Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Abstract

Many real-world processes tend to be chaotic and also do not lead to satisfactory analytical modelling. It has been shown here that for such chaotic processes represented through short chaotic noisy time-series, a multi-input and multi-output recurrent neural networks model can be built which is capable of capturing the process trends and predicting the future values from any given starting condition. It is further shown that this capability can be achieved by the Recurrent Neural Network model when it is trained to very low value of mean squared error. Such a model can then be used for constructing the Bifurcation Diagram of the process leading to determination of desirable operating conditions. Further, this multi-input and multi-output model makes the process accessible for control using open-loop/closed-loop approaches or bifurcation control etc. All these studies have been carried out using a low dimensional discrete chaotic system of Hénon Map as a representative of some real-world processes.

Modelling and control of chaotic processes through their bifurcation diagrams generated with the help of recurrent neural network models. Part 2: An industrial study

“…While the proper identification of issues have been addressed in subsequent sections, using proposition outlined in [10], passive identification of chaos has been done by observing the increase of scatter in the outputs, whenever the operation is in the chaotic region. In addition the presence of large oscillations in the outputs without any visible change in input conditions has also been taken as a sign of chaotic conditions in the plant.…”

“…However, it has been shown in [10] that it is possible to build a multi-input and multi-output recurrent neural networks model of the system by observing the inputs and outputs of a given system or process and training the network to low levels of mean squared error (MSE) (typically of the order of 10 À2 or better). This multi-input and multi-output (MIMO) model can be used either for bifurcation control or parameter based control of chaos or for implementing other strategies like stabilisation, etc.…”

“…It has also been demonstrated that these MIMO models are robust against Gaussian noise and thus suggesting that these could be used in noisy process plant environments. The feasibility of the propositions have been demonstrated in [10] using a well known two dimensional discrete chaotic system of Hénon map. It is proposed here to examine the applicability of this technique, through suitable adaptation, for modelling real-world process plants that can operate in stable and chaotic operating conditions.…”

“…In [14] it has been shown that using reinforcement learning techniques, control of chaotic systems can be achieved from observed data based methods without formal models, but the approach has been limited to stabilising an unstable fixed orbit. Therefore, the proposition brought out in [10], where modelling the system through multiinput and multi-output recurrent neural networks, training it to very low MSE levels using back-propagation algorithm, to achieve a capability to predict number of subsequent steps, has been adopted here, for the study of real-world systems. It is to be noted that in long terms all chaotic systems remain unpredictable [4].…”

“…Thus, to study the operation and control of SAF, or any plant which is likely to be chaotic, an approach based on the strategies developed in [10] and the issues discussed above the following sequential procedure is adopted: operations of the system, for any particular set of initial conditions the largest Lyapunov exponent is computed based on the model developed, and the degree and the nature of chaos is established. (VI) Next, the bifurcation control strategy is implemented through the input parameter control using the knowledge from the above mentioned model and the possibility of changing the Bifurcation Diagram or the operating point on it, is examined to operate the process in desirable conditions.…”