genetic algorithm maintains a population of strings that each represent a different controller(digits on the strings characterize parameters of the controller ), and it uses a fitness measure that characterizes the closed-loop specifications Suppose, for instance, that the closed-loop specifications indicate that you want, for a step input, a(stable)response with a rise-time of t a percent overshoot of m and a settling time of t we need to define the fitness function so that it ires how close each individual in the population at time k(i.e, each controller candidate) is to meeting these specifications Suppose that we let tr Mp, and Is, denote the rise-time, overshoot, and settling time, respectively, for a given individual(we compute these for an individual in the population by performing a simulation of the closed-loop system with the candidate controller and a model of the plant ). Given these values, we let (for each individual and every time step k) 7=(1-1)+12(Mn-M2)+(1-2) where,i-1, 2, 3, are positive weighting factors. The function J characterizes how well the candidate controller meets the closed-loop specifications where if J=0 it meets the specifications perfectly. The weighting factors can be used to prioritize the importance of meeting the various specifications(e.g, a high value of w2 relative to the others indicates that the percent overshoot specification is more important to meet than the others) Now, we would like to minimize J, but the genetic algorithm is a maximization routine. To minimize J with the genetic algorithm we can choose the fitness function where 8>0 is a small positive number Maximization of can only be achieved by minimization of J, so the desired effect is achieved. Another way to define the fitness function is to let J((k)=-/((k)+max{/(e(k) 0(k) The minus sigh in front of the J(e(k)) term turns the minimization problem into a maximization problem(to see this,consider J(0)=02, where 0 is a scalar, as an example). The maxon J(0(k))) term is needed to shift the function up so that J(e(k)) is always positive. We need it positive since in selection, Equation(4.7)defines a probability that must al ways be positive and between one and zero This completes the definition of how to use a genetic algorithm for computer-aided control system design. Note that the above approach depends in no way on whether the Controller that is evolved is a conventional controller(e.g, a PID controller)or a fuzzy system or neural network. For instance, you could use a Takagi-Sugeno fuzzy system or a standard fuzzy system for the controller and let the genetic algorithm tune the appropriate parameters. Moreover, we could take any of the controllers and parameterize them and use the above approach to tune these adaptive or supervisory controllers. We have used the genetic algorithm to tune direct, adaptive, and supervisory controllers for several applications, and while this approach is computationally intensive, and we did have to make some application-depender modifications to the above fitness evaluation approach, it did produce successful resul The above approach can also be used in system identification and for the construction of estimators and predictors The genetic algorithm can be used for the tuning of fuzzy system parameters that enter in a nonlinear fashion and can be used in conjunction with other methods. To use the genetic algorithm for tuning fuzzy systems as estimators, we could PDF文件使用" pdffactory Pro"试用版本创建ww. fineprint,com,cngenetic algorithm maintains a population of strings that each represent a different controller (digits on the strings characterize parameters of the controller), and it uses a fitness measure that characterizes the closed-loop specifications. Suppose, for instance, that the closed-loop specifications indicate that you want, for a step input, a (stable) response with a rise-time of * r t , a percent overshoot of * M p , and a settling time of * s t . We need to define the fitness function so that it measures how close each individual in the population at time k (i.e., each controller candidate) is to meeting these specifications. Suppose that we let tr , Mp, and ts , denote the rise-time, overshoot, and settling time, respectively, for a given individual (we compute these for an individual in the population by performing a simulation of the closed-loop system with the candidate controller and a model of the plant). Given these values, we let (for each individual and every time step k) * 2 * 2 * 2 1 2 3 ( ) ( ) ( ) r r p p s s J = w t -t + w M - M + - w t t where , i=1,2,3, are positive weighting factors. The function J characterizes how well the candidate controller meets the closed-loop specifications where if J = 0 it meets the specifications perfectly. The weighting factors can be used to prioritize the importance of meeting the various specifications (e.g., a high value of w2 relative to the others indicates that the percent overshoot specification is more important to meet than the others). Now, we would like to minimize J , but the genetic algorithm is a maximization routine. To minimize J with the genetic algorithm, we can choose the fitness function 1 J J e = + where e > 0 is a small positive number. Maximization of J can only be achieved by minimization of J , so the desired effect is achieved. Another way to define the fitness function is to let { } ( ) ( ( )) ( ( )) max ( ( )) k J k J k J k q q = - + q q The minus sigh in front of the J k (q ( )) term turns the minimization problem into a maximization problem (to see this, consider 2 J ( ) q q = , where θ is a scalar, as an example). The maxq ( ) k {J k (q ( ))} term is needed to shift the function up so that J k (q ( )) is always positive. We need it positive since in selection, Equation (4.7) defines a probability that must always be positive and between one and zero. This completes the definition of how to use a genetic algorithm for computer-aided control system design. Note that the above approach depends in no way on whether the Controller that is evolved is a conventional controller (e.g., a PID controller) or a fuzzy system or neural network. For instance, you could use a Takagi-Sugeno fuzzy system or a standard fuzzy system for the controller and let the genetic algorithm tune the appropriate parameters. Moreover, we could take any of the controllers and parameterize them and use the above approach to tune these adaptive or supervisory controllers. We have used the genetic algorithm to tune direct, adaptive, and supervisory controllers for several applications, and while this approach is computationally intensive, and we did have to make some application-dependent modifications to the above fitness evaluation approach, it did produce successful results. The above approach can also be used in system identification and for the construction of estimators and predictors. The genetic algorithm can be used for the tuning of fuzzy system parameters that enter in a nonlinear fashion and can be used in conjunction with other methods. To use the genetic algorithm for tuning fuzzy systems as estimators, we could PDF 文件使用 "pdfFactory Pro" 试用版本创建 www.fineprint.com.cn