Higher-order tracking properties of nonlinear adaptive control systems


Adaptive control systems can detect the variations in the functioning of the process to automatically modify the controlling parameters and compensate for the variations. In doing so, such systems achieve the best mode of operation suitable for the prevailing conditions. Typically, adaptive control is founded on parameter estimation and classified into different categories depending on the operation. Among these categories, the adaptive control of nonlinear systems has attracted significant research attention. With the help of linearization, differential geometry theory, and backstepping, various adaptive control challenges have been solved, enabling control designs for higher-order normal nonlinear systems with improved output tracking performance and closed-loop system stabilization.

The higher-order convergence properties of adaptive systems are important for many practical systems. In particular, it enables smooth output tracking desirable for numerous practical applications. In aircraft design, for example, the knowledge of higher-order convergence of the tracking error is critical in preventing harmful aircraft oscillations. Nevertheless, despite the good progress, most literature research has concentrated on exploring the higher-order convergence properties for linear adaptive control systems with little attention to nonlinear adaptive control systems. Additionally, whereas the asymptotic output tracking properties have been demonstrated for nonlinear adaptive controls, research on their higher-order tracking property remains sparse.

To this note, Dr. Liyan Wen, Professor Gang Tao and Dr. Ge Song developed a feedback linearization design to investigate the higher-order tracking properties of nonlinear adaptive control systems. Their intention was to show that under normal system conditions, some higher-order derivatives of the output tracking error converge to zero without persistent excitation. The work is currently published in the research journal, Systems and Control Letters.

In their approach, the study was performed in a feedback linearization framework. First, the researchers proved and specified, for a MIMO nonlinear adaptive control system, the orders of certain time derivative components of output tracking error that converge to zero to assess their adaptive control performance. Next, the adaptive control of the MIMO nonlinear systems was extended by developing unmatched input disturbances to expand its disturbance rejection capacity and higher-order tracking capabilities. The additional properties of the adaptive nonlinear controls were identified and discussed. Finally, the feasibility of the newly established high-order tracking properties was validated through an illustrative simulation of a twin otter model control simulation.

Results showed that the error components and their kth-order time-derivative of the nonlinear system converged to zero asymptotically. The author found additional nonlinear adaptive control system properties that are of great importance for both theoretical research and practical applications. As demonstrated by the simulation results, the resulting system exhibited an improved disturbance rejection capacity and higher-order tracking capabilities. Additionally, a smooth output tracking performance was achieved.

In summary, the research team established the high-order tracking properties of multivariable nonlinear adaptive systems. The additional high-order tracking properties implied that the overall performance of the adaptive control systems was much higher than expected. Moreover, extending the MIMO nonlinear system control using the unmatched input disturbances significantly enhanced the disturbance rejection capacity and high-order tracking performance of the system. The convergence analysis simulations of the twin aircraft illustrated not only the feasibility of the approach but also its engineering significance. In a statement to Advances in Engineering, the authors stated that the convergence analysis would benefit both theoretical research and practical applications.


Wen, L., Tao, G., & Song, G. (2020). Higher-order tracking properties of nonlinear adaptive control systemsSystems & Control Letters, 145, 1-8.

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