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In many practical cases the theoretical conditions required by the “one-shot” solutions are not met. Moreover, operational constraints of the process often require that only small adjustments to existing parameter settings can be made, which precludes the application of VRFT in these situations. In such cases the data-driven control design must be performed through iterative procedures in which each iteration requires collecting more data, each time with a different controller in the loop. In Chap. 4 a general review of basic optimization theory is given, setting the stage for the chapters to follow. The basic convergence properties of the basic optimization algorithms—steepest descent and Newton-Raphson in particular—are analyzed. Some robustness properties, that is, convergence under imprecise information, of these algorithms are also demonstrated.
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Of course similar definitions can be made for maxima.
Note that the Hessian is symmetric and thus all its eigenvalues are real.
This second constraint is not present when the algorithm is an autonomous system.
For a more complete proof the reader is referred to standard optimization books. Such proofs get somewhat technical, so we prefer to give here only a sketch that provides insight into the convergence mechanisms.
D. Eckhard, A.S. Bazanella, Robust convergence of the steepest descent method for data-based control. Int. J. Syst. Sci. (2011 in press). http://www.informaworld.com/10.1080/00207721.2011.563874
D. Eckhard, A.S. Bazanella, Data-based controller tuning: Improving the convergence rate, in Decision and Control (CDC), 2010 49th IEEE Conference on, (2010), pp. 4801–4806 CrossRef
M. Vidyasagar, Nonlinear Systems Analysis, 2nd edn. (Prentice Hall, New York, 1992) MATH
- Iterative Optimization
Alexandre Sanfelice Bazanella
- Springer Netherlands
- Chapter 4