Although I know that analytic functions are infinitely differentiable and the taylor series converges to the function but I want to know why they are so useful from the perspective of robust control theory? Can a non-analytic function perform as well as an analytic function?
Dear Biswajit,
I suggest you to see links and attached files in topic.
-Analytic Functions I/Complex Analysis | SMC
https://www.math-stockholm.se/.../analytic-functions-1-1.337573 -WHY REAL ANALYTICITY IS IMPORTANT IN CONTROL THEORY HJ ...
citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50...
- Perspectives in Control Theory: Proceedings of the Sielpia ...
https://books.google.dz/books?isbn=1475721056
- Interpolation and Sampling in Spaces of Analytic Functions
https://books.google.dz/books?isbn=0821835548
Best regards
In nonlinear control, when the focus is on the performance of the system around an equilibrium point, we may want to design suboptimal controllers that match the optimal design in terms of the Taylor series expansion up to arbitrary order of precision. Matching high order terms in the optimal design is useful in its own right. But, one may wonder whether by matching more and more terms of the optimal control, does the suboptimal controller ever converge to the optimal design. That question's answer requires that the optimal design is an analytic function to begin with. Then, the suboptimal controller, if properly designed, will converge to the optimal design (at least locally around the equilibrium point) by matching higher and higher order control terms. Some relevant papers in this topic is
@article{luk69,
Author = {Lukes, D.~L.},
Journal = {SIAM Journal on Control},
Keywords = {Nonlinear control systems;Optimal control;Taylor series expansion},
Month = {Febuary},
Number = {1},
Pages = {75--100},
Title = {Optimal regulation of nonlinear dynamical systems},
Volume = {7},
Year = {1969}}
@article{panezakrekok2001b,
Author = {Pan, Zigang and Ezal, Kenan and Krener, Arthur J. and Kokotovi{\'c}, Petar V.},
Journal = {IEEE Transactions on Automatic Control},
Month = {July},
Number = {7},
Pages = {1014--1027},
Title = {Backstepping design with local optimality matching},
Volume = {46},
Year = {2001}}
The second paper above had a mistake which was corrected in an RG posting with the mark up on the copy of this paper, which is attached below.
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