## 1. Introduction

The VSS with the SMC can provide the effective means to the control of uncertain nonlinear
dynamical systems under parameter variations and external disturbances^{[1-}^{4]}. 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, s=0^{[5-}^{7]}. To take the advantages of the sliding mode on the predetermined sliding surface,
the precise existence condition of the sliding mode, $s \cdot \dot{s} < 0$ for the
SI case as well as $s_{i} \cdot \dot{s}_{i} < 0, \quad i=1,2, \ldots m$ for the
MI(Multi Input) case should be satisfied^{[8}^{24]}. Therefore the precise existence condition of the sliding mode must be proved completely
for linear plants moreover for nonlinear plants. Utkin in [4]^{[4]} presented the two nonlinear methodologies without the complete proofs in order to
prove the precise existence condition of the sliding mode on the pre-selected sliding
surface. It is so called the invariance theorem, that is the equation of the sliding
mode is invariant with respect to the two nonlinear transformations. Those are the
control input transformation and sliding surface transformation, so called the two
diagonalization methods. The essential feature of both nonlinear transformation methods
is the conversion of a multi-input design problem into m single-input design problems^{[4,}^{24]}. Those were only reviewed in [5]^{[5]}. DeCarlo, Zak, and Matthews tried to prove Utkin's invariance theorem. But, the proof
also is incomplete, and only the same as that of Utkin. In [9]^{[9]}, Su, Drakunov, and Ozguner mentioned the sliding surface transformation, which would
diagonalize the control coefficient matrix to the dynamics for the sliding surface
$s$. But they did not completely prove the precise existence condition of the sliding
mode on the predetermined sliding surface. Even for a SI linear case, that proof is
hardly reported. For MI uncertain linear plants, instead of proving the precise existence
condition of the sliding mode, some design methods were studied, those are, including
the two diagonalization methods [4][5], the hierarchical control methodology^{[4,}^{6]}, simplex algorithm^{[14]}, Lyapunov approach^{[1,}^{9,}^{18,}^{22]}, and so on. Until now in MIMO(Multi input multi output) VSSs, it is difficult to
prove the precise existence condition of the sliding mode on the predetermined sliding
surface theoretically, but in [9]^{[9]}, [18]^{[18]}, [22]^{[22]}, only the result that the derivative of the Lyapunov candidate function is negative,
i,e. $\dot{V} < 0$ is obtained when the Lyapunov candidate function is taken as
$V=1 / 2 s^{T} s$. In SI systems, both the VSS existence condition of the sliding
mode and the Lyapunov stability are the same when the Lyapunov candidate function
is taken as $V=1 / 2 s^{T} s$. However, the VSS existence condition of the sliding
mode in multi input systems is the more strict condition than the Lyapunov stability
because if the VSS existence condition of the sliding mode is satisfied, then the
Lyapunov stability is did but the reverse argument does not hold generally^{[24]}. In [23]^{[23]}, for MI uncertain linear plants, the proof of Utkin’s theorem is given comparatively,
the precise existence condition of the sliding mode is proved completely, and the
complete formulation of the multivariable VSS is possible from the design of the sliding
surface to the proof of the precise existence of the sliding mode and the proof of
the stability of the closed loop system. For SI uncertain nonlinear plants, the proof
of Utkin’s theorem is given in [24]^{[24]}. Thus, there are the two methodologies, i.e., sliding surface transformation and
control input transformation to prove the existence condition of the sliding mode
in the SMC for uncertain nonlinear systems, but the controlled output is not predictable.

By the way, in order to improve the output performances. the integral action has been
augmented to the linear as well as nonlinear VSS or SMC so called IVSS(integral variable
structure system) or ISMC in the mainly three aspects, i.e., improvement of the steady
state performance^{[16,}^{25-}^{28]}, removal of the reaching phase^{[8,}^{29-}^{36]}, and remedy of the chattering problems^{[37,}^{38]}, etc^{[39]}. In [16]^{[16]}, [25]-[28]^{[25-}^{28]}, a simple integral state with the zero initial value is added to the conventional
linear sliding surface to improve the steady state performance. However the inevitable
overshoot problems exist because the integral state accumulated from the zero should
be re-regulated to the zero. In order to remove the reaching phase problems, the non
zero special initial value for the integral state is introduced to in [8]^{[8]} for the first time. The reaching phase is completely removed and the controlled system
slides from the initial time. The sliding surface can be defined from any given initial
state to the origin without the reaching phase and the existence condition of the
sliding mode is satisfied. Thus the robustness can be guaranteed for the entire trajectory
from any given initial state to the origin and the output is predictable by means
of obtaining the solution of the ideal sliding dynamics of the integral sliding surface.
Other versions of [8]^{[8]} are reported in [29]-[36]^{[19-}^{36]}. By using the low pass filter property of the integral action, the chattering problems
are improved in [37][38]^{[37,}^{38]}. In the ISMC for uncertain nonlinear systems of [21]^{[21]}, [30]^{[30]}, [34]^{[34]}, [36]^{[36]}, [39]^{[39]}, the Lyapunov stability is used to proving the closed loop stability, instead of
proving the exact precise existence condition of the sliding mode.

Until now, for SI uncertain integral nonlinear systems, a rigorous proof of Utkin's
theorem is necessary to completely prove the existence condition of the sliding mode
but hardly reported, while the proofs of Utkin invariant theorem for MI uncertain
linear plants and for SI uncertain nonlinear plants are given.

In this note, a complete proof of Utkin's theorem is presented for the ISMC of SI
uncertain nonlinear plants. The invariance theorem with respect to the two nonlinear
transformation methods so called the two diagonalization methods is proved clearly
and comparatively. To remove the reaching phase problems, the integral sliding surface
is adopted. The guideline for obtaining the ideal sliding dynamics of the integral
sliding surface is given and after obtaining the solution of the ideal sliding dynamics
of the integral sliding surface, the output can be predicted which is shown in the
design example and simulation study. The complete formulation from the formulation
of the objective plants to the proof of the existence condition of the sliding mode
and the stability of the closed loop system is possible in the IVSS for uncertain
nonlinear systems. If the control input matrix $g_{0}(X, t)$ in the model of nonlinear
plants is constant(not function of $X$ or $t$ i.e. $g_{0}(X, t)=B_{0}$, then both
transformation(diagonalization) methods of Utkin’s theorem can be used to, otherwise,
only the control input transformation can be applied to the proof of the existence
condition of the sliding mode on the predetermined integral sliding surface in the
nonlinear IVSS. A design example and simulation study shows the usefulness of the
main results.

## 2. Main Results of Proof of Utkin's Theorem

The invariant theorem of Utkin is as follows^{[4,}^{5]}:

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 trans-
formation:

where $H _{u} (X,t)$ and $H _{s} (X,t)$ are the nonlinear trans- formation matrices
for $\operatorname{det} H_{u} \neq 0$ and $\operatorname{det} H_{s} \neq 0$.

In order to prove this theorem, consider a SI affine uncertain nonlinear system with
matched uncertainties and disturbances

where $X \in R^{n}$ is the state vector, $X(0)$ is its initial state, $X \in R^{1}$
is the control input, $f(X, t) \in C^{k}$ and $g(X, t) \in C^{k}, k \geq 1$, $g(X,
t) \neq 0,$ for all $X \in R^{n}$ and for $all t \geq 0$ are of suitable dimensions,
and $d^{\prime}(X, t)$ implies bounded matched external disturbances.

Assumption

A1: $f(X, t)$ is continuously differentiable.

Then, the uncertain nonlinear system (1) can be represented in the more affine nonlinear system of the state dependent coefficient
form as [24]^{[24]}

where $f_{0}(x, t)$ and $g_{0}(x, t)$ is each nominal value such that

respectively, $\triangle f_{1}(X, t)$ and $\triangle g(X, t)$ are matched uncertainties,
$\triangle f_{2}(X, t)$ is matched uncertainty, $d^{\prime}(X, t)$ is matched external
disturbance, and $d(X, t)$ is the totally matched lumped uncertainties, respectively.
By using the state dependent coefficient form in (3), the nonlinear controller design and the proof of the closed loop stability for uncertain
integral nonlinear systems become easy [34]^{[34]}.

Assumption:

A2:The pair $\left(f_{0}(X, t), g_{0}(X, t)\right)$ is controllable for $a l l X \in
R^{n}$ and for all $t \geq 0$

A3: The lumped uncertainties $d(X, t)$ is bounded

A4: $C^{T} g(X, t)$ and $C^{T} g_{0}(X, t)$ have the full rank and invertible

For later use, an integral state for uncertain nonlinear systems is augmented as follows:

where for non zero element $C=\left[c_{i}\right]$ and $C_{0}=\left[c_{0_ {i}}\right]$

The integral sliding surface is a linear combination of the full state variable and
the integral state as [8]^{[8]}

where $C$ is a non zero element constant coefficient column vector for the integral
sliding surface. Because of the non zero initial value $X_{0}(0)$ in (5) or (6) for the integral state, the integral sliding surface of (7) $s$ is zero at $t=0$ and there is no reaching phase^{[8]}. From $\dot{s}=0$, the n-th order equation of the ideal sliding dynamics of the integral
sliding surface is derived as

Combining (8) with the n-1 differential equations in (1) or (3) analytically leads to the ideal sliding dynamics and its solution of the ideal sliding
dynamics coincides with the integral sliding surface of (7)^{[8]}, which will be shown in the design example and simulation study. By using this solution,
the output can be predictable in the nonlinear IVSS under matched uncertainties and
disturbance^{[8]}. If the mismatched uncertainties and disturbances exist, the asymptotic stabilization
is possible but the control output is disturbed by them and the output is not predictable^{[21]}, thus we assume that the uncertainties and disturbances are matched. In [8]^{[8]}, the asymptotic stabilization to the sliding surface finally the origin is possible,
but the prediction of the output is not feasible due to the reaching phase. The ISMC
control input is as follows:

where $K(X)$ is a static nonlinear feedback gain, $\Delta K$ is a state dependent
switching gain, and $G$ is a switching gain in order to conceal out the effect of
the uncertainties and disturbances to the output.

### 2.1 Sliding surface transformation^{[4,}^{5,}^{9]}

The sliding surface transformation matrix is selected as $H_{s}(X, t)=\left[C^{T}
g_{0}(X, t)\right]^{-1}$. In [5]^{[5]}, the proof is not sufficient. Now, the IVSS control input for the new integral sliding
surface is taken as follows:

The real dynamics of the integral sliding surface, i.e. the time derivative of $s^{*}$
becomes

Since $g_{0}(X, t)$ is a function of the state vector $X$, the further formulation
of (12) is very difficult. In [3]^{[3]} and [4]^{[4]}, the proofs of the sliding surface transformation are the same and stopped here.
Only an example to show the proof is given. However if $g_{0}(X, t)$ is constant,
i.e. $g_{0}(X, t)=B_{0}\left(H_{s}(X, t)=\left[C^{T} B_{0}\right]^{-1}\right)$, which
is effective in this paragraph, then it is possible to go the further steps of the
formulation with an assumption

A5: $\left[C^{T} B_{0}\right]^{-1} C^{T} \triangle g(X, t)=\triangle \dot {I}$ and
$|\triangle \dot {I}| \leq \gamma < 1$.

Even though the objective plant is uncertain and nonlinear, it is natural and convenient
that the modelling of $g_{0}(X, t)$ is to be constant and only the maximum bound of
the parameter variation is found and used in most of the VSS controller design for
uncertain nonlinear plants. Therefore the first term of the right hand side of (12) equation is zero, the real dynamics of the integral sliding surface leads to

By letting the static nonlinear feedback gain as

which is proposed in this paper. Then the real dynamics of $S^{*}$ becomes

In [9]^{[9]}, without uncertainty and disturbance, it is mentioned that the sliding surface transformation
would diagonalize the control coefficient matrix to the dynamics for $s$ and the $\dot{V}(x)
< 0$ is proved when $V(x)=x^{T} P_{X} > 0$. In [23]^{[23]}, for MI uncertain linear plants, the proof of the sliding surface transformation
is given.
If one takes the switching gains as follows:

then

The existence condition of the sliding mode is proved. One takes the Lyapunov candidate
function as $V\left(s^{*}\right)= 1 / 2 s^{* T} s^{*} > 0$, then the time derivative
of the Lyapunov candidate function is negative from (18), that is $\dot{V}\left(s^{*}\right)=s^{*} \cdot \dot{s}^{*} < 0$. Therefore the
asymptotic stability of the closed loop system is satisfied in the sense of Lyapunov.
If the sliding mode equation $s^{*}=0$, then $s=0$ since $C B_{0} \neq 0$ and invertible.
The inverse augment also holds, therefore the two integral sliding surfaces both are
equal i.e. $s=0=s^{*}$, which completes the proof of the sliding surface transformation
part of Theorem 1.

### 2.2 Control input transformation^{[4,}^{5]}

The following assumption is made before going into the control input transformation
step.

A6: $C^{T} \triangle g(X, t)\left[C^{T} g_{0}(X, t)\right]^{-1}=\triangle I$ and $|\Delta
I | \leq \delta < 1$.

The nonlinear control input transformation matrix is selected as $H_{u}=\left(C^{T}
g_{0}(X, t)\right)^{-1}$ for SI uncertain integral nonlinear plants.

In [4]^{[4]} and [5]^{[5]}, the proof for the nonlinear control input transformation is not complete. The real
dynamics of $s$, i.e. the time derivative of $s$ is as follows:

If $C$ is a function of the state vector $x$ i.e. $C(X, t)$, then the formulation
of (20) can not be obtained easily. By letting the static nonlinear feedback gain

which is proposed in this paper. Then the real dynamics of $s$ becomes

If one takes the switching gains as the design parameters

then one can obtain the following equation

The precise existence condition of the sliding mode is proved for SI uncertain nonlinear
systems. The robustness for the entire sliding trajectory from a given initial condition
to the origin. The equation of the sliding mode, i.e. the sliding surface is invariant
to the nonlinear control input transformation. One takes the Lyapunov candidate function
as $V(s)=1 / 2 s^{T} s > 0$, then the time derivative of the Lyapunov candidate
function is negative from (25), that is $\dot{V}(s)=s \cdot \dot{s} < 0$. Therefore the asymptotic stability of
the closed loop system is satisfied in the sense of Lyapunov.

The sliding mode equation i.e. the sliding surface $s=0$ is the same as that of $s^{*}=0$.
To compare the control inputs, $u_{1}$ and $u_{2}$, the form is the same but the gains
of $u_{1}$ are multiplied by $\left[C^{T} g_{0}(X, t)\right]$. To compare the control
input, $u^{*}$ and $u_{2}$, the form and the gain is the same. The two transformation
methods equivalently diagonalize the system, so those are called the two diagonalization
methods.

## 3. Design Examples and Simulation Studies

Consider a second order affine uncertain nonlinear system with matched uncertainties
and disturbances

Since (26) satisfy the Assumption A1, (26) is represented in the state dependent coefficient form as

where the nominal parameter $f_{0}(X, t)$ and $g_{0}(X, t)$ and mismatched uncertainties
$\triangle f_{1}(X, t)$ and $\triangle g(X, t)$, and matched uncertainty $\triangle
f_{2}(X, t)$ are

For later use, the integrals of the states are augmented as follows

where the non zero initial values for the integral states are chosen later.

### 3.1 Sliding surface transformation

The nonlinear feedback gain $K(X)$ is selected as

By the relationship of (14), the coefficients of the integral sliding surface are determined as

those are not unique. From $\dot{S}=0$ and the first differential equation in (26), the equation (8) i.e. 2nd equation of the ideal sliding dynamics is derived as

Combining (33) with the first differential equation in (26), then the ideal sliding dynamics is analytically obtained as^{[8]}

The ideal sliding dynamics (34) is stable for all finite $X_{1}$^{[34]}. The solution of (34), $X_{s}(t)$, coincides with the integral sliding surface of (7)^{[8]}. By using that, the output can be predictable because of the complete strong robustness
for the entire trajectory from any given initial state to the origin by the IVSS^{[8]}. The sliding surface transformation is designed as

Now, the IVSS control input is taken as follows:

The real dynamics of the sliding surface, i.e. the time derivative of $S^{*}$ becomes

where

If one takes the switching gains as follows:

then

if $s^{*}=0$, then $s=0$. The inverse augment also holds. The simulation is carried
out under 0.1[msec] sampling time and with $x(0)=\left[ \begin{array}{ll}{10} &
{2}\end{array}\right]^{T}$ initial state. The initial condition for the integral states
of (29) and (30) are calculated by (6) as

Fig. 1 shows the two output responses, $x_{1}$ and $x_{2}$ by $u^{*}$ with $s$ and the ideal
sliding output, i.e., the solution of the ideal sliding dynamics (34). Both the outputs are identical to each other. Thus, the output can be predicted
by means of the ideal siding output, i.e. the numerical solution of (34). The real phase trajectory(i) and ideal sliding trajectory(ii) are depicted in Fig. 2. As can be seen, there is no reaching phase and both the trajectories are same because
of the strong robustness for the entire trajectory from the given initial condition
to the origin. The sliding surface $s^{*}(t)$ is shown in Fig. 3. The control input $u_{2}(t)$ is depicted in Fig. 4.

그림. 1. 슬라이딩 면 변환에 의한 두 출력 응답 $x_{1}$과 $x_{2}$

Fig. 1. Two output responses, $x_{1}$ and $x_{2}$ for sliding surface transformation
by $u_{2}$ with $s^{*}$ and ideal sliding output

그림. 2. 실제 상 궤적과 이상 슬라이딩 궤적

Fig. 2. Real phase trajectory(i) and ideal sliding trajectory(ii)

그림. 3. 슬라이딩 면 $s^{*}(t)$

Fig. 3. Sliding surface $s^{*}(t)$

그림. 4. 제어입력 $u_{2}(t)$

Fig. 4. Control input $u_{2}(t)$

### 3.2 Control input transformation

The nonlinear feedback gain $K(X)$ is selected as

By the relationship of (21), the coefficients of the integral sliding surface are determined as

those are not unique. Because the coefficient vectors in (43) are the same as those of (32), the ideal sliding dynamics are also the same as that of (34). The initial condition for the integral states of (29) and (30) are calculated by (6) as the same of (41a) and (41b)

The control input transformation is designed as

Then, the real dynamics of $s$, i.e. the time derivative of $s$ is as follows:

where

If one takes the switching gain as design parameters

then one can obtain the following equation

The existence condition of the sliding mode is proved precisely. The switching gains
in (48) can be obtained also from (39) by multiplying $\left[C^{T} B_{0}\right]=2$. Fig. 5 shows the two output responses, $x_{1}$ and $x_{2}$ by $u^{*}(t)$ with $s$ and the
ideal sliding output that is the solution of (34). Fig. 5 is almost identical to Fig. 1 because the integral sliding surfaces $s=0=s^{*}$ are equal and the continuous gains
and discontinuous gains of the two controls, $u^{*}(t)$ and $u_{2}$, both are equal.
The output of this case is also predictable. The real phase trajectory(i) and ideal
sliding trajectory(ii) are depicted in Fig. 6. As can be seen, both the trajectories are the same because of the strong robustness
for the entire trajectory and there is no reaching phase like the case of the sliding
surface transformation. In Fig. 7, the sliding surface $s(t)$ is shown which is a $\left[C^{T} B_{0}\right]=2$ multiplied
value of $s^{*}(t)$ in Fig. 3. The control input $u^{*}(t)$ is depicted in Fig. 8 which is the same as $u_{2}(t)$ in Fig. 4.

그림. 5. 제어입력 변환에 의한 두 출력 응답

Fig. 5. Two output responses, $x_{1}$ and $x_{2}$ for control input transformation
by $u^{*}$ with $s$ and ideal sliding output

그림. 6. 실제 상 궤적과 이상 슬라이딩 궤적

Fig. 6. Real phase trajectory(i) and ideal sliding trajectory(ii)

그림. 7. 슬라이딩 면 $s(t)$

Fig. 7. Sliding surface $s(t)$

그림. 8. 제어입력 $u^{*}(t)$

Fig. 8. Control input $u^{*}(t)$