Uncertainty modeling via frequency domain model validation

The majority of literature on robust control assumes that a design model is available and that the uncertainty model bounds the actual variations about the nominal model. However, methods for generating accurate design models have not received as much attention in the literature. The influence of the level of accuracy of the uncertainty model on closed loop performance has received even less attention. The research reported herein is an initial step in applying and extending the concept of model validation to the problem of obtaining practical uncertainty models for robust control analysis and design applications. An extension of model validation called 'sequential validation' is presented and applied to a simple spring–mass–damper system to establish the feasibility of the approach and demonstrate the benefits of the new developments. Introduction Robust control theory guarantees that a feedback control system can be designed that will maintain desired levels of stability and performance subject to modeling errors and uncertainties. [1,2] There is, however, an underlying assumption that the uncertainty model used in the design effectively characterizes the differences between the responses of the true system and the nominal design model. This means that the family of responses associated with the design model contain the responses of the true system. It is impossible to conclusively prove that this assumption is satisfied for any real system. However, if there is sufficient knowledge of the response characteristics of the true system then a model can be generated that, subject to the available knowledge, characterizes the possible range of responses that can be produced by the system. The goal is to systematically generate such a model with the added * Senior Research Engineer. Senior Member AIAA. † Associate Professor, School of Aeronautics and Astronautics. Senior Member AIAA. Copyright © 1999 by the American Institute of Aeronautics and Astronautics, Inc. No copyright i s asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. property that it be accomplished with minimal conservatism. That is, the model should characterize responses of the real system that are possible but should not characterize responses that are not possible. Of course, it is impossible to conclusively prove this as well. However, if the available knowledge is comprehensive enough it is possible to achieve this goal within some qualitative level of confidence. Model Validation A concept called "model validation" has recently been developed to attack this problem. [3–8] The idea behind model validation is that given input and output data for a system that is otherwise unknown, a model can be generated that driven by the same input (and possibly some additional inputs) can exactly reproduce the output of the true system. The frequency domain version of this statement is depicted in block diagram form in Figure 1. The input to the true system u s ( ) produces the output of the true system y s ( ). The same input drives the model along with an external disturbance w s ( ) . Another external input v s ( ) is added to the response of the model to produce the validation output ( ) y s . The external inputs are included because the true system output includes artifacts of the method(s) by which it is obtained including external disturbances, estimation errors, and quantization effects. The external inputs provide a mechanism to account for these effects. Validation is achieved by choosing the model and the external inputs w s ( ) and v s ( ) so that the difference between the validation output and the true system output e s ( ) is identically zero for the available input/output data. (The Laplace variable s will be omitted henceforth for ease of discussion.)