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Intelligent Control for MISO Nonlinear Systems
Published in Yung C. Shin, Chengying Xu, Intelligent Systems Modeling, Optimization, and Control, 2017
where ∘ is the composition operator of fuzzy relations. Suppose the number of MFs assigned for each input variable is N, and then a total of N3 fuzzy rules will be required to cover N × N × N possible combinations of the input fuzzy sets. The rule base with three inputs becomes rather large as N increases. Because of the multidimensionality of the fuzzy relation R, the compositional fuzzy infer-encing is difficult to perform and the computational time will be too long for real-time implementation. To overcome these difficulties, it is proposed to break up the inferencing of the multidimensional rule base into parts, so that the fuzzy inferencing is easier to perform. The fuzzy PID structure is analyzed and a fuzzy PD–PI controller is formed by the following fuzzy derivation:
Models and Tools for Complex Embedded Software and Systems
Published in Luciano Lavagno, Igor L. Markov, Grant Martin, Louis K. Scheffer, Electronic Design Automation for IC System Design, Verification, and Testing, 2017
Composing a set of components means applying a set of rules at each of the three levels. The rules defined in Reference 118 define an associative and commutative composition operator. Further, the composition model allows for composability (properties of components are preserved after composition) and compositionality (properties of the compound can be inferred from properties of the components) with respect to deadlock freedom (liveness).
DETONATE: Nonlinear Dynamic Evolution Modeling of Time-dependent 3-dimensional Point Cloud Profiles
Published in IISE Transactions, 2023
Michael Biehler, Daniel Lin, Jianjun Shi
Using latent encoding, the dynamics can be approximated using a linear model, which alleviates the curse of dimensionality and allows the integration of prior physics knowledge about the data sequence into the training process. This is a direct implication from Koopman theory, which states that there exists a data transformation for any nonlinear dynamical system so that the future states can be predicted via a linear mapping from to Formally, the Koopman operator is defined as where the dynamics induce the Koopman operator that acts on scalar functions where is a finite-dimensional manifold on (Koopman 1931). Throughout this article, we use the following notations: The function composition takes two functions and and produces a function such that A composition operator with symbol is a linear operator defined by the rule where denotes the function composition. The Koopman operator is a composition operator, as the function is composed with the map In other words, is defined as The operator models the evolution of the scalar function by utilizing future values: at is the value of evaluated at the future state We note that our data is equally spaced in time, i.e., is the sampling interval, which is the same for each time although extensions to this setting can be accommodated (Tu et al., 2014). Intriguingly, it is easy to show that is linear for any