Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability
Abstract
1. Introduction
- (i)
- A new formal proof of stability of the controller developed in [12] is developed;
- (ii)
- The new formal proof of stability is different from the existing ones in the literature for switched systems, which are typically based on multiple Lyapunov functions [16];
- (iii)
- A study on how stability regions are influenced by different controller parameter choices and mission requirements is presented, using a depth-controlled sensor platform previously developed by the authors.
2. Underwater Device Description and Model
3. Controller Description
4. Proof of Stability of the Closed-Loop System
4.1. Outline of the Proof of Stability
4.2. Part 1 of the Controller Proof of Stability
4.3. Part 2 of the Controller Proof of Stability
4.4. Part 3 of the Controller Proof of Stability
4.5. Quantifying the Controller Action Limits
4.5.1. Quantifying the Upper Limits of
- 1.
- From to , for the purposes of estimating . Since is the maximum value takes between and , this will contribute to an overestimation of , leading to a conservative estimation of .
- 2.
- The integral factor . Since is always positive, assuming that its integral is zero contributes to a conservative estimation.
- 3.
- From to , . The difference corresponds to the required time for to increase from to . Since from to , , it will be considered, in a conservative scenario, that .
- 4.
- From to , for the purposes of estimating . Since is the maximum value takes between and , this will contribute to an overestimation of , leading to a conservative estimation of .
- 5.
- To calculate the integral factor, will be assumed to take its maximum value, , in a given mission, thus considering a conservative estimation.
- 6.
- From to , . The difference corresponds to the required time for to decrease from to . Since from to , , it will be considered, in a conservative scenario, that .
4.5.2. Quantifying the Lower Limit of
5. Closed-Loop System Stability: Case Study
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Regions | Decision | i | |||
---|---|---|---|---|---|
OFF | 1 | ||||
ON | 2 | ||||
OFF | 1 | ||||
OFF | 1 | ||||
ON | 2 |
Parameter | Value | Unit |
---|---|---|
7.7 × 10−3 | [m2] | |
9.81 | [ms−2] | |
4.44 × 10−4 | [ms−1V−1] | |
7.9355 × 103 | [ms−1m−3] | |
3.31 × 10−2 | [Vm−1] | |
36.3 | [s] | |
0.5 | [m] | |
1 × 103 | [kgm−3] | |
3.4 × 10−7 | [m3m−1] | |
100 | [m] |
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Bravo Pinto, J.; Falcão Carneiro, J.; Gomes de Almeida, F.; Cruz, N.A. Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability. Actuators 2025, 14, 340. https://doi.org/10.3390/act14070340
Bravo Pinto J, Falcão Carneiro J, Gomes de Almeida F, Cruz NA. Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability. Actuators. 2025; 14(7):340. https://doi.org/10.3390/act14070340
Chicago/Turabian StyleBravo Pinto, João, João Falcão Carneiro, Fernando Gomes de Almeida, and Nuno A. Cruz. 2025. "Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability" Actuators 14, no. 7: 340. https://doi.org/10.3390/act14070340
APA StyleBravo Pinto, J., Falcão Carneiro, J., Gomes de Almeida, F., & Cruz, N. A. (2025). Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability. Actuators, 14(7), 340. https://doi.org/10.3390/act14070340