Disturbance Attenuation for Guidance and Control

The formulation of a disturbance attenuation problem provides a conceptual structure for the development of robust high performance guidance and control schemes in the presence of large uncertainties in the process and measurement disturbances. The disturbance attenuation problem is to find a feedback controller based upon the measurement output which guarantees that the disturbance attenuation function is bounded below some value for all admissible input disturbances. The disturbance attenuation function is an input-output relationship between a norm of a desired output, such as tracking error and/or control effort, to a norm associated with the input disturbance. This problem is usually solved by converting it to a zero-sum differential game between the controller, attempting to minimize a functional, and the disturbances acting as adversaries that maximize the performance index. By using a dynamic programming approach to differential game associated with the disturbance attenuation problem can be divided naturally into two problems. From the present time into the future and up to the terminal time, a game problem based upon perfect information naturally occurs since no actual measurements can be made, i.e., the system is causal. The solution to the dynamic programming problem is the optimal value function in terms of the state variables and time. This optimal value function is, in general, not easily generated, but in certain guidance problems it can be obtained approximately by using perturbation theory, producing solutions of high accuracy. The second half of the problem, from the initial time to the current time, is essentially viewed as a state estimation problem since the control sequence used in the past cannot be modified. The estimation problem can be viewed as dissipative with respect to a generalized energy function. An optimal accumulation function, which is the solution to the estimation dynamic programming problem, is a function of the current state and time. Some suggestions for determining suboptimal estimators are given. The final step is to maximize, with respect to the current state, the sum of the filter optimal accumulation function and the controller optimal value function. The resulting worst case state, which in the applications considered is a function of the state estimate and curvature, is used in the controller. Note that even though the worst case state is only a function of the measurement history and apriori values of the parameters, no certainty equivalence assumption is made.