**4. Conventional control system for synchronous generator's rotor excitation equipment**

Two principal control systems affect a synchronous generator directly: a governor control system and an excitation control system. The governor control system controls the mechanical power from a steam or water turbine by opening or closing valves regulating the steam or water flow. The response of the governor control system is too slow to damp the synchronous generator's oscillations, which are mainly in the frequency range 0.5–2.5 Hz. Damping the oscillations is possible only with the excitation control system. The excitation control system (also called an automatic voltage regulator) changes the rotor field voltage (and current) in such a way that the generator's output voltage is the same (or close enough) to the reference voltage. In modern power plants, the thyristor or transistor rectifiers are used mainly to generate the required voltage for rotor winding. The electrical power flow from the excitation system is much smaller than the mechanical power flow. This, and the fact that semiconductor components are used in the excitation system instead of mechanical ones, is the reason that the excitation system is significantly faster than the governor system. Therefore, an exciter is used for the damping of the oscillations.

A conventional linear PSS approach is based on utilization of the static excitation system. Through this system, the PSS changes the field excitation voltage of a synchronous generator. An additional component of an electrical torque is generated as a consequence. This torque must be in phase with the rotor speed and thus increases damping of the synchronous generator [9]. **Figure 20** presents a block diagram of the Heffron-Phillips model of synchronous generator equipped with an excitation system, voltage controller, and power system stabilizer [10]. The generator's output voltage is compared with a reference voltage, and the calculated error is driven to the rectifier with an integrated voltage controller. The rectifier with the voltage controller is presented with the first-order model. The PSS input represents one or more signals in which oscillations are visible. The PSS generates an additional signal, which is added to the voltage error.

#### **Figure 20.**

*Block diagram of the Heffron-Phillips model of synchronous generator equipped with excitation system, voltage controller, and power system stabilizer.*

From the analysis of the effect of different loadings on the synchronous generator dynamic characteristics, it can be concluded that the variations in the machine dynamics are considerable in the entire operating range, and, therefore, a control system is necessary for damping of the oscillations. From the comparison of the responses across different operating points, it is obvious that the present system is nonlinear and that the conventional linear control theory does not provide adequate damping throughout the entire operating area. Therefore, the implementation of a

*Synchronous generator outputs' trajectories: Rotor speed* ω*(*t*) [pu] and rotor angle* δ*(*t*) [degrees], operating*

*Synchronous generator outputs' trajectories: Generated electrical power* P*e(*t*) [pu] and stator terminal voltage*

robust or adaptive control theory is meaningful.

V*t(*t*) [pu], operating point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

**Figure 19.**

**256**

**Figure 18.**

*Automation and Control*

*point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

The symbols in **Figure 20** represent the following: *k*es and *T*es are the excitation system gain [pu.] and the time constant [s], respectively, *E*fd,ref is the reference for field excitation voltage *E*fd (both in [pu]), while *u*PSS, *y*PSS, and *G*PSS(s) are the PSS input, the output (both in [pu]), and the transfer function, respectively. As for the variables of the Heffron-Phillips model, subscript <sup>Δ</sup> denotes the deviation of the variables from the steady-state operating points, and *s* is the Laplace complex variable.

classes of control problems, new mathematical methods, and new prospects of

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement*

A modification of the sliding mode control based on the decoupling principle will be used for the proposed PSS design. The mathematical model of the controlled

where **A**RFij (i,j = 1,2) and **B**RF2 are constant matrices of relevant dimensions, **<sup>x</sup>**RF1ð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*n*�*<sup>m</sup>* and **<sup>x</sup>**RF2ð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>m</sup>* are state-space vectors, and **<sup>u</sup>**p(*t*) is a controlled

For the PSS design being based on the simplified linearized model of synchronous generator, a state-space vector in regular form **x**RF(*t*), where n = 3 and m = 1,

Sliding mode control for implementation in PSS requires knowledge of all statespace variables of the synchronous generator's regular form model. Measurements of electrical power, rotor speed, and terminal voltage are feasible only at the synchronous generator. For the sliding mode control, the state-space variables for the regular form model need to be calculated from the measured variables. To calculate regular form state-space variables, firstly, variables *δ*Δ(*t*) and *E*'qΔ(*t*) can be calcu-

*δ*Δð Þ*t* \_ *δ*Δð Þ*t*

> *<sup>K</sup>*<sup>6</sup> ‐*K*<sup>2</sup> ‐*K*<sup>5</sup> *<sup>K</sup>*<sup>1</sup>

<sup>2</sup>*<sup>H</sup>* � *<sup>K</sup>*2*ω*<sup>s</sup> 2*H*

**<sup>s</sup>**ðÞ¼ *<sup>t</sup>* **Dx**RF1ðÞþ*<sup>t</sup>* **<sup>x</sup>**RF2ð Þ*<sup>t</sup>* , **<sup>s</sup>**ð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>m</sup>* (26)

**x**\_ RF1 ¼ ð Þ **A**RF11 � **A**RF12**D x**RF1 (27)

� � *Pe*Δð Þ*t*

2 6 4

*Vt*Δð Þ*t*

*δ*Δð Þ*t ω*Δð Þ*t E*0 *<sup>q</sup>*Δð Þ*t* 3 7

<sup>5</sup> (25)

� � (24)

� � and **<sup>x</sup>**RF2ðÞ¼ *<sup>t</sup>* €*δ*Δð Þ*<sup>t</sup>* (23)

**x**\_ RF1ðÞ¼ *t* **A**RF11**x**RF1ð Þþ*t* **A**RF12**x**RF2ð Þ*t* (21)

**x**\_ RF2ðÞ¼ *t* **A**RF21**x**RF1ðÞþ*t* **A**RF22**x**RF2ðÞþ*t* **B**RF2**u**pð Þ*t* (22)

implementation [13–15].

could be selected as

**x**RF1ð Þ*t* **x**RF2ð Þ*t*

lated by inverting (Eq. (19)), such as

*δ*Δð Þ*t E*0 *<sup>q</sup>*Δð Þ*t* " #

*δ*Δð Þ*t* \_ *δ*Δð Þ*t* €*δ*Δð Þ*<sup>t</sup>*

the system behavior is governed by (*n-m*)-th-order equation

2 6 4

the sliding surface was selected:

<sup>¼</sup> <sup>1</sup>

� *<sup>K</sup>*1*ω*<sup>s</sup>

*K*1*K*<sup>6</sup> � *K*2*K*<sup>5</sup>

Finally, state-space variables **x**RF1(*t*) and **x**RF2(*t*) can be calculated with trans-

100 0 ω<sup>r</sup> 0

<sup>2</sup>*<sup>H</sup>* � *<sup>D</sup>ω*<sup>s</sup>

In such a way, the state-space variables could be obtained without explicit

A sliding surface was selected, such that the rotor's angle deviation and rotor's speed deviation converged exponentially to zero. For this aim, a linear equation of

When the sliding mode appears on manifold **s**(*t*) = 0 where **x**RF2(*t*) = –**Dx**RF1(*t*),

To obtain the required dynamic properties of the control system, we assigned eigenvalues of a closed-loop system with a linear feedback. For the controllable

**x**RFðÞ¼ *t*

formation

differentiation.

**259**

plant must be transformed to a regular form:

*DOI: http://dx.doi.org/10.5772/intechopen.92558*

plant input vector. Matrix **B**RF2 must be nonsingular.

� � where : **<sup>x</sup>**RF1ðÞ¼ *<sup>t</sup>*

For PSS input *u*PSSΔ, the variables must be used which contain information about oscillations. These variables are electrical power, rotor angle, rotor speed, frequency, terminal voltage, and acceleration torque. The electrical power is selected commonly as the input to the PSS. The output of the PSS, *y*PSSΔ, is the control signal for the excitation system. A transfer function of a conventional linear PSS is represented as follows:

$$\mathcal{G}\_{\rmPSS}(s) = \frac{y\_{\rmPSS}(s)}{u\_{\rmPSS}(s)} = k\_{\rmPSS} \left( \frac{sT\_1 + 1}{sT\_2 + 1} \right) \left( \frac{sT\_3 + 1}{sT\_4 + 1} \right) \left( \frac{sT\_w}{sT\_w + 1} \right) G\_{\rm{auf}}(s) \tag{20}$$

where *k*PSS denotes the stabilizer gain [pu]; *T*1,*T*2,*T*3, and *T*<sup>4</sup> are time constants of the stabilizers lead–lag compensators [s]; *T*<sup>w</sup> is the time constant of the high-pass (washout) filter [s]; and *G*aaf(*s*) is the transfer function of the low-pass (antialiasing) filter.

Based on the block diagram in **Figure 20** and the transfer function in (Eq. (6)), the IEEE Association established the IEEE Standard for the PSS studies [11]. The Standard enables the unification of commercial applications of PSS. The Standard sets out four basic types of PSS, which differ mainly with regard to the available input and degree of the transfer function. Most of the commercial PSS are realized on the standardized proposals.

For synthesis of a PSS, knowledge is required of a mathematical model of a synchronous generator with an excitation system. The required model is calculated from the known data of a synchronous generator, or by means of identification. Usage of systematic methods for tuning parameters of conventional PSS assures effective damping for the nominal operating point, though with a significantly decreased damping for some non-nominal operating points. The other disadvantages of these methods are the requirement of the synchronous generator mathematical model's parameters and the time-consuming tuning. Therefore, in practice, the systematic methods are rarely implemented. Hence, neither optimal damping in the nominal operating point nor stable operation is secured in the entire operating range. The implementations of an incorrectly tuned PSS could be harmful. Such PSS are, in practice, often turned off [12].

Due to a mathematical model of a synchronous generator not being available, sophisticated and time-consuming synthesis of the conventional linear PSS, and its proven non-optimum damping in the entire operating range of synchronous generator, advanced control theories are recommended for the PSS implementation.

## **5. Robust PSS**

Among many robust control approaches, the sliding mode control is one of the most interesting. The main advantages of this control are its insensitivity to parameter variations, rejection of disturbances, a decoupling design procedure, and simple implementations by means of power converters [13].

The fundamentals of the sliding mode control theory date back to the late 1950s. Since that time, new research directions emerged, due to the appearance of new

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement DOI: http://dx.doi.org/10.5772/intechopen.92558*

classes of control problems, new mathematical methods, and new prospects of implementation [13–15].

A modification of the sliding mode control based on the decoupling principle will be used for the proposed PSS design. The mathematical model of the controlled plant must be transformed to a regular form:

$$
\dot{\mathbf{x}}\_{\rm RF1}(t) = \mathbf{A}\_{\rm RF11} \mathbf{x}\_{\rm RF1}(t) + \mathbf{A}\_{\rm RF12} \mathbf{x}\_{\rm RF2}(t) \tag{21}
$$

$$
\dot{\mathbf{x}}\_{\rm RF2}(t) = \mathbf{A}\_{\rm RF21}\mathbf{x}\_{\rm RF1}(t) + \mathbf{A}\_{\rm RF22}\mathbf{x}\_{\rm RF2}(t) + \mathbf{B}\_{\rm RF2}\mathbf{u}\_{\rm p}(t) \tag{22}
$$

where **A**RFij (i,j = 1,2) and **B**RF2 are constant matrices of relevant dimensions, **<sup>x</sup>**RF1ð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*n*�*<sup>m</sup>* and **<sup>x</sup>**RF2ð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>m</sup>* are state-space vectors, and **<sup>u</sup>**p(*t*) is a controlled plant input vector. Matrix **B**RF2 must be nonsingular.

For the PSS design being based on the simplified linearized model of synchronous generator, a state-space vector in regular form **x**RF(*t*), where n = 3 and m = 1, could be selected as

$$\mathbf{x}\_{\rm RF}(t) = \begin{bmatrix} \mathbf{x}\_{\rm RF1}(t) \\ \mathbf{x}\_{\rm RF2}(t) \end{bmatrix} \quad \text{where} \; \mathbf{: } \mathbf{x}\_{\rm RF1}(t) = \begin{bmatrix} \delta\_{\Delta}(t) \\ \dot{\delta}\_{\Delta}(t) \end{bmatrix} \quad \text{and} \quad \mathbf{x}\_{\rm RF2}(t) = \ddot{\delta}\_{\Delta}(t) \tag{23}$$

Sliding mode control for implementation in PSS requires knowledge of all statespace variables of the synchronous generator's regular form model. Measurements of electrical power, rotor speed, and terminal voltage are feasible only at the synchronous generator. For the sliding mode control, the state-space variables for the regular form model need to be calculated from the measured variables. To calculate regular form state-space variables, firstly, variables *δ*Δ(*t*) and *E*'qΔ(*t*) can be calculated by inverting (Eq. (19)), such as

$$
\begin{bmatrix} \delta\_{\Delta}(t) \\ E\_{q\Delta}'(t) \end{bmatrix} = \frac{1}{K\_1 K\_6 - K\_2 K\_5} \begin{bmatrix} K\_6 & \text{-} K\_2 \\ \text{-} K\_5 & K\_1 \end{bmatrix} \begin{bmatrix} P\_{t\Delta}(t) \\ V\_{t\Delta}(t) \end{bmatrix} \tag{24}
$$

Finally, state-space variables **x**RF1(*t*) and **x**RF2(*t*) can be calculated with transformation

$$
\begin{bmatrix}
\delta\_{\Delta}(t) \\
\dot{\delta}\_{\Delta}(t) \\
\ddot{\delta}\_{\Delta}(t)
\end{bmatrix} = \begin{bmatrix}
1 & 0 & 0 \\
0 & \alpha\_{\rm r} & 0 \\
\end{bmatrix} \begin{bmatrix}
\delta\_{\Delta}(t) \\
\alpha\_{\Delta}(t) \\
E\_{q\Delta}'(t)
\end{bmatrix} \tag{25}
$$

In such a way, the state-space variables could be obtained without explicit differentiation.

A sliding surface was selected, such that the rotor's angle deviation and rotor's speed deviation converged exponentially to zero. For this aim, a linear equation of the sliding surface was selected:

$$\mathbf{s}(t) = \mathbf{D}\mathbf{x}\_{\text{RF1}}(t) + \mathbf{x}\_{\text{RF2}}(t), \quad \mathbf{s}(t) \in \mathfrak{R}^m \tag{26}$$

When the sliding mode appears on manifold **s**(*t*) = 0 where **x**RF2(*t*) = –**Dx**RF1(*t*), the system behavior is governed by (*n-m*)-th-order equation

$$
\dot{\mathbf{x}}\_{\text{RF1}} = (\mathbf{A}\_{\text{RF11}} - \mathbf{A}\_{\text{RF12}} \mathbf{D}) \mathbf{x}\_{\text{RF1}} \tag{27}
$$

To obtain the required dynamic properties of the control system, we assigned eigenvalues of a closed-loop system with a linear feedback. For the controllable

The symbols in **Figure 20** represent the following: *k*es and *T*es are the excitation system gain [pu.] and the time constant [s], respectively, *E*fd,ref is the reference for field excitation voltage *E*fd (both in [pu]), while *u*PSS, *y*PSS, and *G*PSS(s) are the PSS input, the output (both in [pu]), and the transfer function, respectively. As for the variables of the Heffron-Phillips model, subscript <sup>Δ</sup> denotes the deviation of the variables from the steady-state operating points, and *s* is the Laplace complex

For PSS input *u*PSSΔ, the variables must be used which contain information about

oscillations. These variables are electrical power, rotor angle, rotor speed, frequency, terminal voltage, and acceleration torque. The electrical power is selected commonly as the input to the PSS. The output of the PSS, *y*PSSΔ, is the control signal for the excitation system. A transfer function of a conventional linear PSS is

> *sT*<sup>1</sup> þ 1 *sT*<sup>2</sup> þ 1

(washout) filter [s]; and *G*aaf(*s*) is the transfer function of the low-pass

*sT*<sup>3</sup> <sup>þ</sup> <sup>1</sup>

where *k*PSS denotes the stabilizer gain [pu]; *T*1,*T*2,*T*3, and *T*<sup>4</sup> are time constants of the stabilizers lead–lag compensators [s]; *T*<sup>w</sup> is the time constant of the high-pass

Based on the block diagram in **Figure 20** and the transfer function in (Eq. (6)), the IEEE Association established the IEEE Standard for the PSS studies [11]. The Standard enables the unification of commercial applications of PSS. The Standard sets out four basic types of PSS, which differ mainly with regard to the available input and degree of the transfer function. Most of the commercial PSS are realized

For synthesis of a PSS, knowledge is required of a mathematical model of a synchronous generator with an excitation system. The required model is calculated from the known data of a synchronous generator, or by means of identification. Usage of systematic methods for tuning parameters of conventional PSS assures effective damping for the nominal operating point, though with a significantly decreased damping for some non-nominal operating points. The other disadvantages of these methods are the requirement of the synchronous generator mathematical model's parameters and the time-consuming tuning. Therefore, in practice, the systematic methods are rarely implemented. Hence, neither optimal damping in the nominal operating point nor stable operation is secured in the entire operating range. The implementations of an incorrectly tuned PSS could be harmful. Such PSS

Due to a mathematical model of a synchronous generator not being available, sophisticated and time-consuming synthesis of the conventional linear PSS, and its proven non-optimum damping in the entire operating range of synchronous generator, advanced control theories are recommended for the PSS implementation.

Among many robust control approaches, the sliding mode control is one of the most interesting. The main advantages of this control are its insensitivity to parameter variations, rejection of disturbances, a decoupling design procedure, and sim-

The fundamentals of the sliding mode control theory date back to the late 1950s. Since that time, new research directions emerged, due to the appearance of new

ple implementations by means of power converters [13].

*sT*<sup>4</sup> þ 1

*sTw*

*sTw* þ 1 

*G*aafð Þ*s* (20)

variable.

represented as follows:

*Automation and Control*

(antialiasing) filter.

*<sup>G</sup>*PSSðÞ¼ *<sup>s</sup> <sup>y</sup>*PSSð Þ*<sup>s</sup>*

on the standardized proposals.

are, in practice, often turned off [12].

**5. Robust PSS**

**258**

*<sup>u</sup>*PSSð Þ*<sup>s</sup>* <sup>¼</sup> *<sup>k</sup>*PSS

system described with (Eqs. (21), (22)), there exists matrix **D**, which ensures the desired eigenvalues of the system in (Eq. (27)).

In the first stage of design of the sliding mode, we chose the desired eigenvalues of the system described with (Eq. (27)). From the desired eigenvalues, we determined matrix **D** as the solution to the (*n-m*)-th-order eigenvalue task. Matrix **D** determines the equation of discontinuous sliding surfaces (Eq. (27)).

The second stage of the design procedure is the selection of the discontinuous control law, such that the sliding mode always arises at manifold **s**(*t*) = 0, which is equivalent to the stability of the origin in *m*-dimensional space **s**(*t*). The dynamics on the **s**(*t*) space are described by the equation

$$\begin{split} \dot{\mathbf{s}}(t) &= [\mathbf{D} \mathbf{A}\_{\text{RF1}1} + \mathbf{A}\_{\text{RF2}1}] \mathbf{x}\_{\text{RF1}}(t) + [\mathbf{D} \mathbf{A}\_{\text{RF1}2} + \mathbf{A}\_{\text{RF2}2}] \mathbf{x}\_{\text{RF2}}(t) + \mathbf{B}\_{\text{RF2}} \mathbf{u}(t) \\ &= \mathbf{E} \mathbf{x}\_{\text{RF}}(t) + \mathbf{B}\_{\text{RF2}} \mathbf{u}\_{\text{p}}(t) \end{split} \tag{28}$$

An appropriate choice of the control law represents the discontinuous control described with

$$\mathbf{u}\_{\rm p}(t) = -\mathbf{g}|\mathbf{x}\_{\rm RF}(t)|\mathbf{B}\_{\rm RF2}^{-1}\operatorname{sgn}\mathbf{s}(t)\tag{29}$$

where **<sup>x</sup>**pð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>n</sup>* is the controlled plant state-space vector, **<sup>u</sup>**pð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*m*is the controlled plant input vector, **<sup>y</sup>**pð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>r</sup>* is the controlled plant output vector, and

• All possible pairs **A**<sup>p</sup> and **B**<sup>p</sup> are controllable and output stabilizable.

where **x**m(*t*) is the model state vector, **u**m(*t*) is the model command vector, **y**m(*t*) is the model output vector, and **A**m, **B**m, and **C**<sup>m</sup> are matrices of appropriate dimensions. The model is assumed to be stable. The dimension of the model state

The control **u**p(*t*) for the plant output vector **y**p(t) to approximate "reasonably well" the output of the reference model **y**m(*t*) without explicit knowledge of **A**p, **B**p,

<sup>y</sup> ð Þ*<sup>t</sup>* , **<sup>x</sup>**<sup>T</sup>

The adaptive gains **K**(*t*) are obtained as a combination of the "proportional" and

where σ term is introduced in order to avoid divergence of the integral gains in the presence of disturbance and **T** and **T** are positive definite and positive semi-

The necessary condition for asymptotic tracking when **u**m(*t*) is a step command is that the controlled plant is almost strictly positive real (ASPR) [19]. If the controlled plant is not ASPR, the augmenting of the plant with a feedforward

<sup>m</sup>ð Þ*<sup>t</sup>* , **<sup>u</sup>**<sup>T</sup>

**x**\_ <sup>m</sup>ðÞ¼ *t* **A**m**x**mð Þþ*t* **B**m**u**mð Þ*t* (33)

**y**mðÞ¼ *t* **C**m**x**mð Þ*t* (34)

**e**yðÞ¼ *t* **y**mðÞ�*t* **y**pð Þ*t* (35)

**u**pðÞ¼ *t* **K**ð Þ*t* **r**ð Þ*t* (37)

h i*:* (39)

**K**ðÞ¼ *t* **K**PðÞþ*t* **K**Ið Þ*t* (40)

**<sup>K</sup>**PðÞ¼ *<sup>t</sup>* **<sup>e</sup>**yð Þ*<sup>t</sup>* **<sup>r</sup>***<sup>T</sup>*ð Þ*<sup>t</sup>* **<sup>T</sup>** (41)

**<sup>K</sup>**\_ <sup>I</sup>ðÞ¼ *<sup>t</sup>* **<sup>e</sup>**yð Þ*<sup>t</sup>* **<sup>r</sup>***<sup>T</sup>*ð Þ*<sup>t</sup>* **<sup>T</sup>** � *<sup>σ</sup>***K**Ið Þ*<sup>t</sup>* (42)

**K**ðÞ¼ *t* ½ � **K**eð Þ*t* , **K**xð Þ*t* , **K**uð Þ*t* (38)

<sup>m</sup>ð Þ*t*

**u**pðÞ¼ *t* **K**eð Þ*t* **e**yðÞþ*t* **K**xð Þ*t* **x**mðÞþ*t* **K**uð Þ*t* **u**mð Þ*t* (36)

**A***p*, **B***p*, and **C***<sup>p</sup>* are the matrices of the appropriate dimensions.

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement*

• The range of the plant matrices parameters is bounded.

• All possible pairs **A**<sup>p</sup> and **C**<sup>p</sup> are observable.

may be less than the dimension of the plant state. The output tracking error is defined as

and **C**<sup>p</sup> is generated by the adaptive algorithm:

**<sup>r</sup>**<sup>T</sup>ðÞ¼ *<sup>t</sup>* **<sup>e</sup>**<sup>T</sup>

definite adaptation coefficient matrices, respectively.

where

"integral" terms

**261**

The reference model is described by

*DOI: http://dx.doi.org/10.5772/intechopen.92558*

It is assumed that:

where j j **x**RFð Þ*t* is the sum of vector **x**RFð Þ*t* component moduli and g is the constant.

The selected discontinuous control leads to

$$\dot{\mathbf{s}}(t) = \mathbf{E}\mathbf{x}\_{\text{RF}}(t) - \mathbf{g}|\mathbf{x}\_{\text{RF}}(t)|\operatorname{sgn}\mathbf{s}(t)\tag{30}$$

There exists such positive value of g that the functions **s**ð Þ*t* and **s**\_ð Þ*t* have different signs. It means that the sliding mode will occur on a discontinuity surface. The influence of discontinuity of the control signal is reduced by varying the amplitude of the control signal.
