Varaiya, Pravin

University of California Berkeley, Berkeley, CA, United States

A key problem of control theory is to construct feedback control laws based on available data and to determine how the level of uncertainty in the model and in the information arriving on-line, through noisy measurements, would affect the system performance. In the absence of statistical description such problems of Measurement Output Feedback Control (MOFC) were investigated mostly within the H(inf) setting, with soft-type integral costs, while problems with hard bounds on the unknowns were less developed. The present proposal is aimed to produce a fairly complete investigation of the problem of output feedback control constructed through available measurements under unknown but bounded disturbances subject to hard bounds on the uncertain items. The proposal covers topics from basic theoretical problems to computational methods. The novelty of the suggested solution schemes, applicable to nonlinear systems, with greater specifics for the linear case, lies in the combination of dynamic programming, set-valued analysis and minmax approaches. It is well understood that the overall problem under consideration is a combination of two ? a finite-dimensional problem of guaranteed state estimation and an infinite-dimensional problem of feedback control under set-membership uncertainty. The second problem is especially difficult to formalize and solve. In the proposed solution, based on earlier work, the aim is to reduce the second problem to a finite-dimensional one, which would facilitate calculation. For systems with linear structure and convex constraints the computational procedure is to be based on using and also generalizing the ellipsoidal calculus which proved effective for many problems and allows development of complementary software. It is expected that such approach will produce solutions "to the end," with illuminating examples. For the nonlinear case calculations may be facilitated by using the specifics of the problem and applying modifications of the earlier suggested comparison principles that allow one to relax the original equations or variational inequalities of the Hamilton-Jacobi type through simpler, finite-dimensional relations.

The Measurement Output Feedback Control (MOFC) problem has been thoroughly studied in a stochastic setting as a combination of stochastic filtering theory within the theory of stochastic control. However a considerable number of problems in control design have to deal with systems subjected to information conditions that are other than stochastic. Such problems of MOFC are increasingly motivated by applied issues, given the progress in design of high-tech complex systems arising in automation, navigation and the cyber-physical field (including hybrid, impulsive, time-lag, multi-agent, communication-oriented processes and the like). They naturally require new techniques, new types of models and their mathematical formalization, as well as new numerical methods, algorithms and software. The present project is designed in response to the indicated demand.

This research produced a complete theory for the problem of output feedback control based on available measurements under unknown but bounded disturbances subjected to hard bounds on the uncertain items. The research results range from basic theoretical problems to computational methods. The novelty of the derived solution schemes lies in the combination of Hamiltonian methods in the form dynamic programming with set-valued analysis and minmax approaches. The overall general solution for the considered problem is a combination of the solutions of a finite-dimensional problem of guaranteed state estimation and an infinite- dimensional problem of feedback control under set-membership uncertainty. For the first problem new types of set-valued observers were introduced. For the second problem which is especially difficult to formalize and solve, the achieved solution is reduced to a finite- dimensional one, which facilitates calculation. For systems with linear structure and convex constraints the solution is more detailed. Here the computational procedure is based on new developments in ellipsoidal calculus, which proved to be effective and allowed to design complemental software, producing solutions o the end", with examples including out put feedback for impulse control. For the nonlinear case the calculations are facilitated by using the specifics of the problem in combination with comparison principles that allow to relax the original equations or variational inequalities of the Hamilton-Jacobi type through simpler, nite-dimensional relations. The broader impact of this research indicates a class of important applied problems solvable to the end through mathematics of control; improves existing computational methods for feedback control under complete and incomplete information and set-membership uncertainty; indicates new approaches to specific classes of systems (bilinear, with sampled measure- ments, impulse, etc.) and new techniques for existing approaches like model-predictive control; triggers new development of observer-based control in complex systems (hybrid, time-lag, communication, multiagent, etc), which typically operate under incomplete feedback and set-membership uncertainty.

- Agency
- National Science Foundation (NSF)
- Institute
- Division of Mathematical Sciences (DMS)
- Application #
- 0807771
- Program Officer
- Mary Ann Horn

- Project Start
- Project End
- Budget Start
- 2008-09-01
- Budget End
- 2011-08-31
- Support Year
- Fiscal Year
- 2008
- Total Cost
- $200,000
- Indirect Cost

- Name
- University of California Berkeley
- Department
- Type
- DUNS #

- City
- Berkeley
- State
- CA
- Country
- United States
- Zip Code
- 94704