Innovation of new Nussbaum functions: adaptive control with unknown direction and time-varying impact


Adaptive control systems have been proved as substantial techniques for enhancing various operations in industrial settings. Technically, adaptive control involves the use of a controller that can modify the operation of the system based on the estimation of unknown parameters to achieve the best and efficient mode of operation. It detects the behaviors of the dynamic system and adjusts the controller characteristics to automatically compensate for the estimation errors. Presently, some control systems have an unknown sign of control coefficients, therefore, making it hard to estimate the control coefficients and design an effective controller as this may result in wrong direction which may shift the system from the desired behavior.

To minimize the system degradation, several approaches including increasing the controller gain by deliberately changing the sign of the controller force have been employed. However, handling of unknown control of directions has remained quite challenging.

Recent work has shown the significance of using the Nussbaum gain technique for increasing the controller gain, especially for unknown control directions. However, additional fundamental tools including the use of lemma tools and Lyapunov functions for analyzing the bounded derivatives are necessary. This works perfectly well with systems with appropriately designed controllers. The formulation lemma functions have become more popular amongst scientists. For instance, they have been used to describe the role of Nussbaum function in analyzing the boundedness of Lyapunov functions. Owing to the recent increase in the systems with unknown time-varying control coefficients, the efficiency of using lemma techniques in solving the boundedness of Lyapunov-like functions has been questioned.

Professor Zhiyong Chen from The University of Newcastle explored the feasibility of using the general Nussbaum functions in time-varying or multivariable unknown control systems. In particular, the author opposed the instincts base on an example. Therefore, a new improved version of Nussbaum function was developed and its main characteristics suitable for time-varying and multivariable unknown control coefficients examined. However, the analysis was not only based on the explicit calculation of a particular function but also a general class of functions were equally considered. Eventually, he cross-examined the necessity of using the new version of Nussbaum function for adaptive controller design. The work is currently published in the research journal, Automatica.

Professor Zhiyong Chen opposed the assumption that the general Nussbaum function were suitable for both time-varying and multivariable control coefficients with unknown signs. As a prove, two new types of Nussbaum functions were introduced and their significant characteristics for the various scenarios in adaptive control system presented. Type A Nussbaum functions were noted to be unsuitable for multivariable and time-varying control coefficients with unknown signs while type B, on the other hand, proved suitable for the aforementioned scenarios. This was attributed to the geometrically increasing rate observed in type B as compared to the arithmetically one observed in type A. Additionally, the Nussbaum function provided the desired control gain based on the changing sign and growing magnitude. Therefore, application of the new Nussbaum functions will advance adaptive control problems in various systems.


“The control coefficient varies between 1.9 and 2.1 (left frame) in a deliberately designed way that can fool a traditional Nussbaum function (called type A in the paper). A counter example shows that a Nussbaum function (type A) fails in handling time-varying control coefficients. A new Nussbaum function (called type B) is able to do deal with this case, with which the control gain is successfully aligned with the correct control direction (a negative gain is aligned with a positive control coefficient in a negative feedback stabilization scenario) after a couple of iterations of searching (right frame). The control coefficient is unknown to the gain designer. ”

About the author

Zhiyong CHEN received the B.E. degree from the University of Science and Technology of China, and the M.Phil. and Ph.D. degrees from the Chinese University of Hong Kong, in 2000, 2002 and 2005, respectively. He worked as a research associate at the University of Virginia during 2005-2006. He joined the University of Newcastle, Australia, in 2006, where he is currently a professor. He is also a Changjiang chair professor with Central South University, Changsha, China.

His research interests include nonlinear systems and control, biological systems, and networked systems. He is/was an associate editor of Automatica, IEEE Transactions on Automatic Control and IEEE Transactions on Cybernetics.

E-mail: [email protected].


Chen, Z. (2019). Nussbaum functions in adaptive control with time-varying unknown control coefficientsAutomatica, 102, 72-79.

Go To Automatica

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