A bimatrix co-evolution algorithm is proposed to solve the parameter robust control problem subject to design requirements. The parameter robust control problem is to design the controller having the robust performance against the worst, or the most performance degrading, uncertainties. This is often expressed as a game between the controller and uncertain parameters and the game is formulated as a minimax problem whose solution is known as a saddle-point solution. It is difficult to obtain the saddle-point solution by using conventional methods but if the solution is once secured, the resulting controller made from the solution always guarantees the robustness against bounded parameter uncertainties. In addition, when there are various kinds of design requirements such as parameter limits or condition functions, it is expressed as a constrained minimax problem, which is extremely hard to be solved using existing optimization methods.
In this dissertation, a bimatrix co-evolution algorithm is proposed as a new approach addressing the constrained minimax problem by combining a bimatrix game concept and the co-evolution algorithm. In this approach, each of two players, the controller and the uncertain parameters, has an independent fitness measure which is composed of a performance term and a penalty term derived from the constraints. Because the constraints should be satisfied by the robust solution or the optimized parameters, both of the players cooperate to minimize the penalty terms while they fight for the performance trade-off. These two contrary actions of two players are conducted simultaneously in the bimatrix game, in which each of two players has a fitness measure matrix calculated by its population. In general, the robust solution can not be obtained at the first bimatrix game. The proposed algorithm finds the robust solution by using the parallel evolutions of two populations. Each of two evolving populations has an independent evolution process except the fitness measure process where the bimatrix game is conducted. Then, two populations converge to the robust solution through successive bimatrix games.
In the proposed algorithm, the cooperative Stackelberg game method is adopted, in which the optimizing parameter is the leader who determines its strategy first and the uncertainty is the follower who reacts to the leader`s decision. Moreover, due to the prematuring nature of Evolution Algorithms, the proposed.
algorithm may not converge to the solution which satisfies the given constraints for every possible uncertainty deviation. To exclude this possibility, some random populations of the uncertainty are generated and distributed over the given uncertainty range at every generation. These secondary populations are quite effective to find the robust solution satisfying all constraints. This is verified by solving a simple constrained minimax problem.
The proposed algorithm is applied to the robust control design benchmark problem. The benchmark plant is a two-mass-spring system with noncollocated sensor and actuator. The performance of the proposed algorithm is compared with 10 solutions based on analytic approaches such as $H_∞$, nonlinear constrained optimization, and so on. The bimatrix coevolution algorithm results a satisfactory robustness of performance and stability.
In order to show more practical application, the proposed algorithm is applied to the robust attitude control design of a flexible launch vehicle with design constraints, which is a representative parameter robust control problem as the structural elasticity makes the vehicle easily unstable by small change of parameters. The equations of motion in pitch plane are derived and a linearized state equation containing 3rd-order bending mode is formulated. A lower-order classical type controller composed of PID and notch filter is used and optimized by using the proposed algorithm. The numerical results show that it can find the robust controller and handle design requirements without difficulty.
