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A note on the Takagi–Sugeno control algorithm for a class of distributed parameter system

Abstract

The paper treats the control problem of a class of hyperbolic PDE with nonlinear components by using the three-dimensional fuzzy control since the spatial nature of a distributed parameter system requires an extension of the traditional fuzzy set by adding the third coordinate for spatial information. A fuzzy transform is proposed in order to obtain a spatial compression of the distributed model, and for this reason, the linguistic expression is reformulated by the compression of the spatial information, in terms of the new fuzzy sets. The effect of this spatial compression on the 3D fuzzy components, on the premise and consequence parts, is presented and discussed. A Takagi–Sugeno fuzzy mechanism that operates on the spatial compressed variables is proposed. The reconstruction of variables is obtained as a spatial decompression by the inverse F-transform. The control quality is analysed, and the conditions that ensure the algorithm convergence are determined. The proposed method is validated by numerical simulations. The behaviour of system is examined, and numerical results illustrate the control algorithms.

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Correspondence to Nirvana Popescu.

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Appendix

Appendix

In order to derive the compressed form of dynamic model (1), in terms of error (7), (8), consider (12) rewritten as:

$$\begin{aligned} \begin{aligned} e^{j*}(t)&= \frac{p}{l} \int _0^l e(t, s) W^{j} (s) \mathrm{d}s\\&= \frac{p}{l} \int _{L_{j-1}}^{L_{j+1}} e(t, s) W^j(s) \mathrm{d}s, \quad j = 1, \dots , p \end{aligned} \end{aligned}$$
(A1)

where

$$\begin{aligned} \int _{L_{j-1}}^{L_{j+1}} W^j(s) \mathrm{d}s = \frac{1}{p} \end{aligned}$$
(A2)

Multiplying the both sides of (1), expressed in terms of error e(ts), by \(W^j(s)\) and integrating on [0, l], it yields:

$$\begin{aligned} \int _{L_{j-1}}^{L_{j+1}} \ddot{e}(t,s) W^j(s) \mathrm{d}s =&\, a^2 \int _{L_{j-1}}^{L_{j+1}} e_{ss} (t,s) W^j(s) \mathrm{d}s \nonumber \\&- b \int _{L_{j-1}}^{L_{j+1}} \dot{e}(t,s) W^j(s) \mathrm{d}s\nonumber \\&- c \int _{L_{j-1}}^{L_{j+1}} e(t,s) W^j(s) \mathrm{d}s \nonumber \\&- \int _{L_{j-1}}^{L_{j+1}} h(e, \dot{e}) W^j(s) \mathrm{d}s \end{aligned}$$
(A3)

The first term of the second side is integrating by parts by using the relation

$$\begin{aligned} \begin{aligned} e_{ss}(t,s) W^j(s) =&\, (e_s(t,s) W^j(s))_s \\&- (e(t,s) W_s^j(s))_s + e(t,s) W_{ss}^j (s) \end{aligned} \end{aligned}$$
(A4)

where \(W^j(s)\) satisfies the following conditions:

$$\begin{aligned} W_{ss}^j(s)= & {} \frac{p^2 \pi ^2}{2t^2} - \frac{\pi ^2 p^2}{t^2} W^j(s) \end{aligned}$$
(A5)
$$\begin{aligned} W^j(L_{j-1})= & {} W^j(L_{j+1})=0, \ W^j(L_j)=1; \nonumber \\&W_s^j(L_{j-1}) = W_s^j(L_{j+1}) = 0 \end{aligned}$$
(A6)
$$\begin{aligned} \ddot{e}^{j*}(t)= & {} -b \dot{e}^{j*}(t) - (a^2 \frac{\pi ^2}{l^2}+c) e^{j*}(t) \nonumber \\&+\,\varDelta h^*(e^{j*}, \dot{e}^{j*}) + u^{j*}(t); \ \ j=1, \dots , p\nonumber \\ \end{aligned}$$
(A7)

where

$$\begin{aligned} \begin{aligned} \varDelta h^*(e^{j*}, \dot{e}^{j*}=&\frac{p}{l} \int _{L_{j-1}}^{L_{j+1}} \Big ( a^2 \frac{p^2 \pi ^2}{2l^2}e \\&+ \Big ( \frac{\partial h(\theta )}{\partial \theta } \Big )_{( \theta = \theta ^d)} W^j(s) e \Big ) \mathrm{d}s \end{aligned} \end{aligned}$$
(A8)

Nonlinear model (A8), width \(\varDelta h^* \in \varOmega _h^*\), can be converted in a linear model, by dividing \(\varOmega _k^*\) in sub-domains \(\varDelta \varOmega _h^*\).

$$\begin{aligned} \begin{aligned} \dot{e}^{j*}(t)&=A^\mathrm{T} e^{j*}(t) + B^\mathrm{T} u^{j*}(t) \\ y^{j*}(t)&= C^\mathrm{T} e^{j*}(t) \end{aligned} \end{aligned}$$
(A9)

with the initial condition

$$\begin{aligned} e^{j*}(0) = e_0^{j*} \end{aligned}$$
(A10)

where \(e^{j*}, \dot{e}^{j*} \in \varDelta \varTheta _v, u^{j*} \in \varDelta U_v\), and \(A^\mathrm{T} \in R^{4x4}, B^r \in R^{4x1}, C^\mathrm{T} \in R^{1x4}\) define the linear relations on the local areas \(\varDelta \varOmega _h^*\)

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Popescu, N., Popescu, D., Ciobanu, V. et al. A note on the Takagi–Sugeno control algorithm for a class of distributed parameter system. Soft Comput 23, 7207–7214 (2019). https://doi.org/10.1007/s00500-018-3366-8

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Keywords

  • Takagi–Sugeno control
  • Distributed parameter systems
  • 3D fuzzy components
  • Spatial variable compression