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State Space Representation01:27

State Space Representation

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
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A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. Due to its adaptability and capacity to withstand complex loads, the space truss is widely used in various construction projects.
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The escape velocity of an object is defined as the minimum initial velocity that it requires to escape the surface of another object to which it is gravitationally bound and never to return. For example, what would be the minimum velocity at which a satellite should be launched from the Earth's surface such that it just escapes the Earth's gravitational field?
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LSTM: A Search Space Odyssey.

Klaus Greff, Rupesh K Srivastava, Jan Koutnik

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    This study analyzed eight Long Short-Term Memory (LSTM) variants for machine learning tasks. Results show standard LSTM with its forget gate and output activation function remains optimal, offering guidelines for hyperparameter tuning.

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    Area of Science:

    • Artificial Intelligence
    • Machine Learning
    • Deep Learning

    Background:

    • Long Short-Term Memory (LSTM) networks are advanced recurrent neural networks.
    • Various LSTM architectures have been developed since 1995.
    • These networks are state-of-the-art for many machine learning tasks, driving interest in their components.

    Purpose of the Study:

    • To conduct a large-scale analysis of eight LSTM variants.
    • To evaluate the utility of different computational components within LSTM architectures.
    • To compare LSTM variants on speech recognition, handwriting recognition, and music modeling tasks.

    Main Methods:

    • Performed a large-scale analysis of 5400 experimental runs across eight LSTM variants.
    • Optimized hyperparameters for each variant and task using random search.
    • Assessed hyperparameter importance using functional ANalysis Of VAriance (fANOVA).

    Main Results:

    • No LSTM variant significantly outperformed the standard LSTM architecture.
    • The forget gate and output activation function were identified as the most critical LSTM components.
    • Studied hyperparameters demonstrated near-complete independence.

    Conclusions:

    • Standard LSTM architecture is highly effective for speech, handwriting, and music modeling.
    • Forget gate and output activation function are key to LSTM performance.
    • Derived guidelines for efficient hyperparameter adjustment in LSTMs.