2.1 Descriptions of plants
An n-th order uncertain general linear plant is described by
where $z(\cdot) \in R^{n}$ is the original state, $u(\cdot) \in R^{1}$ is the control
input, $f \in R^{r}$ is the external disturbance, respectively, $A_{0}$ and $B_{0}$
is the nominal parameter matrices, $\Delta A$, $\Delta B$ and $D$ are the bounded
matrix uncertainties and those satisfy the matching condition as follows
Moreover the assumption on $\Delta B$ is made.
AssumptionA1:It is assumed the following equation is satisfied for a non zero element
coefficient vector $C_{z 1} \in R^{1 \times n}$
where $\eta$ is a positive constant less than 1.
The assumption 1 means that the value of uncertainty $\Delta B$ is less than the nominal
value $B_{0}$, which is acceptable in practical situations.
The purpose of the controller design is to control of the state of (1) to follow the predetermined intermediate sliding dynamics (trajectory) from a given
initial state to the origin. By the state transformation, $x=Pz$, a weak canonical
form [9][9] of (1) is obtained as
where
where $x(0)$ is the initial condition transformed from $z(0)$ and $d(t)$ is the lumped
uncertainty in the transformed system as
In (5), $b$ is 1, then the system (4) is the standard canonical form, otherwise, then the (4) is the weak canonical one[9].
2.2 Design of Transformed Integral sliding Surfaces
To design the ICTISMC, the transformed[49] integral sliding surfaces[6,15,16] are suggested to the following form having an integral of the state with a certain
initial condition as
where the coefficient matrices and the initial conditions for the integral states
are expressed as shown
The initial conditions (10a) and (10b) for the integral states in (7) and (8) are selected so that the transformed integral sliding surfaces are the zeros at $t=0$
for removing the reaching phase[6,16], which is stemmed from the idea in [15][15] Without these initial conditions, the reaching phase still exists and the overshoot
problem maybe exist because the integral state starting from the zero will be re-regulated
to the zero[6,8,27]. From
the differential equation for $x_{n}$ is obtained as
where
Combing (12) with the first n-1 differential equation in the system (4) leads to the ideal sliding dynamics
and
where
which is considered as a dynamic representation of the trans- formed integral sliding
surfaces (7) or (8)[6]. The solutions of (14) and (15), $x_{s}^{*}$ and $z_{s}^{*}$ coincide with and predetermine the transformed integral
sliding surfaces (7) and (8)(the sliding trajec- tories) from a given initial condition to the origin[15]. By using the solutions of (14) and (15), the output is predetermined and predicted. To design the transformed integral sliding
surfaces (7) and (8), the system matrix $\Lambda_{c}$ is to be stable or Hurwitz, that is all the eignvalues
of $\Lambda_{c}$ have the negative real parts. To choose the coefficient vectors of
the transformed integral sliding surfaces by means of the well known linear regulator
theories, (14) and (15) are transformed to the each nominal system form of (1) and (4)
where
and expressed with the original state as
where
After determining $K$ or $G$ to have the desired ideal sliding dynamics, the coefficient
vectors of the transformed integral sliding surfaces (7) or (8) can be directly chosen from the relationship
which is derived from (18). If this regulation control problem is designed by using the nominal plants (17) or (19), then the transformed integral sliding surface having exactly the same performance
can be effectively chosen by using (21). If $\Lambda_{c}$ is designed to be Hurwitz, then which guarantees the exponential
stability of the system (14) and there exist the positive scalar constants $K_{1}$ and $k$ such that
where $\|\cdot\|$ is the induced Euclidean norm as $\sqrt{\operatorname{trace}\left(e^{\Lambda_{c}
t^{T}} \cdot e^{\Lambda_{c} t}\right)}$.
Now, define $\overline{E_{0}}(t)$ and $\overline{E_{1}}(t)$ are the modified error
vector from the ideal sliding trajectory and its derivative, ie. the error vector,
respectively as
where
If the transformed integral sliding surface is the zero for all time, naturally this
defined error and its derivative are also the zeros. The transformed integral sliding
surfaces may be not exactly zeros if the control input of the ICTISMC is continuously
imple- mented. Hence the effect of the non-zero value of the transfor-med integral
sliding surface to the error to the sliding trajectory is analyzed in the following
Theorem 1[19] as a prerequisite to the main theorem.
Theorem 1 : If the transformed integral sliding surfaces defined by equation (7) or (8) satisfy $\|s(t)\| \leqq \gamma$ for any $t \geqq 0$ and $\left\|\overline{E}_{0}(0)\right\|
\leqq \gamma / \kappa$ is satisfied at the initial time, then
is satisfied for all $t \geqq 0$ where $\epsilon_{1}$ and $\epsilon_{2}$ are the positive
con- stants defined as follows:
Proof: The transformed integral sliding surface can be re-written as
and can be re-expressed in a differential matrix from as
In (28), the transformed integral sliding surface may be con- sidered as the bounded disturbance
input, $\|s(t)\| \leqq \gamma$. The solution of (28) is expressed as
From the boundness of the transformed sliding surface and (22), the Euclidean norm of the vector $\overline{E_{0}}$ becomes
for all time, $t \geqq 0$. From (28), the following equation is obtained
which completes the proof of Theorem 1.
The above Theorem 1 implies that the modified error vector and error vector from the
ideal sliding trajectory are uniformly bounded provided the transformed integral sliding
surface is bounded for all time $t \geqq 0$. Using this result of Theorem 1, we can
give the specifications on the norm of the error vector from the ideal sliding trajectory
being dependent upon the value of the transformed integral sliding surface, (7). In the next section, we will design the discontinuous and continuous variable structure
regulation controllers which can guarantee the boundedness of s(t), i.e., $\|s(t)\|
\leqq \gamma$ for a given $\gamma$, then the error vector to the ideal sliding trajectory
is bounded by $\epsilon_{2}$ in virtue of Theorem 1.
2.3 Transformed Discontinuous and Continuous Control Inputs
As the second design phase of the ICTISMC, a following cor- responding discontinuous
control input to generate the perfect sliding mode on the every point of the pre-selected
transformed integral sliding surface from a given initial state to the origin is proposed
as composing of the continuous and discontinuously switching terms
where
The $G_{1} \cdot s$ in the continuous feedback term can reinforce the controlled systems
in more closer tracking to the pre-selected ideal transformed integral sliding surface
from a given initial condition to the origin[6,15,27] in order to increase the control accuracy and steady state performance. By this discontinuous
control input, the real dynamics of $s$, i.e. the time derivative of $s$ becomes
The closed loop stability and existence of the sliding mode on the preselected transformed
integral sliding surface by the proposed discontinuous control input will be investigated
in the next theorem.
Theorem 2 : The proposed integral variable structure controller with the discontinuous
input (32) and the transformed integral sliding surface (7) can exhibit the exponential stability to the ideal transformed integral sliding surface
and the ideal output of the sliding dynamics for all the uncertainties exactly defined
by the transformed integral sliding surface (7).
Proof: Take a Lyapunov candidate function as
Differentiating (38) with time leads to
Substituting (37) into (39) and by (33)-(36), one can obtain the following equation
From (40), the following equation is obtained as
which completes the proof.
As can be seen in (40) and (41), because is included in the decay rate parameter, the larger , the fast closer tracking
to the transformed integral sliding surface. The term can increase the steady state
performance and control accuracy to the ideal transformed sliding surface including
the zero(origin) within the boundary layer. The exponential stability to the transformed
integral sliding surface and the existence condition of the sliding mode on the every
point of the transformed integral sliding surface is proved, while in the previous
works on the VSS, only the asymptotic stability is guaranteed[1,7,8,22,47]. The sliding mode on the every point of the transformed integral sliding surface
from a given initial state to the origin is guaranteed. Hence the sliding output from
a given initial state to the origin is insensitive to the matched uncertainties and
external disturbances by the proposed discontinuous VSS input (32). By using the solution of the ideal sliding dynamics (15), the controlled output from a given initial state to the origin can be predicted
and predetermined, as an attractive performance in the theoretic aspect, because the
reaching phase is removed and the existence condition of the sliding mode is proved.
The discontinuous input (32) can regulate the transformed integral sliding surface to be zero theore- tically.
However, the control input is discontinuous which results in the chattering problems[5,26]. So for practical applications, the discontinuous input term is essentially approximated
to be continuous. By using the modified fixed boundary layer method[51], the discontinuous input (32) has changed to the following form
where $MBLF(s)$ is defined as a modified fixed boundary layer function as follows:
Because the switching terms in (42) are stable itself which is shown through Theorem 2, the $MBLF(s)$ function can not
influence on the closed loop stability and only can modify the magnitude of the switching
terms within the fixed boundary layer instead of the sign function when $s$ is positive
as well as negative. If $l_{+}=l_{-}$, then the $MBLF(s)$ function is symmetric, otherwise
it is asymmetric, which is suitable in case of the unbalanced uncertainty and disturbance
and unbalanced chattering inputs.
Theorem 3 : The proposed integral variable structure controller with the suggested
continuous input (42) and the transformed integral sliding surface (7) can exhibit the bounded stability for all the uncertainties and external disturbances.
Proof: Take a Lyapunov candidate function as
From the proof of Theorem 2, we can obtain the following equation
as long as $\left.\right|_{S}(z, t) | \geqq l=\max \left(l_{+}, l_{-}\right)$. From
(45), the following equation is obtained as
as long as $|s(z, t)| \geqq l$, which completes the proof.
As can be seen in (45) and (46), outside the boundary layer, the exponential stability is still guaranteed and inside
the boundary layer the $G_{1} \cdot s$ term can increase the control accuracy and
steady state performance. The larger $G_{1}$, the closer tracking to the ideal transformed
sliding surface from a given initial condition to the origin. By Theorem 3, the continuously
imple- mented control input (42) can guarantee that the transformed integral sliding surface (7) is bounded by $l$. Hence it is possible to design that $l$ is less than $\gamma$,
that is $l \leqq \gamma$. Thus the trans- formed integral sliding surface is bounded
by $\gamma$ which satisfies the condition of Theorem 1. Then by Theorem 1, the fact
that the norm of the error vector to the ideal transformed sliding surface is bounded
by $\epsilon_{2}$ is possible as the prescribed control performance.