1. Introduction
The sliding mode control(SMC) can be divided into the two parts:. linear sliding mode
control(LSMC)(1)(2) and terminal sliding mode control(TSMC)(3)(4).
In the LSMC, the variable structure system(VSS) with the SMC can provide the effective
means to control of uncertain systems under parameter variations and external disturbances(5). One of its essential advantages is the robustness of the controlled system against
matched parameter uncertainties and external disturbances in the sliding mode on the
predetermined sliding surface. In order to fully utilize the advantages of the sliding
mode on the predetermined sliding surface, the precise existence condition of the
sliding mode should be satisfied and proved completely for the complete formulation
of the design of the LSMC(11)(12) as well as the TSMC. Utkin presented the two methodologies to prove the existence
condition of the sliding mode on the sliding surface(1). These are known as the invariance theorem, where the equation of the sliding mode
is invariant with respect to the two nonlinear transformations: the control input
transformation and sliding surface transformation. These methods are also called the
diagonalization methods. Those were reviewed in (2). DeCarlo, Zak, and Matthews tried to prove Utkin's invariance theorem. But, the proof
is not clear. In (5), Su, Drakunov, and Ozguner mentioned the sliding surface transformation, which would
diagonalize the control coefficient matrix to the dynamics for the sliding surface.
But they did not prove the existence condition of the sliding mode on the predetermined
sliding surface. However the proofs of Utkin’s theorem are given in (6) for single input(SI) uncertain linear plants, in (10) for SI uncertain integral linear plants without the reaching phase problems, in (14) for multi input(MI) uncertain linear plants, in (15) for MI uncertain integral linear plants, in (16) for SI uncertain nonlinear plants, and in (17) for integral augmented uncertain nonlinear systems without the reaching phase problems(11)(12). The researches on the proof of Utkin’s theorem is sufficient to prove Utkin’s theorem
for various uncertain plants in the LSMC. However, in the TSMC, the proof of Utkin’s
theorem is rare except (32).
In the TSMC, for the first time, Haimo developed a finite time controller with a finite
time stabilization in 1986(3). Haimo invented the finite time stabilization rather than the asymptotic convergence
of the LSMC. The TSMC has the advantages over the LSMC, for example, convergence in
finite time and high control precision. Zak presented the terminal attractors with
the finite time convergence in 1988(4). After those, a lot of the researches on the TSMCs for example many theoretical developments
and application examples are reported until now(18)-(32). In (18), Zhihong et. al. studied the TSMC which is applied to the control of multi-input
multi-output(MIMO) robot manipulators. Zhihong and Yu reported the TSMCs for higher
order single-input single-output(SISO) linear systems with the hierarchical terminal
sliding surface and regular MIMO systems with the fractional order sliding surface
in (19). In 1997, Yu et. al. suggested a TSMC having the fast transient performance with
the recursive sliding surface for higher order SISO systems in (20). The acceleration term is added to the terminal sliding manifold in order to speed
up the output response. Yu et. al suggested a nonsingular terminal sliding mode control
of a class of nonlinear dynamical systems in (21). The conventional TSMCs until now have the singularity problem that the control input
becomes infinity in certain domain. However, there is no singular problem in (21). Feng et. al investigated a second order TSMC for uncertain multivariable systems
for chattering-free performance and nonsingularity in (22). Zong et. al proposed a higher order sliding mode control with self-tuning law based
on the integral sliding mode when the uncertainty in the input matrix is not the zero
that is $\triangle b\ne 0$ in (23). The method can be viewed as the finite stabilization based on the higher sliding
mode with the geometric homogeneity and the integral sliding surface with no reaching
phase. A derivative and integral TSMC for a class of MIMO nonlinear systems is suggested
by Chiu in (24). The recursive sign or fractional integral terminal sliding manifolds are proposed
to remove the reaching phase. Pen et. al in 2015 designed an integral terminal sliding
surface for uncertain nonlinear systems without the singularity by means of the saturation
on the singular component of the control for temporary avoiding the singularity in
(25). For the noncanonical plants of the interceptors, a fast robust guidance and control
is designed based on a fast fractional integral terminal sliding surface for removing
the reaching phase in (26). Hu et. al analyzed a dynamic sliding mode manifold based continuous fractional order
nonsingular terminal sliding mode control for a class of second order nonlinear systems
when in 2020 and in (27). In the algorithm, the uncertainty term in the input matrix is treated as the total
lumped uncertainty. To cope with the reaching phase, the time varying sliding hyperplanes
are proposed in (28). But the real output is not predicted. A study of nonsingular fast terminal sliding
mode fault tolerant control based on the nonsingular sliding surface is presented
by Xu et. al in 2015 and in (30). In (31), the discontinuous and continuous control input transformation integral TSMCs by
using the integral sliding surface without the reaching phase and with the output
prediction performance as the one approach are presented for second order uncertain
plants when $\triangle b\ne 0$. Applying the idea of (9) for the LSMC, the integral sliding surface without the reaching phase is suggested
for the TSMCs. The exponent of the power function can be any positive numbers satisfying
$q>p>0$ such that $0<p/q<1$. The ideal sliding dynamics of the integral sliding surface
is derived and the real robust output can be predicted, predesigned, predetermined
by means of the solution of the ideal sliding dynamics. Based on defining a new auxiliary
nonlinear state, the closed loop exponential stability together with the existence
condition of the sliding mode on the predetermined sliding surface is investigated
theoretically for the complete formulation of the TSMC design for the output prediction
performance. As a remedy of the singularity, a certain limit is imposed on the new
auxiliary nonlinear state. In (32), the proof of Utkin’s theorem is given for second order SI non-integral uncertain
linear plants when $\triangle b\ne 0$. About only the two transformations, the control
input one and sliding surface one, Utkin’s theorem is proved rigorously in the TSMCs.
The researches on the proof of Utkin’s theorem in the TSMC are rare except (32). Only the design approaches of the control input transformation SMCs are in (24)-(26)(27)(29)-(31). In (31), the algorithm is first called as the control input transformation SMC. About the
sliding surface part transformation, the research does not exist. The integral action
is augmented to the TSMC for removing the reaching phase in (31). To the integral TSMC systems, the proof of Utkin theorem is essentially needed.
In this paper, a complete proof of Utkin's theorem is presented for the ITSMC of second-order
SI uncertain linear systems when $\triangle b\ne 0$. The paper presents five approaches
for designing ITSMCs for second-order uncertain linear plants, namely: control input
transformation, sliding surface full transformation, and three sliding surface part
transformations. The invariance theorem is demonstrated clearly and comparatively
for the first two transformation methods, which are known as the two diagonalization
methods, in the context of second-order SI integral uncertain linear systems when
$\triangle b\ne 0$. Although the control input transformation has been previously
discussed in works such as (31) and (32), it is explicitly named here for the first time. Similarly, the sliding surface transformation
is introduced for the first time in the TSMC, except for (32), and the sliding surface part transformations make their debut in the TSMC through
this research. This paper covers the first three transformations, and further studies
will include the last two sliding surface part transformations. The output prediction,
predetermination, and predesign capabilities of the first three transformations yield
the same performance. The design examples and simulation studies are provided to illustrate
the practical value of the main results.
2. Main Results of Proof of Utkin's Theorem for TSMCs
The invariant theorem of Utkin is as follows(1)(2):
Theorem 1: The equation of the sliding mode is invariant with respect to the two nonlinear
transformations, i.e. the control input transformation and sliding surface transformation:
for $\det H_{u}\ne 0{and}\det H_{s}\ne 0$.
For a second order SI uncertain canonical linear system(31):
where $x_{1}\in R \quad{and}\quad x_{2}\in R$ are the state variables, $u\in R$ is
the control input, $a_{10},\:a_{20},\: {and} \quad b_{0}\in R$ are the nominal values,
$\triangle a_{1},\: \triangle a_{2},\: {and} \quad \triangle b$ are the uncertainties,
those are assumed to be matched and bounded, and $\Delta d(x,\:t)$ is the external
disturbance which is also assumed to be matched and bounded.
Assumption 1:
$(b_{0})^{-1}\triangle b=\triangle b(b_{0})^{-1}=\triangle I$, and $\vert\triangle
I |\le\rho <1$ where $\rho$ is a positive constant less than 1.
An integral state $x_{0}\in R$ with a special initial condition is augmented for use
later in the integral terminal sliding surface as follows:
where $x_{0}(0)$ is the special initial condition for the integral state which is
determined later.
Based on the idea of (9) of the LSMC, for removing the reaching phase completely, the integral terminal sliding
surface for the TSMC $s\in R$ is proposed as follows(31):
where $p \quad {and} \quad q$ are any positive numbers satisfying $q>p>0$ such that
$p/q$ is real fractional that is $0<p/q<1$, in which any positive number such that
$0<p/q<1$ are first mentioned except (31) and (32). The $C_{0}$ and $C_{1}$ are designed such that the polynomial $r^{2}+C_{1}r+C_{0}=0$
should be Hurwitz. The special initial condition $x_{0}(0)$ in Eq. (3) for the integral state is determined so that the integral terminal sliding surface
Eq. (4) is the zero at $t=0$ for any initial condition $x_{1}(0) \quad {and} \quad x_{2}(0)$
as
With the initial condition Eq. (5) for the integral state, the integral terminal sliding surface is zero at the initial
time $t=0$ that is $s(t)_{t=0}=0$. Hence, the integral sliding surface Eq. (4) can define the surface from any given initial condition finally to the origin in
the state space, and the controlled system slides from the initial time $t=0$. The
first condition of removing reaching phase problems is satisfied(11)(12). In the sliding mode, the equation $s=0=\dot s$ is satisfied. Then from Eq. (2), Eq. (3), and Eq. (4) the ideal sliding dynamics is derived as
which is a dynamic representation of the integral terminal sliding surface Eq. (4). The solution of Eq. (6) is identical to the ideal integral terminal sliding surface and the real robust controlled
output itself(11)(12). Therefore, the output can be pre-designed, pre-determined, and predicted.
Now, the suggested discontinuous ITSMC input for the uncertain plant Eq. (2) and the integral terminal sliding surface Eq. (4) is taken as follows:
where an auxiliary nonlinear state $x_{3}$ is defined as
which is first defined in (31). Based on defining the auxiliary state $x_{3}$ in Eq. (8), the discontinuous input is chattering according to the condition of $sx_{3}$ in
Eq. (15). Since that, it is easily shown that the existence condition of the sliding mode
is clearly satisfied when $\triangle b\ne 0$. One takes the constant gains as
and takes the discontinuously switching gains as follows:
where $sign(s)$ is the $sig\nu m(s)$ function as
2.1 Control input transformation
where for the second order SI uncertain integral terminal linear case, the control
input transformation is selected as $H_{u}=(b_{0})^{-1}$ while it is chosen as $H_{u}=(CB_{0})^{-1}$
in the LSMC. Most of the TSMCs commonly use this transformation(24)-(26)(27)(29)-(30) without explicitly naming it. However, the name of the control input transformation
is first introduced in (31) and (32). $b_{0}^{-1}$ is multiplied to all the components of the discontinuous input Eq. (7) because the transformation of the control input for easy proving that the existence
condition of the sliding mode is satisfied as one approach among the five approaches
of the transformation (diagonalization)s(1)(2)(31). Since $\triangle b\ne 0$, the effect of $\triangle b\ne 0$ is considered in the
selection of the discontinuous chattering gains Eq. (13)-Eq. (16). The results of $\triangle b\ne 0$ is the increase of the magnitude of the discontinuous
chattering gains compared with the case when $\triangle b=0$. In the discontinuous
input Eq. (18), the integral terminal sliding surface itself is one of the feedback elements and
that brings the controlled system closer to the ideal predetermined terminal sliding
surface(9).
From Eq. (9)-(11), the real dynamics of $s$ becomes finally
The original design problem of the ITSMC is finally converted to the stabilization
problem against uncertainties and external disturbances by means of the discontinuously
chattering input components and the feedback of the integral sliding surface. The
performance(output) designed in the integral sliding surface becomes the real performance(output)
for the output prediction, predetermination, and predesign(11)(12), and hence is completely separated with the performance robustness problem. The total
closed loop stability with the transformed discontinuous control input Eq. (18) and the integral terminal sliding surface Eq. (4), along with the precise existence condition of the sliding mode, will be investigated
in Theorem 1.
Theorem 1: If the integral terminal sliding surface Eq. (4) is designed to be stable, the transformed discontinuous control input Eq. (18) with the integral terminal sliding surface Eq. (4) satisfies the existence condition of the sliding mode on the pre-designed integral
terminal sliding surface and closed loop exponential stability to the integral terminal
sliding surface $s=0$ including the origin.
Proof: Take a Lyapunov function candidate as
Differentiating Eq. (21) with time leads to
Substituting Eq. (20) into Eq. (22) leads to
Since the uncertainty and external disturbance terms in Eq. (23) are canceled out due to the chattering discontinuous input terms by means of the
switching gains in Eq. (13)-Eq. (16), one can obtain the following equation(11)(12)
The existence condition of the sliding mode on the predetermined integral terminal
sliding surface by the transformed discontinuous control input is proved theoretically
for the complete formulation of the TSMC design for the output prediction. By only
through the proof of the existence condition of the sliding mode, the strong robustness
of every point on the whole trajectory of the predetermined integral terminal sliding
surface from a given initial condition to the origin is guaranteed. Hence, the controlled
robust output can be predicted, predesigned, and predetermined. The second condition
of removing reaching phase problems is satisfied(11)(12). From Eq. (24) the following equation is obtained.
From Eq. (25), the following equation is obtained
which completes the proof of Theorem 1.
The existence condition of the sliding mode is proved with respect to the control
input transformation for second order uncertain linear systems. The equation of the
sliding mode, i.e. the sliding surface is invariant with respect to the control input
transformation.
2.2 Sliding surface (full) transformation
The transformation is selected as $H_{s}(x,\:t)=(b_{0})^{-1}$ as the same way as the
control input transformation described above, while in the LSMC it is chosen as $H_{s}(x,\:t)=(CB_{0})^{-1}$.
Except (32), this transformation appears for the first time in the TSMC. Most TSMCs typically
use only the well known control input transformation without giving it a specific
name. The special initial condition $x_{0}(0)$ in Eq. (2) for the integral state is determined to ensure the integral sliding surface Eq. (27) is the zero at $t=0$ for any initial condition $x_{1}(0) \quad {and} \quad x_{2}(0)$
as
With the initial condition Eq. (28) for the integral state, the integral terminal sliding surface is zero at the initial
time $t=0$ that is $s(t)_{t=0}=0$. Hence, the transformed integral terminal sliding
surface Eq. (27) can define the surface from any given initial condition finally to the origin in
the state space, as a result, the controlled system slides from the initial time $t=0$.
The first condition of removing reaching phase problems is satisfied(11)(12). In the sliding mode, the equation $s=0=\dot s$ is satisfied. Then from Eq. (2), Eq. (3), and Eq. (27), the ideal sliding dynamics is derived as
which is a dynamic representation of the transformed integral terminal sliding surface
Eq. (27) and the same as that of the control input transformation since $s=0=s*$ when $b_{0}\ne
0or\infty$. The solution of Eq. (29) is identical to the set of the ideal integral terminal sliding surface and the real
robust controlled output itself(11)(12). Therefore, the output can be pre-designed, predetermined, and predicted. The prediction
of the controlled output is possible by means of the solution of Eq. (29).
Now, the suggested discontinuous ITSMC input for uncertain plant Eq. (2) and the transformed integral terminal sliding surface Eq. (27) is taken as follows:
where one takes the constant gains as
and takes the discontinuously switching gains as follows:
Then the real dynamics of the transformed integral sliding surface by the discontinuous
control input, i.e. the time derivative of $s*$ becomes
From Eq. (31)-(33), the real dynamics of $s*$ becomes finally
From Eq. (40), the original design problem of the ITSMC is finally converted to the stabilization
problem against the uncertainties and external disturbances by means of the discontinuously
chattering input and the feedback of the transformed integral terminal sliding surface.
The total closed loop stability with the discontinuous control input Eq. (30) and the transformed integral terminal sliding surface Eq. (27) together with the precise existence condition of the sliding mode will be investigated
in Theorem 2.
Theorem 2: If the transformed integral terminal sliding surface Eq. (27) is stably designed, the discontinuous control input Eq. (30) with the stable transformed integral terminal sliding surface Eq. (27) satisfies the existence condition of the sliding mode on the pre-designed integral
terminal sliding surface and closed loop exponential stability to the integral terminal
sliding surface $s=0$ including the origin.
Proof: Take a Lyapunov function candidate as
Differentiating Eq. (41) with time leads to
Substituting Eq. (40) into Eq. (42) leads to
Since the uncertainty and external disturbance terms in Eq. (43) are canceled out due to the chattering control input terms by means of the switching
gains in Eq. (35)-Eq. (38), one can obtain the following equation(11)(12)
The theoretical proof of the existence condition for sliding mode on a predetermined
transformed integral terminal sliding surface, implemented by a discontinuous control
input, is crucial for establishing a complete formulation of TSMC design for output
prediction performance. This proof ensures strong robustness at every point along
the entire trajectory of the predetermined integral sliding surface, from a given
initial condition to the origin without the reaching phase, by satisfying the existence
condition of the sliding mode. As a result, we can predict, pre-design, and pre-determine
controlled robust output. Moreover, this proof also satisfies the second condition
of removing reaching phase problems, as demonstrated in previous works (11)(12). From Eq. (44), we can derive the following equation.
From Eq. (45), the following equation is obtained
which completes the proof of Theorem 2.
If the sliding mode equation $s*=0$, then $s=0$ since $b_{0}\ne 0$ or $b_{0}\ne\infty$.
The inverse augment also holds, therefore the both ideal sliding surfaces are equal
i.e. $s=0=s*$, which completes the proof of Theorem 2.
Until the first two transformations, Utkin's invariant theorem can be applied. However,
for the next transformation, the theorem cannot be used. Nevertheless, this transformation
can serve as an alternative design method for the ITSMC and LSMC. It is possible to
prove the existence condition of the sliding mode and stabilization using this method.
2.3 Sliding surface part transformations
The part transformation is selected as $(b_{0})^{-1}$ that is multiplied to only $x_{2}$
term in the integral terminal sliding surface. This part transformation appears for
the first time in the TSMC. The property of the Utkin’s invariant theorem can not
be applicable since $s=0$ is not equal to $s^{+1}=0$ except $b_{0}=1$. Besides this
part transformation, there are the two more part transformation sliding surfaces as
The research on these two remaining part transformation sliding surfaces, $s^{+2}$
and $s^{+3}$, is dropped in this paper and will be included in the further study because
of the limited allowed space.
The special initial condition $x_{0}(0)$ in Eq. (47) for the integral state is determined so that the part transformation integral sliding
surface Eq. (47) is the zero at $t=0$ for any given initial condition $x_{1}(0){and}x_{2}(0)$ as
With the initial condition Eq. (50) for the integral state, the integral terminal sliding surface is zero at the initial
time $t=0$ that is $s^{+}(t)_{t=0}=0$. Hence, the part transformation integral sliding
surface Eq. (47) can define the surface from any given initial condition finally to the origin in
the state space, and the controlled system can slide from initial time $t=0$. The
first condition of removing reaching phase problems is satisfied(11)(12). In the sliding mode, the equation $s^{+1}=0=\dot s^{+1}$ is satisfied. Then from
Eq. (2) and Eq. (47) the ideal sliding dynamics is derived as
which is the dynamic representation of the part transformation integral terminal sliding
surface Eq. (47). The solution of Eq. (51) is identical to the set of the ideal part transformed integral terminal sliding surface
and the real robust controlled output itself(11)(12). Therefore, the output can be pre-designed, predetermined, and predicted.
Now, the suggested discontinuous ITSMC input for uncertain plant Eq. (2) and the partially transformed integral terminal sliding surface Eq. (47) is taken as follows:
where one takes the constant gains as
and takes the discontinuously switching gains as follows:
Then the real dynamics of the partially transformed integral sliding surface by the
discontinuous control input, i.e. the time derivative of $s$ becomes
From Eq. (53)-Eq. (55), the real dynamics of $s^{+1}$ becomes finally
From Eq. (62), the original design problem of the TSMC is finally converted to the stabilization
problem against the uncertainties and external disturbances by means of the discontinuously
chattering input and the feedback of the transformed integral sliding surface. The
total closed loop stability with the discontinuous control input Eq. (52) and the transformed integral sliding surface Eq. (47) together with the precise existence condition of the sliding mode will be investigated
in Theorem 3.
Theorem 3: If the stable partially transformed integral terminal sliding surface Eq. (47) is designed, the discontinuous control input Eq. (52) with the stable partially transformed integral sliding surface Eq. (47) satisfies the existence condition of the sliding mode on the pre-designed integral
sliding surface and closed loop exponential stability to the integral sliding surface
$s=0$ including the origin.
Proof: Take a Lyapunov function candidate as
Differentiating Eq. (63) with time leads to
Substituting Eq. (62) into Eq. (63) leads to
Since the uncertainty and external disturbance terms in Eq. (65) are canceled out due to the chattering control input terms by means of the switching
gains in Eq. (57)-Eq. (59), one can obtain the following equation(11)(12)
The theoretical proof of the existence condition for sliding mode on a predetermined
transformed integral sliding surface, implemented by a discontinuous control input,
is essential to establish a complete formulation of TSMC design for output prediction
performance. Through this proof, we can guarantee strong robustness at every point
along the whole trajectory of the predetermined integral sliding surface, from a given
initial condition to the origin. This, in turn, enables us to predict, pre-design,
and pre-determine the controlled robust output. Moreover, the second condition of
removing reaching phase problems is also satisfied, as described in previous works
(11)(12). From equation Eq. (22), we can derive the following result.
From Eq. (67), the following equation is obtained
which completes the proof of Theorem 3.
Although the invariance property of Utkin's theorem may not be applicable to this
particular transformation, it still serves as a valuable design and stabilization
approach for ITSMCs. In light of the results from Theorem 1, Theorem 2, and Theorem
3, there are now five approaches or design methods for ITSMCs: control input transformation,
full sliding surface transformation, and three sliding surface part transformations.
While the first approach is already well-known (though not always referred to by name),
having been discussed in previous works such as (31) and (32), the second transformation is introduced here for the first time in the context of
TSMCs except (32). As for the last three cases, they also make their debut in TSMCs through this research.
3. Design Example and Illustrative Simulation Study
Consider a second order uncertain canonical system
where the nominal parameter $a_{10}$, $a_{20}$, and $b_{0}$, matched uncertainties
$\triangle a_{1}$, $\triangle a_{2}$, and $\triangle b$, and external disturbance
$\triangle d(x,\:t)$ are
The $\triangle I$ in assumption 1 becomes
which satisfies the assumption 1
To design the proposed ITSMC with the integral sliding surface and transformed control
input, first the stable coefficient in the suggested integral sliding surface is determined
as
such that the polynomial is Hurwitz
The $p$ and $q$ are selected as
which satisfies the terminal condition. The $p$ is positive real not positive odd
integer because any positive number is possible such that $0<p/q<1$. Then the integral
sliding surface becomes
where
and the ideal sliding dynamics becomes
which is the design goal for the same outputs with respect to the three transformations
3.1 Control input transformation
From Eq. (9)-Eq. (11), the constant feedback gains are accordingly designed as
If one takes the switching gains as follows:
The equation Eq. (24) becomes
The simulation is carried out under 1[msec] sampling time and with $x(0)=[3 1.5]^{T}$
initial state and by Eq. (4), the initial condition for the integral state becomes
Fig. 1 shows the two finite time stabilization output responses, $x_{1}$ and $x_{2}$ for
the two cases those are the ideal sliding dynamics output and the real output with
uncertainty and external disturbance. Those outputs are identical, which means that
the prediction, predetermination, predesign of the output, and design separation of
the performance design and robustness problem are possible, by means of removing the
reaching phase. Fig. 2 shows the phase trajectories with the ideal trajectory and the real trajectory. The
real trajectory is identical to the ideal one as can be shown in Fig. 2. The terminal sliding surface time trajectory and the corresponding control input
are depicted in Fig. 3 and Fig. 4, respectively. The abrupt change in the terminal sliding surface and control input
in Fig. 3 and Fig. 4 is due to the singular problem, which requires further investigation.
3.2 Sliding surface (full) transformation
For the comparison of both transformations, the same stable coefficient of the sliding
surface and the same parameters of the power function are determined.
Thus the new transformed integral terminal sliding surface of the sliding surface
transformation becomes
그림. 1. 제어입력 변환에 의한 두가지 응답 $x_{1}$과 $x_{2}$
Fig. 1. Output responses, $x_{1}$ and $x_{2}$ by control input transformation
그림. 2. 제어입력 변회에 의한 이상과 실제 플랜트의 두가지 상 궤적
Fig. 2. Phase trajectories by control input transformation
그림. 3. 제어입력 변환애 의한 슬라이딩 면
Fig. 3. Sliding surface time trajectory by control input transformation
그림. 4. 제어입력 변환애 의한 제어입력
Fig. 4. Control input by control input transformation
The ideal sliding dynamics is the same as that of the control input transformation
and the initial condition $x_{0}(0)$ is also the same as that of the above transformation.
Now, the integral TSMC control input is taken as follows:
From Eq. (31)-Eq. (33), by letting the constant gain
If one take the switching gain as the design parameters
then one can obtain the following equation
if $s*=0$, then $s=0$. The inverse augment is also true. The switching gains in Eq. (93) can be obtained also from Eq. (83) by multiplying $(b_{0})^{-1}=2^{-1}$.
The simulation was conducted using a sampling time of 1[msec] and with the same initial
condition. Fig. 5 illustrates the two output responses, $x_{1}$ and $x_{2}$, for the ideal and real
cases. These responses are almost identical to those shown in Fig. 1 because the sliding surface is identical and the continuous and discontinuous gains
resulting from both control inputs, $u*$ and $u_{2}$, are the same. It is possible
to predict, predesign, and
그림. 5. 슬라이딩면 변환에 의한 두가지 응답 $x_{1}$과 $x_{2}$
Fig. 5. Output responses, $x_{1}$ and $x_{2}$ by sliding surface transformation
그림. 6. 슬라이딩면 변회에 의한 이상과 실제 플랜트의 두가지 상 궤적
Fig. 6. Phase trajectories by sliding surface transformation
그림. 7. 슬라이딩면 변환애 의한 슬라이딩 면
Fig. 7. Sliding surface time trajectory by sliding surface transformation
그림. 8. 슬라이딩면 변환애 의한 제어입력
Fig. 8. Control input by sliding surface transformation
predetermine the real robust output by designing and solving the ideal sliding dynamics,
just like with the above control input transformation. Fig. 6displays the two phase trajectories for the ideal and real cases, those are identical.
The sliding surface time trajectory and the corresponding control input are shown
in Fig. 7 and Fig. 8, respectively. The value of the sliding surface in Fig. 7 is about half of that in Fig. 3 since the new sliding surface in the sliding surface transformation is multiplied
by $\dfrac{1}{2}$.
3.3 Sliding surface part transformation
For the comparison of the three transformations and the same ideal sliding dynamics
with that of the above first two transformations for the same output (performance),
the coefficients in the part transformation sliding surface are set to
These coefficients are half of those used in the control input transformation and
share the same parameters in the power function of the part transformation sliding
surface as the first two transformations. As a result, the new partially transformed
sliding surface of the sliding surface part transformation becomes
Now, the integral TSMC control input is taken as follows:
From Eq. (21)-Eq. (23), by letting the constant gain
그림. 9. 슬라이딩 면 부분 변환에 의한 두가지 응답 $x_{1}$과 $x_{2}$
Fig. 9. Output responses, $x_{1}$ and $x_{2}$ by sliding surface part transformation
그림. 10. 슬라이딩면 부분 변회에 의한 이상과 실제 플랜트의 두가지 상 궤적
Fig. 10. Phase trajectories by sliding surface part transformation
그림. 11. 슬라이딩 면 부분 변환애 의한 슬라이딩 면
Fig. 11. Sliding surface time trajectory by sliding surface part transformation
그림. 12. 슬라이딩면 부분 변환애 의한 제어입력
Fig. 12. Control input by sliding surface transformation
If one take the switching gain as the design parameters
then one can obtain the following equation
The switching gains in Eq. (103) can be obtained also from Eq. (93).
Simulations are performed with a sampling time of 1[msec] and using the same initial
condition as before. Fig. 9displays the output responses for the two cases: the ideal case and the real case.
These responses are nearly identical to those shown in Fig. 1 and Fig. 5, respectively, as the same ideal sliding dynamics are employed. Consequently, the
real robust output can be predicted, predetermined, and predesigned in the same way
as those of the first two transformations. Fig. 10depicts the phase trajectories for the ideal and real cases, with the real trajectory
being the same as that of the ideal case. Additionally, Fig. 11 and Fig. 12 show the sliding surface time trajectory and the corresponding control input, respectively.
The outputs of the three transformations are identical to the three ideal outputs,
indicating that the three transformation outputs exhibit the same level of performance
because of the identical ideal dynamics employed. Therefore, the three real outputs
can be predicted, predesigned, and predetermined in the same way as designed and expected.