**6. VSS example**

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

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

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:

$$\begin{cases} T^2 \ddot{\mathbf{x}} + K|\mathbf{x}| \text{sign} \mathbf{S} = \mathbf{0} \\ \mathbf{S} = T\_1 \dot{\mathbf{x}} + \mathbf{x} \end{cases} \tag{7}$$

**Figure 7.** *Block diagram of the VSS of the second order.*

#### **Figure 8.** *Replacement block diagram of the VSS of the second order.*

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

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

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

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

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

channel does not need accuracy, that is, in large number of discharges.

formulas (4)–(6) of the frequency response of the link.

**Figure 5.**

*Control Theory in Engineering*

**Figure 6.**

**334**

*in other channels.*

$$\text{Let } \mathbf{x} > \mathbf{0} \, \frac{kT\_1}{T^2} - \frac{\mathbf{1}}{T\_1} \ge \mathbf{0}; T\_1 \ge \sqrt{\frac{T^2}{k}}\tag{8}$$

1. Ideal sliding: the condition (8) is met, equivalent phase shifts of elements TG and circuit K is <sup>90</sup><sup>0</sup> minimum, frequency characteristics are presented in

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

2.Unstable mode: when the condition is not satisfied in the area of the cutoff

3. Imperfect glide: when the condition is not satisfied only in the high-frequency zone, the "slow" processes are stable. But around the sliding path, fast

movements have finite amplitude and frequency. This option is most often found in real SPS and satisfies the technical requirements in systems with sliding.

Delay links primarily affect fast movements. This was dealt with in detail in all

*Frequency characteristics of VSS: (a) with "perfect" slip, (b) if the conditions for "slow" slip are violated, (c) if*

**Figure 12a**.

**Figure 11.**

**Figure 12.**

**337**

*the conditions for "fast" slip are not met.*

*Block diagram of a VSS model with discretization elements.*

frequency, **Figure 12b**.

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

fundamental works on TAC [14, 16, 17].

The final condition links transfer functions of the controlled member ( <sup>1</sup> *<sup>T</sup>*2*p*<sup>2</sup> Þ, switch trajectory generator (*T*1*p* þ 1), and controller (К).

Along with that, the element with cutoff frequency ω ¼ ffiffiffiffi *K T*2 q is a controlled member engaged in the feedback with controller with *К* coefficient.

The slip condition turned out to be equivalent to the stability condition of the equivalent circuit with a relay element.

These conditions were extended to a system with arbitrary links with frequency characteristics: WCO for the control object, WTG for the shaper of the switching path ("slip"), and WC for the amplifier.

The corresponding replacement block diagram is shown in **Figure 10**.

The condition (8) may be "transferred" to the frequency characteristics of EMS elements as follows: The condition of ideal sliding is met when two elements—the sliding trajectory generator and the circuit formed by the controller and controlled member—are connected in series with equivalent phase characteristic of �90° minimum, and the value of �90° is reached at ω ! **∞**.

The suggested frequency condition is met if the real part of frequency characteristics under consideration transferred to the complex space is positive:

$$\begin{aligned} \operatorname{Re}\left[\boldsymbol{W}\_{K}\cdot\boldsymbol{W}\_{TG}\right] &> \mathbf{0} \\ \boldsymbol{\varrho}[\boldsymbol{W}\_{K}\cdot\boldsymbol{W}\_{TG}] &> -\mathbf{90}^{\circ} \\ \boldsymbol{W}\_{K} &= \frac{\boldsymbol{W}\_{C}\cdot\boldsymbol{W}\_{CO}}{\mathbf{1} + \boldsymbol{W}\_{C}\cdot\boldsymbol{W}\_{CO}} \end{aligned} \tag{9}$$

**Figure 11** shows the direct correlation between the condition (8) and this assumption.

**Figure 9.**

*Block diagram of the VSS of an arbitrary order system.*

**Figure 10.** *Replacement block diagram of the VSS of an arbitrary order system.*

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


Delay links primarily affect fast movements. This was dealt with in detail in all fundamental works on TAC [14, 16, 17].

#### **Figure 11.**

*at x*<sup>&</sup>gt; <sup>0</sup> *kT*<sup>1</sup>

member engaged in the feedback with controller with *К* coefficient.

switch trajectory generator (*T*1*p* þ 1), and controller (К). Along with that, the element with cutoff frequency ω ¼

minimum, and the value of �90° is reached at ω ! **∞**.

equivalent circuit with a relay element.

("slip"), and WC for the amplifier.

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assumption.

**Figure 9.**

**Figure 10.**

**336**

*Block diagram of the VSS of an arbitrary order system.*

*Replacement block diagram of the VSS of an arbitrary order system.*

*<sup>T</sup>*<sup>2</sup> � <sup>1</sup> *T*1

The final condition links transfer functions of the controlled member ( <sup>1</sup>

The slip condition turned out to be equivalent to the stability condition of the

The corresponding replacement block diagram is shown in **Figure 10**.

teristics under consideration transferred to the complex space is positive:

These conditions were extended to a system with arbitrary links with frequency characteristics: WCO for the control object, WTG for the shaper of the switching path

The condition (8) may be "transferred" to the frequency characteristics of EMS elements as follows: The condition of ideal sliding is met when two elements—the sliding trajectory generator and the circuit formed by the controller and controlled member—are connected in series with equivalent phase characteristic of �90°

The suggested frequency condition is met if the real part of frequency charac-

*Re W*½ � *<sup>K</sup>* � *WTG* >0 *<sup>φ</sup>*½ � *WK* � *WTG* <sup>&</sup>gt; � <sup>90</sup>*<sup>o</sup> WK* <sup>¼</sup> *WC* � *WCO*

**Figure 11** shows the direct correlation between the condition (8) and this

1 þ *WC* � *WCO*

≥0; *T*<sup>1</sup> ≥

ffiffiffiffiffi *T*2 *k*

> ffiffiffiffi *K T*2 q

(8)

(9)

*<sup>T</sup>*2*p*<sup>2</sup> Þ,

is a controlled

s

*Frequency characteristics of VSS: (a) with "perfect" slip, (b) if the conditions for "slow" slip are violated, (c) if the conditions for "fast" slip are not met.*

**Figure 12.** *Block diagram of a VSS model with discretization elements.*

#### *Control Theory in Engineering*

Let us consider how slip conditions change with the introduction of suppression units into the structures. Consider the VSS model with slip with some modifications (**Figure 12**). An integrating channel is added to the amplification channel, hysteresis is introduced into the relay element so that the slip frequency is finite. It can be assumed that the presence of suppression links in the regulator channels will violate ideal slip conditions. As it comes from the frequency characteristics of the links with the discretizer, to ensure sufficient slip parameters, fast quantization will be required in only one of the channels—differential.

Diagram shows a 1-reference signal link «step» **Figure 12**, 2-adjustable coordinate, 3-derivative of this coordinate-link «hold2», 4-signal at the output of the proportional channel controller-link «hold», 5-output of the integrated channel-link

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

When introducing discretization links into the channels of a regulator, the following results were obtained; with sampling over all channels in 0.1 s, the slip was destroyed (**Figure 15a**). At discretization of the differential channel of 0.001 s, in the remaining discrete channels the following—0.1 s and 0.3 s the process in

At the same time, fast movements do not correspond to perfect gliding, while slow movements completely correspond to a monotonous process. Qualitatively, the processes fully comply with the theoretical principles obtained from the analysis of the frequency characteristics of the system links for compliance with the sliding

This simulation is yet another confirmation of the effectiveness of the analysis methodology for the suppression link and controllers with different discreteness and timing of the calculations. From the time and nature of the processes, it can be seen that the sliding processes are preserved at the necessary speed of the channel for the formation of the slip function, which is determined by the fast discretization of the differential channel. Performance enhancement channels are not required. But accuracy is required. In this case, the slip condition is violated at high frequencies; it

*Diagrams of processes in VSS with continuous elements (a) and fast discrete elements (b).*

*Diagrams of processes: (a) in VSS with slow discrete elements, (b) in VSS with fast D-channel and slow*

«hold1», 6-proportional channel controller-link «hold3».

**Figure 15b** is optimal both in accuracy and speed.

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

conditions.

**Figure 14.**

**Figure 15.**

**339**

*remaining channels.*

**Figure 13** shows the frequency characteristics of the links of the original circuit. The slip condition is satisfied in the absence of discrete elements (**Figure 13a**). If they distort the frequency characteristics of the links, as shown in **Figure 13c**, that is, in the zone of slow movements, then the process becomes unstable; in the highfrequency zone (**Figure 13b**), conditions of ideal slip are violated (infinitely high frequency and infinitesimal amplitude of "slip"), but "slow movements are stable. As can be seen from the **Figure 13b**, for the existence of "real" slip, a sufficiently high discrete frequency of only the differential channel forming the slip path

$$\pi\_i \le \frac{1}{o\nu\_1}; \pi\_p \le \frac{1}{o\nu\_2}; \pi\_d \to T\_{min}$$

To confirm these provisions and verify the effect of discretization and suppression links on them simulation was carried out (**Figures 14** and **15**).

#### **Figure 13.**

*Sliding conditions in VSS: (a) for continuous links, (b) for "fast" discretization elements in the D-channel, (c) for "slow" discretization in the D-channel.*

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

Diagram shows a 1-reference signal link «step» **Figure 12**, 2-adjustable coordinate, 3-derivative of this coordinate-link «hold2», 4-signal at the output of the proportional channel controller-link «hold», 5-output of the integrated channel-link «hold1», 6-proportional channel controller-link «hold3».

When introducing discretization links into the channels of a regulator, the following results were obtained; with sampling over all channels in 0.1 s, the slip was destroyed (**Figure 15a**). At discretization of the differential channel of 0.001 s, in the remaining discrete channels the following—0.1 s and 0.3 s the process in **Figure 15b** is optimal both in accuracy and speed.

At the same time, fast movements do not correspond to perfect gliding, while slow movements completely correspond to a monotonous process. Qualitatively, the processes fully comply with the theoretical principles obtained from the analysis of the frequency characteristics of the system links for compliance with the sliding conditions.

This simulation is yet another confirmation of the effectiveness of the analysis methodology for the suppression link and controllers with different discreteness and timing of the calculations. From the time and nature of the processes, it can be seen that the sliding processes are preserved at the necessary speed of the channel for the formation of the slip function, which is determined by the fast discretization of the differential channel. Performance enhancement channels are not required. But accuracy is required. In this case, the slip condition is violated at high frequencies; it

#### **Figure 14.**

Let us consider how slip conditions change with the introduction of suppression units into the structures. Consider the VSS model with slip with some modifications (**Figure 12**). An integrating channel is added to the amplification channel, hysteresis is introduced into the relay element so that the slip frequency is finite. It can be assumed that the presence of suppression links in the regulator channels will violate ideal slip conditions. As it comes from the frequency characteristics of the links with the discretizer, to ensure sufficient slip parameters, fast quantization will be

**Figure 13** shows the frequency characteristics of the links of the original circuit. The slip condition is satisfied in the absence of discrete elements (**Figure 13a**). If they distort the frequency characteristics of the links, as shown in **Figure 13c**, that is, in the zone of slow movements, then the process becomes unstable; in the highfrequency zone (**Figure 13b**), conditions of ideal slip are violated (infinitely high frequency and infinitesimal amplitude of "slip"), but "slow movements are stable. As can be seen from the **Figure 13b**, for the existence of "real" slip, a sufficiently high discrete frequency of only the differential channel forming the slip path

> 1 *ω*2

To confirm these provisions and verify the effect of discretization and suppres-

*Sliding conditions in VSS: (a) for continuous links, (b) for "fast" discretization elements in the D-channel,*

; *τ<sup>d</sup>* ! *Tmin*

required in only one of the channels—differential.

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*τ<sup>i</sup>* ≤ 1 *ω*1 ; *τ<sup>p</sup>* ≤

**Figure 13.**

**338**

*(c) for "slow" discretization in the D-channel.*

sion links on them simulation was carried out (**Figures 14** and **15**).

*Diagrams of processes in VSS with continuous elements (a) and fast discrete elements (b).*

#### **Figure 15.**

*Diagrams of processes: (a) in VSS with slow discrete elements, (b) in VSS with fast D-channel and slow remaining channels.*

#### *Control Theory in Engineering*

means, that fast movements are imperfect, which matches the model. This confirms the validity of the previously derived criteria for sliding along the frequency response and the effectiveness of the proposed frequency response suppression links for assessing the dynamics of even complex nonlinear control systems.
