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Theory and Application of a Novel Optimal Robust Control: A Fuzzy Approach

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Abstract

A new optimal robust control is proposed for mechanical systems with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty in the mechanical system. The desirable system performance is deterministic (assuring the bottom line) and also fuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if–then rules-based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the Lyapunov minimax approach. A performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. The unique closed-form optimal gain and the cost are explicitly shown. The resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost. In the end, an example is chosen for demonstration.

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Acknowledgments

The authors would like to express sincere thanks to National Natural Science Foundation of China No.51275147, who have supported this research work.

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Authors and Affiliations

  1. School of Mechanical and Automotive Engineering, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Kuibang Zhang, Jiang Han, Lian Xia, Qingyan Yang & Shengchao Zhen

Authors
  1. Kuibang Zhang
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  2. Jiang Han
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  3. Lian Xia
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  4. Qingyan Yang
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  5. Shengchao Zhen
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Corresponding author

Correspondence to Shengchao Zhen.

Appendix: fuzzy mathematics

Appendix: fuzzy mathematics

We briefly review some preliminaries regarding fuzzy numbers and their operations [14]:

1.1 Fuzzy number

Let G be a fuzzy set in R, the real number. G is called a fuzzy number if (i) G is normal, (ii) G is convex, (iii) the support of G is bounded, and (iv) all α-cuts are closed intervals in R.

Throughout, we shall always assume the universe of discourse of a fuzzy set number to be its 0-cut.

1.2 Fuzzy arithmetic

Let G and H be two fuzzy numbers and \( G_{\alpha } = [g_{\alpha }^{ - } ,g_{\alpha }^{ + } ] \), \( H_{\alpha } = [h_{\alpha }^{ - } ,h_{\alpha }^{ + } ] \) be their α-cuts, \( \alpha \in [0,1] \). The addition, subtraction, multiplication, and division of G and H are given by, respectively,

$$ (G + H)_{\alpha } = [\overline{g}_{\alpha }^{ - } + h_{\alpha }^{ - } ,g_{\alpha }^{ + } + h_{\alpha }^{ + } ] $$
(9.1)
$$ (G - H)_{\alpha } = [\hbox{min} (g_{\alpha }^{ - } - h_{\alpha }^{ - } ,g_{\alpha }^{ + } - h_{\alpha }^{ + } ),\hbox{max} (g_{\alpha }^{ - } + h_{\alpha }^{ - } ,g_{\alpha }^{ + } + h_{\alpha }^{ + } )] $$
(9.2)
$$ (G.H)_{\alpha } = [\hbox{min} (g_{\alpha }^{ - } h_{\alpha }^{ - } ,g_{\alpha }^{ - } h_{\alpha }^{ + } ,g_{\alpha }^{ + } h_{\alpha }^{ - } ,g_{\alpha }^{ + } h_{\alpha }^{ + } ),\hbox{max} (g_{\alpha }^{ - } h_{\alpha }^{ - } ,g_{\alpha }^{ - } h_{\alpha }^{ + } ,g_{\alpha }^{ + } h_{\alpha }^{ - } ,g_{\alpha }^{ + } h_{\alpha }^{ + } )] $$
(9.3)
$$ (G / H)_{\alpha } = [\hbox{min} (g_{\alpha }^{ - } /h_{\alpha }^{ - } ,g_{\alpha }^{ - } /h_{\alpha }^{ + } ,g_{\alpha }^{ + } /h_{\alpha }^{ - } ,g_{\alpha }^{ + } /h_{\alpha }^{ + } ),\hbox{max} (g_{\alpha }^{ - } /h_{\alpha }^{ - } ,g_{\alpha }^{ - } /h_{\alpha }^{ + } ,g_{\alpha }^{ + } /h_{\alpha }^{ - } ,g_{\alpha }^{ + } /h_{\alpha }^{ + } )]. $$
(9.4)

1.3 Decomposition theorem

Define a fuzzy set \( \tilde{V}_{\alpha } \) in U with the membership function \( \mu_{{\tilde{V}_{\alpha } }} = I_{{\tilde{V}_{\alpha } }} (x), \) where \( I_{{\tilde{V}_{\alpha } }} (x) = 1 \) if \( x \in \tilde{V}_{\alpha } \) and \( I_{{\tilde{V}_{\alpha } }} (x) = 0 \) if \( x \in U - \tilde{V}_{\alpha } \). Then the fuzzy set V is obtained as

$$ V = \bigcup\limits_{\alpha \in [0,1]} {\tilde{V}_{\alpha } }, $$
(9.5)

where \( \cup \) is the union of the fuzzy sets (that is, sup over \( \alpha { \in }[0,1] \)).

Based on these, after the operation of two fuzzy numbers via their α-cuts, one may apply the decomposition theorem to build the membership function of the resulting fuzzy number.

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Zhang, K., Han, J., Xia, L. et al. Theory and Application of a Novel Optimal Robust Control: A Fuzzy Approach. Int. J. Fuzzy Syst. 17, 181–192 (2015). https://doi.org/10.1007/s40815-015-0026-3

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