**4. Formula of transfer function and frequency characteristics of suppression link**

The desired formula may look like this:

At the same time, it is said that the sampling time is "small," although it is not

In this case, the phase characteristic of the delay link shown in **Figure 2** allows a formal possibility of its correction by successive links, but this is only in the case if the correction frequencies are far from the quantization frequency, and since there is a "veto" for their "rapprochement"—the prohibition of the original methodology—this possible correction is simply excluded by the method itself. Over the past years, naturally, a lot of works on these topics have been written and published. But, practically, in almost all approaches remained the same. All methods are based on discrete Laplace transforms. The operator in these transformations is replaced by the exponential function of the delay unit, and the sampling time is included in these transformations by a parameter. Frequencies close to the

"The cutoff frequency of the system must be less than the quantization frequency by at least 10 times, otherwise nothing can be guaranteed." Moreover, the delay link does not change the amplitude frequency response, that is, when the condition of "smallness" of the quantization interval is satisfied, one can

Thus, the "traditional" ACS theory offers two fundamental approaches:

1.Go to discrete transformations and translate ALL ACS elements into discrete formats, inevitably simplifying nonlinearities, high-order links are complex structural relationships and then operate with discrete criteria and methods.

2."Work" only in the frequency range of 10 or more times less, than sampling

It does not take a lot of imagination to understand that the second approach is

One of the most commonly used devices in electromechanical systems is pulsed power amplifiers—frequency converters for asynchronous drives and voltage converters for DC drives. The switching frequency of power elements is usually in the range from 4 to 16 kHz. Mechanical processes in these systems range from 0 to 20 Hz. That is, the condition of "smallness" of the switching period of pulse elements is fulfilled. The frequency of clocking of control signals in microprocessors is

Many years of experience with electromechanical systems, theoretical research,

and simulation showed that reducing the discretization links to the delay links according to the methods mentioned above is ineffective. This inefficiency is reflected in the inability to describe the influence of discrete links on processes in ACS, especially complex and nonlinear, and in the fact that they do not allow the formation of an effective correction of such systems. It seems appropriate to distinguish two features of the traditional representation of discretization links: firstly, the formal possibility of correcting the phase shift, and secondly, the invariability of the amplitude characteristics. To overcome these problems, it was proposed to introduce a suppression link into systems with signal sampling, the main property of which is the complete suppression of input signals with a frequency higher than

specified how small it should be and how wrong if "not small." For engineers, this condition sounds something like this:

completely forget about it.

*Control Theory in Engineering*

clock frequency are not considered [3–10].

rate neglecting her at all.

**3. Suppression link**

**330**

chosen more often in engineering calculations and studies.

most often not mentioned even in advertising materials for converters.

$$W = A(\alpha)e^{j\varphi(w)}$$

$$\varphi(w) = \begin{cases} -\frac{K\_1 \cdot (\pi \alpha)}{1 - \alpha \pi}, & \text{if } \alpha \le \frac{1}{\pi} \\\\ -\infty, & \text{if } \alpha > \frac{1}{\pi} \end{cases} \tag{4}$$

$$A(\alpha) = \begin{cases} K\_2 \cdot e^{\frac{1}{\alpha \pi - 1}}, & \text{if } \alpha \le \frac{1}{\pi} \\\\ 0, & \text{if } \alpha > \frac{1}{\pi} \end{cases} \tag{5}$$

$$\text{Log}[A(\alpha)] = \begin{cases} \frac{K\_3}{\alpha \pi - 1}, & \text{if } \alpha < \frac{1}{\pi} \\\\ & 1 \end{cases} \tag{6}$$

1

$$\left| \begin{array}{c} -\infty, \quad \text{if } \ a > \frac{1}{\tau} \end{array} \right.$$
 A graphical interpretation of the suppression link is shown in Figure 2. A of the formula (4)-(6) are phase- and amplitude-frequency characteristics. They differ from formula (3), especially in the frequency zone close to the clock frequency and show that in this frequency zone a signal is suppressed, which cannot be overcome by sequential correction, since no serial link can overcome the amplitude suppression by formula (3), the phase shift (2), at the lower frequencies, similar to the shift of the delay unit in the zone of the clock frequency, increases sharp and also

cannot be seriously corrected. **Figure 3** shows the logarithmic characteristics of the suppression link—

amplitude and phase characteristics.

According to these characteristics, the features of the proposed suppression link are very clearly visible.

At the clock frequency and higher, in the ACS "after" the suppression link, no sequential correction and feedback of the system will work. Disturbances at these frequencies will also not be worked out by the regulators.

Since at a frequency equal to the clock frequency, the phase takes the conditional value (� ∞), no sequential correction can overcome this limitation, unlike the phase characteristics of the links proposed in the sources [1, 2], which can theoretically be corrected.

As follows from the formulas and frequency characteristics of the proposed suppression link, for any sequential correction at a frequency below the quantization frequency, the phase shift will reach a critical value of �180° and lead to instability of the closed loop. Depending on other parts of the system, how far from the quantization frequency this will happen?

It should be noted that the negative phase shift increases much faster than the suppression of the amplitude coefficient. So, at a frequency three times lower than

In this knob, the P-channel is responsible for the speed of the system and for the overall dynamics of the control loop, the D-channel provides system stability, and

If we imagine the frequency characteristics of the controller as a combination of the frequency characteristics of the channels and links of suppression, it turns out, that the equivalent characteristic does not change if the discreteness of the proportional channel and the integrator is significantly slowed down (**Figure 4**). Since the links in the PID controller are connected in parallel, their resulting frequency characteristics can be determined by the "top-notch" rule. Thus you can see that the decisive role in this controller is played only by the quantization frequency of the differential channel; with its decrease (**Figure 4c**), the differentiating properties of

The simplicity of the model makes it easy to repeat this simulation and make sure it is correct. The control object was represented by a double integrator with an integration constant of 1 s. Here "Gain" is the channel of proportional gain with K = 10, "Deriv" is a differentiating channel with a time constant of 2.2 s, and

The parameters of the PID controller, in the continuous version of the model,

In **Figures 5** and **6**, a diagram shows a 1–reference signal, 2–adjustable coordinate, 3–derivative of this coordinate, 4–signal at the output of the proportional channel of the PID controller, 5–output of the integrated channel. Continuous links

Then, three quantizers were introduced into the control channels. At quantiza-

With an increase in the quantization time (0.3 s), the processes became oscilla-

At high speeds, the channel for differentiating the discreteness of the proportional

tion values of 0.01 s, the processes did not differ from continuous systems.

tory. The PID controller becomes equivalent to the PI controller (**Figure 3c**). Further, in the differential channel, the discreteness is significantly reduced (0.01 s). And in other channels this discreteness still increased; so, in the proportional channel this discreteness is 0.1 s and in the integrator 0.3 s. The results are

and integral channels practically does not affect the stability of ACS. If you pay attention to the process diagrams, the following can be noted: the time of transients in

*Block diagram of a model of AСS with a PID controller with discrete elements.*

the I-integrator provides high static accuracy of the control system.

*Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems…*

On a fairly simple model, these provisions are fully confirmed.

"Trans" is an integrating channel with a time constant of 15 s.

synthesized a process bordering on the oscillations.

the controller deteriorate significantly.

*DOI: http://dx.doi.org/10.5772/intechopen.91467*

simulate processes, shown in **Figure 5a**.

shown in **Figure 6b**.

**Figure 4.**

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**Figure 3.**

*Frequency characteristics of the PID controller: (a) without discrete elements, (b) with a "fast" discrete element in the D-channel, and (с) with a "slow" discrete element in the D-channel.*

the clock frequency, the suppression coefficient is 0.8, and the phase shift is 90°, that is, it already significantly affects the stability of a closed system with a sampling unit.

It should be noted that according to their transfer functions, the suppression links for stability analysis of a closed loop can be converted in the same way as other dynamic links. In addition, this discreteness representation allows us to consider systems with several links, and with different sampling clocks and does not to offer cumbersome transformations. This significantly distinguishes the proposed mathematical apparatus from discrete transformations, in which each circuit of links required its own calculations of discrete transfer functions [1, 2].

Let us consider several examples of applications of these links in the structures of widely known ACS variants.

These will be proportional-integral-differential controllers (PID controllers) of control systems, variable structure systems (VSS), in which ideal sliding modes (SM) and asynchronous electric drive control systems are synthesized.
