of the uncertain models to be included in the robust analysis. This is an advantage over current norm-bounded robust control methods, which yield only the worst-case performance of the uncertain system as the answer to the robust analysis question, with no information as to how likely the worst case is to occur in practice. Hence, the probability-based approach adds an "extra dimension" to the robust analysis, where the extra dimension is the probability over the uncertain model set and the probable (rather than worst-case) performance measure.
A disadvantage of the probabilistic robust analysis method is that the total prob- ability integral over the uncertain model set must be performed, which can be very expensive computationally. An efficient asymptotic expansion method is used to approximate the integral that appears to work well for the problems that have been studied. The method is particularly effective for the problems studied in this work because the peak of the integrand is generally close to the most-probable model of the system, so the most-probable model is a good initial guess for the search for the maximum of the integrand, which is required for the asymptotic expansion. In Chapter 5, the asymptotic approximation is applied to the probabilistic robust anal- ysis of a system with 18 uncertain parameters (9 uncertain modal frequencies and damping ratios), and the computation time required is approximately an hour (run using MATLAB on a DEC/ Alpha workstation). This computation time is reasonable for an analysis problem of this size, but it does preclude optimizing the performance for robust control design.
The total reliability (or failure probability) of an uncertain system can be deter- mined from the methods outlined in Chapter 2, then used in the controller design discussed in Chapter 3. Using the structural reliability as a performance objective for the controller design is desirable from a structural engineering perspective, as the system's reliability is often the quantity used to measure structural performance.
In Chapter 3, the post-data analysis technique that uses Bayes's theorem to up- date the probabilities of the models in the uncertain model set is useful for modifying the description of a system's performance in the presence of new information from the system. This is important in earthquake engineering, as the actual behavior of
a structure during an earthquake is often quite different from that predicted by an initial finite element model of the system. The initial probabilities for the model uncertainty class are generally not sharply peaked near the most probable model of the system (which corresponds to the best pre-data model). Earthquake response data from a structure can be used to modify the "best" model and to make the probability distribution for the uncertain model class more highly peaked around the most-probable (post-data) model.
The post-data analysis of the probable robust performance is applied to the Caltech Flexible Structure in Chapter 5, where large differences are observed in the performance of the controllers designed for the post-data model relative to the pre-data model. The poor performance of the controllers designed for the pre-data model would presumably be improved with robust controller design. Unfortunately, the probabilistic robust control optimization described in Chapter 3 is intractable for this system. Furthermore, the performance differences between the pre-data and post-data controllers provide a strong argument for updating the controller design when new data is available from the system.
6.2 Future work
This thesis has laid out the framework for a probabilistic approach to control of uncertain civil engineering structures. Several directions exist for possible future work on the probabilistic robust analysis and control design methodology.
The first is to consider application of the control design approach to passive or semi-active systems for vibration control. The analysis methodology remains the same, and all that would be required is to choose a controller class that corre- sponds to the particular application, then optimize over the design parameters for the passive or semi-active system.
In addition, alternative performance measures could be considered for the struc- ture, rather than the inter-story drift failure probability. For example, probability of instability for the uncertain system is of particular interest, as a controller design
that could lead to instability would be unacceptable.
One shortcoming of the approach that is presented herein is that the computa- tion time required to solve the design problem for the uncertain-model controller is considerable. An area for further research is to explore whether an alternative formulation of the probabilistic analysis problem or other probability-based per- formance objectives and uncertainty descriptions can be found that would lead to more efficient solutions for both the analysis and control design. As experience is gained with the probabilistic robust analysis approach, computational procedures that exploit the structure of particular problems are likely to be developed.
For the Caltech Flexible Structure, several analysis and control design issues could be explored more fully under the probabilistic robust control framework. The first is to improve the modal identification method to obtain better models for the system. One idea is to use a modal identification algorithm that incorporates a physical basis for the modeshape components and modal participation factors for the closely spaced flexural modes. Then, uncertainty in the modeshape components and modal participation factors could be included more easily. Also, the effect of uncertainty in the model for the Gaussian process used to describe the modeling error and sensor noise should be studied for the structure and for other systems.
Another interesting task for the flexible structure would be to use the proof-mass actuators for control. These actuators are analogous to active mass driver actuators, which are common in civil engineering structural control applications. The proof mass actuators have more internal dynamics in the frequency range of the structural modes than do the voice coil actuators, which would provide additional control design challenges. A final task for the flexible structure (and other complicated systems) is to explore methods for making the probabilistic robust control design more tractable.