# A Novel Evolutionary Algorithm for Designing Robust Analog Filters

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Methods: A Robust Analog Filter Design Using Evolutionary Algorithms

#### 3.1. Analog Filter Design using Bond Graphs and GP

#### 3.1.1. Bond Graphs

#### 3.1.2. Evolving Analog Filters Using Bond Graphs and GP: The GPBG Framework

- The lowpass filter synthesis problem is extracted from [13], in which the frequency response performance of a candidate filter is defined as the weighted sum of deviations from an ideal frequency response magnitude over 101 points:$${F}_{koza}(t)={\displaystyle \sum _{i=0}^{100}}[W(d({f}_{i}),{f}_{i})\times d({f}_{i})]$$
- The highpass filter synthesis problem has a similar configuration to the lowpass filter except for the complementary definitions of the pass and stop bands. The pass band is now defined as [2K, 10K] Hz, while the stop band is [1, 1K] Hz.

#### 3.2. Evolving Robust Analog Filters With Respect to Parameter Variation: The Unified Approach

#### 3.3. Genetic Algorithms for Robust Analog Filter Design: The Traditional Robust Design

#### 3.4. GPRD: Robust Analog Filter Design Using GP

#### 3.5. Evaluation Criteria

## 4. Experiments and Results

- standard genetic programming (GP) without considering robustness requirements;
- genetic programming with robustness-by-multi-simulation (GPRMS);
- genetic programming with robustness-by-perturbed-evaluation (GPRPE);
- hybrid GP/GA robust design method (GPGARMS).

#### 4.1. Evolving Analog Filters Using GP without Considering Robustness in the Fitness Function

#### 4.2. Evolving Robust Analog Filters Using Genetic Algorithms: The Classical Robust Design

#### 4.3. Evolving Robust Analog Filters Using Genetic Programming: Open-Ended Topology Innovation for Robust Design

#### 4.4. Statistical Results of the Three Methods for Evolving Robust Filters

#### 4.4.1. GA-RMS Improves the Robustness of Standard GP Results

#### 4.4.2. Topological Innovation Using GP-Evolved Filters with Higher Robustness than Parametric Robust Design Using GA

#### 4.4.3. GP with Robustness Requirements Constrains Bloating

#### 4.4.4. Comparison with Other Approaches for Robust Filter Design

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A bond graph and its equivalent electrical circuit. The dotted boxes in the left bond graph indicate modifiable sites at which further topological manipulations can be applied (to be explained in the next section).

**Figure 2.**The bond graph structure of a lowpass filter evolved by the GPBG method with 500,000 function evaluations. This filter has 39 components without counting the embryo components. (Component sizing values are omitted in the figure for simplicity).

**Figure 3.**An example of a GP tree, composed of topology operators applied to an embryo, generating a bond graph after depth-first execution (numeric ERC nodes are omitted). Note that the 010 and 001 are the flag bits showing the presence or absence of attached C/I/R components.

**Figure 4.**The topology of the best highpass analog filters evolved with standard GP with 500,000 function evaluations without considering a robustness requirement in the fitness function(parameters are omitted). This filter has 27 C/I/R components without counting the original embryo components. The best evolved lowpass filter is shown in Figure 2. These topologies are the results of a simplification procedure that removes redundancy in the original evolved bond graphs while their functional behaviors are maintained.

**Figure 5.**Frequency responses of best lowpass and highpass filters evolved using GP, GPGARMS, GPRMS, and GPRPE with 20% Gaussian perturbation of their components: (

**a**) Frequency response distribution of the best four lowpass filters. The total sums of 101-point deviations from the target response curve are 6.43 (GP), 33.03 (GPGARMS), 9.61 (GPRMS), and 4.13 (GPRPE). (

**b**) Frequency response distribution of the best four highpass filters. The total deviations from the target response curve are 0.32 (GP), 42.69 (GPGARMS), 2.53 (GPRMS), and 0.19 (GPRPE).

**Figure 6.**Frequency responses of the best lowpass and highpass filters evolved using GP, GPGARMS, GPRMS, and GPRPE in the perturbed-parametric environment: (

**a**) Frequency response distribution of the best filter evolved using standard GP. (

**b**) Frequency response distribution of the best filter evolved using standard GPGARMS. (

**c**) Frequency response distribution of the best filter evolved using standard GPRMS. (

**d**) Frequency response distribution of the best filter evolved using standard GPRPE.

**Figure 7.**Robustness of highpass filters evolved by GP, GPGARMS, GPRMS, and GPRPE. For (

**a**,

**b**), lower values correspond to higher robustness. (

**a**) The average Type-I robustness defined in Section 3.5. (

**b**) The average Type-II robustness defined in Equation (5).

**Figure 8.**Evolved robust lowpass and highpass filters with complexity much lower than the filters evolved without considering robustness in the fitness in Figure 2 and Figure 4. (

**a**) Topology of the most robust lowpass filter with only 13 evolved components using GPRMS. (

**b**) Topology of the most robust lowpass filter with only 16 evolved components using GPRMS. (

**c**) Equivalent analog circuit of the most robust lowpass filter in (

**b**).

Total Population Size: 2000 (400/400/400/400/400) | Number of Subpopulations: 5 |
---|---|

Migration interval: 5 generations | Migration size: 30 individuals |

Max tree depth: 8 | Crossover probability: 0.7 |

InitTreeDepth: 3–5 | Standard mutation probability: 0.1 |

Flag bit mutation rate: 0.1 | Swapping-tree mutation rate: 0.1 |

Tournament size: 7 | Parametric mutation probability: 0.5 |

Max evaluations: 1,000,000 | Flag mutation probability: 0.3 |

Pool size of elite individuals: 20 | Elite pool update frequency: 5 generations |

Total Population Size: 200 | Max Evaluations: 500,000 |
---|---|

Number of parents in crossover: 3 | Family size: 2 |

${\sigma}_{\zeta}$: 0.1 | ${\sigma}_{\eta}$: 0.1 |

SPI: 10 | Perturbation noise percentage: 20% |

**Table 3.**The performance and robustness of evolved lowpass filters using GP, GPGARMS, GPRMS, and GPRPE.

Perturbation Magnitude | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Stderr | Mean | Stderr | Mean | Stderr | Mean | Stderr | Mean | Stderr | ||

${f}_{robustI}$ | GP | 36.57 | 5.55 | 40.04 | 5.00 | 45.33 | 4.56 | 54.68 | 4.25 | 61.91 | 4.06 |

GPGARMS | 65.16 | 6.60 | 50.08 | 6.05 | 64.33 | 3.66 | 73.85 | 3.84 | 79.81 | 5.28 | |

GPRMS | 48.83 | 3.99 | 51.36 | 3.67 | 56.09 | 3.46 | 63.91 | 3.52 | 72.71 | 3.72 | |

GPRPE | 14.08 | 2.65 | 19.64 | 2.22 | 27.25 | 1.82 | 37.51 | 1.81 | 48.98 | 2.21 | |

${f}_{robustII}$ | GP | 2.44 | 0.40 | 5.59 | 0.79 | 14.37 | 1.61 | 30.20 | 2.83 | 42.78 | 3.76 |

GPGARMS | 5.42 | 1.85 | 10.81 | 1.72 | 18.18 | 2.12 | 33.57 | 1.85 | 54.05 | 4.45 | |

GPRMS | 1.35 | 0.34 | 4.58 | 0.54 | 13.58 | 1.05 | 28.46 | 1.95 | 40.88 | 2.64 | |

GPRPE | 3.92 | 0.46 | 8.13 | 0.78 | 20.85 | 3.75 | 30.89 | 2.36 | 43.11 | 2.92 |

**Table 4.**The sizes of filters evolved using four algorithms with different numbers of fitness evaluations. Each algorithm was run 10 times. * A total of 1,000,000 function evaluations are required to do 100,000 actual fitness evaluations.

Algorithm (Fitness Evaluations) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean | Stdev |
---|---|---|---|---|---|---|---|---|---|---|---|---|

GP (100,000) | 34 | 26 | 35 | 24 | 14 | 20 | 24 | 25 | 26 | 46 | 27 | 8.9100 |

GP (500,000) | 46 | 47 | 37 | 47 | 57 | 40 | 40 | 45 | 48 | 79 | 51.7500 | 15.3200 |

GPRMS (100,000) * | 28 | 15 | 36 | 21 | 28 | 14 | 43 | 26 | 31 | 20 | 26.2000 | 9.1100 |

GPRPE (1,000,000) | 67 | 36 | 46 | 36 | 37 | 57 | 29 | 63 | 27 | 36 | 43.4000 | 14 |

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**MDPI and ACS Style**

Li, S.; Zou, W.; Hu, J. A Novel Evolutionary Algorithm for Designing Robust Analog Filters. *Algorithms* **2018**, *11*, 26.
https://doi.org/10.3390/a11030026

**AMA Style**

Li S, Zou W, Hu J. A Novel Evolutionary Algorithm for Designing Robust Analog Filters. *Algorithms*. 2018; 11(3):26.
https://doi.org/10.3390/a11030026

**Chicago/Turabian Style**

Li, Shaobo, Wang Zou, and Jianjun Hu. 2018. "A Novel Evolutionary Algorithm for Designing Robust Analog Filters" *Algorithms* 11, no. 3: 26.
https://doi.org/10.3390/a11030026