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Embedded Systems Control

This overview explores control theory concepts relevant to embedded systems, emphasizing open and closed-loop control mechanisms. Key topics include the Laplace Transform, with applications to dynamics such as the spring-mass-damper system, and the feedback mechanisms involving α-motor and γ-motor neurons. The importance of negative feedback in muscle control and the historical context of cybernetics is discussed. This comprehensive guide serves as a foundation for understanding time and frequency-domain responses in motor circuits and control stability in dynamic systems. ###

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Embedded Systems Control

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  1. Embedded SystemsControl

  2. Basic Tools of Control Theory • Open- and Closed-Loop Control • Laplace Transform (just briefly) • The Canonical Spring-Mass-Damper • Equilibrium Setpoint Control • Qualitative Second-Order Response • Closed-Loop Transfer Function • Time- and Frequency-Domain Response

  3. Motor Circuits • α-motor neurons initiate overt motion---they’re fast • on average, each will innervate approximately 200 individual muscle fibers. • relatively slow γ-motor neuron regulates muscle tone by setting the reference length of the spindle receptor. • Golgi tendon organ measures the tension in the tendon and inhibits the α-motor neuron if it exceeds safe levels

  4. Negative Feedback • If the measured spindle length is greater than the reference, the α-motor neuron cause a contraction of the muscle tissue • if the measured spindle length is less than the reference, the α-motor neuron is inhibited, allowing the muscle to extend ...the α-motor neuron changes its output so as to cancel some of its input... Negative Feedback

  5. Negative Feedback • first submitted for a patent in 1928 by Harold S. Black • Black’s patent application was met with great skepticism, reportedly associated with a perpetual motion machine • subsequently, it was the basis for Watt’s governor • spawned the field of cybernetics • now heralded as a fundamental principle of stability in compensated dynamical systems

  6. The Muscle Stretch Reflex

  7. Open-Loop Control an initiating stimulus causes a response without further stimulation SR reflexes - e.g. withdrawl reflex

  8. Closed-Loop Control • a (time-varying) setpoint is achieved by constantly measuring and correcting in order to actively reject disturbances • autopilot • Norbert Weiner - cybernetics (helmsman), homeostasis, endocrine system living things as feedback regulators

  9. Laplace Transform assume: x(t) ~ Cest • L( d/dt (x(t)) ) = s L( x(t) ) • L( d2/dt2 (x(t)) ) = s2L( x(t) ) ...a linear differential equation with constant coefficients and a finite number of terms is Laplace-transformable...they transform into polynomials in s…

  10. Laplace Transform assume: x(t) ~ Cest • L( d/dt (x(t)) ) = s L( x(t) ) • L( d2/dt2 (x(t)) ) = s2L( x(t) ) an Nth order differential equation anDn + an−1Dn−1+ …+ a0 = 0 into an Nth order polynomial ansn + an−1sn−1+ …+ a0 = 0

  11. Example: and RC Circuit

  12. Example: and RC Circuit “transfer” function s

  13. Spring-Mass-Damper

  14. Spring-Mass-Damper

  15. Wall Following

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