1. Introduction

The sliding mode control(SMC) is divided into the two parts, i.e. linear sliding mode control(LSMC)^{(1)(2)(27)-(35)} and terminal sliding mode control(TSMC)^(3-26). The TSMC has the advantages over the LSMC for example convergence in finite time and high control precision. For the first time, Haimo developed a finite time controller with the finite time stabilization in 1986⁽³⁾. Haimo invented the finite time stabilization rather than the asymptotic convergence of the LSMC. Zak presented the terminal attractors with the finite time convergence in 1988⁽⁴⁾. To be the finite time stabilization, the exponent of the power function should be $0<k<1$ with $1$ over the positive odd numbers. After those, a lot of the researches on the TSMCs are reported until now^(5)-(26). Venkataraman and Gulati first studied the TSMC with a nonlinear sliding surface and finite time convergence⁽⁵⁾⁽⁶⁾. The exponent of the power function in the terminal attractors must be fractional with the positive odd numbers over the positive odd numbers. The TSMC is applied to the control of robot manipulators. Successively, many theoretical developments and application examples of the TSMCs have been reported in literatures^(7)-(26). In ⁽⁷⁾, 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 ⁽⁸⁾. In 1997, Yu et. al. suggested a TSMC having the fast transient performance with the recursive sliding surface for higher order SISO systems in ⁽⁹⁾. The acceleration term is added to the terminal sliding manifold in order to speed up the output response. Terminal sliding mode controls for high order SISO systems and for MIMO linear systems are proposed by Zhihong and Yu in ⁽¹⁰⁾. Yu et. al suggested a nonsingular terminal sliding mode control of a class of nonlinear dynamical systems in ⁽¹¹⁾. The conventional TSMCs until now have the singularity problems that the control input becomes infinity in certain domain. However, there is no singular problem in ⁽¹¹⁾. For the control of rigid manipulators, a terminal SMC with the nonsingular sliding surface is suggested by Feng et al in ⁽¹²⁾. Feng et. al investigated a second order TSMC for uncertain multivariable systems for chattering-free performance and nonsingularity in ⁽¹³⁾. A terminal sliding mode observer is designed by Feng et. al and applied to control of permanent magnet synchronous motor systems with an nonsingular sliding manifold in ⁽¹⁴⁾. Jo et. al designed a terminal sliding mode control system for second order systems in ⁽¹⁵⁾. It is shown that the fractional rational number can be used to the exponent of a power function of the terminal sliding surface, and also the exponent with an odd number over an even number can be used. In ⁽¹⁶⁾ and in 2015, a discrete time integral TSMC of precision micro motion systems is investigated. 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 ⁽¹⁷⁾. 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. But, the real output can not be predicted. A derivative and integral TSMC for a class of MIMO nonlinear systems is suggested by Chiu in ⁽¹⁸⁾. The recursive sign or fractional integral terminal sliding manifolds are proposed to remove the reaching phase. However the real output can not be predictable. 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 ⁽¹⁹⁾. There exists the reaching phase problem. For noncanonical plants of interceptors, a fast robust guidance and control is designed based on a fast fractional integral terminal sliding surface for removing the reaching phase in ⁽²⁰⁾. It is not possible to predict the real output. A continuous TSMC for a class of uncertain nonilinear systems is proposed by means of the integral terminal sliding surface for removing the reaching phase and finite time disturbance observer for coping with the chattering problems in ⁽²¹⁾. For the tracking control of noncanonical unmanned underwater vehicles with an adaptive dynamic compensation of uncertainties and disturbances, a double loop adaptive integral terminal sliding mode control is designed by using an integral terminal sliding manifold or fast integral terminal sliding manifold for removing the reaching phase in ⁽²²⁾. The real output can not be predictable. Hu et. al analyzed a dynamic sliding mode manifold based continuous fractional order noonsingular terminal sliding mode control for a class of second order nonlinear systems when $\triangle b\ne 0$ in 2020 and in ⁽²³⁾. 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 ⁽²⁴⁾. But the real output is not predicted. By using the saturation on the singular component of the control input, a nonsingular terminal sliding mode control of nonlinear systems is suggested to avoid the singularity temporary in ⁽²⁵⁾. 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 ⁽²⁶⁾.

Utkin presented his invariant theorem by means of the transformation(diagonalization)s without the complete proof⁽¹⁾⁽²⁾. In ⁽²⁷⁾, the Utkin’s invariance theorem is completely proved for MIMO uncertain linear plants. There are the two approaches or three approaches when the integral(compensator) augmentation, to the design of the variable structure system(VSS)s, the two are the sliding surface transformation VSS^(28)(30) and control input transformation VSS^{(16)(18)-(20)(23)(25)(26)(29)(31)}, and the last is the sliding surface part transformation when the compensator augmentation⁽³⁵⁾. To remove the reaching phase problems, the two conditions must be satisfied, those are that the sliding surface is defined from any initial condition to the origin and the existence condition of the sliding mode should be satisfied and proved for the whole trajectory^(30)(31).

In most of TSMCs until now except ⁽¹⁷⁾ and ⁽²³⁾, the TSMC is designed when $\triangle b=0$. When $\triangle b\ne 0$, the the problem of the TSMC design becomes complex and it is difficult to satisfy the existence condition of the sliding mode that is $s·\dot s < 0$ for SISO plants and $s_{i}·\dot s_{i}<0,\: i=1,\:2,\:...,\:m$ for MIMO plants. To remove the reaching phase in ^{(17), (18), (20)-(22),} and ⁽²⁴⁾ of the integral TSMCs, the first requirement is satisfied that is $s(t)_{t=0}=0$ but the second condition is hardly satisfied to completely remove the reaching phase. Then, the strong robustness of the controlled real output is not guaranteed for the whole trajectory. Hence the real robust output can not be predicted.

In this paper, the discontinuous and continuous control input transformed integral TSMCs by using the integral sliding surface without the reaching phase problems and with the output prediction performance as the one approach among the three approaches are presented for second order uncertain plants. Theoretically discontinuous and practically continuous control input transformed TSMCs are proposed when $\triangle b\ne 0$. The integral sliding surface idea of ⁽³²⁾ for the LSMC is applied to the integral sliding surface for the TSMC in this paper. And the exponent of the power function can be fractional such that $p$ and $q$ are 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 predesigned, predicted, and predetermined by means of the solution of the ideal sliding dynamics of the integral sliding surface. Based on defining a new auxiliary nonlinear state and chattering according to the the condition of that, a transformed control input is suggested for easily satisfying the existence condition of the sliding mode when $\triangle b\ne 0$. 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. To temporary avoid the singularity of the new auxiliary nonlinear state, a certain limit is imposed on. For practical applications, a continuous approximation of the discontinuous TSMC is made by means of the modified boundary layer function^(29)-(31). In addition to, the closed loop bounded stability together with the existence condition of the sliding mode by the continuous TSMC is analyzed. The discontinuity of the control input as the inherent property of the VSS is much improved in view of the practical applications. Through a design example and simulation studies, the usefulness of the proposed discontinuous and continuous control input transformed TSMC controllers is verified.

2. Discontinuous and Continuous Integral TSMCs

A second uncertain phase canonical linear system is considered

where $x_{1}\in R$ and $x_{2}\in R$ are the state variables, $u\in R$ is the control input to be designed, $a_{10},\:a_{20},\: {and}b_{0}\in R$ are the nominal values, $\triangle a_{1},\: \triangle a_{2},\: {and}\triangle b$ 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. Until now, most of the TSMCs except ⁽¹⁷⁾ and ⁽²³⁾ are studied when $\triangle b=0$, while in this paper the proposed TSMCs are designed when $\triangle b\ne 0$ as the extension studies of the previous TSMCs of ^(3)-(16) and ^(18)-(22).

Assumption 1:

$\triangle b(b_{0})^{-1}=\triangle I$, and $|\triangle I|\le\rho <1$ where $\rho$ is the constant.

The first aim of the integral TSMC design is to maintain the integral sliding surface $s$ to be the zero value from the initial time without the reaching phase when $\triangle b\ne 0$ and finally to regulate the state $x$ to be the zero value in a finite time from any initial condition by the discontinuous input with the output prediction. By the continuous input for practical applications, the second aim of the integral TSMC controller design is to maintain the integral sliding surface $s$ to be bounded near to the zero and finally to regulate the state $x$ to be bounded in a finite time with almost output prediction by the continuous input.

An integral state $x_{0}\in R$ with a special initial condition is augmented for use later in the integral 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 ⁽³²⁾ of the LSMC, for removing the reaching phase, the integral terminal sliding surface for the TSMC $s\in R$ is proposed as follows:

where $p$ and $q$ 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 numbers such that $0<p/q<1$ are first mentioned. 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(2) for the integral state is determined so that the integral sliding surface eq(3) is the zero at $t=0$ for any initial condition $x_{1}(0)$ and $x_{2}(0)$ as

With the initial condition eq(4) 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(3) 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^(30)(31). In the sliding mode, the equation $s=0=\dot s$ is satisfied. Then from eq(2) and eq(3) the ideal sliding dynamics is derived as

which is a dynamic representation of the integral sliding surface eq(3). The solution of eq(5) is identical to the integral sliding surface and the real robust controlled output itself. Therefore, the output can be pre-designed, predetermined, and predicted.

Now, the suggested transformed discontinuous TSMC input for the integral sliding surface and uncertain plant eq(1) is taken as follows:

where a new auxiliary nonlinear state $x_{3}$ is defined as

and one takes the constant gains as

and takes the discontinuously switching gains as follows:

where $sign(s)$ is $sig\nu m(s)$ function as

$b_{0}^{-1}$ is multiplied to all the components of the discontinuous input eq(6) because the transformation of the control input for easy proving that the existence condition of the sliding mode is realized as one approach among the three approaches of the transformation(diagonalization)s^(1)(2)(27). Based on defining the auxiliary state $x_{3}$ for the first time, the discontinuous input is chattering according to the condition of $sx_{3}$ in eq(14). Since that, it is easily shown that the existence condition of the sliding mode is clearly satisfied when $\triangle b\ne 0$. Since $\triangle b\ne 0$, the effect of $\triangle b\ne 0$ is considered in the selection of the discontinuous chattering gains eq(12)-(15). 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(6), the sliding surface itself is one of the feedback elements which makes the controlled system be closer to the ideal predetermined sliding surface⁽³²⁾. Then the real dynamics of the integral sliding surface by the transformed discontinuous control input, i.e. the time derivative of $s$ becomes

From eq(8)-(10), the real dynamics of $s$ becomes finally

From eq(18), 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 components and the feedback of the integral sliding surface. The total closed loop stability with the transformed discontinuous control input eq(6) and the integral sliding surface eq(3) together with the precise existence condition of the sliding mode will be investigated in Theorem 1.

Theorem 1: If the integral sliding surface eq(3) is designed to be stable, the transformed discontinuous control input eq(6) with the integral sliding surface eq(3) 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(19) with time leads to

]Substituting eq(18) into eq(20) leads to

Since the uncertainty and external disturbance terms in eq(21) are canceled out due to the chattering input terms by means of the switching gains in eq(12)-(15), one can obtain the following equation^(30)(31)

The existence condition of the sliding mode on the predetermined integral 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 on every point on the whole trajectory of the predetermined integral 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^(30)(31). From eq(22) the following equation is obtained.

From eq(23), the following equation is obtained

which completes the proof of Theorem 1.

Due to the proof of Theorem 1, the following is concluded by means of the transformed discontinuous control input

\begin{align*} ma\int a\in V(x(t))as 0 \\ => ma\int a\in s(t)as 0\\ => x(t)arrow 0 \end{align*}

To avoid the singularity for the auxiliary nonlinear state $x_{3}$ when $x_{1}=0$ and $x_{2}\ne 0$, a certain heuristic limit value is imposed on $x_{3}$ as

which may sacrifice the existence condition of the sliding mode at the moment, but effectively avoids the singular problems. In ⁽¹⁹⁾ and ⁽²⁵⁾, there are some limits on the singular control input component. The saturation on the singular state is more easier than the saturation on the singular control input component.

The transformed control input eq(6) may have the chattering problems because of the high frequency switching of the discontinuous part of the control input eq(6) due to the switching of the sign function in eq(16) according to the value of the sliding surface which may be harmful to practical real plants^(33)(34). Hence, the continuous approximation of the discontinuous TSMC is essentially necessary for practical applications to real uncertain plants without a severe performance loss. Applying the idea in ⁽²⁹⁾ of the modified fixed boundary layer method, the transformed discontinuous input eq(6) is modified to the following form

where $mblf(s)$ is defined as a modified fixed boundary layer function as follows^(30)(31):

Because the switching parts in eq(27) are stable itself which is shown through Theorem 1, 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. If $l_{+}=l_{-}$, then the $mblf(s)$ function is symmetric with respect to y-axis, otherwise it is asymmetric, which is suitable in case of the unbalanced uncertainty and disturbance and unbalanced chattering gains.

Theorem 2: The integral TSMC with the transformed continuous input eq(27) and the proposed integral sliding surface eq(3) can exhibit the bounded stability for all the uncertainties and external disturbances.

Proof: Take a Lyapunov candidate function as

If $s>l_{+}$ or $s<l_{-}$, then $mblf(s)=1$ from eq(28) and the continuous input eq(27) becomes the discontinuous input eq(6). Therefore from the proof of Theorem 1, we can obtain the following equation

as long as $vert vert s(x,\:t)vert vert\ge l=\max(l_{+},\:l_{-})$, which means that over the boundary layer, the existence condition of the sliding mode is the same as that of the discontinuous input case. From eq(30), the following equation is obtained as

as long as $vert vert s(X,\:t)vert vert\ge l$, which completes the proof.

Due to the proof of Theorem 2, the following statement is concluded by the continuous control input as $t arrow\infty$

$V(x(t)) \rightarrow$ bounded by $l^{2} / 2$

implies $s(t) \rightarrow$ bounded byl

implies $x_{1}(t) \rightarrow$ bounded

The major contributions of this paper are as follows:

1. The designs of the theoretically discontinuous and practically continuous control input transformed integral TSMCs as one approach among the three approaches of the transformations are presented when $\triangle b\ne 0$.

2. By means of applying the idea of the integral sliding surface of ⁽³²⁾ for the LSMC to that of the TSMC, the reaching phase is completely removed by means of satisfying the two requirements.

3. The ideal sliding dynamics as the dynamic representation of the integral sliding surface is derived.

4. The performance that the real robust output can be predetermined, pre-designed, and predicted is obtained first in the TSMCs by using the solution of the ideal sliding dynamics.

5. It is first pointed out that the any number exponent of the power function satisfying $0<p/q<1$ is possible for the more degree of freedom.

6. A new auxiliary nonlinear state $x_{3}$ is defined and the chattering according to the condition of $sx_{3}$ is made in order to satisfy the existence condition of the sliding mode clearly when $\triangle b\ne 0$.

7. A certain heuristic limit on the new auxiliary nonlinear state is imposed for temporary avoiding the singularity

3. Design Example and Comparative Simulation Studies

Consider a second-order SISO uncertain linear system with a single input described by the state equation

where the nominal parameters $a_{10}$, $a_{20}$, and $b_{0}$, matched uncertainties $\triangle a_{1}$, $\triangle a_{2}$ and $\triangle b$, and disturbance $\triangle d(x,\:t)$ are

$a_{10}=0$, $a_{20}=-3$, $b_{0}=2$, $\triangle a_{1}= +- 0.01$,

To design the proposed integral TSMC 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

$r^{2}+C_{1}r+C_{0}=r^{2}+6r+9=(r+3)^{2}$

The $p$ and $q$ are selected as

The $p$ is even not odd because any number is possible such that $0<p/q<1$. Then the integral sliding surface becomes

where

and the ideal sliding dynamics becomes

The equation in Assumption 1 is calculated as

The constant feedback gains are accordingly designed as

If one takes the switching gains as follows:

The eq(21) and eq(29) become

The existence condition of the sliding mode is satisfied for the every point on the integral sliding surface i.e. for the entire sliding trajectory from a given initial to the origin. To avoid the singular problems of the auxiliary nonlinear state when $x_{1}=0$ and $x_{2}\ne 0$, the limits on $x_{3}$ are selected as follows:

By using Fortran software, the simulation is carried out under $0.1[m\sec]$ sampling time and with $x(0)=\left[x_{1}(0)x_{2}(0)\right]^{T}=[2.0 1. 0]^{T}$ initial condition. Then by eq(4) the initial condition for the integral state becomes

For comparison, the simulation results of the previous TSMC of ⁽²⁵⁾ are given from Fig 1 to Fig 4. Fig 1 shows the two state output responses, $x_{1}$ and $x_{2}$ without uncertainty and disturbance as a model and with uncertainty and disturbance as a real plant of the reference ⁽²⁵⁾ for comparison. The two responses are different because during the reaching phase the outputs are disturbed. The three trajectories with the ideal, model, and real plant are shown in Fig 2. There are the reaching phases as can be seen. The two sliding surfaces and two continuous control inputs are depicted in Fig 3 and Fig 4, respectively. As can be seen, from the beginning, the control input can not be chattered because of the reaching phase.

The simulation results of the proposed integral TSMCs are given from Fig 5 to Fig 12, in which from Fig 5 to Fig 8 the results by the proposed discontinuous input are shown and from Fig 9 to Fig 12 the results by the continuous input are depicted. Fig 5 shows the three state output responses $x_{1}$ and $x_{2}$ with the ideal sliding, model, and real plant by the discontinuous control input. The ideal sliding output is the time solution of the ideal sliding dynamics eq(38) for the given initial condition. As can be seen, the ideal sliding output, model output, and real plant output are almost identical. The robust real output can be exhibited as designed in the integral sliding surface and the robust real output can be predetermined and predicted. The three trajectories are shown in Fig 6. The ideal sliding trajectories are depicted by the solution of eq(38). As can be seen, therefore is no the reaching phase. The two sliding surfaces and two discontinuous control inputs are depicted in Fig 7 and Fig 8, respectively. As can be seen, the controlled system slides from the initial time without the reaching phase by the discontinuous input. The large chattering of the discontinuous input is shown, which is harmful to real plants. The continuous approximation by the transformed continuous control inputs eq(27) with the proper layer

$l_{-1}=l_{+1}=0.1$ is made. Fig 9 shows the three state output responses with the same performance of Fig 5. The three trajectories are shown in Fig 10. The two continuous sliding surfaces and two continuous control inputs are depicted in Fig 11 and Fig 12, respectively. As can be seen, the large chattering of the two control inputs is much improved with almost the same output performance as that of the proposed discontinuous TSMC.

4. Conclusion

In this note, when $\triangle b\ne 0$, the discontinuous and continuous control input transformation TSMCs are proposed with the output prediction and predetermination performance for all uncertainties and disturbances as the one approach of the transformation. Based on the idea in ⁽³²⁾ of the LSMC, the integral sliding surface is suggested for removing the reaching phase in the proposed TSMCs. The ideal sliding dynamics for whole trajectory is derived. The exponent of the power function in the integral sliding surface $p$ and $q$ are any positive numbers satisfying $0<p/q<1$. By means of defining the new auxiliary nonlinear state, the discontinuous terminal control input is suggested for satisfying the existence condition of the sliding mode when $\triangle b\ne 0$. To present the complete formulation of the TSMC design for the output prediction and predetermination performance, the closed loop exponential stability together with the existence condition of the sliding mode on the selected sliding surface by the proposed transformed discontinuous control input is investigated in Theorem 1. By using the solution of the ideal sliding dynamics, the real robust output can be predicted, pre-designed, and predetermined. The output(performance) designed in the sliding surface is completely guaranteed in the real output for all uncertainties and disturbances as the performance robustness that is how can control designers decrease the difference between the design output(performance) and real output(performance), which is the problem in the previous TSMCs. As a remedy of the singularity, a certain limit is imposed on the new auxiliary nonlinear stste. For practical application of the proposed transformed integral TSMC, the harmful chattering of the discontinuous input is effectively much improved without the severe performance loss. The bounded stability together with the existence condition of the sliding mode by the continuous transformed integral TSMC is investigated in Theorem 2. Through a design example and comparative simulation studies, the effectiveness of the proposed discontinuous and continuous transformed integral TSMC controllers is verified. Theoretically the discontinuous input is considered for the discontinuous control of simulation plants, and practically the continuous integral TSMC can be applied to the continuous control of real uncertain plants.