Algorithm of adaptive control, based on the modified parametrization of the plant’s equation, страница 3

It means that if the system begin to work from some domain of initial values, than will exist domain as

with some domain of attraction , for which the target condition (3) fair according to a lemma [10].

However preservation of domain of dissipative does not guarantee, that in the singular-disturbing system   many attraction  remain the same.

Let in (13) . Let’s will consider that movement of system begin in initial number of entry conditions , consequently all trajectories of the system will be is in domain of dissipative  according to lemma [10].

Let’s consider Lyapunov function as (15) again, and will take a derivative from Lypunov function on trajectories of the systems (12) and (13), taking into account the result in (16)

.              (17)

Take into attention correlations

,

, where , ,

, - single matrix,

for derivative from Lyapunov function (17), will receive

.

One can see that changing , , ,  and  can change the value  in the target condition (3). And, choosing  as big enough, value of function  can make as such as small, that is reducing trajectories (12) to small enough neighborhood of .

3. Example

Let’s consider unstable system  and reference model, which set by the following equation , where .

In the algorithm of estimation (9) choose such parameters as: , ; . As , then filter on an output and an input are expelled and then vector will be formed as . The law of control is formed as the equation , and in the algorithm of adjustment (12) parameters  and  will choose as  and .

In the figure the results of modeling of transitional processes of tracking error for reference signal  and the law of control  at the following meaning of parameters of plant (1): . All other entry values in the system will accept equal zero too.

4. Conclusion

In the work for the linear plant, set in the form of an input-output with unknown parameters and with scalar input and output, which accessible to measurement, is considered the construction of adaptive law of control on the base of offered scheme of parameterization of control plant (1), resulting to SPR-function and on the base of modified algorithm of adaptation. In the tie of what filters through which all components of vector are passed, are expelled. The offered approach allows to simplify the construction of the closed control system and to reduce the order of closed-loop system. It gives concerning small algorithm. Modeling on the computer shows good results, both, on tracking error and on control influence.

5. REFERENCES

[1]  V. O. Nikiforov, and A. L. Fradkov, “The scheme of adaptive control with an extended error signal”, Avtomatika i telemekhanika, no. 9, 1994, pp. 3-22 (in Russian).

[2]  A.S. Morse, A. Isidori, “High-order parameter tuners for adaptive control on nonlinear system”, Tarn T. I. (eds). Systems, Models and Feedback: Theory and Applications. Birkhanser, 1992, pp. 339-364.

[3]  V.O. Nikiforov, “Robust high-order tuner of simplified structure”, Automatica, vol. 35, no 8, 1999, pp. 1409-1417.

[4]  I. V. Miroshnik, V. O. Nikiforov, and A. L. Fradkov, Nonlinear and adaptive control of complex Dynamical systems,  St. - Petersburg: Nauka, 2000 (in Russian).

[5]  A. Feuer, and A.S. Morse, “Adaptive control of single input, single-output linear systems”, IEEE Trans. on Automat. Control, vol. 45, no. 3, 2000, pp. 490-494.

[6]  H.K. Khalil, “Universal integral controllers for minimum-phase nonlinear systems”, IEEE Trans. on Automat. Control, vol. 45, no. 3, 2000, pp. 490-494.

[7]  H.K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models”, IEEE Trans. On Automatic Control, vol. 41, no. 2, 1996, pp. 177-188.

[8]  N.A. Mahmoud, and H.K. Khalil, “Robust control for a nonlinear servomechanism problem”, Int. J. Control,  vol. 66, no. 6, 1997, pp. 779-802.

[9]  A.L. Fradkov, “Synthesis of adaptive system of stabilization of linear dynamic plants”, Avtomatika i telemekhanika, no.12, 1974, pp. 96-103 (in Russian).

[10]  V.A. Brusin, “About one class of singular-disturbing adaptive systems. 1”, Avtomatika i telemekhanika, no.4, 1995, pp. 119-127 (in Russian).