authors).

$\mathbf{LQR}$ | $\mathbf{LQR}-\mathit{I}$ | $\mathbf{RCTPC}$ | |
---|---|---|---|

$\mathbf{ISE}$ (m) | $2.431$ | $1.843$ | $1.638$ |

$\mathbf{RT}$ (s) | $2.4\times {10}^{2}$ | $1.67\times {10}^{2}$ | $1.58\times {10}^{2}$ |

MOE (m^{3}/s) | $3.655\times {10}^{-6}$ | $4.999\times {10}^{-6}$ | $1.514\times {10}^{-7}$ |

$\mathbf{LQR}$ | $\mathbf{LQR}-\mathit{I}$ | $\mathbf{RCTPC}$ | |
---|---|---|---|

$\mathbf{ISE}$ (°C) | $6.749\times {10}^{4}$ | $5.949\times {10}^{4}$ | $5.208\times {10}^{4}$ |

$\mathbf{RT}$ (s) | $6.27\times {10}^{2}$ | $3.86\times {10}^{2}$ | $2.38\times {10}^{2}$ |

$\mathbf{MOE}$ (W) | $7.784\times {10}^{9}$ | $8.877\times {10}^{9}$ | $9.599\times {10}^{9}$ |

**Table 3.**The parameters of the sensor-aided servomechanism system [47].

Symbol | Description | Unit |
---|---|---|

$v\left(t\right)$ | input voltage | V |

$i\left(t\right)$ | armature current | [A] |

$\omega \left(t\right)$ | angular velocity of the rotor | [rad/s] |

R | armature resistance | [Ω] |

$\beta $ | damping factor | - |

${K}_{e}\omega \left(t\right)$ | electromagnetic field | - |

RCTPC | 100 [°] | 300 [°] |
---|---|---|

$\mathbf{ISE}\phantom{\rule{3.33333pt}{0ex}}{e}_{r}\left(t\right)$ [°] | ||

$1\phantom{\rule{3.33333pt}{0ex}}test$ | $8.17\times {10}^{6}$ | $4.81\times {10}^{8}$ |

$2\phantom{\rule{3.33333pt}{0ex}}test$ | $8.15\times {10}^{6}$ | $4.77\times {10}^{8}$ |

$3\phantom{\rule{3.33333pt}{0ex}}test$ | $8.16\times {10}^{6}$ | $4.78\times {10}^{8}$ |

$\mathbf{ISE}\phantom{\rule{3.33333pt}{0ex}}{e}_{m}\left(t\right)$ [°] | ||

$1\phantom{\rule{3.33333pt}{0ex}}test$ | $7.76\times {10}^{6}$ | $4.93\times {10}^{8}$ |

$2\phantom{\rule{3.33333pt}{0ex}}test$ | $7.76\times {10}^{6}$ | $4.93\times {10}^{8}$ |

$3\phantom{\rule{3.33333pt}{0ex}}test$ | $7.76\times {10}^{6}$ | $4.93\times {10}^{8}$ |

$\mathbf{ISE}\phantom{\rule{3.33333pt}{0ex}}{e}_{c}\left(t\right)$ [°] | ||

$1\phantom{\rule{3.33333pt}{0ex}}test$ | $5.12\times {10}^{4}$ | $2.31\times {10}^{6}$ |

$2\phantom{\rule{3.33333pt}{0ex}}test$ | $5.36\times {10}^{4}$ | $2.32\times {10}^{6}$ |

$3\phantom{\rule{3.33333pt}{0ex}}test$ | $5.57\times {10}^{4}$ | $2.35\times {10}^{6}$ |

$\mathbf{RT}\phantom{\rule{3.33333pt}{0ex}}{e}_{r}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}$[s] | ||

$1\phantom{\rule{3.33333pt}{0ex}}test$ | $2.65\times {10}^{1}$ | $4.25\times {10}^{1}$ |

$2\phantom{\rule{3.33333pt}{0ex}}test$ | $2.64\times {10}^{1}$ | $4.27\times {10}^{1}$ |

$3\phantom{\rule{3.33333pt}{0ex}}test$ | $2.64\times {10}^{1}$ | $4.28\times {10}^{1}$ |

$\mathbf{RT}\phantom{\rule{3.33333pt}{0ex}}{e}_{m}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}$[s] | ||

$1\phantom{\rule{3.33333pt}{0ex}}test$ | $3.3\times {10}^{1}$ | $3.93\times {10}^{1}$ |

$2\phantom{\rule{3.33333pt}{0ex}}test$ | $3.3\times {10}^{1}$ | $3.93\times {10}^{1}$ |

$3\phantom{\rule{3.33333pt}{0ex}}test$ | $3.3\times {10}^{1}$ | $3.93\times {10}^{1}$ |

PID | 100 [°] | 300 [°] |

$\mathbf{ISE}\phantom{\rule{3.33333pt}{0ex}}{e}_{r}\left(t\right)$ [°] | ||

$PID\phantom{\rule{3.33333pt}{0ex}}test$ | $6.4\times {10}^{8}$ | $5.24\times {10}^{8}$ |

$\mathbf{RT}\phantom{\rule{3.33333pt}{0ex}}{e}_{r}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}$[s] | ||

$PID\phantom{\rule{3.33333pt}{0ex}}test$ | $2.28\times {10}^{1}$ | $3.85\times {10}^{1}$ |

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**MDPI and ACS Style**

Majewski, P.; Hunek, W.P.; Pawuś, D.; Szurpicki, K.; Wojtala, T.
A Sensor-Aided System for Physical Perfect Control Applications in the Continuous-Time Domain. *Sensors* **2023**, *23*, 1947.
https://doi.org/10.3390/s23041947

**AMA Style**

Majewski P, Hunek WP, Pawuś D, Szurpicki K, Wojtala T.
A Sensor-Aided System for Physical Perfect Control Applications in the Continuous-Time Domain. *Sensors*. 2023; 23(4):1947.
https://doi.org/10.3390/s23041947

**Chicago/Turabian Style**

Majewski, Paweł, Wojciech P. Hunek, Dawid Pawuś, Krzysztof Szurpicki, and Tomasz Wojtala.
2023. "A Sensor-Aided System for Physical Perfect Control Applications in the Continuous-Time Domain" *Sensors* 23, no. 4: 1947.
https://doi.org/10.3390/s23041947