Abstract
This paper discusses the problem of global adaptive finite-time control for a class of stochastic nonlinear systems with parametric uncertainty. Under the assumption that the drift and diffusion terms satisfy lower-triangular growth conditions, a continuous adaptive controller is designed based on the adding one power integrator technique and parameter separation principle. By constructing an adaptive law to counteract the effects of uncertain parameters, it is proved that system states can be regulated to the origin almost surely in a finite time. Two simulation examples are given to demonstrate the effectiveness of the proposed control procedure.
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Acknowledgments
This work was supported in part by National Natural Science Foundation of China (61473082, 61104068, 61273119), Scientific Innovation Research of College Graduates in Jiangsu Province (KYLX_0134), Fundamental Research Funds for the Central Universities (2242013R30006), and Six Talents Peaks Program of Jiangsu Province (2014-DZXX-003) and PAPD.
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Appendix
Appendix
For convenience, some generic functions \(\beta _i(\bar{x}_i,\hat{\varTheta })\), \(i=1,\ldots ,n\), are used throughout the paper to stand for any nonnegative smooth functions with respect to their variables and may be implicitly changed in different places.
Proof of Proposition 1
The estimate of \(|\partial (x_k^{*1/r_k})/\partial x_i|\) can be done by an inductive argument. Note that
Assume that, for \(i=1,\ldots ,k-2\),
Therefore, it can be verified that
which implies that (41) also holds for the \(k\)th virtual controller. According to the definition of \(\xi _i\), \(i=1,\ldots ,k\), and Lemma 4, one gets
which leads to \(\forall i=1,\ldots ,k-1\)
Clearly, Proposition 1 follows from (45). \(\square \)
Proof of Proposition 2
Under Assumption 1, the drift terms can be estimated as
According to Lemmas 4 and 5, one has \(\forall i=1,\ldots ,k\)
under which Proposition 2 holds naturally. \(\square \)
Proof of Proposition 3
Based on Assumption 1, one has \(\forall i=1,\ldots ,k\)
Similar to the proof in Proposition 1, the estimate of \(|\partial ^2(x_k^{*\mu /r_k})/\partial x_i^2|\) can also be done inductively. Specifically, for \(i=1,\ldots ,k-1\)
Combining (48) and (49), it yields that for \(i=1,\ldots ,k-1\),
where \(\bar{h}_{k3}(\cdot )\ge 0\) is a smooth function of \(x_1,\ldots ,x_{k-1},\hat{\varTheta }\). Moreover, for a nonnegative smooth function \(\tilde{h}_{k3}(\bar{x}_k,\hat{\varTheta })\)
It is clear that Proposition 3 follows from (50) and (51), by letting \(h_{k3}(\cdot )\!=(k\!-1)\bar{h}_{k3}(\cdot )+\tilde{h}_{k3}(\cdot )\). \(\square \)
Proof of Proposition 4
In a similar way, for \(i,j=1,\ldots ,k-1\) and \(i\ne j\),
under which
where \(\bar{h}_{k4}(\cdot )\) is a nonnegative smooth function. For \(i=k\) and \(j\ne k\), there exists a smooth function \(\tilde{h}_{k4}(\bar{x}_k,\hat{\varTheta })\) satisfying
which leads to Proposition 4 by combining it with (53). \(\square \)
Proof of Proposition 5
Using (15) and the definition of \(\sigma _k\), one can get
for a nonnegative smooth function \(h_{k5}(\cdot )\). In addition, it is easy to obtain that
By choosing \(h_{k6}(\bar{x}_k,\hat{\varTheta })=\sqrt{1+(\sum _{i=1}^{k-1}\frac{\partial W_i}{\partial \hat{\varTheta }}\delta _k(\cdot ))^2}\), we complete the proof. \(\square \)
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Zha, W., Zhai, J. & Fei, S. Global Adaptive Finite-Time Control for Stochastic Nonlinear Systems via State Feedback. Circuits Syst Signal Process 34, 3789–3809 (2015). https://doi.org/10.1007/s00034-015-0043-3
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DOI: https://doi.org/10.1007/s00034-015-0043-3