## 4 Applications in Optimization
### Particle Swarm Optimization (PSO)
#### 4.1.1 Thermal Vacuum System  
In [55], the PSO approach was intertwined with Lozi map chaotic sequences to obtain a Takagi-Sugeno (TS) fuzzy model for representing dynamical behaviors. This proposed chaotic PSO (CPSO) approach was employed to optimize the premises in the IF-THEN rules of the TS fuzzy model, and were utilized in an experimental application, specifically, a thermal vacuum system employed for space environmental emulation and satellite qualification. The essential idea of this model is to compute a global nonlinear model by partitioning the input space into fuzzy areas and approximating each area through a linear model. The TS models consist of linguistic IF-THEN rules that can be represented by the following general form:  
\[\left\{ \begin{array}{l}R^{(j)}:IF\left(z_{1}\;is\;A_{1}^{j}\right)\;AND\; \cdots\;AND\;(z_{m}\;is\;A_{j}^{m}),\\ THEN\;y_{j}=b_{0}^{j}+b_{1}^{j}x_{1}^{j}+b_{q_{j}}^{j}x_{q_{j}}^{j}.\end{array} \right. \tag{25}\]  
The IF statements define the premise part, while the THEN functions constitute the subsequent part of the fuzzy system; here, \(A_{i}^{j}\) represents the labels of the fuzzy sets and \(z=[z_{1},z_{2},\ldots,z_{m}]^{T}\) with \(i=1,\ldots,m\) is the input vector of premise \(p\). PSO is an optimization tool based on a population in which the position of each member particle is a potential solution to an analyzed problem. Each particle is associated with a randomized velocity that moves throughout the problem space. The coordinates of each particle are tracked in the problem space, and are associated with the best solution (fitness) achieved thus far. This value is called _pbest_. Another _'best'_ value that is tracked by the global version of the particle swarm optimizer is the overall best value. Its location, called _gbest_, can be obtained by any particle in the population. The past best position and the entire best overall position of the group are employed to minimize (maximize) the solution. At each time step, the PSO algorithm changes the velocity (acceleration) of each particle moving toward its _pbest_ and _gbest_ locations. Acceleration is weighted by random terms, with separate random numbers being generated for acceleration toward the _pbest_ and _gbest_ locations (see Figure 8). Araujo and Coelho defined the following three CPSO approaches based on the Lozi map, considering that the velocity equation of each particle is updated at each step \(t\) by  
\[v_{i}(t+1)=K\big{[}v_{i}(t)+c_{1}ud_{i}(t)(p_{i}(t)-x_{i}(t))+c_{2}ud_{i}(t)(p _{g}(t)-x_{i}(t))\big{]}. \tag{26}\]  
Figure 8: Geometric core of Particle Swarm Optimization (PSO) algorithm. _Approach 1_ (CPSO1): the parameter \(ud_{i}(t)\) is modified to  
\[v_{i}(t+1)=K\big{[}v_{i}(t)+c_{1}Y_{i}(t)(p_{i}(t)-x_{i}(t))+c_{2}ud_{i}(t)(p_{g} (t)-x_{i}(t))\big{]}. \tag{27}\]  
_Approach 2_ (CPSO2): the parameter \(ud_{i}(t)\) is modified to  
\[v_{i}(t+1)=K\big{[}v_{i}(t)+c_{1}ud_{i}(t)(p_{i}(t)-x_{i}(t))+c_{2}Y_{i}(t)(p_{ g}(t)-x_{i}(t))\big{]}. \tag{28}\]  
_Approach 3_ (CPSO3): the parameter \(ud_{i}(t)\) is modified to  
\[v_{i}(t+1)=K\Big{[}v_{i}(t)+c_{1}Y_{i}(t)(p_{i}(t)-x_{i}(t))+c_{2}Y_{i}^{ \prime}(t)(p_{g}(t)-x_{i}(t))\Big{]}, \tag{29}\]  
where \(Y_{i}(t)\) are obtained from the chaotic attractor (3) with the parameter values \(a=1.7\) and \(b=0.5\) by scaling the variable \(y\in[-0.6418,0.6716]\) within 0 and 1 at each iteration, as follows:  
\[Y_{i}(t)=\frac{y_{i}(t)-\sigma}{\delta-\sigma}, \tag{30}\]  
with \([\sigma,\delta]=(-0.6418,0.6716)\). The \(Y_{i}^{\prime}(t)\) are obtained in the same way as \(Y_{i}(t)\), though with different initial values in (3). The constriction coefficient \(K\) is usually set to \(K=0.729\). After their launch and during their operational life, satellites are exposed to difficult physical conditions due to sunshine, albedo, earth radiation, and earthshine infrared in alternative shadow/eclipse conditions.