Zero dynamics

In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems.[1]

History

The idea was introduced thirty years ago as the nonlinear approach to the concept of transmission of zeros. The original purpose of introducing the concept was to develop an asymptotic stabilization with a set of guaranteed regions of attraction (semi-global stabilizability), to make the overall system stable.[2]

Initial working

Given the internal dynamics of any system, zero dynamics refers to the control action chosen in which the output variables of the system are kept identically zero.[3] While, various systems have an equally distinctive set of zeros, such as decoupling zeros, invariant zeros, and transmission zeros. Thus, the reason for developing this concept was to control the non-minimum phase and nonlinear systems effectively.[4]

Applications

The concept is widely utilized in SISO mechanical systems, whereby applying a few heuristic approaches, zeros can be identified for various linear systems.[5] Zero dynamics adds an essential feature to the overall system’s analysis and the design of the controllers. Mainly its behavior plays a significant role in measuring the performance limitations of specific feedback systems. In a Single Input Single Output system, the zero dynamics can be identified by using junction structure patterns. In other words, using concepts like bond graph models can help to point out the potential direction of the SISO systems.[6]

Apart from its application in nonlinear standardized systems, similar controlled results can be obtained by using zero dynamics on nonlinear discrete-time systems. In this scenario, the application of zero dynamics can be an interesting tool to measure the performance of nonlinear digital design systems (nonlinear discrete-time systems).[7]

Before the advent of zero dynamics, the problem of acquiring non-interacting control systems by using internal stability was not specifically discussed. However, with the asymptotic stability present within the zero dynamics of a system, static feedback can be ensured. Such results make zero dynamics an interesting tool to guarantee the internal stability of non-interacting control systems.[8]

References
  1. Van de Straete, H.J.; Youcef-Toumi, K. (June 1996). "Physical Meaning of Zeros and Transmission Zeros from Bond Graph Models". IFAC Proceedings Volumes. 29 (1): 4422–4427. doi:10.1016/s1474-6670(17)58377-9. hdl:1721.1/11140. ISSN 1474-6670.
  2. Isidori, Alberto (September 2013). "The zero dynamics of a nonlinear system: From the origin to the latest progresses of a long successful story". European Journal of Control. 19 (5): 369–378. doi:10.1016/j.ejcon.2013.05.014. ISSN 0947-3580. S2CID 15277067.
  3. Youcef-Toumi, K.; Wu, S-T (June 1991). "Input/Output Linearization using Time Delay Control". 1991 American Control Conference. IEEE: 2601–2606. doi:10.23919/acc.1991.4791872. ISBN 0-87942-565-2. S2CID 20562917.
  4. "Control Theory", Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, vol. 1768, Springer Berlin Heidelberg, 2001, pp. 91–149, doi:10.1007/3-540-45436-5_5, ISBN 978-3-540-42626-4
  5. Miu, D. K. (1991-09-01). "Physical Interpretation of Transfer Function Zeros for Simple Control Systems With Mechanical Flexibilities". Journal of Dynamic Systems, Measurement, and Control. 113 (3): 419–424. doi:10.1115/1.2896426. ISSN 0022-0434.
  6. Huang, S.Y.; Youcef-Toumi, K. (June 1996). "Zero Dynamics of Nonlinear MIMO Systems from System Configurations - A Bond Graph Approach". IFAC Proceedings Volumes. 29 (1): 4392–4397. doi:10.1016/s1474-6670(17)58372-x. ISSN 1474-6670.
  7. Monaco, S.; Normand-Cyrot, D. (September 1988). "Zero dynamics of sampled nonlinear systems". Systems & Control Letters. 11 (3): 229–234. doi:10.1016/0167-6911(88)90063-1. ISSN 0167-6911.
  8. Isidori, A.; Grizzle, J.W. (October 1988). "Fixed modes and nonlinear noninteracting control with stability". IEEE Transactions on Automatic Control. 33 (10): 907–914. doi:10.1109/9.7244. ISSN 0018-9286.
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