In control design, the quality of the mathematical model of the process is the key to accurate closed-loop operation, especially when combined with model-based controllers. As the analytical procedure for deriving such a model proves to be very time-consuming, the blooming field of identification focuses on extracting dynamic process properties directly from the data. Identification can drastically reduce the time and required expertise in model development.
In the identification of non-linear systems, whether static or dynamic, Takagi-Sugeno fuzzy models are successfully used, primarily due to their capabilities in modelling complex non-linear systems despite their simple and interpretable structure. The idea behind the Takagi-Sugeno models is to split the problem domain into smaller partitions, within which the otherwise complex process can be described with simpler models. These models are later fuzzily fused together into single model, which now describes the behaviour of the system across the entire problem domain.
Despite the exponential development in the field of deep neural networks, Takagi-Sugeno models still offer some advantages that are critical to the field of control and identification. These advantages include demonstrable stability, simple insight into the system’s operation, and the possibility of local identification. In particular, the latter benefit -– local identification -– is a feature that we believe, in our opinion, ensure the enduring relevance of Takagi-Sugeno models in the field of identification research. Takagi-Sugeno models provides a much higher sample efficiency, as each sample affects a smaller number of parameters, resulting in a correspondingly faster convergence of the parameters. This faster convergence, in turn, allows the model to describe changes in the system more quickly and accurately, making them suitable for time-variant dynamical systems. Proficient local identification also enables the usage of evolving fuzzy models.
The main challenge in the identification of Takagi-Sugeno models is the problem of domain partitioning. The main task of the model designer is to find a good balance between the ability to model complex systems (the model can describe more complex processes if more partitions are used) and the accuracy of local model parameters (the larger the partitions, the more samples can be used in parameter estimation, which decreases the parameter variance).
In this thesis, the author presents a combined membership function in which information about the quality of local models is innovatively incorporated into the sharing process. The combined membership function alters the criteria for a sample to belong to a specific local model, as the sample needs to be close both in the problem domain and in the output space. This approach enhances the detection of linear areas.
However, the results with the combined membership function were not uniquely better. The final result shows that a model with single local model is the best description of the shape of load consumption. The final result is not intuitive, since we expected that it has sufficient samples for at least two local models if not more. After reviewing the literature, the author concludes that this is a consequence of well-known property of Takagi-Sugeno models, namely the trade-off between accuracy and transparency.
In the following part of the thesis, the trade-off between the accuracy and the transparency of the fuzzy models is analysed in detail. The author identifies and proves that the structure of the Takagi-Sugeno model itself is the primary source of the trade-off between model accuracy and transparency. It is concluded that the structure of Takagi-Sugeno model remains the foremost reason why the partitioning of the problem domain in fuzzy models has persisted as a significant research challenge for the past five decades, remaining unresolved to this day. Since fuzzy modelling is still successfully used in many fields despite the existence of the trade-off, this thesis pointed out the exceptions where fuzzy models are still a accurate choice for identification. Nevertheless, the author assumes that the research of the problem domain partitioning of Takagi-Sugeno models in the classical sense will not drastically contribute to the improved quality of these models, as the error caused by the limitations of the structure of the Tagaki-Sugeno model outweighs the error caused by poor partitioning of the problem domain.
As a solution to the limitation of Takagi-Sugeno model, the author presents a new structure of fuzzy models based on the fusion in parameter space, called the fuzzy parameter structure. The objectives in designing this new structure were to preserve the main principles of fuzzy models -- partitioning into smaller domains in which simpler models can be designed and the possibility of using local identification methods -- but without the trade-off between accuracy and transparency. The main distinction between the Takagi-Sugeno models and the fuzzy parameter structure lies in the space of the local model fusion. The Takagi-Sugeno models fuse local models in the output space, while the fusion in the fuzzy parameter structure is relocated to the parameter space of the affine model.
The fuzzy parameter structure is defined by representing the local models by a probability distribution over the function space, where the influence of the model at a given input is implicitly defined by means of the probability distribution. The key step of the fuzzy parameter structure is the mapping of the function space into a smaller parametric space of an affine model by linearising the basis functions of the function space. This allows local models to be combined in a single and relatively low-dimensional space, while at the same time not restricting the designer’s freedom in the choice of local models. This approach has many advantages: an arbitrary description for each local model separately, the local influence of local models and easy incorporation of expert knowledge about the system (for example, incorporation of uncertainties due to noise or bias present), and the possibility of local identification, as each parameter has a limited area of influence on the model output. However, the flexibility of the structure is also its biggest problem, as arbitrary basis functions and probability distributions cannot guarantee an analytical solution to the output.
Therefore the first model built on a fuzzy parameter structure is introduced, which is called the fuzzy parameter model with a rectified linear unit. The main feature of the model is its simplicity, as the designer only needs to configure the parameters of the linear models and specify their valid centres. Other properties, such as probability distributions in the function space, are derived from the given parameters of the other local models. The fuzzy parameter model with the rectified linear unit is compared with different Takagi-Sugeno models in experiments, where the transparent parameters and the centres of the local models are known in advance. The fuzzy parameter model, while adhering to the principle of fuzzy models and preserving the possibility of local identification, significantly enhances the performance of all Takagi-Sugeno models in simple cases. However, in more complex scenarios, we observe a negative impact originating in some of the assumptions that can be relaxed with the future research. We would like to emphasize that one of these assumptions pertains to the description of the model domain using only the center without incorporating a covariance matrix, thus constraining the shapes of the local domain.
The new structure also needs a new way of partitioning the problem domain in order to find the most linear parts of the local domain. The two goals in developing the new partitioning method were to preserve the locality of the partitioning (which allows parallelizable execution and easy adaptation of the partitioning to evolving systems) and the accuracy of the local affine models.
The final result is the next contribution of the thesis: bottom-up clustering with the introduction of model quality. The preservation of locality is achieved by the bottom-up partitioning of the problem domain, while accuracy is achieved by a clustering criterion based entirely on minimising the expected value of the square of the model error. The clustering criterion finds the optimal ratio between the error due to the variance of the identified parameters, and the error due to the structural bias of the affine models.
This is achieved through the introduction of three novelties. The first novelty is the introduction of external structural bias, based on which the model locality is incorporated into the computation of the expected squared error. This allows better ranking between different cluster merges. The second novelty is the introduction of the desired usage area and internal structural bias, through which a better estimation of the squared error of the model is achieved, especially for local models with few samples. The last novelty is a new way of accounting for the input noise. While the results of the developed method are only comparable to existing methods, worse than anticipated results are attributed to the poor fusion capabilities of Takagi-Sugeno and fuzzy parameter models. The Takagi-Sugeno fusion performs unexpectedly due to the trade-off between accuracy and transparency, while the fusion of the fuzzy parameter model does not take into account the shape of the extracted domains yet, which is critical especially for dynamical systems in this thesis. Regardless of the choice of clustering, the fuzzy parameter model achieves much better results compared to Takagi-Sugeno models, provided that the parameters are identified by local identification.
Neither the fuzzy parameter model nor the partitioning with the introduction of model quality is specifically adapted for modelling time-varying processes. Therefore, in the last part of the thesis, an innovative partitioning of the space in the vertical direction is presented, where a simpler model, whose model parameters can be adapted more quickly, is shrunk by more complex models. This combines the ability of more complex models to model non-linear processes and faster parameter convergence of simpler models. The identification of the affine constant is performed based on the errors of the affine model on samples. In the identification of the affine model, the locality is in the time space, so only samples that are close to the current moment in time are used for estimation. The algorithm, how many samples are needed for estimation with the least squared error is also proposed.
The cascade model is evaluated in combination with a predictive controller and compared to other recursive models. Using cascade model improve process control in the presence of abrupt changes in process dynamics compared to existing modelling approaches for linear processes.
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