# A Simple Soft Computing Structure for Modeling and Control

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## Abstract

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## 1. Introduction

## 2. The Model Structure, the Activation Function and the Teaching Process

- Each neuron investigates if the input belongs to its range of competence. If $q\in [{q}_{mi{n}_{i}},{q}_{ma{x}_{i}})\wedge \dot{q}\in [{\dot{q}}_{mi{n}_{i}},{\dot{q}}_{ma{x}_{i}})$ then the given neuron (number i) is competent to make operation. In this case, in the learning phase, it
- associates the appropriate $\ddot{q}$ input value with the given grid element;
- augments the vector $b=[q,\dot{q}]$ into $B=[q,\dot{q},{B}_{d}]$, $a=[\ddot{q},0]$ into $A=[\ddot{q},0,{A}_{d}]$ in which ${A}_{d}$ and ${B}_{d}$ are the “dummy components” that guarantee the common norm $\parallel A\parallel =\parallel B\parallel =R$;
- calculates the unit vectors ${e}_{A}$, ${e}_{B}$ and the angle of abstract rotation, $\phi $, and finally
- if the place of the appropriate node is still empty, simply stores the computed data in the hyper-matrix as $Node[{i}_{q},{i}_{\dot{q}},1:3]={e}_{A}$, $Node[{i}_{q},{i}_{\dot{q}},4:6]={e}_{B}$, and $Node[{i}_{q},{i}_{\dot{q}},7]=\phi $ where the subscripts ${i}_{q}$ and ${i}_{\dot{q}}$ now denote the number of the given cell/node.
- If the node already is filled in, it either does nothing (simple skips the operation), or applies “incremental learning”.

- 1.
- For a given pair $(q,\dot{q})$ each neuron determines whether the input belongs to the box associated with its “competence of operation”: if not, the output value will be zero, otherwise it completes the following calculations:
- 2.
- the neuron retrieves the parameters of the activation function as ${e}_{A}$, ${e}_{B}$, and $\phi $,
- 3.
- computes the orthogonal matrix O in (4);
- 4.
- augments the vector $[q,\dot{q}]$ into $[q,\dot{q},d]$, where d is the physically not interpreted “dummy component”;
- 5.
- computes the rotated vector $O[q,\dot{q},d]$, and uses its first component as $\ddot{q}$ obtained from this special model, and
- 6.
- optionally it can refresh the cell content via “incremental learning”.

## 3. Teaching Example: The Free Motion of the van der Pol Oscillator

`VDP_free_motion.jl`”. (It is available at the link given in Section entitled “Supplementary Materials”.) This code is a simple text file edited by the use of

`Atom 1.57.0 x64`that allows the use of very special characters in the variable names. The initial state of the free motion was ${q}_{ini}=0.0$ m, ${\dot{q}}_{ini}=2.0$ m·s${}^{-1}$, the discrete time resolution of the simple Euler integration was $dt={10}^{-3}$ s. The resolution of the grid was determined as follows: the cell size for q was $\delta q=0.125$ m and the interval $[-2.0,2.0]$ m was covered by the model; for $\dot{q}$$\delta \dot{q}=1.25$ m·s${}^{-1}$ was chosen for covering the interval $[-20.0,20.0]$ m·s${}^{-1}$. In the teaching process the already filled in cell remained invariant, which theoretically corresponds to the aggregation weight $\alpha =1$, $\beta =0$ in (10).

## 4. Controlled Motion of the van der Pol Oscillator Using the Novel Neural Model

#### 4.1. The Computed Torque Control and the Robust VS/SM Schemata

#### 4.2. Simulation Investigations

`VDP_machines_learning.jl`” exemplified the CTC control, and the program “

`VDP_machines_learning_VSSM.jl`” represented the VS/SM control. (The codes are available at the link given in Section entitled “Supplementary Materials”.) Both programs used the parameters $dt={10}^{-3}$ s discrete time resolution in the Euler integration, $\Lambda =6.0$ s${}^{-1}$, $\lambda =25.0$ s${}^{-1}$, the norm of the augmented vectors was $R=5000$, for the aggregation of the unit vectors $\alpha =0.9$ was chosen. In the VS/SM controller the further parameters in (14) were $K=100.0$ m·s${}^{-2}$, and $w=0.1$ m·s${}^{-1}$.

## 5. Conclusions

- Its activation function’s operation has simple geometric interpretation: it executes abstract rotations in higher-dimensional Euclidean spaces;
- The parameters of this function are encoded in two orthogonal unit vectors and in the angle of the abstract rotation that is executed by this function;
- The “extrapolation ability” of this function originates from the fact that with this rotation operator an array has to be transformed that conveys information on the “absolute value” of the modeled state;
- As with the polytopic models, the system model is so coded that each cell in a grid has its own activation function with its parameters;
- In contrast to the multilayer perceptrons or recurrent neural networks, no complicated “connecting wire structure” must be realized in its implementation;
- In contrast to the teaching phase of a multilayer perceptron, in which the error backpropagation requires the modification of the parameters of each function and the weight parameters in a massively parallel process, in this system only one neuron is active in a given time instant that is responsible for the necessary rotation. The other cells only must determine whether they must make any computation;
- In a similar manner, in the operating phase, when the model is in use, only one neuron executes the necessary transformation in a given time; the other ones only must check their competence. In contrast to that, the fuzzy inference systems must make massive distributed operations by executing the necessary “AND” and “OR” operations, fuzzification of the input and defuzzification of the output. Additionally, in the case of a multilayer perceptron, each neuron has to essentially take part in the computation of the final result that can be obtained by collecting the output of each neuron of a given layer, and forwarding their output to the neurons of the next layer;
- Due to the fact that the parameters of the functions are interpreted as orthogonal unit vectors, incremental adaptive improvement or “further teaching” of this model during its operation is possible by the use of a simple approximate “aggregation technique” that was elaborated for unit vectors.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**The structure of the nodes for an ${\mathbb{R}}^{2}\mapsto \mathbb{R}$ mapping used for checking the mapping abilities in the case of the function $\ddot{q}\left(q,\dot{q}\right)$ of the free motion of the van der Pol oscillator.

**Figure 2.**Learning the free motion of the van der Pol oscillator: (

**a**) The phase trajectory. (

**b**) The $\ddot{q}$ values provided by the exact dynamic model. (

**c**) The learned $\ddot{q}$ values retrieved from the neural model. (

**d**) Comparison of the exact and the learned $\ddot{q}$ values.

**Figure 3.**The CTC and the robust VS/SM control schemata (the difference between these schemata consists of the contents of the box “Kinematic Block”).

**Figure 4.**The structure of the nodes for an ${\mathbb{R}}^{3}\mapsto \mathbb{R}$ mapping used for control purposes, i.e., for the approximation of the function $Q\left(q,\dot{q},{\ddot{q}}^{Des}\right)$.

**Figure 5.**Various projections of the stored control surfaces expressed by the angle of abstract rotations in the case of the CTC control.

**Figure 6.**Simulation results for trajectory tracking for amplitude $A=1.5$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 7.**Simulation results for amplitude $A=1.5$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

**Figure 8.**Simulation results for trajectory tracking for amplitude $A=1.5$ m and circular frequency $\omega =8.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 9.**Simulation results for amplitude $A=1.5$ m and circular frequency $\omega =8.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

**Figure 10.**Simulation results for trajectory tracking for amplitude $A=1.5$ m and circular frequency $\omega =4.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 11.**Simulation results for amplitude $A=1.5$ m and circular frequency $\omega =4.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

**Figure 12.**Simulation results for trajectory tracking for amplitude $A=1.5$ m and circular frequency $\omega =1.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 13.**Simulation results for amplitude $A=1.5$ m and circular frequency $\omega =1.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

**Figure 14.**Simulation results for trajectory tracking for amplitude $A=1.0$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 15.**Simulation results for amplitude $A=1.0$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

**Figure 16.**Simulation results for trajectory tracking for amplitude $A=0.5$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 17.**Simulation results for amplitude $A=0.5$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

**Figure 18.**Simulation results for trajectory tracking for amplitude $A=1.6$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Trajectory tracking of the CTC controller. (

**b**) Trajectory tracking of the VS/SM controller. (

**c**) Trajectory tracking error of the CTC controller. (

**d**) Trajectory tracking error of the VS/SM controller.

**Figure 19.**Simulation results for amplitude $A=1.6$ m and circular frequency $\omega =10.0$ s${}^{-1}$: (

**a**) Phase trajectory tracking of the CTC controller. (

**b**) Phase trajectory tracking of the VS/SM controller. (

**c**) Control force of the CTC controller. (

**d**) Control force of the VS/SM controller.

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Redjimi, H.; Tar, J.K.
A Simple Soft Computing Structure for Modeling and Control. *Machines* **2021**, *9*, 168.
https://doi.org/10.3390/machines9080168

**AMA Style**

Redjimi H, Tar JK.
A Simple Soft Computing Structure for Modeling and Control. *Machines*. 2021; 9(8):168.
https://doi.org/10.3390/machines9080168

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Redjimi, Hemza, and József Kázmér Tar.
2021. "A Simple Soft Computing Structure for Modeling and Control" *Machines* 9, no. 8: 168.
https://doi.org/10.3390/machines9080168