# A Novel Evolutionary Algorithm for Designing Robust Analog Filters
## 2 Related Work
's approach, each individual is simulated \(t\) times to estimate its expected loss function (fitness) defined as \(\frac{1}{t}\cdot\sum\limits_{i=1}^{t}{(\tau-\tilde{f}(\vec{x},\vec{\delta}_{i}))^{2}}\), where \(\vec{\delta}_{i},\ i=1,\ldots,t\) are normally distributed noise with zero mean and standard deviation \(\sigma\); \(\tau\) is the ideal target value. Their experiments showed that multilayer optical coating designs found by the evolutionary algorithm were substantially more robust to parameter variations than the reference design, but usually at the cost of reduced performance in undisturbed situations. This observation motivated the later work of using an evolutionary multi-objective approach to figure out the trade-off map between robustness and optimal functional performance [26; 27; 28]. [PERSON] [26] introduced an interval computation method to avoid artificial insertion of Gaussian noise to parameter variables in order to build tolerance against internal or external perturbations. In this approach, each design variable is represented as an interval. The robustness of a candidate solution is defined as the signal-to-noise ratio \(S/N\), which considers both target performance and variations in face of perturbations. As a minimization criterion, they used  
\[S/N=\eta_{s}=-10\log(\frac{\sum\limits_{i=1}^{n}{y_{i}^{2}}}{n})\]  
and for maximization criterion,  
\[S/N=\eta_{l}=-10\log(\frac{\sum\limits_{i=1}^{n}{1/y_{i}^{2}}}{n})\]  
where \(n\) is the number of simulations, and \(y_{i}\) is the performance of the \(i\)-th simulation. This robustness measure ensures that either improvement in the mean result or a reduction in variability will lead to greater \(S/N\) values. They then used a multi-objective GA to evolve a solution with multiple robustness criteria using a design-of-experiments (DOE) method with orthogonal arrays to evaluate the fitness. In a more explicit way of addressing the balance of performance and robustness, [PERSON] [27] expressed the robust design problem as a three-criterion multi-objective problem, simultaneously optimizing an individual's performance without perturbation, the mean performance of its neighbors resulting from perturbations, and the standard deviation of its neighbors' performances. The perturbations in their experiments are simply modeled by deviations of \(\pm 1\%\) beyond the variable values. They also observed a degradation in performance of the evolved robust solutions. In the most existing evolutionary robust design, multiple simulations are needed to evaluate the robustness of a single individual, which severely increases the computational burden. To address this problem, [PERSON] et al. [28] used two methods for estimating the robustness measures of an individual--by exploiting its neighbor individuals in the current population as first proposed by [PERSON] [29] and by using all evaluated individuals. They then evolved robust solutions with both performance and robustness as criteria in a two-criterion optimization as suggested by [PERSON] [30]. [PERSON]'s robustness estimation approach can greatly reduce the number of function evaluations, when it is applicable. However, it is difficult to apply this method for evolving robust designs with variable structures as in topologically open-ended automated synthesis using GP. In these cases, it is difficult to define a neighborhood for a given individual, since most other individuals have different numbers and types of components. A related research area to robust design focuses on evolving robustness in a noisy environment [31; 32; 33]. In these problems, the variation in the environment leads to uncertainty in the fitness function evaluation and the true fitness of a candidate solution needs to be evaluated based on sampling in multiple environments. In the evolutionary robotics area, [PERSON] et al. [34] and [PERSON] [23] evolved robust mobile robot controllers by training them in multiple trials of simulation, using genetic programming and a genetic algorithm, respectively.