Current Projects
Project 1: Robust adaptive control
My main goal is to study robust adaptive control theory in detail, give it a solid foundation, solve for MIMO systems, and generalize to nonlinear and possibly infinite-dimensional systems. Currently, I am advancing toward this goal from many fronts.
First of all, it is envisioned that the approach adopted by the project works on finite-dimensional minimum phase linear systems (necessary and sufficient condition). The characterization of minimum phase linear systems is studied in the conference paper by me and Prof. Tamer Başar that appeared in 57th IEEE Conference on Decision and Control. Minimum phase property for multiple-input and multiple-output linear systems have also be obtained in the conference paper by Prof. Tamer Başar and me in the IFAC World Congress 2020. These papers, together with the vectorized version of the SISO adaptive control paper linadaptpub.pdf, then solves the robust adaptive control problem for finite-dimensional continuous-time square MIMO linear time invariant system completely in the paper MIMOlinadapt-IJACSP-2023-08-21.pdf (which had appeared in the International Journal of Adaptive Control and Signal Processing, October, 2023.) The companion paper of robust adaptive control for MIMO LTI systems with uniform vector relative degree of zero has been published in Automatica MIMOlinadapt0rd.pdf. There was some concern that the bounds obtained in the papers are only applicable for finitely many unknown systems. We have looked into the concern and found that the definition of minimum phase has to be modified so the bounds obtained in the papers are applicable to a continuum of unknown systems as prescribed in the paper. This modification is completed and results are reported in the following working papers: MinPhaseMIMO-IFAC2020-Update.pdf, MinPhase2ACC-Update.pdf, MinPhaseICS-Update.pdf, MinPhaseMIMO2-Update.pdf, where all results have been upgraded to MIMO LTI systems as well. The robust adaptive control papers (the IJACSP paper and the Automatica paper) are correct as they are.
A new result on parallel interconnected linear systems where each one of the subsystem admit uniform vector relative degree and uniform observability indices (and thus is robust adaptive control design ready) but the subsystems do not share a uniform vector relative degree is studied, and the extended zero dynamics canonical form for the composite system is obtained. This result is summarized in the working paper EZDCF-MIMO-PI-IFAC2026.pdf. (Also has been updated). Another result is on the minimum phase property for MIMO LTI systems in tandem with an integration block on the output of the system, which is summarized in the working paper MIMOMinPhaseICS1.pdf. (Also has been updated). Thus, if one can show that the composite system is minimum phase, then robust adaptive control design can be performed on the composite system with nonuniform vector relative degrees. This work is ongoing, and the only thing left to be completed is an example. For some reason, Mathematica has been refusing to do the computation of the controller, which requires 20GB’s RAM that is plenty on my computer.
As a precursor to inverse optimal adaptive control designs for MIMO LTI systems, a paper has been published by Dynamic Games and Applications, that solves the H-infinity optimal control problem under imperfect state measurements in an estimator and controller design sequential manner. It is shown that this sequential design method recovers all of the well-known results of the problem and further establishes that the H-infinity control problem is solvable only if it is solvable when the desired performance level is infinite. This suggests that the robust adaptive control design methodology that we have been working on is optimal. Comments are very welcome.
Project 2: Teaching notes development
To set the foundation for robust adaptive control, reading notes and teaching notes are input into latex documents and revised. The book “Measure-Theoretic Calculus in Abstract Spaces: On the Playground of Infinite-Dimensional Spaces” has been published by Birkhäuser, an imprint of Springer-Nature in January 2024. Comments are very welcome. This book is the Holy Grail in analysis and now it is available for the public.
I have begun to write a reading note on “Nonlinear Systems”. The note is tentatively titled “Nonlinear and Inverse Optimal Systems – a game theoretic view point.” Currently, it contains a complete proof of my 2001 IEEE Transaction on Automatic Control paper coauthored with Ezal, Krener, and Kokotović. I am extending the result to MIMO systems in block lower triangular form. This extension will broaden the applicable systems for the approach. The simulation results are very encouraging. This book is under development.
Based on my book “Measure-Theoretic Calculus in Abstract Spaces: on the playground of infinite-dimensional spaces”, my earlier research for linear stochastic system seemed to be correct, but those for nonlinear stochastic systems are missing key technical assumptions, and therefore considered not correct. Those results for nonlinear stochastic systems can only serve as pointers to what one might achieve for those systems.
