# Experimental Design of an Adaptive LQG Controller for Battery Charger/Dischargers Featuring Low Computational Requirements

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## Abstract

**:**

## 1. Introduction

## 2. Modeling of the Power Stage

## 3. Adaptive LQG Voltage Controller

#### 3.1. LQI Controller Design

#### 3.2. State Observer

#### 3.3. Adaptive Parameters of the LQG Controller

#### 3.3.1. Adaptive Lookup Table Method

#### 3.3.2. Adaptive Polynomial Interpolation Method

#### 3.4. Structure of the Adaptive LQG Voltage Controller

## 4. Design Example and Simulation Results

#### 4.1. Calculation of Controller Parameters

#### 4.1.1. Adaptive Lookup Table

#### 4.1.2. Adaptive Polynomial Interpolation

#### 4.2. Simulations Results

^{®}from Matlab

^{®}using the Simscape™ Electrical™ toolbox [42]. The adaptive parameter methods were implemented using a three-dimensional lookup table (first method), and using Matlab

^{®}functions for the polynomial interpolation (second method). The simulations evaluate the solution performance in two scenarios: voltage regulation for a fixed reference value, which is the most common application; and reference tracking, which introduces a strong perturbation to test the system stability.

#### 4.2.1. Bus Voltage Regulation for a Fixed Reference Value

#### 4.2.2. Reference Tracking

## 5. Experimental Validation

#### 5.1. Experimental Implementation

#### 5.2. Experimental Validation of the Bus Voltage Regulation

#### 5.3. Experimental Validation of Reference Tracking

#### 5.4. Performance and Cost Comparison

^{®}report, the processor load for each of the adaptive solutions was extracted, which includes the online approach reported in [38]. This information is summarized in Table 6, where the two proposed (offline) solutions require almost the same CPU utilization (MIPS required). Instead, the online method [38] requires between 7 and 8 times the MIPS for the adaptive process.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 0.02513 | 0.02462 | 0.02421 | 0.02388 | 0.02359 | 0.02334 | 0.02311 | 0.02289 | 0.02268 | 0.02248 | |

10 | 0.02569 | 0.02520 | 0.02480 | 0.02447 | 0.02419 | 0.02394 | 0.02371 | 0.02350 | 0.02330 | 0.02310 | |

12 | 0.02600 | 0.02555 | 0.02518 | 0.02487 | 0.02461 | 0.02438 | 0.02417 | 0.02397 | 0.02378 | 0.02359 | |

14 | 0.02614 | 0.02573 | 0.02541 | 0.02514 | 0.02490 | 0.02470 | 0.02450 | 0.02432 | 0.02415 | 0.02398 | |

16 | 0.02618 | 0.02582 | 0.02554 | 0.02530 | 0.02510 | 0.02492 | 0.02475 | 0.02459 | 0.02443 | 0.02428 | |

18 | 0.02616 | 0.02584 | 0.02560 | 0.02540 | 0.02522 | 0.02507 | 0.02492 | 0.02478 | 0.02465 | 0.02452 | |

20 | 0.02612 | 0.02582 | 0.02561 | 0.02544 | 0.02529 | 0.02516 | 0.02504 | 0.02493 | 0.02481 | 0.02470 | |

22 | 0.02606 | 0.02578 | 0.02559 | 0.02545 | 0.02533 | 0.02522 | 0.02512 | 0.02503 | 0.02493 | 0.02483 | |

24 | 0.02600 | 0.02572 | 0.02555 | 0.02543 | 0.02533 | 0.02525 | 0.02517 | 0.02509 | 0.02501 | 0.02493 | |

26 | 0.02594 | 0.02566 | 0.02550 | 0.02539 | 0.02532 | 0.02525 | 0.02519 | 0.02513 | 0.02507 | 0.02500 | |

28 | 0.02588 | 0.02559 | 0.02544 | 0.02535 | 0.02529 | 0.02524 | 0.02519 | 0.02515 | 0.02510 | 0.02505 |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 0.06072 | 0.06116 | 0.06110 | 0.06075 | 0.06025 | 0.05967 | 0.05905 | 0.05842 | 0.05780 | 0.05720 | |

10 | 0.05981 | 0.06010 | 0.05996 | 0.05956 | 0.05903 | 0.05843 | 0.05779 | 0.05716 | 0.05653 | 0.05593 | |

12 | 0.05888 | 0.05906 | 0.05886 | 0.05844 | 0.05789 | 0.05728 | 0.05664 | 0.05601 | 0.05538 | 0.05478 | |

14 | 0.05796 | 0.05807 | 0.05782 | 0.05738 | 0.05682 | 0.05621 | 0.05558 | 0.05495 | 0.05433 | 0.05373 | |

16 | 0.05708 | 0.05712 | 0.05684 | 0.05639 | 0.05583 | 0.05522 | 0.05460 | 0.05397 | 0.05336 | 0.05277 | |

18 | 0.05622 | 0.05623 | 0.05593 | 0.05546 | 0.05490 | 0.05430 | 0.05369 | 0.05307 | 0.05247 | 0.05189 | |

20 | 0.05541 | 0.05538 | 0.05506 | 0.05459 | 0.05404 | 0.05345 | 0.05284 | 0.05224 | 0.05165 | 0.05107 | |

22 | 0.05462 | 0.05458 | 0.05425 | 0.05378 | 0.05323 | 0.05265 | 0.05205 | 0.05146 | 0.05088 | 0.05032 | |

24 | 0.05387 | 0.05381 | 0.05348 | 0.05301 | 0.05247 | 0.05190 | 0.05131 | 0.05073 | 0.05016 | 0.04961 | |

26 | 0.05314 | 0.05309 | 0.05276 | 0.05229 | 0.05176 | 0.05119 | 0.05062 | 0.05005 | 0.04949 | 0.04895 | |

28 | 0.05244 | 0.05240 | 0.05207 | 0.05161 | 0.05109 | 0.05053 | 0.04997 | 0.04941 | 0.04886 | 0.04833 |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 0.00814 | 0.00790 | 0.00768 | 0.00754 | 0.00748 | 0.00748 | 0.00753 | 0.00763 | 0.00775 | 0.00789 | |

10 | 0.00845 | 0.00815 | 0.00787 | 0.00766 | 0.00753 | 0.00746 | 0.00746 | 0.00750 | 0.00758 | 0.00768 | |

12 | 0.00862 | 0.00831 | 0.00799 | 0.00774 | 0.00756 | 0.00745 | 0.00740 | 0.00740 | 0.00744 | 0.00750 | |

14 | 0.00868 | 0.00838 | 0.00806 | 0.00778 | 0.00757 | 0.00743 | 0.00735 | 0.00731 | 0.00732 | 0.00736 | |

16 | 0.00865 | 0.00839 | 0.00807 | 0.00779 | 0.00756 | 0.00740 | 0.00730 | 0.00724 | 0.00722 | 0.00724 | |

18 | 0.00855 | 0.00834 | 0.00804 | 0.00776 | 0.00754 | 0.00736 | 0.00724 | 0.00717 | 0.00713 | 0.00713 | |

20 | 0.00839 | 0.00824 | 0.00798 | 0.00771 | 0.00749 | 0.00731 | 0.00718 | 0.00709 | 0.00704 | 0.00703 | |

22 | 0.00819 | 0.00812 | 0.00789 | 0.00764 | 0.00743 | 0.00725 | 0.00711 | 0.00702 | 0.00696 | 0.00693 | |

24 | 0.00796 | 0.00797 | 0.00778 | 0.00756 | 0.00735 | 0.00718 | 0.00704 | 0.00694 | 0.00688 | 0.00684 | |

26 | 0.00770 | 0.00780 | 0.00765 | 0.00746 | 0.00726 | 0.00710 | 0.00697 | 0.00687 | 0.00680 | 0.00675 | |

28 | 0.00742 | 0.00762 | 0.00752 | 0.00734 | 0.00717 | 0.00701 | 0.00689 | 0.00679 | 0.00672 | 0.00667 |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 0.04164 | 0.04310 | 0.04443 | 0.04562 | 0.04666 | 0.04758 | 0.04838 | 0.04910 | 0.04973 | 0.05029 | |

10 | 0.04490 | 0.04608 | 0.04717 | 0.04815 | 0.04902 | 0.04979 | 0.05047 | 0.05107 | 0.05160 | 0.05207 | |

12 | 0.04762 | 0.04860 | 0.04950 | 0.05032 | 0.05105 | 0.05169 | 0.05227 | 0.05278 | 0.05323 | 0.05364 | |

14 | 0.04990 | 0.05073 | 0.05149 | 0.05218 | 0.05280 | 0.05335 | 0.05384 | 0.05428 | 0.05467 | 0.05501 | |

16 | 0.05185 | 0.05256 | 0.05321 | 0.05380 | 0.05433 | 0.05480 | 0.05523 | 0.05560 | 0.05593 | 0.05623 | |

18 | 0.05351 | 0.05415 | 0.05471 | 0.05522 | 0.05567 | 0.05608 | 0.05645 | 0.05677 | 0.05706 | 0.05732 | |

20 | 0.05495 | 0.05553 | 0.05603 | 0.05647 | 0.05687 | 0.05722 | 0.05754 | 0.05782 | 0.05807 | 0.05829 | |

22 | 0.05620 | 0.05675 | 0.05719 | 0.05758 | 0.05793 | 0.05824 | 0.05852 | 0.05876 | 0.05898 | 0.05917 | |

24 | 0.05730 | 0.05782 | 0.05822 | 0.05857 | 0.05888 | 0.05915 | 0.05940 | 0.05961 | 0.05980 | 0.05997 | |

26 | 0.05827 | 0.05878 | 0.05915 | 0.05946 | 0.05974 | 0.05998 | 0.06019 | 0.06038 | 0.06055 | 0.06070 | |

28 | 0.05913 | 0.05964 | 0.05998 | 0.06027 | 0.06051 | 0.06073 | 0.06092 | 0.06109 | 0.06123 | 0.06136 |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 7.26$\text{}\times \text{}{10}^{3}$ | 8.08$\text{}\times \text{}{10}^{3}$ | 8.88$\text{}\times \text{}{10}^{3}$ | 9.68$\text{}\times \text{}{10}^{3}$ | 1.05$\text{}\times \text{}{10}^{4}$ | 1.13$\text{}\times \text{}{10}^{4}$ | 1.21$\text{}\times \text{}{10}^{4}$ | 1.29$\text{}\times \text{}{10}^{4}$ | 1.37$\text{}\times \text{}{10}^{4}$ | 1.46$\text{}\times \text{}{10}^{4}$ | |

10 | 8.11$\text{}\times \text{}{10}^{3}$ | 8.96$\text{}\times \text{}{10}^{3}$ | 9.78$\text{}\times \text{}{10}^{3}$ | 1.06$\text{}\times \text{}{10}^{4}$ | 1.14$\text{}\times \text{}{10}^{4}$ | 1.22$\text{}\times \text{}{10}^{4}$ | 1.30$\text{}\times \text{}{10}^{4}$ | 1.39$\text{}\times \text{}{10}^{4}$ | 1.47$\text{}\times \text{}{10}^{4}$ | 1.55$\text{}\times \text{}{10}^{4}$ | |

12 | 8.95$\text{}\times \text{}{10}^{3}$ | 9.82$\text{}\times \text{}{10}^{3}$ | 1.07$\text{}\times \text{}{10}^{4}$ | 1.15$\text{}\times \text{}{10}^{4}$ | 1.23$\text{}\times \text{}{10}^{4}$ | 1.31$\text{}\times \text{}{10}^{4}$ | 1.40$\text{}\times \text{}{10}^{4}$ | 1.48$\text{}\times \text{}{10}^{4}$ | 1.56$\text{}\times \text{}{10}^{4}$ | 1.65$\text{}\times \text{}{10}^{4}$ | |

14 | 9.77$\text{}\times \text{}{10}^{3}$ | 1.07$\text{}\times \text{}{10}^{4}$ | 1.15$\text{}\times \text{}{10}^{4}$ | 1.24$\text{}\times \text{}{10}^{4}$ | 1.32$\text{}\times \text{}{10}^{4}$ | 1.41$\text{}\times \text{}{10}^{4}$ | 1.49$\text{}\times \text{}{10}^{4}$ | 1.57$\text{}\times \text{}{10}^{4}$ | 1.66$\text{}\times \text{}{10}^{4}$ | 1.74$\text{}\times \text{}{10}^{4}$ | |

16 | 1.06$\text{}\times \text{}{10}^{4}$ | 1.15$\text{}\times \text{}{10}^{4}$ | 1.24$\text{}\times \text{}{10}^{4}$ | 1.33$\text{}\times \text{}{10}^{4}$ | 1.41$\text{}\times \text{}{10}^{4}$ | 1.49$\text{}\times \text{}{10}^{4}$ | 1.58$\text{}\times \text{}{10}^{4}$ | 1.66$\text{}\times \text{}{10}^{4}$ | 1.75$\text{}\times \text{}{10}^{4}$ | 1.83$\text{}\times \text{}{10}^{4}$ | |

18 | 1.14$\text{}\times \text{}{10}^{4}$ | 1.24$\text{}\times \text{}{10}^{4}$ | 1.32$\text{}\times \text{}{10}^{4}$ | 1.41$\text{}\times \text{}{10}^{4}$ | 1.50$\text{}\times \text{}{10}^{4}$ | 1.58$\text{}\times \text{}{10}^{4}$ | 1.67$\text{}\times \text{}{10}^{4}$ | 1.75$\text{}\times \text{}{10}^{4}$ | 1.84$\text{}\times \text{}{10}^{4}$ | 1.92$\text{}\times \text{}{10}^{4}$ | |

20 | 1.22$\text{}\times \text{}{10}^{4}$ | 1.32$\text{}\times \text{}{10}^{4}$ | 1.41$\text{}\times \text{}{10}^{4}$ | 1.50$\text{}\times \text{}{10}^{4}$ | 1.59$\text{}\times \text{}{10}^{4}$ | 1.67$\text{}\times \text{}{10}^{4}$ | 1.76$\text{}\times \text{}{10}^{4}$ | 1.84$\text{}\times \text{}{10}^{4}$ | 1.93$\text{}\times \text{}{10}^{4}$ | 2.01$\text{}\times \text{}{10}^{4}$ | |

22 | 1.30$\text{}\times \text{}{10}^{4}$ | 1.40$\text{}\times \text{}{10}^{4}$ | 1.49$\text{}\times \text{}{10}^{4}$ | 1.58$\text{}\times \text{}{10}^{4}$ | 1.67$\text{}\times \text{}{10}^{4}$ | 1.76$\text{}\times \text{}{10}^{4}$ | 1.85$\text{}\times \text{}{10}^{4}$ | 1.93$\text{}\times \text{}{10}^{4}$ | 2.02$\text{}\times \text{}{10}^{4}$ | 2.10$\text{}\times \text{}{10}^{4}$ | |

24 | 1.38$\text{}\times \text{}{10}^{4}$ | 1.49$\text{}\times \text{}{10}^{4}$ | 1.58$\text{}\times \text{}{10}^{4}$ | 1.67$\text{}\times \text{}{10}^{4}$ | 1.76$\text{}\times \text{}{10}^{4}$ | 1.85$\text{}\times \text{}{10}^{4}$ | 1.93$\text{}\times \text{}{10}^{4}$ | 2.02$\text{}\times \text{}{10}^{4}$ | 2.11$\text{}\times \text{}{10}^{4}$ | 2.19$\text{}\times \text{}{10}^{4}$ | |

26 | 1.47$\text{}\times \text{}{10}^{4}$ | 1.57$\text{}\times \text{}{10}^{4}$ | 1.66$\text{}\times \text{}{10}^{4}$ | 1.76$\text{}\times \text{}{10}^{4}$ | 1.85$\text{}\times \text{}{10}^{4}$ | 1.93$\text{}\times \text{}{10}^{4}$ | 2.02$\text{}\times \text{}{10}^{4}$ | 2.11$\text{}\times \text{}{10}^{4}$ | 2.20$\text{}\times \text{}{10}^{4}$ | 2.28$\text{}\times \text{}{10}^{4}$ | |

28 | 1.55$\text{}\times \text{}{10}^{4}$ | 1.65$\text{}\times \text{}{10}^{4}$ | 1.75$\text{}\times \text{}{10}^{4}$ | 1.84$\text{}\times \text{}{10}^{4}$ | 1.93$\text{}\times \text{}{10}^{4}$ | 2.02$\text{}\times \text{}{10}^{4}$ | 2.11$\text{}\times \text{}{10}^{4}$ | 2.20$\text{}\times \text{}{10}^{4}$ | 2.28$\text{}\times \text{}{10}^{4}$ | 2.37$\text{}\times \text{}{10}^{4}$ |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 5.51$\text{}\times \text{}{10}^{3}$ | 6.46$\text{}\times \text{}{10}^{3}$ | 7.38$\text{}\times \text{}{10}^{3}$ | 8.28$\text{}\times \text{}{10}^{3}$ | 9.18$\text{}\times \text{}{10}^{3}$ | 1.01$\text{}\times \text{}{10}^{4}$ | 1.09$\text{}\times \text{}{10}^{4}$ | 1.18$\text{}\times \text{}{10}^{4}$ | 1.27$\text{}\times \text{}{10}^{4}$ | 1.36$\text{}\times \text{}{10}^{4}$ | |

10 | 6.19$\text{}\times \text{}{10}^{3}$ | 7.17$\text{}\times \text{}{10}^{3}$ | 8.12$\text{}\times \text{}{10}^{3}$ | 9.05$\text{}\times \text{}{10}^{3}$ | 9.96$\text{}\times \text{}{10}^{3}$ | 1.09$\text{}\times \text{}{10}^{4}$ | 1.18$\text{}\times \text{}{10}^{4}$ | 1.26$\text{}\times \text{}{10}^{4}$ | 1.35$\text{}\times \text{}{10}^{4}$ | 1.44$\text{}\times \text{}{10}^{4}$ | |

12 | 6.87$\text{}\times \text{}{10}^{3}$ | 7.89$\text{}\times \text{}{10}^{3}$ | 8.87$\text{}\times \text{}{10}^{3}$ | 9.81$\text{}\times \text{}{10}^{3}$ | 1.07$\text{}\times \text{}{10}^{4}$ | 1.17$\text{}\times \text{}{10}^{4}$ | 1.26$\text{}\times \text{}{10}^{4}$ | 1.35$\text{}\times \text{}{10}^{4}$ | 1.44$\text{}\times \text{}{10}^{4}$ | 1.52$\text{}\times \text{}{10}^{4}$ | |

14 | 7.57$\text{}\times \text{}{10}^{3}$ | 8.62$\text{}\times \text{}{10}^{3}$ | 9.61$\text{}\times \text{}{10}^{3}$ | 1.06$\text{}\times \text{}{10}^{4}$ | 1.15$\text{}\times \text{}{10}^{4}$ | 1.24$\text{}\times \text{}{10}^{4}$ | 1.34$\text{}\times \text{}{10}^{4}$ | 1.43$\text{}\times \text{}{10}^{4}$ | 1.52$\text{}\times \text{}{10}^{4}$ | 1.61$\text{}\times \text{}{10}^{4}$ | |

16 | 8.27$\text{}\times \text{}{10}^{3}$ | 9.35$\text{}\times \text{}{10}^{3}$ | 1.04$\text{}\times \text{}{10}^{4}$ | 1.13$\text{}\times \text{}{10}^{4}$ | 1.23$\text{}\times \text{}{10}^{4}$ | 1.32$\text{}\times \text{}{10}^{4}$ | 1.42$\text{}\times \text{}{10}^{4}$ | 1.51$\text{}\times \text{}{10}^{4}$ | 1.60$\text{}\times \text{}{10}^{4}$ | 1.69$\text{}\times \text{}{10}^{4}$ | |

18 | 8.97$\text{}\times \text{}{10}^{3}$ | 1.01$\text{}\times \text{}{10}^{4}$ | 1.11$\text{}\times \text{}{10}^{4}$ | 1.21$\text{}\times \text{}{10}^{4}$ | 1.31$\text{}\times \text{}{10}^{4}$ | 1.40$\text{}\times \text{}{10}^{4}$ | 1.50$\text{}\times \text{}{10}^{4}$ | 1.59$\text{}\times \text{}{10}^{4}$ | 1.68$\text{}\times \text{}{10}^{4}$ | 1.77$\text{}\times \text{}{10}^{4}$ | |

20 | 9.68$\text{}\times \text{}{10}^{3}$ | 1.08$\text{}\times \text{}{10}^{4}$ | 1.19$\text{}\times \text{}{10}^{4}$ | 1.29$\text{}\times \text{}{10}^{4}$ | 1.39$\text{}\times \text{}{10}^{4}$ | 1.48$\text{}\times \text{}{10}^{4}$ | 1.58$\text{}\times \text{}{10}^{4}$ | 1.67$\text{}\times \text{}{10}^{4}$ | 1.76$\text{}\times \text{}{10}^{4}$ | 1.86$\text{}\times \text{}{10}^{4}$ | |

22 | 1.04$\text{}\times \text{}{10}^{4}$ | 1.16$\text{}\times \text{}{10}^{4}$ | 1.26$\text{}\times \text{}{10}^{4}$ | 1.37$\text{}\times \text{}{10}^{4}$ | 1.47$\text{}\times \text{}{10}^{4}$ | 1.56$\text{}\times \text{}{10}^{4}$ | 1.66$\text{}\times \text{}{10}^{4}$ | 1.75$\text{}\times \text{}{10}^{4}$ | 1.85$\text{}\times \text{}{10}^{4}$ | 1.94$\text{}\times \text{}{10}^{4}$ | |

24 | 1.11$\text{}\times \text{}{10}^{4}$ | 1.23$\text{}\times \text{}{10}^{4}$ | 1.34$\text{}\times \text{}{10}^{4}$ | 1.44$\text{}\times \text{}{10}^{4}$ | 1.55$\text{}\times \text{}{10}^{4}$ | 1.64$\text{}\times \text{}{10}^{4}$ | 1.74$\text{}\times \text{}{10}^{4}$ | 1.83$\text{}\times \text{}{10}^{4}$ | 1.93$\text{}\times \text{}{10}^{4}$ | 2.02$\text{}\times \text{}{10}^{4}$ | |

26 | 1.18$\text{}\times \text{}{10}^{4}$ | 1.31$\text{}\times \text{}{10}^{4}$ | 1.42$\text{}\times \text{}{10}^{4}$ | 1.52$\text{}\times \text{}{10}^{4}$ | 1.62$\text{}\times \text{}{10}^{4}$ | 1.72$\text{}\times \text{}{10}^{4}$ | 1.82$\text{}\times \text{}{10}^{4}$ | 1.92$\text{}\times \text{}{10}^{4}$ | 2.01$\text{}\times \text{}{10}^{4}$ | 2.10$\text{}\times \text{}{10}^{4}$ | |

28 | 1.26$\text{}\times \text{}{10}^{4}$ | 1.38$\text{}\times \text{}{10}^{4}$ | 1.50$\text{}\times \text{}{10}^{4}$ | 1.60$\text{}\times \text{}{10}^{4}$ | 1.70$\text{}\times \text{}{10}^{4}$ | 1.80$\text{}\times \text{}{10}^{4}$ | 1.90$\text{}\times \text{}{10}^{4}$ | 2.00$\text{}\times \text{}{10}^{4}$ | 2.09$\text{}\times \text{}{10}^{4}$ | 2.19$\text{}\times \text{}{10}^{4}$ |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | −1.06$\text{}\times \text{}{10}^{3}$ | −3.46$\text{}\times \text{}{10}^{2}$ | 3.00$\text{}\times \text{}{10}^{2}$ | 8.92$\text{}\times \text{}{10}^{2}$ | 1.44$\text{}\times \text{}{10}^{3}$ | 1.95$\text{}\times \text{}{10}^{3}$ | 2.43$\text{}\times \text{}{10}^{3}$ | 2.89$\text{}\times \text{}{10}^{3}$ | 3.32$\text{}\times \text{}{10}^{3}$ | 3.73$\text{}\times \text{}{10}^{3}$ | |

10 | −1.92$\text{}\times \text{}{10}^{3}$ | −1.17$\text{}\times \text{}{10}^{3}$ | −4.95$\text{}\times \text{}{10}^{2}$ | 1.20$\text{}\times \text{}{10}^{2}$ | 6.89$\text{}\times \text{}{10}^{2}$ | 1.22$\text{}\times \text{}{10}^{3}$ | 1.72$\text{}\times \text{}{10}^{3}$ | 2.19$\text{}\times \text{}{10}^{3}$ | 2.63$\text{}\times \text{}{10}^{3}$ | 3.06$\text{}\times \text{}{10}^{3}$ | |

12 | −2.74$\text{}\times \text{}{10}^{3}$ | −1.95$\text{}\times \text{}{10}^{3}$ | −1.25$\text{}\times \text{}{10}^{3}$ | −6.12$\text{}\times \text{}{10}^{2}$ | −2.54$\text{}\times \text{}{10}^{1}$ | 5.20$\text{}\times \text{}{10}^{2}$ | 1.03$\text{}\times \text{}{10}^{3}$ | 1.52$\text{}\times \text{}{10}^{3}$ | 1.97$\text{}\times \text{}{10}^{3}$ | 2.41$\text{}\times \text{}{10}^{3}$ | |

14 | −3.53$\text{}\times \text{}{10}^{3}$ | −2.70$\text{}\times \text{}{10}^{3}$ | −1.97$\text{}\times \text{}{10}^{3}$ | −1.31$\text{}\times \text{}{10}^{3}$ | −7.07$\text{}\times \text{}{10}^{2}$ | −1.46$\text{}\times \text{}{10}^{2}$ | 3.79$\text{}\times \text{}{10}^{2}$ | 8.74$\text{}\times \text{}{10}^{2}$ | 1.34$\text{}\times \text{}{10}^{3}$ | 1.79$\text{}\times \text{}{10}^{3}$ | |

16 | −4.29$\text{}\times \text{}{10}^{3}$ | −3.41$\text{}\times \text{}{10}^{3}$ | −2.66$\text{}\times \text{}{10}^{3}$ | −1.98$\text{}\times \text{}{10}^{3}$ | −1.36$\text{}\times \text{}{10}^{3}$ | −7.85$\text{}\times \text{}{10}^{2}$ | −2.48$\text{}\times \text{}{10}^{2}$ | 2.57$\text{}\times \text{}{10}^{2}$ | 7.37$\text{}\times \text{}{10}^{2}$ | 1.19$\text{}\times \text{}{10}^{3}$ | |

18 | −5.03$\text{}\times \text{}{10}^{3}$ | −4.11$\text{}\times \text{}{10}^{3}$ | −3.32$\text{}\times \text{}{10}^{3}$ | −2.62$\text{}\times \text{}{10}^{3}$ | −1.99$\text{}\times \text{}{10}^{3}$ | −1.40$\text{}\times \text{}{10}^{3}$ | −8.52$\text{}\times \text{}{10}^{2}$ | −3.36$\text{}\times \text{}{10}^{2}$ | 1.52$\text{}\times \text{}{10}^{2}$ | 6.16$\text{}\times \text{}{10}^{2}$ | |

20 | −5.75$\text{}\times \text{}{10}^{3}$ | −4.78$\text{}\times \text{}{10}^{3}$ | −3.96$\text{}\times \text{}{10}^{3}$ | −3.24$\text{}\times \text{}{10}^{3}$ | −2.59$\text{}\times \text{}{10}^{3}$ | −1.99$\text{}\times \text{}{10}^{3}$ | −1.43$\text{}\times \text{}{10}^{3}$ | −9.08$\text{}\times \text{}{10}^{2}$ | −4.12$\text{}\times \text{}{10}^{2}$ | 5.99$\text{}\times \text{}{10}^{1}$ | |

22 | −6.46$\text{}\times \text{}{10}^{3}$ | −5.43$\text{}\times \text{}{10}^{3}$ | −4.58$\text{}\times \text{}{10}^{3}$ | −3.84$\text{}\times \text{}{10}^{3}$ | −3.18$\text{}\times \text{}{10}^{3}$ | −2.56$\text{}\times \text{}{10}^{3}$ | −2.00$\text{}\times \text{}{10}^{3}$ | −1.46$\text{}\times \text{}{10}^{3}$ | −9.58$\text{}\times \text{}{10}^{2}$ | −4.79$\text{}\times \text{}{10}^{2}$ | |

24 | −7.15$\text{}\times \text{}{10}^{3}$ | −6.07$\text{}\times \text{}{10}^{3}$ | −5.19$\text{}\times \text{}{10}^{3}$ | −4.42$\text{}\times \text{}{10}^{3}$ | −3.74$\text{}\times \text{}{10}^{3}$ | −3.12$\text{}\times \text{}{10}^{3}$ | −2.54$\text{}\times \text{}{10}^{3}$ | −2.00$\text{}\times \text{}{10}^{3}$ | −1.49$\text{}\times \text{}{10}^{3}$ | −1.00$\text{}\times \text{}{10}^{3}$ | |

26 | −7.84$\text{}\times \text{}{10}^{3}$ | −6.69$\text{}\times \text{}{10}^{3}$ | −5.77$\text{}\times \text{}{10}^{3}$ | −4.99$\text{}\times \text{}{10}^{3}$ | −4.29$\text{}\times \text{}{10}^{3}$ | −3.66$\text{}\times \text{}{10}^{3}$ | −3.07$\text{}\times \text{}{10}^{3}$ | −2.52$\text{}\times \text{}{10}^{3}$ | −2.00$\text{}\times \text{}{10}^{3}$ | −1.51$\text{}\times \text{}{10}^{3}$ | |

28 | −8.52$\text{}\times \text{}{10}^{3}$ | −7.30$\text{}\times \text{}{10}^{3}$ | −6.35$\text{}\times \text{}{10}^{3}$ | −5.54$\text{}\times \text{}{10}^{3}$ | −4.83$\text{}\times \text{}{10}^{3}$ | −4.18$\text{}\times \text{}{10}^{3}$ | −3.58$\text{}\times \text{}{10}^{3}$ | −3.02$\text{}\times \text{}{10}^{3}$ | −2.50$\text{}\times \text{}{10}^{3}$ | −2.00$\text{}\times \text{}{10}^{3}$ |

${\mathit{v}}_{\mathit{b}}\left(\mathit{V}\right)$ | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{dc}}\left(\mathit{V}\right)$ | |||||||||||

8 | 5.78$\text{}\times \text{}{10}^{3}$ | 6.26$\text{}\times \text{}{10}^{3}$ | 6.69$\text{}\times \text{}{10}^{3}$ | 7.09$\text{}\times \text{}{10}^{3}$ | 7.46$\text{}\times \text{}{10}^{3}$ | 7.81$\text{}\times \text{}{10}^{3}$ | 8.15$\text{}\times \text{}{10}^{3}$ | 8.47$\text{}\times \text{}{10}^{3}$ | 8.77$\text{}\times \text{}{10}^{3}$ | 9.07$\text{}\times \text{}{10}^{3}$ | |

10 | 6.12$\text{}\times \text{}{10}^{3}$ | 6.59$\text{}\times \text{}{10}^{3}$ | 7.02$\text{}\times \text{}{10}^{3}$ | 7.41$\text{}\times \text{}{10}^{3}$ | 7.77$\text{}\times \text{}{10}^{3}$ | 8.11$\text{}\times \text{}{10}^{3}$ | 8.44$\text{}\times \text{}{10}^{3}$ | 8.75$\text{}\times \text{}{10}^{3}$ | 9.06$\text{}\times \text{}{10}^{3}$ | 9.35$\text{}\times \text{}{10}^{3}$ | |

12 | 6.45$\text{}\times \text{}{10}^{3}$ | 6.92$\text{}\times \text{}{10}^{3}$ | 7.33$\text{}\times \text{}{10}^{3}$ | 7.71$\text{}\times \text{}{10}^{3}$ | 8.07$\text{}\times \text{}{10}^{3}$ | 8.40$\text{}\times \text{}{10}^{3}$ | 8.72$\text{}\times \text{}{10}^{3}$ | 9.03$\text{}\times \text{}{10}^{3}$ | 9.33$\text{}\times \text{}{10}^{3}$ | 9.61$\text{}\times \text{}{10}^{3}$ | |

14 | 6.77$\text{}\times \text{}{10}^{3}$ | 7.23$\text{}\times \text{}{10}^{3}$ | 7.63$\text{}\times \text{}{10}^{3}$ | 8.01$\text{}\times \text{}{10}^{3}$ | 8.36$\text{}\times \text{}{10}^{3}$ | 8.69$\text{}\times \text{}{10}^{3}$ | 9.00$\text{}\times \text{}{10}^{3}$ | 9.30$\text{}\times \text{}{10}^{3}$ | 9.59$\text{}\times \text{}{10}^{3}$ | 9.87$\text{}\times \text{}{10}^{3}$ | |

16 | 7.08$\text{}\times \text{}{10}^{3}$ | 7.53$\text{}\times \text{}{10}^{3}$ | 7.93$\text{}\times \text{}{10}^{3}$ | 8.29$\text{}\times \text{}{10}^{3}$ | 8.63$\text{}\times \text{}{10}^{3}$ | 8.96$\text{}\times \text{}{10}^{3}$ | 9.27$\text{}\times \text{}{10}^{3}$ | 9.56$\text{}\times \text{}{10}^{3}$ | 9.85$\text{}\times \text{}{10}^{3}$ | 1.01$\text{}\times \text{}{10}^{4}$ | |

18 | 7.37$\text{}\times \text{}{10}^{3}$ | 7.82$\text{}\times \text{}{10}^{3}$ | 8.21$\text{}\times \text{}{10}^{3}$ | 8.57$\text{}\times \text{}{10}^{3}$ | 8.91$\text{}\times \text{}{10}^{3}$ | 9.22$\text{}\times \text{}{10}^{3}$ | 9.53$\text{}\times \text{}{10}^{3}$ | 9.82$\text{}\times \text{}{10}^{3}$ | 1.01$\text{}\times \text{}{10}^{4}$ | 1.04$\text{}\times \text{}{10}^{4}$ | |

20 | 7.66$\text{}\times \text{}{10}^{3}$ | 8.10$\text{}\times \text{}{10}^{3}$ | 8.48$\text{}\times \text{}{10}^{3}$ | 8.84$\text{}\times \text{}{10}^{3}$ | 9.17$\text{}\times \text{}{10}^{3}$ | 9.48$\text{}\times \text{}{10}^{3}$ | 9.78$\text{}\times \text{}{10}^{3}$ | 1.01$\text{}\times \text{}{10}^{4}$ | 1.03$\text{}\times \text{}{10}^{4}$ | 1.06$\text{}\times \text{}{10}^{4}$ | |

22 | 7.94$\text{}\times \text{}{10}^{3}$ | 8.37$\text{}\times \text{}{10}^{3}$ | 8.75$\text{}\times \text{}{10}^{3}$ | 9.10$\text{}\times \text{}{10}^{3}$ | 9.43$\text{}\times \text{}{10}^{3}$ | 9.73$\text{}\times \text{}{10}^{3}$ | 1.00$\text{}\times \text{}{10}^{4}$ | 1.03$\text{}\times \text{}{10}^{4}$ | 1.06$\text{}\times \text{}{10}^{4}$ | 1.08$\text{}\times \text{}{10}^{4}$ | |

24 | 8.21$\text{}\times \text{}{10}^{3}$ | 8.64$\text{}\times \text{}{10}^{3}$ | 9.01$\text{}\times \text{}{10}^{3}$ | 9.36$\text{}\times \text{}{10}^{3}$ | 9.68$\text{}\times \text{}{10}^{3}$ | 9.98$\text{}\times \text{}{10}^{3}$ | 1.03$\text{}\times \text{}{10}^{4}$ | 1.05$\text{}\times \text{}{10}^{4}$ | 1.08$\text{}\times \text{}{10}^{4}$ | 1.11$\text{}\times \text{}{10}^{4}$ | |

26 | 8.47$\text{}\times \text{}{10}^{3}$ | 8.90$\text{}\times \text{}{10}^{3}$ | 9.27$\text{}\times \text{}{10}^{3}$ | 9.61$\text{}\times \text{}{10}^{3}$ | 9.92$\text{}\times \text{}{10}^{3}$ | 1.02$\text{}\times \text{}{10}^{4}$ | 1.05$\text{}\times \text{}{10}^{4}$ | 1.08$\text{}\times \text{}{10}^{4}$ | 1.10$\text{}\times \text{}{10}^{4}$ | 1.13$\text{}\times \text{}{10}^{4}$ | |

28 | 8.72$\text{}\times \text{}{10}^{3}$ | 9.15$\text{}\times \text{}{10}^{3}$ | 9.52$\text{}\times \text{}{10}^{3}$ | 9.85$\text{}\times \text{}{10}^{3}$ | 1.02$\text{}\times \text{}{10}^{4}$ | 1.05$\text{}\times \text{}{10}^{4}$ | 1.07$\text{}\times \text{}{10}^{4}$ | 1.10$\text{}\times \text{}{10}^{4}$ | 1.13$\text{}\times \text{}{10}^{4}$ | 1.15$\text{}\times \text{}{10}^{4}$ |

Coefficient | ${\mathit{K}}_{1}$ | ${\mathit{K}}_{2}$ | ${\mathit{K}}_{3}$ | ${\mathit{K}}_{4}$ |
---|---|---|---|---|

$p00$ | 2.70$\text{}\times \text{}{10}^{1}$ | 4.75$\text{}\times \text{}{10}^{1}$ | 4.70 | 6.08 |

$p10$ | 7.29$\text{}\times \text{}{10}^{-1}$ | 1.29$\text{}\times \text{}{10}^{-1}$ | 7.08$\text{}\times \text{}{10}^{-1}$ | 4.10 |

$p01$ | −9.49$\text{}\times \text{}{10}^{-1}$ | 3.35 | 3.07$\text{}\times \text{}{10}^{-1}$ | 2.17 |

$p20$ | −4.28$\text{}\times \text{}{10}^{-2}$ | −1.18$\text{}\times \text{}{10}^{-2}$ | −4.24$\text{}\times \text{}{10}^{-2}$ | −1.20$\text{}\times \text{}{10}^{-1}$ |

$p11$ | 6.81$\text{}\times \text{}{10}^{-3}$ | −9.29$\text{}\times \text{}{10}^{-2}$ | −1.15$\text{}\times \text{}{10}^{-2}$ | −1.46$\text{}\times \text{}{10}^{-1}$ |

$p02$ | 4.78$\text{}\times \text{}{10}^{-2}$ | −2.09$\text{}\times \text{}{10}^{-1}$ | −3.77$\text{}\times \text{}{10}^{-2}$ | −4.28$\text{}\times \text{}{10}^{-2}$ |

$p30$ | 7.20$\text{}\times \text{}{10}^{-4}$ | 1.41$\text{}\times \text{}{10}^{-4}$ | 4.65$\text{}\times \text{}{10}^{-4}$ | 1.29$\text{}\times \text{}{10}^{-3}$ |

$p21$ | 5.49$\text{}\times \text{}{10}^{-4}$ | 1.73$\text{}\times \text{}{10}^{-3}$ | 2.34$\text{}\times \text{}{10}^{-3}$ | 4.15$\text{}\times \text{}{10}^{-3}$ |

$p12$ | −3.83$\text{}\times \text{}{10}^{-4}$ | 2.78$\text{}\times \text{}{10}^{-3}$ | −1.79$\text{}\times \text{}{10}^{-3}$ | 1.70$\text{}\times \text{}{10}^{-3}$ |

$p03$ | −1.24$\text{}\times \text{}{10}^{-3}$ | 5.41$\text{}\times \text{}{10}^{-3}$ | 2.12$\text{}\times \text{}{10}^{-3}$ | 4.83$\text{}\times \text{}{10}^{-4}$ |

$p31$ | −1.74$\text{}\times \text{}{10}^{-5}$ | −1.30$\text{}\times \text{}{10}^{-5}$ | −2.18$\text{}\times \text{}{10}^{-5}$ | −3.43$\text{}\times \text{}{10}^{-5}$ |

$p22$ | 9.53$\text{}\times \text{}{10}^{-6}$ | −2.07$\text{}\times \text{}{10}^{-5}$ | −1.73$\text{}\times \text{}{10}^{-5}$ | −3.75$\text{}\times \text{}{10}^{-5}$ |

$p13$ | −1.10$\text{}\times \text{}{10}^{-6}$ | −2.97$\text{}\times \text{}{10}^{-5}$ | 3.54$\text{}\times \text{}{10}^{-5}$ | 1.59$\text{}\times \text{}{10}^{-6}$ |

$p04$ | 1.36$\text{}\times \text{}{10}^{-5}$ | −5.28$\text{}\times \text{}{10}^{-5}$ | −3.40$\text{}\times \text{}{10}^{-5}$ | −4.72$\text{}\times \text{}{10}^{-6}$ |

Coefficient | ${\mathit{\ell}}_{1}$ | ${\mathit{\ell}}_{2}$ | ${\mathit{\ell}}_{3}$ | ${\mathit{\ell}}_{4}$ |
---|---|---|---|---|

$p00$ | −3.73$\text{}\times \text{}{10}^{-1}$ | −2.21 | −1.82 | 9.83$\text{}\times \text{}{10}^{-1}$ |

$p10$ | 3.96$\text{}\times \text{}{10}^{-1}$ | 2.50$\text{}\times \text{}{10}^{-1}$ | −6.31$\text{}\times \text{}{10}^{-1}$ | 2.32$\text{}\times \text{}{10}^{-1}$ |

$p01$ | 4.43$\text{}\times \text{}{10}^{-1}$ | 5.70$\text{}\times \text{}{10}^{-1}$ | 6.13$\text{}\times \text{}{10}^{-1}$ | 4.03$\text{}\times \text{}{10}^{-1}$ |

$p20$ | −2.29$\text{}\times \text{}{10}^{-3}$ | 6.64$\text{}\times \text{}{10}^{-4}$ | 5.90$\text{}\times \text{}{10}^{-3}$ | −2.35$\text{}\times \text{}{10}^{-3}$ |

$p11$ | 8.02$\text{}\times \text{}{10}^{-3}$ | 1.04$\text{}\times \text{}{10}^{-2}$ | 1.39$\text{}\times \text{}{10}^{-2}$ | −2.81$\text{}\times \text{}{10}^{-3}$ |

$p02$ | −4.83$\text{}\times \text{}{10}^{-3}$ | −8.88$\text{}\times \text{}{10}^{-3}$ | −1.90$\text{}\times \text{}{10}^{-2}$ | −8.43$\text{}\times \text{}{10}^{-3}$ |

$p30$ | 3.73$\text{}\times \text{}{10}^{-5}$ | 6.63$\text{}\times \text{}{10}^{-6}$ | −5.50$\text{}\times \text{}{10}^{-5}$ | 1.70$\text{}\times \text{}{10}^{-5}$ |

$p21$ | −2.58$\text{}\times \text{}{10}^{-5}$ | −4.04$\text{}\times \text{}{10}^{-5}$ | −9.80$\text{}\times \text{}{10}^{-6}$ | 2.59$\text{}\times \text{}{10}^{-5}$ |

$p12$ | −1.22$\text{}\times \text{}{10}^{-4}$ | −1.52$\text{}\times \text{}{10}^{-4}$ | −2.45$\text{}\times \text{}{10}^{-4}$ | 1.19$\text{}\times \text{}{10}^{-5}$ |

$p03$ | 1.05$\text{}\times \text{}{10}^{-4}$ | 1.58$\text{}\times \text{}{10}^{-4}$ | 2.94$\text{}\times \text{}{10}^{-4}$ | 1.01$\text{}\times \text{}{10}^{-4}$ |

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**Figure 5.**Performance with ${v}_{b}=12$ V and reference voltage ${v}_{d{c}_{ref}}=10$ V using adaptive LQG with lookup table (buck mode).

**Figure 6.**Performance with ${v}_{b}=12$ V and reference voltage ${v}_{d{c}_{ref}}=12$ V using adaptive LQG with lookup table (unitary gain mode).

**Figure 7.**Performance with ${v}_{b}=12$ V and reference voltage ${v}_{d{c}_{ref}}=16$ V using adaptive LQG with lookup table (boost mode).

**Figure 8.**Performance with ${v}_{b}=24$ V and reference voltage ${v}_{d{c}_{ref}}=20$ V using adaptive LQG with lookup table (buck mode).

**Figure 9.**Performance with ${v}_{b}=24$ V and reference voltage ${v}_{d{c}_{ref}}=24$ V using adaptive LQG with lookup table (unitary gain mode).

**Figure 10.**Performance with ${v}_{b}=24$ V and reference voltage ${v}_{d{c}_{ref}}=26$ V using adaptive LQG with lookup table (boost mode).

**Figure 11.**Performance with ${v}_{b}=12$ V and reference voltage ${v}_{d{c}_{ref}}=10$ V using adaptive lqg with polynomial interpolation (buck mode).

**Figure 12.**Performance with ${v}_{b}=12$ V and reference voltage ${v}_{d{c}_{ref}}=12$ V using adaptive LQG with polynomial interpolation (unitary gain mode).

**Figure 13.**Performance with ${v}_{b}=12$ V and reference voltage ${v}_{d{c}_{ref}}=16$ V using adaptive LQG with polynomial interpolation (boost mode).

**Figure 14.**Performance with ${v}_{b}=24$ V and reference voltage ${v}_{d{c}_{ref}}=20$ V using adaptive LQG with polynomial interpolation (buck mode).

**Figure 15.**Performance with ${v}_{b}=24$ V and reference voltage ${v}_{d{c}_{ref}}=24$ V using adaptive LQG with polynomial interpolation (unitary gain mode).

**Figure 16.**Performance with ${v}_{b}=24$ V and reference voltage ${v}_{d{c}_{ref}}=26$ V using adaptive LQG with polynomial interpolation (boost mode).

**Figure 19.**Reference tracking performance at ${v}_{b}=12$ V using adaptive LQG with polynomial interpolation.

**Figure 20.**Reference tracking performance at ${v}_{b}=24$ V using adaptive LQG with polynomial interpolation.

**Figure 24.**Bus regulation with ${v}_{b}=12$ V and reference ${v}_{d{c}_{ref}}=10$ V (buck mode) using a LQG by (

**a**) lookup table and (

**b**) polynomial interpolation. Magenta: battery voltage ${v}_{b}$; yellow: duty cycle d; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 25.**Bus regulation with ${v}_{b}=12$ V and reference ${v}_{d{c}_{ref}}=12$ V (unitary gain mode) using adaptive LQG by (

**a**) lookup table and (

**b**) polynomial interpolation. Magenta: battery voltage ${v}_{b}$; yellow: duty cycle d; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 26.**Bus regulation with ${v}_{b}=12$ V and reference ${v}_{d{c}_{ref}}=16$ V (boost mode) using adaptive LQG by (

**a**) lookup table and (

**b**) polynomial interpolation. Magenta: battery voltage ${v}_{b}$; yellow: duty cycle d; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 27.**Bus regulation with ${v}_{b}=24$ V and reference ${v}_{d{c}_{ref}}=20$ V (buck mode) using adaptive LQG by (

**a**) lookup table and (

**b**) polynomial interpolation. Magenta: battery voltage ${v}_{b}$; yellow: duty cycle d; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 28.**Bus regulation with ${v}_{b}=24$ V and reference ${v}_{d{c}_{ref}}=24$ V (unitary gain mode) using adaptive LQG by (

**a**) lookup table and (

**b**) polynomial interpolation. Magenta: battery voltage ${v}_{b}$; yellow: duty cycle d; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 29.**Bus regulation with ${v}_{b}=24$ V and reference ${v}_{d{c}_{ref}}=26$ V (boost mode) using adaptive LQG by (

**a**) lookup table and (

**b**) polynomial interpolation. Magenta: battery voltage ${v}_{b}$; yellow: duty cycle d; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 30.**Reference tracking performance at ${v}_{b}=12$ V using adaptive LQG with lookup table: (

**a1**) discharge mode, (

**b1**) standby mode, and (

**c1**) charge mode; and with polynomial interpolation: (

**a2**) discharge mode, (

**b2**) standby mode, and (

**c2**) charge mode. Magenta: duty cycle d; yellow: bus voltage reference ${v}_{dcref}$; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

**Figure 31.**Reference tracking performance at ${v}_{b}=24$ V using adaptive LQG with lookup table: (

**a1**) discharge mode, (

**b1**) standby mode, and (

**c1**) charge mode; and with polynomial interpolation: (

**a2**) discharge mode, (

**b2**) standby mode, and (

**c2**) charge mode. Magenta: duty cycle d; yellow: bus voltage reference ${v}_{dcref}$; cyan: bus voltage ${v}_{dc}$; green: bus current ${i}_{o}$.

Acronyms | Significance |
---|---|

LQG | linear–quadratic–Gaussian |

MPC | model predictive control |

DSP | digital signal processor |

LQR | linear quadratic regulator |

LQI | linear quadratic integral |

PID | proportional–integral–derivative |

RMSE | root mean square error |

ADCs | analog-to-digital converters |

MIPS | millions of instructions per second |

Variable | Significance |
---|---|

${i}_{o}$ | Output current, DC bus current of a microgrid |

${v}_{dc}$ | Output voltage, DC bus voltage of a microgrid |

${v}_{d{c}_{ref}}$ | Desired DC bus voltage |

${v}_{b}$ | Battery side voltage |

${v}_{ci}$ | Coupling capacitor voltage |

${v}_{c{i}_{e}}$ | Coupling capacitor voltage on stationary state |

${i}_{L1}$ | Inductor current of battery side |

${i}_{L2}$ | Inductor current of DC bus side |

${i}_{L{1}_{e}}$ | Inductor current on stationary state of battery side |

${i}_{L{2}_{e}}$ | Inductor current on stationary state of DC bus side |

u | Binary control signal |

d | Continuous duty cycle |

${d}_{e}$ | Continuous duty cycle on stationary state |

${T}_{sw}$ | Switching period |

Test Conditions | Overshoot (%) | Settling Time (ms) | |||||
---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{b}}\phantom{\rule{4pt}{0ex}}\left(\mathit{V}\right)$ | ${\mathit{v}}_{\mathit{dc}}\phantom{\rule{4pt}{0ex}}\left(\mathit{V}\right)$ | Lookup Table | Polynomial Interp. | LQG Online | Lookup Table | Polynomial Interp. | LQG Online |

12 | 10 | 10.9 | 10.85 | 9.7 | 3.3 | 3.65 | 3.83 |

12 | 8.8 | 8.76 | 8.75 | 3.2 | 3.45 | 3.46 | |

16 | 6.25 | 6.21 | 6.19 | 2.9 | 3 | 3.02 | |

24 | 20 | 4.13 | 4.1 | 4.1 | 0.75 | 0.75 | 0.75 |

24 | 3.3 | 3.29 | 3.29 | 0.671 | 0.672 | 0.67 | |

26 | 3 | 2.98 | 3 | 0.63 | 0.624 | 0.63 |

${\mathit{v}}_{\mathit{b}}\phantom{\rule{4pt}{0ex}}\left(\mathit{V}\right)$ | ${\mathit{v}}_{\mathit{dc}}\phantom{\rule{4pt}{0ex}}\left(\mathit{V}\right)$ | Mean Error (%) | Relative Standard Deviation (%) | ||
---|---|---|---|---|---|

Lookup Table | Polynomial Interp. | Lookup Table | Polynomial Interp. | ||

12 | 10 | $1.33\times {10}^{-5}$ | $1.04\times {10}^{-5}$ | 0.5674 | 0.5720 |

12 | $6.02\times {10}^{-5}$ | $4.69\times {10}^{-5}$ | 0.4632 | 0.4667 | |

16 | $4.91\times {10}^{-6}$ | $9.09\times {10}^{-7}$ | 0.3440 | 0.3460 | |

24 | 20 | $1.05\times {10}^{-5}$ | $8.43\times {10}^{-6}$ | 0.1797 | 0.1812 |

24 | $6.21\times {10}^{-5}$ | $5.93\times {10}^{-5}$ | 0.1443 | 0.1455 | |

26 | $6.75\times {10}^{-6}$ | $5.16\times {10}^{-6}$ | 0.1367 | 0.1379 |

Test Conditions | Overshoot (%) | Settling Time (ms) | |||||
---|---|---|---|---|---|---|---|

${\mathit{v}}_{\mathit{b}}\phantom{\rule{4pt}{0ex}}\left(\mathit{V}\right)$ | ${\mathit{v}}_{\mathit{dc}}\phantom{\rule{4pt}{0ex}}\left(\mathit{V}\right)$ | Lookup Table | Polynomial Interp. | LQG Online | Lookup Table | Polynomial Interp. | LQG Online |

12 | 10 | 9.28 | 9.45 | 8.75 | 9.25 | 9.16 | 9.00 |

12 | 8.07 | 7.91 | 7.00 | 9.03 | 8.93 | 8.80 | |

16 | 6.48 | 6.09 | 5.85 | 9.19 | 8.85 | 8.70 | |

24 | 20 | 2.85 | 2.80 | 3.16 | 6.04 | 6.47 | 7.00 |

24 | 2.68 | 2.33 | 2.63 | 6.37 | 5.80 | 6.43 | |

26 | 2.30 | 2.23 | 2.63 | 5.31 | 4.76 | 6.00 |

Adaptive Method | Max CPU Utilization (MIPS) |
---|---|

Lookup table | 94 |

Polynomial interpolation | 93 |

Online LQG | 714 |

Adaptive Method | Suitable Control Card | Performance Index | CPU Utilization Index | Cost Index |
---|---|---|---|---|

Lookup table | TMDSCNCD2800157 | 0.934 | 0.989 | 1 |

Polynomial interpolation | TMDSCNCD2800157 | 0.970 | 1 | 1 |

Online LQG | TMDSCNCD28379D | 0.946 | 0.13 | 0.371 |

Adaptive Method | Performance–Cost Relation | Performance–CPU Relation |
---|---|---|

Lookup table | 1.93 | 1.92 |

Polynomial interpolation | 1.97 | 1.97 |

Online LQG | 1.32 | 1.08 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Montenegro-Oviedo, J.A.; Ramos-Paja, C.A.; Orozco-Gutierrez, M.L.; Franco-Mejía, E.; Serna-Garcés, S.I.
Experimental Design of an Adaptive LQG Controller for Battery Charger/Dischargers Featuring Low Computational Requirements. *World Electr. Veh. J.* **2023**, *14*, 142.
https://doi.org/10.3390/wevj14060142

**AMA Style**

Montenegro-Oviedo JA, Ramos-Paja CA, Orozco-Gutierrez ML, Franco-Mejía E, Serna-Garcés SI.
Experimental Design of an Adaptive LQG Controller for Battery Charger/Dischargers Featuring Low Computational Requirements. *World Electric Vehicle Journal*. 2023; 14(6):142.
https://doi.org/10.3390/wevj14060142

**Chicago/Turabian Style**

Montenegro-Oviedo, Jhoan Alejandro, Carlos Andres Ramos-Paja, Martha Lucia Orozco-Gutierrez, Edinson Franco-Mejía, and Sergio Ignacio Serna-Garcés.
2023. "Experimental Design of an Adaptive LQG Controller for Battery Charger/Dischargers Featuring Low Computational Requirements" *World Electric Vehicle Journal* 14, no. 6: 142.
https://doi.org/10.3390/wevj14060142