On the differentiation of a composite function with a generalized vector argument on homogeneous time scales; pp. 309–322Full article in PDF format | https://doi.org/10.3176/proc.2017.3.04
The paper proves a theorem on the differentiation of a composite function with a generalized vector argument. The theorem is formulated in terms of the delta derivative, which in the case of homogeneous time scales incorporates both the ordinary derivative and the difference operator. The term “generalized vector argument” implies that a composite function is allowed to depend not only on some variables but also on their delta derivatives. A formula in the theorem shows how the higher-order delta and partial derivatives of a composite function commute. Moreover, it enables reducing the order of the delta derivative, making computations simpler and more efficient. The computational efficiency of the formula was analysed on the basis of experiments in the symbolic computation software Mathematica.
1. Bartosiewicz, Z., Kotta, Ü., Pawłuszewicz, E., and Wyrwas, M. Control systems on regular time scales and their differential rings. Math. Control Signals Systems, 2011, 22, 185–201.
2. Belikov, J., Kotta, Ü., and Tõnso, M. NLControl: symbolic package for study of nonlinear control systems. In The IEEE Multi-Conference on Systems and Control. IEEE, Hyderabad, India, 2013, 322–327.
3. Bohner, M. and Peterson, A. Dynamic Equations on Time Scales: An Introduction with Applications. Birkh¨auser, Boston, 2001.
4. Ciulkin, M., Kaparin, V., Kotta, Ü., and Pawłuszewicz, E. Linearization by input–output injections on homogeneous time scales. Proc. Estonian Acad. Sci., 2014, 63, 387–397.
5. Ciulkin, M., Pawłuszewicz, E., Kaparin, V., and Kotta, Ü. Linearization by generalized input–output injections on homogeneous time scales. In The 21st International Conference on Methods and Models in Automation and Robotics. IEEE, Miedzyzdroje, Poland, 2016, 48–53.
6. Kaparin, V. and Kotta, Ü. Necessary and sufficient conditions in terms of differential-forms for linearization of the state equations up to input-output injections. In The 8th UKACC International Conference on Control (Burnham, K. J. and Ersanilli, V. E., eds), IET, Coventry, UK, 2010, 507–511.
7. Kaparin, V. and Kotta, Ü. Theorem on the differentiation of a composite function with a vector argument. Proc. Estonian Acad. Sci., 2010, 59, 195–200.
8. Kaparin, V. and Kotta, Ü. Transformation of nonlinear state equations into the observer form: necessary and sufficient conditions in terms of one-forms. Kybernetika, 2015, 51, 36–58.
9. Middleton, R. H. and Goodwin, G. C. Digital Control and Estimation: A Unified Approach. Prentice Hall, Englewood Cliffs, 1990.
10. Mishkov, R. L. Generalization of the formula of Faa di Bruno for a composite function with a vector argument. Int. J. Math. Math. Sci., 2000, 24, 481–491.
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