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2.3 Elementary links

Lecture






  2.3 Elementary links



Elementary links are called the simplest components (blocks) of a system whose behavior is described by algebraic equations or differential equations of the 1st – 2nd order:



(2.50)   2.3 Elementary links ,



Where   2.3 Elementary links - output variable   2.3 Elementary links - input variable   2.3 Elementary links - constant coefficients (parameters). Equation (2.50) can be written in operator form:



  2.3 Elementary links ,



those. the transfer function of the link is



(2.51)   2.3 Elementary links   2.3 Elementary links .



  2.3 Elementary links



Proportional (inertia-free) link . The link is described by an algebraic equation.



(2.52)   2.3 Elementary links ,



  2.3 Elementary links



Where   2.3 Elementary links - coefficient of proportionality, which (due to the lack of inertial properties of the block) coincides with the static characteristic. Proportional transition function -



(2.53)   2.3 Elementary links .



Examples: measuring potentiometers, gearboxes, voltage amplifiers (   2.3 Elementary links ) etc.



Aperiodic link. The link is described by a differential equation



(2.54)   2.3 Elementary links



or, in reduced form, by the equation



  2.3 Elementary links



(2.55)   2.3 Elementary links ,



Where   2.3 Elementary links - coefficient,   2.3 Elementary links - time constant, a = K / T, b = 1 / K. The operator form of the link has the form



(2.56)   2.3 Elementary links



or accordingly



(2.57)   2.3 Elementary links ,



The transition function of the link is determined by the expression



  2.3 Elementary links







Fig. 2.10. Transition function aperiodic link





(2.58)   2.3 Elementary links ,





  2.3 Elementary links



and the static characteristic is



(2.59)   2.3 Elementary links .



Examples: power amplifiers, thermal processes, engine acceleration process   2.3 Elementary links - chain (see example 1.1), LR chain.





Integrating link. The link is described by a differential equation



  2.3 Elementary links



(2.60)   2.3 Elementary links



or, in operator form



(2.61)   2.3 Elementary links .



Transition function integrator



(2.62)   2.3 Elementary links .



The link belongs to astatic blocks and therefore has no static characteristic.



  2.3 Elementary links



Fig. 2.11. Transition function integrator







Examples: elements of mechanical systems (see the movement of a material point, example 2.3), described by the equations of the dynamics of a type



  2.3 Elementary links



  2.3 Elementary links ,



and kinematic equations



  2.3 Elementary links ;



electronic integrators (   2.3 Elementary links ) etc.





Differentiating link (perfect). The link is described by a differential equation



  2.3 Elementary links



(2.63)   2.3 Elementary links



or, in operator form,



(2.64)   2.3 Elementary links .



Differential Link Transition Function -



(2.65)   2.3 Elementary links ,



and the response of the link to the linearly increasing signal x 2 = t -



(2.66)   2.3 Elementary links .



When x 2 = const for any t> 0 ,   2.3 Elementary links and, therefore, the static characteristic of the link is a direct   2.3 Elementary links .



  2.3 Elementary links



Fig. 2.12. Differentiator response   2.3 Elementary links on linearly increasing impact   2.3 Elementary links



  2.3 Elementary links



Examples: tachogenerator (electric speed sensor), electronic differentiator (   2.3 Elementary links ).



Remark 2.4 . The output of the differentiating link is the derivative of the input signal, i.e. its instant speed is dx 2 / dt . The operation of finding the current value of the velocity x 1 (t) = dx 2 (t) / dt only from the information about the signal at the given time t currently x 2 (t) is not physically realizable and therefore there are no ideal differentiating links. However, the derivative can be approximated as   2.3 Elementary links 1 (t) = D x 2 (t) / D t , where D t is the time interval, D x 2 is the corresponding signal increment x 2 . By reducing the interval D t you can get the value   2.3 Elementary links 1 (t) , arbitrarily close to the current speed value x 1 (t) . Consequently, despite the unrealizability (with absolute precision) of the operation of differentiation, it is theoretically possible to build a link that ensures finding the derivative dx 2 (t) / dt with arbitrarily high accuracy.







Real differentiating link. The link is described by the equation



(2.67)   2.3 Elementary links .



or, in operator form,



  2.3 Elementary links



(2.68)   2.3 Elementary links



The transition function of the link is





  2.3 Elementary links



Fig. 2.13. Transition function of a real differentiator



(2.69)   2.3 Elementary links ,



and the response of the link to the linearly increasing signal x 1 = t coincides with the transition function of the aperiodic link, i.e.



(2.70)   2.3 Elementary links .



When x 2 = const and   2.3 Elementary links performed   2.3 Elementary links that corresponds to the static characteristic of the link.



With sufficiently small time constants T , the characteristics of the link approach the characteristics of the ideal differentiating link (see Remark 2.4).



  2.3 Elementary links



Fig. 2.14. Real Differential Link Reaction   2.3 Elementary links on linearly increasing impact   2.3 Elementary links



Examples: CR and RL chains.



  2.3 Elementary links   2.3 Elementary links





Oscillating link . The link is described by a 2nd order differential equation.



(2.71)   2.3 Elementary links ,



  2.3 Elementary links - time constant   2.3 Elementary links - attenuation parameter, or operator equation (2.50), where the transfer function is



(2.72)   2.3 Elementary links .



The roots of the characteristic equation take values



  2.3 Elementary links   2.3 Elementary links ,



Where   2.3 Elementary links - attenuation coefficient,   2.3 Elementary links - The angular frequency of oscillation.



The transition function of the link is



(2.73)   2.3 Elementary links ,



Where   2.3 Elementary links ;   2.3 Elementary links , and the static characteristic is



  2.3 Elementary links



Fig. 2.15. Oscillating transition function

(2.74)   2.3 Elementary links .





  2.3 Elementary links



Examples: pendulum in a viscous medium,   2.3 Elementary links - chain.



Remark 2.5. In the limiting case,   2.3 Elementary links At the output of the link, continuous oscillations occur, and   2.3 Elementary links - monotonous (aperiodic) process, which corresponds to the conservative and double aperiodic link considered below.



Conservative link (oscillator). The link is described by a differential equation



(2.75)   2.3 Elementary links



  2.3 Elementary links



or operator equation (2.50), where



(2.76)   2.3 Elementary links ,



and is obtained from the oscillatory level when   2.3 Elementary links . Conservative link has pure imaginary poles.



  2.3 Elementary links



and transition function



  2.3 Elementary links



Fig. 2.16. Oscillating transition function



(2.77)   2.3 Elementary links ,



Where   2.3 Elementary links . The link has no static characteristic.



  2.3 Elementary links



Examples: pendulum in vacuum; ideal oscillatory (LC) circuit.







Double aperiodic link. The link is described by the equation



(2.78)   2.3 Elementary links



  2.3 Elementary links



or operator equation (2.50), where



(2.79)   2.3 Elementary links .



The link has equal real roots of the characteristic equation



  2.3 Elementary links ,



and transient function



(2.80)   2.3 Elementary links .



  2.3 Elementary links



Fig. 2.17. Double aperiodic transition function



Static Link Characterization



(2.81)   2.3 Elementary links .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control