Analyse Temporalis Systematis Controlis

In systemate de controllo, elementa quaedam conservativa ad eum adiuncta esse possunt. Elementa conservativa sunt generaliter inductores et condensatores in casu systematis electrici. Owing to the presence of these energy storing elements, if the energy state of the system is disturbed, it will take a certain time to change from one energy state to another. The exact time taken by the system for changing one energy state to another is known as transient time and the value and pattern voltages and currents during this period are known as the transient response.

Responsus transitorius est normaliter cum oscillatione associatus, quae sustentari aut decrescere potest natura. Natura exacta systematis dependet a parametri systematis. Omne systema per aequationem differentialem linearem repraesentari potest. Solutio huius aequationis differentialis linealis dat responsionem systematis. Repraesentatio systematis controlis per aequationem differentialem linealem functionum temporis et solutio eius collectiva dicitur analyse in temporalibus systematis controlis.

Functio Gradualis

Ducamus fontem tensionis independentem vel bateriam quae ad voltmeter per commutatorem s iuncta est. Clarum est ex figura infra, quando commutator s apertus est, tensio inter terminos voltmeter nullus apparet. Si tensio inter terminos voltmeter v (t) representetur, haec situatio mathematica sic repraesentari potest

Nunc consideremus t = 0, commutator clausus est et instantanea tensio bateriae V voltis apparuit inter terminos voltmeter et haec situatio sic repraesentari potest,

Combining the above two equations we get

In the above equations if we put 1 in place of V, we will get a unit step function which can be defined as

Now let us examine the Laplace transform of unit step function. Laplace transform of any function can be obtained by multiplying this function by e-st and integrating multiplied from 0 to infinity.
Fig 6.2.1

If input is R(s), then

Functio Inclinata

Functio quae linea recta inclinata originem secans repraesentatur, functio inclinata dicitur. Hoc est, haec functio ab zero incipit et lineariter cum tempore crescendo vel decrescendo. Functio inclinata sic repraesentari potest,

Here in this above equation, k is the slope of the line.
Fig 6.2.2
Now let us examine the Laplace transform of ramp function. As we told earlier Laplace transform of any function can be obtained by multiplying this function by e-st and integrating multiplied from 0 to infinity.

Functio Parabolica

Hic, valor functionis est nullus quando tempus t<0 et est quadraticus quando tempus t > 0. Functio parabolica sic definiri potest,

Now let us examine the Laplace transform of parabolic function. As we told earlier Laplace transform of any function can be obtained by multiplying this function by e-st and integrating multiplied from 0 to infinity.
Fig 6.2.3

Functio Impulsus

Signal impulsus producitur quando input subito applicatur ad systema pro duratione temporis infinitesimali. Forma undarum huius signalis repraesentatur ut functio impulsus. Si magnitudo huius functionis est unitas, functio vocatur unit impulsi. Derivatum temporale primum functionis gradualis est functio impulsus. Ergo transformata Laplace functionis unit impulsus nihil aliud est quam transformata Laplace derivati temporales primi functionis graduali.
Fig 6.2.4

Responsum Temporis Systematum Controlorum Primae Ordinis

Cum potentia maxima s in denominatore functionis transferendi sit una, functio transferendi repraesentat systema controlis primae ordinis. Communiter, systema controlis primae ordinis sic repraesentari potest

Responsum Temporis pro Functio Gradualis

Nunc input unit graduali systemati datum est, tum examinemus expressionem output:

Fig 6.3.2It is seen from the error equation that if the time approaching to infinity, the output signal reaches exponentially to the steady-state value of one unit. As the output is approaching towards input exponentially, the steady-state error is zero when time approaches to infinity.

Let us put t = T in the output equation and then we get,

This T is defined as the time constant of the response and the time constant of a response signal is that time for which the signal reaches to its 63.2 % of its final value. Now if we put t = 4T in the above output response equation, then we get,

When the actual value of the response reaches to the 98% of the desired value, then the signal is said to be reached to its steady-state condition. This required time for reaching the signal to 98 % of its desired value is known as setting time and naturally setting time is four times of the time constant of the response. The condition of response before setting time is known as transient condition and condition of the response after setting time is known as steady-state condition. From this explanation, it is clear that if the time constant of the system is smaller, the response of the system reaches its steady-state condition faster.

Responsum Temporis pro Functio Inclinata

In this case, during the steady-state condition, the output signal lags behind the input signal by a time equal to the time constant of the system. If the time constant of the system is smaller, the positional error of the response becomes lesser.

Responsum Temporis pro Functio Impulsus

In the above explanation of time response of the control system, we have seen that the step function is the first derivative of ramp function and the impulse function is the first derivative of a step function. It is also found that the time response of step function is the first derivative of time response of ramp function and time response of impulse function is the first derivative of time response of step function.

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