TEST CASE

To verify the validity of using a genetic algorithm for H₂ optimal controller synthesis, a four disk model was studied. The four disk model represents an apparatus developed for the testing of pointing control systems for flexible space structures. A schematic of the four disk system is shown in Figure 3.3. The four disks are rigidly attached to a flexible axial shaft and control torque is applied to selected disks. The angular displacement of the disks are measured. The equations of motion are written as

or

where the generalized displacements are the angular displacements of the disks, q^T, and the input vector consists of the moments applied to each disk, u^T. Defining the state vector as results in the state-space formulation

where

Figure 3.3  Four disk system.

The implementation of the GA to this H₂ optimal problem will be used to study perturbation compensation for the four disk model. Therefore, the initial controller gain matrix will be known and the GA will optimize toward the control gain perturbations. Since the GA will only be solving for the control gain perturbations, the coding scheme can be simplified. The control gain perturbations can be seen as percent variations from the initial control gain matrix, which will reduce the size of the range of each parameter. A“window”of variation for the elements of the matrix are used, such as negative one percent to positive one percent, negative five percent to positive five percent, etc.

The coding scheme that will be used for this multiparameter optimization problem is a simple floating point coding scheme. The G matrix for the four disk model contains sixteen elements, so each parameter will be an individual element of the control gain perturbation matrix, with a range of the“window”pre-defined. Initially, the“window”of variation for the elements of the G matrix are set to a range of minus five percent to plus five percent. With the floating point scheme, each parameter is represented by a floating point number. A study of the“window”size that allows the genetic algorithm to optimize with both robustness and efficiency has been performed.

The objective function used in the genetic algorithm will contain the minimization of the cost function from Equation (3.15),

This equation is simply a cost function that involves the trace of the matrix in the brackets, which is the sum of the diagonal terms. Since the objective of the H₂ problem is to minimize the H₂ norm on the closed loop transfer function from disturbance inputs to performance inputs, the cost function will be minimized. This corresponds to minimizing the error associated with the disturbance inputs and performance outputs. Minimizing the trace of the product of matrices in the brackets allows the minimization of this error. In order to insure closed-loop stability, the eigenvalues of the matrix must all be negative. This knowledge will be incorporated into the objective function by penalizing the fitness, if any, of the eigenvalues are greater than or equal to zero. The fitness function used in the GA is:

where W equals zero if all of the eigenvalues of are negative, or W equals a positive value, if any, of the eigenvalues of are greater than or equal to zero. A reproduction scheme is used to select strings with high fitness values for future generations. The type of scheme that was used is called tournament selection, where the string with the lowest fitness value of each round of the simulated tournament is reproduced from a mating pool for the new population. There are popsize rounds to the tournament.

The genetic algorithm was applied to the H₂ optimal controller for three separate cases. The first is used to determine the performance of the genetic algorithm with an initial gain matrix that is deviated within the range of ± 5%. The second case incorporates a range of ± 10% and the third incorporates an extreme case with a range of ± 25%. The objective of the genetic algorithm is to obtain solutions that are near the exact gain matrix values, while requiring closed loop stability. The results of this study help determine the validity of genetic algorithms in this area of controller design.