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.

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

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

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

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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

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

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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

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

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

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

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

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$$\begin{aligned} \frac{\partial J}{{\partial \theta l_{i} }} & = \frac{\partial J}{\partial y}\frac{\partial y}{{\partial y_{i} }}\frac{{\partial y_{i} }}{{\partial \theta_{li} }} \\ & = e\left( k \right) \cdot \left( {\frac{{\theta_{li} - \beta_{li} }}{{\mathop \sum \nolimits_{i = 1}^{r} \beta_{li} }}} \right) \\ \end{aligned}$$

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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

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

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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

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

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