What Is Optimization Toolbox?

fgoalattainExamplesConsider a linear system of differential equations.An output feedback controller, K, is designed producing a closed loopsystemThe eigenvalues of the closed loop system are determined from thematrices A, B, C, andK using the command eig(A+B*K*C). Closedloopeigenvalues must lie on the real axis in the complex plane to the left ofthe points [-5,-3,-1]. In order not to saturate the inputs, no elementin K can be greater than 4 or be less than -4.The system is a two-input, two-output, open loop, unstable system,with state-space matrices.The set of goal values for the closed loop eigenvalues is initialized asgoal = [-5,-3,-1];To ensure the same percentage of under- or overattainment in theactive objectives at the solution, the weighting matrix, weight, issettoabs(goal).Starting with a controller, K = [-1,-1; -1,-1], first write an M-file,eigfun.m.function F = eigfun(K,A,B,C)F = sort(eig(A+B*K*C)); % Evaluate objectivesNext, enter system matrices and invoke an optimization routine.A = [-0.5 0 0; 0 -2 10; 0 1 -2];B = [1 0; -2 2; 0 1];8-23