**5. PID controller**

It is known that the PID controller is the most widely used type of controller in industrial automation.

#### *Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems… DOI: http://dx.doi.org/10.5772/intechopen.91467*

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

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 controller deteriorate significantly.

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

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

The parameters of the PID controller, in the continuous version of the model, synthesized a process bordering on the oscillations.

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

Then, three quantizers were introduced into the control channels. At quantization values of 0.01 s, the processes did not differ from continuous systems.

With an increase in the quantization time (0.3 s), the processes became oscillatory. 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 shown in **Figure 6b**.

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

**Figure 4.** *Block diagram of a model of AСS with a PID controller with discrete elements.*

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

*Frequency characteristics of the PID controller: (a) without discrete elements, (b) with a "fast" discrete element*

Let us consider several examples of applications of these links in the structures

These will be proportional-integral-differential controllers (PID controllers) of control systems, variable structure systems (VSS), in which ideal sliding modes

It is known that the PID controller is the most widely used type of controller in

required its own calculations of discrete transfer functions [1, 2].

*in the D-channel, and (с) with a "slow" discrete element in the D-channel.*

(SM) and asynchronous electric drive control systems are synthesized.

of widely known ACS variants.

**5. PID controller**

**Figure 3.**

*Control Theory in Engineering*

industrial automation.

**332**

transient time. Meanwhile, it is a property of discretizers to limit the frequency range of the action of links connected to it by the clock frequency, which can be very useful in correcting systems with nonlinear frequency characteristics. The most widely encountered nonlinear systems at present are asynchronous electric

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

Variable-structure systems (VSSs) are an example of nonlinear control systems, the purpose of which is to obtain maximum performance in control systems. Their implementation in modern microprocessor controllers inevitably faces the problem of discreteness of control signals. Of interest is how the transfer function of the suppression link "manifests" itself in systems with a variable structure with sliding processes. **Figures 7** and **8** show the simplest VSS scheme with a sliding mode (SM). Here, CO is the control object (second-order integrator); TG, the shaper of the

It is known that the sliding process is characterized by infinitely fast structure switching. What happens if a suppression link appears in the channel for calculating the switching path? In [14, 15, 18], the slip condition was given for an arbitrary system whose links are described by non-differential equations and frequency

Consider the ideal slip conditions for a second-order control object—EMS with sliding (the circuit is presented in **Figure 9**) described by the following equation:

*x*€ þ *K x*j j*signS* ¼ 0

(7)

drives, which are discussed below.

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

switching trajectory ("slip"); and C is the amplifier.

response. We briefly recall the main points of this conclusion.

(

*T*2

*S* ¼ *T*1*x*\_ þ *x*

**6. VSS example**

**Figure 7.**

**Figure 8.**

**335**

*Block diagram of the VSS of the second order.*

*Replacement block diagram of the VSS of the second order.*

**Figure 5.**

*Diagrams of processes: (a) in continuous AСS with a PID controller, (b) with "fast" discrete elements.*

#### **Figure 6.**

*Diagrams of processes: (a) with "slow" discrete elements and (b) with fast sampling in the D-channel and slow in other channels.*

all diagrams is approximately the same; it does not depend on the clock frequency. The oscillation period is also unchanged. Only the degrees of vibration of the processes differ—from almost monotonic processes to unstable oscillations. This suggests that the cutoff frequency of the circuit is almost unchanged. But only the phase shifts at this point of the frequency response change. That is, changes in the quantization clocks change the cutoff frequency only slightly, since a sharp decrease in the amplitude characteristic begins near the clock frequency. And at frequencies three times smaller, the phase response shift significantly increases, which corresponds to formulas (4)–(6) of the frequency response of the link.

This shows that the sampling operation can very reasonably allocate controller resources. The integrated channel can have many discharges but a large cycle of calculations, not limited in any way by the cutoff frequency of the circuit as a whole, and the differential channel can have a fast pace of calculations, but this channel does not need accuracy, that is, in large number of discharges.

It is clear that it would hardly have been possible to find and justify such a solution using discrete transformations and related synthesis methods. According to the provisions of the theory of impulse systems set forth in classical works [1, 2, 13] and in their modern interpretations [4–7], it would be necessary to single out one impulse link and all the others "turn" to the option with a simple link. Even less likely is such a solution to be found in the neglect of the discretizer [1, 2] method, which would require a significantly higher sampling frequency compared to the

*Discreteness in Time and Evaluation of the Effectiveness of Automatic Control Systems… DOI: http://dx.doi.org/10.5772/intechopen.91467*

transient time. Meanwhile, it is a property of discretizers to limit the frequency range of the action of links connected to it by the clock frequency, which can be very useful in correcting systems with nonlinear frequency characteristics. The most widely encountered nonlinear systems at present are asynchronous electric drives, which are discussed below.
