I have 5th order nonlinear system of relative degree 5. That is, for feedback linearization this system is completely feedback linearizable where the input-output feedback linearization is equivalent to input-state feedback linearization.
Can any one suggest me relevant theory behind Asymptotic Output tracking problem to track a desired step-signal for this system which is based on feedback linearization?
Furthermore, it will be helpful enough if this theory of Asymptotic Output Regulation problem can be extended to the case where only the output(assume, the first state) is only available for measurement.
Hi Anirudh Nath,
Feedback linearization fundamentals are Lie Algebra, and Asymptotic Output tracking can be analyzed via Lyapunov Theory and Advanced Stability Theory. Both subjects are well explained in the following references:
Applied Nonlinear Control
Jean-Jacques Slotine, Weiping Li
ISBN-13: 978-0130408907
ISBN-10: 0130408905
Nonlinear Systems Paperback
Hassan K. Khalil
ISBN-10: 9332542031
ISBN-13: 978-9332542037
Regards,
Alfonso Gómez
Hello Anirudh Nath,
As mentioned, you are able to achieve the complete linearization of your 5th order system, since the order is equal to the relative degree of the system with the selected output. In such case, you can define an input/control signal such that all the nonlinearities are cancelled and the resulting output dynamics (which, in this case, is the whole closed-loop dynamics) turns out to be linear and time-invariant. Therefore, to ensure that the output trajectories converge to the origin, you are able to apply the available theory for linear systems. That is, you can compute the characteristic polynomial of the constant matrix A, which characterize the output dynamics, and apply the Routh-Hurwitz criterion.
However, if you redefine the output signal such that the relative degree of the system is less than its order, then you have to analyse the resulting internal dynamics, in addition to the external dynamics (the output dynamics).
As the Dr. Gómez-Espinosa mentioned, the best books for the study of feedback linearization theory are the ones by Khalil, and Slotine and Li.
I also can recommend some of my papers, where you can find practical applications of the feedback linearization theory to 4th order systems, which are two-degrees-of-freedom underactuated mechanical systems. I'm sure that you can acquire some insight from them in order to solve your problem.
[1] Carlos Aguilar-Avelar and Javier Moreno-Valenzuela, "New Feedback Linearization-Based Control for Arm Trajectory Tracking of the Furuta Pendulum", IEEE/ASME Transactions on Mechatronics, vol. 21, no. 2, pp. 638-648, 2015.
[2] Carlos Aguilar-Avelar and Javier Moreno-Valenzuela, "A Feedback Linearization Controller for Trajectory Tracking of the Furuta Pendulum", Proc. of the 2014 American Control Conference, Portland, Jun. 2014, pp. 4543-4548.
[3] Carlos Aguilar-Avelar and Javier Moreno-Valenzuela, "A composite controller for trajectory tracking applied to the Furuta pendulum", ISA Transactions, Vol. 57, pp. 286-294, 2015.
Best regards,
Carlos Aguilar-Avelar.
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