How Effective are Smooth Compositions in Predictive Control of TS Fuzzy Models?

Abstract

In this article, we study the structural properties that smooth compositions bring to predictive control of TS fuzzy models and examine how they affect the uncertainties, parameter variations of the system and environmental noises to die out. We have employed the smoothness structure of compositions to convert the MPC cost function of TS fuzzy model of the nonlinear systems to an incremental iterative algorithm. Hence, the proposed algorithm does not linearize the nonlinear dynamics, neither requires solving an NP optimization problem in MPC and, therefore, is very fast and simple. The connectivist identification—MPC approach—can be employed for the systems with the long-range horizons. Therefore, the technique is beneficial to the dead-time and non-minimum phase systems. The stability analysis of the algorithm has been carried out, and the performance of the smooth TS fuzzy identification–controller scheme to the classical ones has been compared on a non-min phase test problem with different uncertainties and working environments, including (a) the normal working conditions, (b) with the additive noises, (c) with the parametric changes, (d) with the additive time-varying disturbances to demonstrate the robust behavior of the smooth compositions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Sadjadi, E.N., Herrero, J., Molina, J., Hatami, Z.: On the approximation properties of smooth fuzzy models. Int. J. Fuzzy Syst. 20(8), 2657–2667 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Sadjadi, E.N., Herrero, J., Molina, J., Borzabadi, A.H., Abchouyeh, M.A.: Fuzzy Model Identification and Self Learning with Smooth Compositions. Int. J. Fuzzy Syst. (under review)

  3. 3.

    Oviedo, E., Jairo, J., Vandewalle, J.P.L., Wertz, V.: Fuzzy Logic, Identification and Predictive Control. Springer, London (2005)

    MATH  Google Scholar 

  4. 4.

    Peeva, K., Kyosev, Y.: Fuzzy Relational Calculus: Theory, Applications and Software. Advances in Fuzzy Systems-Applications and Theory, vol. 22. World Scientific Publishing Co., Pte. Ltd., Singapore (2004)

    MATH  Google Scholar 

  5. 5.

    Campello, R.J.G.B., Amaral, W.C.: Hierarchical fuzzy relational models: linguistic interpretation and universal approximation. IEEE Trans. Fuzzy Syst. 14(3), 446–453 (2006)

    Article  Google Scholar 

  6. 6.

    Campello, R.J.G.B., Amaral, W.C.: Modeling and linguistic knowledge extraction from systems using fuzzy relational models. Fuzzy Sets Syst. 121, 113–126 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Mollov, S., Babuska, R., Abonyi, J., Verbruggen, H.B.: Effective optimization for fuzzy model predictive control. IEEE Trans. Fuzzy Syst. 12(5), 661–675 (2004)

    Article  Google Scholar 

  8. 8.

    Qin, S.J., Badgewell, T.A.: A survey of industrial model predictive control technology. Control Eng. Pract. 11(7), 733–764 (2003)

    Article  Google Scholar 

  9. 9.

    Sadjadi, E.N.: Sufficient conditions for the stability of smooth fuzzy systems (under review)

  10. 10.

    Sadjadi, E.N.: On the monotonicity of smooth fuzzy systems (under review)

  11. 11.

    Courant, R., John, F.: Introduction to Calculus and Analysis, vol. II/1. Springer, New York (1999)

    Book  MATH  Google Scholar 

  12. 12.

    Šostak, A.P.: On a fuzzy topological structure. In: Frolík, Z., Souček, V., Vinárek, J. (eds.) Proceedings of the 13th Winter School on Abstract Analysis. Section of Topology, pp. 89–103. Circolo Matematico di Palermo, Palermo (1985)

    Google Scholar 

  13. 13.

    Ramadan, A.A., Abbas, S.E., Kim, Y.C.: Fuzzy irresolute mappings in smooth fuzzy topological spaces. J. Fuzzy Math. 9, 865–877 (2001)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Amudhambigai, B., Uma, M.K., Roja, E., Nadu, T.: m∗-Fuzzy basically disconnected spaces in smooth fuzzy topological spaces. Math. Bohem. 138(1), 1–13 (2013)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Amudhambigai, B., Uma M.K., Roja E.: Pairwise r-Fuzzy b-Open sets in smooth fuzzy bitolopogical spaces. Int. J. Math. Sci. Appl. 2(2), 663–671 (2012)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Amudhambigai, B., Uma, M.K., Roja, E.: Separations axioms in smooth fuzzy topological spaces. Sci. Magna 5, 90–98 (2009)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Peeters, W.: Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures. Fuzzy Sets Syst. 104, 423–433 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Sadjadi, E.N.: Discussion on accuracy of approximation with smooth fuzzy models (under review)

  19. 19.

    Sadjadi, E.N.: Structural properties of time varying smooth fuzzy models for identification (under review)

  20. 20.

    AmirAskari M., Menhaj, M.B.: Fuzzy model predictive control based on modified fuzzy relational model. In: 13th Iranian Conference on Fuzzy System (IFSC) (2013)

  21. 21.

    Oliveira, J.V., Lemos, J.M.: Long-Term predictive adaptive relational control. Fuzzy Sets Syst. 70, 337–357 (1995)

    Article  Google Scholar 

  22. 22.

    Kandiah, S.: Fuzzy model based predictive control of chemical processes. PhD Thesis, University of Sheffield, Sheffield (1996)

  23. 23.

    Salehfar, H., Bengiamin, N., Huang, J.: A systematic approach to linguistic fuzzy modeling based on input-output data. In: Proceedings of the 2000 Winter Simulation Conference (2000)

  24. 24.

    Vukovic, P.D.: One-step ahead predictive fuzzy controller. Fuzzy Sets Syst. 122, 107–115 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ebrahim Navid Sadjadi.

Ethics declarations

Conflict of interest

Authors declare that there exists no conflict of interest.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Appendices

Appendix 1

In order to drive error derivatives, we study the identification process in more detail. To begin with, we write the gradient descent method formula and define the vectors as follows:

$$\frac{\partial J}{{\partial \rho_{ld} }} = \frac{\partial J}{\partial y}\frac{\partial y}{{\partial y_{i} }}\frac{{\partial y_{i} }}{{\partial {\acute{y}}_{li} }}\frac{{\partial {\acute{y}}_{li} }}{{\partial \xi_{ld} }}\frac{{\partial \xi_{ld} }}{{\partial \beta_{ld} }}\frac{{\partial \beta_{ld} }}{{\partial \rho_{ld} }}$$

But to complete the formulation we need to take partial derivative of each variable separately.

First, we define the fuzzy variables \(\left\{ {{\acute{\xi}}_{1} ,{\acute{\xi}}_{2} , \ldots ,{\acute{\xi}}_{r} } \right\}\) at every time instant as

$${\acute{\xi}}_{l} = \left[ {\xi_{l1} ,\xi_{l2} \ldots ,\xi_{l,m + p} } \right] = \left[ {\beta_{l1} \left( {\xi_{1} } \right),\beta_{l2} \left( {\xi_{2} } \right), \ldots ,\beta_{l,m + p} \left( {\xi_{m + p} } \right)} \right],\;l = 1 \ldots ,r$$

and \({\acute{\xi}} = \left[ {{\acute{\xi}}_{l} } \right]_{l = 1}^{r}\), where \(\beta \left( \cdot \right)\), as stated above, is value of the membership function for the fuzzy set. In general, this function can be written as

$$\beta_{ld} \left( \cdot \right) = \exp \left( {\frac{ - 1}{2}\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ld} }}} \right)^{2} } \right).$$

Therefore, for making up the gradient descent method formula, \(\frac{{\partial \xi_{ld} }}{{\partial \rho_{ld} }}\) can be written as

$$\frac{{\partial \xi_{ld} }}{{\partial c_{ld} }} = \exp \left( {\frac{ - 1}{2}\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ld} }}} \right)^{2} } \right)\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ij}^{2} }}} \right)$$
(26)
$$\frac{{\partial \xi_{ld} }}{{\partial \delta_{ld} }} = \exp \left( {\frac{ - 1}{2}\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ld} }}} \right)^{2} } \right)\left( {\frac{{\left( {\xi_{ld} - c_{ld} } \right)^{2} }}{{\delta_{ld}^{3} }}} \right)$$
(27)

Based on the compositional rule inference, we can say that estimation of the output, according to our notation, is

$${\acute{y}}_{li} = s - {\text{norm}}\left( {t - {\text{norm}}\left( {{\acute{\xi}}_{l} ,R_{l} \left( {{\acute{\xi}},y_{i} } \right)} \right)} \right)$$

for all \(l = 1, \ldots ,r.\) Let us abbreviate \(S:s - {\text{norm}}\) and \(T:t - {\text{norm}}\) in the following.

To facilitate the explanation of the procedure of taking the derivation of \(\frac{{\partial {\acute{y}}_{li} }}{{\partial \xi_{ld} }}\), we assume a simple system and put \({\acute{\xi}}_{l} = \left[ {\xi_{l1} ,\xi_{l2} } \right]\) and \(c = R\left( {{\acute{\xi}},y_{i} } \right).\) Then, based on the properties of t norms, we have

$${\acute{y}}_{li} = S \left( {T\left( {T\left( {\xi_{l1} ,\xi_{l2} } \right),c} \right)} \right) = S\left( {T\left( {\xi_{l1} ,c} \right),T\left( {\xi_{l2} ,c} \right)} \right)$$

We define: \(\varLambda_{1} = T\left( {\xi_{l1} ,c} \right)\) and \(\varLambda_{2} = T\left( {\xi_{l2} ,c} \right),\) then

$$\begin{aligned} {\acute{y}}_{li} & = S\left( {\varLambda_{1} ,\varLambda_{2} } \right) \\ \frac{{\partial {\acute{y}}_{li} }}{{\partial \xi_{l1} }} & = \frac{\partial S}{{\partial \varLambda_{d} }}\frac{{\partial \varLambda_{d} }}{{\partial \xi_{l1} }} = {\acute{S}}^{1} {\acute{T}}^{1} ,\;d = 1,2, \\ \end{aligned}$$
(28)

where \({\acute{S}}^{1}\) and \({\acute{T}}^{1}\) are the first-order derivatives of the compositions, which will be calculated below. If there exist more state variables in the augmented state vector, \(\xi^{\prime}_{l} = \left[ {\xi_{l1} ,\xi_{l2} \cdots ,\xi_{l,m + p} } \right]\) we could continue in the same manner and write as

$$\frac{{\partial {\acute{y}}_{li} }}{{\partial \xi_{ld} }} = {\acute{S}}^{m + p - 1} {\acute{T}}^{m + p - 1} \ldots {\acute{S}}^{1} {\acute{T}}^{1} .$$
(29)

Hence, to derive the gradient descent method formulation, the general formula for the error derivation will be

$$\begin{aligned} \frac{\partial J}{{\partial c_{ld} }} & = \frac{\partial J}{\partial y}\frac{\partial y}{{\partial y_{i} }}\frac{{\partial y_{i} }}{{\partial \xi_{ld} }}\frac{{\partial \xi_{ld} }}{{\partial \beta_{ld} }}\frac{{\partial \beta_{ld} }}{{\partial c_{ld} }} \\ & = e\left( k \right) \cdot \left( {\frac{{\theta_{li} - y_{i} }}{{\mathop \sum \nolimits_{i = 1}^{r} \beta_{li} }}} \right) \cdot \left( {{\acute{S}}^{m + p - 1} {\acute{T}}^{m + p - 1} \ldots {\acute{S}}^{1} {\acute{T}}^{1} } \right) \cdot \exp \left( {\frac{ - 1}{2}\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ld} }}} \right)^{2} } \right)\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ld}^{2} }}} \right) \\ \end{aligned}$$
(30)
$$\begin{aligned} \frac{\partial J}{{\partial \delta_{ld} }} & = \frac{\partial J}{\partial y}\frac{\partial y}{{\partial y_{i} }}\frac{{\partial y_{i} }}{{\partial \xi_{ld} }}\frac{{\partial \xi_{ld} }}{{\partial \beta_{ld} }}\frac{{\partial \beta_{ld} }}{{\partial \delta_{ld} }} \\ & = e\left( k \right) \cdot \left( {\frac{{\theta_{li} - y_{i} }}{{\mathop \sum \nolimits_{i = 1}^{r} \beta_{li} }}} \right) \cdot \left( {{\acute{S}}^{m + p - 1} {\acute{T}}^{m + p - 1} \cdots {\acute{S}}^{1} {\acute{T}}^{1} } \right) \cdot \exp \left( {\frac{ - 1}{2}\left( {\frac{{\xi_{ld} - c_{ld} }}{{\delta_{ld} }}} \right)^{2} } \right) \left( {\frac{{\left( {\xi_{ld} - c_{ld} } \right)^{2} }}{{\delta_{ld}^{3} }}} \right) \\ \end{aligned}$$
(31)
$$\begin{aligned} \frac{\partial J}{{\partial \theta l_{i} }} & = \frac{\partial J}{\partial y}\frac{\partial y}{{\partial y_{i} }}\frac{{\partial y_{i} }}{{\partial \theta_{li} }} \\ & = e\left( k \right) \cdot \left( {\frac{{\theta_{li} - \beta_{li} }}{{\mathop \sum \nolimits_{i = 1}^{r} \beta_{li} }}} \right) \\ \end{aligned}$$
(32)

Appendix 2

Let us take \(B = S_{B} \left( {T_{B} \left( {\xi_{l1} ,c} \right),T_{B} \left( {\xi_{l2} ,c} \right)} \right)\), and, \(\varLambda_{1} = T_{B} \left( {\xi_{l1} ,c} \right), \varLambda_{2} = T_{B} \left( {\xi_{l2} ,c} \right),\) where,

$$\begin{aligned} T_{B} \left( {\xi_{l1} ,c} \right) & = \frac{4}{\pi }\tan^{ - 1} \left( {\tan \left( {\frac{\pi }{4}\xi_{l1} } \right)\tan \left( {\frac{\pi }{4}c} \right)} \right) \\ S_{B} \left( {\varLambda_{1} ,\varLambda_{2} } \right) & = 1 - \frac{4}{\pi }\tan^{ - 1} \left( {\tan \left( {\frac{\pi }{4}(1 - \varLambda_{1} )} \right)\tan \left( {\frac{\pi }{4}(1 - \varLambda_{2} )} \right)} \right) \\ \end{aligned}$$

Now, the first-order derivative will be

$$\begin{aligned} \acute{B} & = {\acute{S}}_{B}^{1} {\acute{T}}_{B}^{1} \\ & = \frac{4}{\pi }\frac{1}{{1 + \left( {\tan \left( {\frac{\pi }{4}\xi_{l1} } \right)\tan \left( {\frac{\pi }{4}c} \right)} \right)^{2} }} \times \frac{\pi }{4}\sec^{2} \left( {\frac{\pi }{4}\xi_{l1} } \right)\tan \left( {\frac{\pi }{4}c} \right) \\ & \quad \times \frac{ - 4}{\pi } \times \frac{1}{{1 + \left( {\tan \left( {\frac{\pi }{4}(1 - \varLambda_{1} )} \right) \times \tan \left( {\frac{\pi }{4}(1 - \varLambda_{2} )} \right)} \right)^{2} }} \times \sec^{2} \left( {\frac{\pi }{4}\left( {1 - \varLambda_{1} } \right)} \right) \times \frac{ - \pi }{4} \times \tan \left( {\frac{\pi }{4} (1 - \varLambda_{2} )} \right) \\ \end{aligned}$$
(33)

Now, similarly, take \(C = S_{C} \left( {T_{C} (\xi_{l1} ,c),T_{C} (\xi_{l2} ,c)} \right)\) with \(\varLambda_{1} = T_{C} (\xi_{l1} ,c), \varLambda_{2} = T_{C} (\xi_{l2} ,\xi_{l1} ),\) and,

$$\begin{aligned} & T_{C} \left( {\mu_{a} \left( \cdot \right),\mu_{b} \left( \cdot \right)} \right) = 1 - \frac{2}{\pi }\cos^{ - 1} \left( {\sin \left( {\xi_{l1} } \right)\sin \left( {\frac{\pi }{2}c} \right)} \right) \\ & S_{C} \left( {\varLambda_{1} ,\varLambda_{2} } \right) = \frac{2}{\pi }\cos^{ - 1} \left( {\cos \left( {\frac{\pi }{2}\varLambda_{1} } \right)\cos \left( {\frac{\pi }{2}\varLambda_{2} } \right)} \right) \\ \end{aligned}$$

Then, the first-order derivative is

$$\begin{aligned} \acute{C} & = {\acute{S}}_{C}^{1} {\acute{T}}_{C}^{1} \\ & = \frac{2}{\pi } \times \frac{ - 1}{{\sqrt {1 - \left( {\cos \frac{\pi }{2}\varLambda_{1} \cos \frac{\pi }{2}\varLambda_{2} } \right)^{2} } }} \times \frac{\pi }{2} \times \left( { - \sin \frac{\pi }{2}c} \right) \times \cos \left( {\frac{\pi }{2}\xi_{l1} } \right) \\ & \quad \times \frac{2}{\pi }\frac{ - 1}{{\sqrt {1 - \left( {\sin \frac{\pi }{2}\xi_{l1} \sin \frac{\pi }{2}c } \right)^{2} } }} \times \frac{\pi }{2} \times \left( { - \sin \frac{\pi }{2}\varLambda_{1} } \right) \times \cos \left( {\frac{\pi }{2}\varLambda_{2} } \right) \\ \end{aligned}$$
(34)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sadjadi, E.N., Menhaj, M.B., Herrero, J.G. et al. How Effective are Smooth Compositions in Predictive Control of TS Fuzzy Models?. Int. J. Fuzzy Syst. 21, 1669–1686 (2019). https://doi.org/10.1007/s40815-019-00676-0

Download citation

Keywords

  • Fuzzy control
  • Fuzzy IF–THEN systems (TSK)
  • Smooth compositions
  • System identification
  • Model predictive control (MPC)
  • Unstable systems