1. Introduction

Terminal sliding mode control (TSMC) was initially conceptualized by Haimo in 1986(3). His work gave rise to a finite-time control mechanism that enabled stabilization within a specific time frame, a departure from the asymptotic convergence observed in linear sliding mode control (LSMC). TSMC boasts several merits over LSMC, most notably finite-time convergence and enhanced control precision. This concept further matured in 1988 when Zak introduced terminal attractors exhibiting finite-time convergence⁽⁴⁾. Subsequently, an array of investigations concerning TSMC emerged, encompassing theoretical advancements and practical illustrations⁽¹⁸⁾-⁽³²⁾. For instance, in ⁽¹⁸⁾, Zhihong et al. delved into the application of TSMC in the realm of controlling multi-input multi-output (MIMO) robot manipulators. Building on this, Zhihong and Yu expanded the scope in ⁽¹⁹⁾, presenting TSMC approaches for higher-order single-input single-output (SISO) linear systems with hierarchical terminal sliding surfaces. They also addressed regular MIMO systems by introducing fractional-order sliding surfaces. In another vein, their work in ⁽²¹⁾ introduced a nonsingular TSMC methodology tailored for a specific class of nonlinear dynamical systems. This is a notable advancement, as conventional TSMCs often grapple with singularity issues, leading to infinite control inputs within certain domains. However, ⁽²¹⁾ successfully circumvented this problem. Furthermore, Zong et al. proposed a novel self-tuning law-based higher-order sliding mode control in ⁽²³⁾. This contribution catered to systems marked by uncertainties in the input matrix, thereby presenting a versatile solution to a practical challenge. This approach attains finite stabilization through the utilization of higher sliding modes characterized by geometric homogeneity, coupled with an integral sliding surface devoid of any reaching phase. In a parallel context, Pen et al. introduced an innovative integral terminal sliding surface design⁽²⁵⁾ targeted at uncertain nonlinear systems. This design uniquely employs saturation on the singular component of the control to temporarily circumvent singularity-related issues. Addressing a distinct scenario, for noncanonical plants of interceptors, an expedited robust guidance and control strategy was formulated⁽²⁶⁾. This strategy draws on a rapid fractional integral terminal sliding surface, effectively eradicating the reaching phase and contributing to enhanced performance. In 2020, Hu et al. conducted an in-depth analysis on dynamic sliding mode manifold-based continuous fractional- order nonsingular terminal sliding mode control. This study centered around a specific category of second-order nonlinear systems⁽²⁷⁾. In their algorithm, the uncertainty term within the input matrix was treated as an aggregated total lumped uncertainty. To address the issue of the reaching phase, a solution was proposed involving the utilization of time-varying sliding hyperplanes⁽²⁸⁾. However, it's important to note that this approach did not incorporate real output prediction. Diving further into the realm of TSMC design, a significant contribution was made in ⁽³¹⁾, where two distinct approaches leveraging integral sliding surfaces were introduced. These approaches aimed to eliminate the reaching phase while also achieving output prediction performance for second-order uncertain plants. Specifically, these methods encompassed the design of discontinuous and continuous control input transformation integral TSMCs. This concept draws inspiration from ⁽⁹⁾, which pertains to LSMC, and extends it to the realm of TSMCs. The introduction of an integral sliding surface devoid of a reaching phase stands as a novel proposition within this domain. The exponent of the power function is flexible and can assume any positive value that satisfies the condition $q>p>0$ such that $0<p/q<1$. The derivation of the integral sliding surface's ideal sliding dynamics enables the prediction, pre-design, or pre-determination of the actual robust output by utilizing the solution derived from these ideal sliding dynamics. In order to attain closed-loop exponential stability and ensure the existence of the sliding mode on the predetermined sliding surface, a novel auxiliary nonlinear state is introduced. This auxiliary state plays a crucial role in the comprehensive formulation of the TSMC design tailored for achieving output prediction performance. The investigation into this design is conducted with a strong theoretical foundation. A pivotal aspect of this design is the avoidance of singularities. To this end, a limit is imposed on the newly introduced auxiliary nonlinear state. This precautionary measure effectively prevents potential singularities from arising within the system. In ⁽³²⁾, the rigorous proof of Utkin's theorem is presented for second-order SI(single input) non-integral uncertain linear plants, specifically in cases where the input gain uncertainty is non-zero (i.e. $\triangle b\ne 0$ ). This proof encompasses Utkin's theorem for the two transformations utilized within TSMCs: control input transformation and sliding surface full transformation. Meanwhile, ⁽³³⁾ introduces a crucial observation by highlighting the existence of five distinct design approaches for ITSMCs. These methodologies include control input transformation, sliding surface full transformation, and three variations of sliding surface part transformations. Furthermore, the invariance theorem proposed by Utkin is clearly and comparatively demonstrated in ⁽³³⁾ for the initial two transformation methods, often referred to as the two diagonalization methods. This demonstration is carried out within the context of second-order SI integral uncertain linear systems, where the input gain uncertainty remains non-zero (i.e. $\triangle b\ne 0$ ). The concept of control input transformation has been extensively explored in prior works such as ⁽³¹⁾, ⁽³²⁾, and ⁽³³⁾. Similarly, the notion of sliding surface transformation is introduced anew within the realm of TSMC, with exceptions being ⁽³²⁾ and ⁽³³⁾. The innovative introduction of sliding surface part transformations is a notable contribution to TSMC, emerging through the research documented in ⁽³³⁾. In the context of the first three transformations, in-depth investigations are thoroughly presented within ⁽³³⁾. These explorations yield a performance characteristic wherein the real output can be forecasted using solutions originating from the ideal sliding dynamics. These solutions are thoughtfully pre-designed and predetermined through the ideal sliding dynamics. Conversely, the research concerning the last two transformations remains an avenue for future exploration. This area is yet to be comprehensively studied, offering fertile ground for further investigation and scholarly contribution. The five transformations serve as pivotal design methodologies for both TSMCs and ITSMCs. These transformations collectively enable the establishment of the sliding mode's existence condition on the sliding surface. Among these five methods, Utkin's theorem finds application solely in the context of the first two transformations, thereby demonstrating their significance. Notably, the ideal sliding surface remains invariant regardless of which of these two transformations is employed. However, the three part transformations pose a distinct scenario. In these cases, the application of Utkin's theorem is not viable, and as a result, the assurance of the invariance property cannot be guaranteed. This distinction highlights the complexity inherent in these three part transformations, and it underscores the challenges in ensuring their invariance property.

Research on the proof of Utkin's theorem within the domain of TSMC is somewhat limited and primarily documented in two key papers, namely ⁽³²⁾ and ⁽³³⁾. The exploration of control input transformation-based SMCs has garnered attention in various works, encompassing ⁽²⁴⁾-^(26)(27)(29)-⁽³¹⁾. It's noteworthy that ⁽³¹⁾ holds the distinction of introducing the algorithm associated with control input transformation SMCs. The realm of sliding surface full transformation has been examined and discussed within the confines of ⁽³²⁾ and ⁽³³⁾. However, it's crucial to emphasize that research into the initial sliding surface part transformation, which encompasses the first two transformations among the set of five, is exclusively presented within ⁽³³⁾. This makes ⁽³³⁾ a significant source of insights in this area.

In this paper, we introduce a novel exploration of the remaining two sliding surface part transformations among the set of five transformations. These investigations are carried out within the realm of ITSMC (Integral Terminal Sliding-Mode Control) for second-order SI uncertain linear systems. Our work serves as an alternative avenue of study to that of ⁽³³⁾, building upon and extending its findings. The primary objective of this paper is to present two distinct approaches for designing ITSMCs targeted at second-order uncertain linear plants, particularly when the input gain uncertainty is non-zero (i.e. $\triangle b\ne 0$). This stands in contrast to the typical scenario in most ITSMCs, where the design assumes zero input gain uncertainty (i.e. $\triangle b=0$). One of the notable outcomes of the last two sliding surface part transformations is their capacity to predict, pre-determine, and pre-design outputs. This culminates in performance levels akin to those detailed in ⁽³³⁾. Additionally, these two transformations take on the pivotal role of design methodologies within the ITSMC framework, thereby serving as alternative options to those outlined in ⁽³³⁾. The practical significance of our findings is underscored through the provision of design examples and comprehensive simulation studies. These demonstrations collectively highlight the tangible value and applicability of the key insights presented in this work.

2. Further Studies of Proof of Utkin's Theorem for ITSMCs

For a second order SI uncertain canonical linear system:

Assumption 1:

$\dfrac{1}{b_{0}}\triangle b=\triangle b\dfrac{1}{b_{0}}=\triangle I$, and $vert\triangle I vert\le\epsilon <1$ where $\epsilon$ is a positive constant.

An integral state $x_{0}\in R^{1}$ with a special initial condition is augmented for use later in the integral terminal sliding surface as follows:

$x_{0}(t)=\int_{0}^{t}x_{1}(\tau)d\tau +\int_{-\infty}^{0}x_{1}(\tau)d\tau$

In reference ⁽³³⁾, five approaches to designing ITSMCs are identified, including control input transformation, sliding surface full transformation, and three sliding surface part transformations. The first three transformations are covered in ⁽³³⁾, while the research on the last two transformations requires further study. The detail researches on the last two part transformations will be studied in this paper. Specifically, the two remaining part transformation sliding surfaces are mentioned as

3. Design Example and Illustrative Simulation Study Consider a second order uncertain canonical system

For comparison, the results of the previous sliding surface part transformation which is already presented in the previous paper ⁽³³⁾ are shown from fig 1 to fig 4 with an initial condition $x(0)=\left[\begin{array}{ll}x_1(0) & x_2(0)\end{array}\right]^T=\left[\begin{array}{ll}3.0 & 1.5\end{array}\right]^T$. By the previous sliding surface part transformation of ⁽³³⁾, fig 1 shows the two output responses, $x_{1}$ and $x_{2}$, for the two cases:ideal and real cases. The real outputs are identical to those of the ideal. Therefore, the real robust output can be predicted, predesigned, and predetermined. The ideal and real phase trajectories are shown in fig 2 by the previous sliding surface part transformation of ⁽³³⁾. Also both are identical. The real sliding surface and control input are depicted in fig 3 and fig 4, respectively.

3.2 Another of remaining sliding surface part transformations

For the same performance as that in the 3.1 paragraph, the same exponents of the power function for the integral sliding surface are selected. The coefficients of the integral terminal sliding surface are chosen for the same sliding dynamics for the same output with all the transformations

(62)

$C_{0}=9.0$ and $C_{1}=3.0$

The resultant integral terminal sliding surface for another of the part transformations becomes

(63)

$s_{+3}=\dfrac{1}{2}9.0x_{0}+3.0x_{1}^{p/q}+\dfrac{1}{2}x_{2}(=0)$

From (29), its ideal sliding dynamics becomes (53) for the purpose of the same ideal sliding dynamics with all the transformations.

Now, the integral TSMC control input is taken as follows:

(64)

\begin{align*} u_{2}=-k_{1}x_{1}-k_{2}x_{2}-k_{3}x_{3}-\Delta k_{1}x_{1}-\triangle k_{2}x_{2}\\ -\triangle k_{3}x_{3}-\triangle k_{4}s_{+3}-\triangle k_{5}sign(s_{+3}) \end{align*}

From (31)-(33), by letting the constant gain

(65)

$k_{1}=(b_{0})^{-1}a_{10}+C_{0}=4.5$

(66)

$k_{2}=(b_{0})^{-1}a_{20}=2^{-1}(-3)=-1.5$

(67)

$k_{3}=C_{1}\dfrac{p}{q}=3.0*0.56=1.68$

(68)

$k_{4}=200.0>0$

If one take the switching gain as the design parameters

(69)

$\Delta k_{1}=\begin{cases} 1.25 {if}s_{+3}x_{1}>0\\ -1.25 {if}s_{+3}x_{1}<0 \end{cases}$ $\Delta k_{2}=\begin{cases} 2.25 {if}s_{+3}x_{2}>0\\ -2.25 {if}s_{+3}x_{2}<0 \end{cases}$ $\Delta k_{3}=\begin{cases} 1.75 {if}s_{+3}x_{3}>0\\ -1.75 {if}s_{+3}x_{3}<0 \end{cases}$ $\triangle k_{5}=\begin{cases} 6.0 {if}s_{+3}>0\\ -6.0 {if}s_{+3}<0 \end{cases}$

then one can obtain the following equation

(70)

$s_{+3}$ \dot $s_{+3}<-170.0s_{+3}^{2}<0$

The simulations were performed with a sampling time of 1[msec] and using the same initial condition as just before. Fig. 9 displays the output responses for two scenarios: the ideal case and the real case. These responses are nearly identical to those shown in Fig. 1 and 5, respectively, as the same ideal sliding dynamics is used. Therefore, the real robust output can be predicted, predetermined, and predesigned in the same way as those of the first two transformations. Figure 10 depicts the phase trajectories for both the ideal and real cases, with the real trajectory being the same as that of the ideal case. Additionally,

그림. 1. ⁽³³⁾의 기존 슬라이딩 면 부분 변환에 의한 두가지 응답 $x_{1}$과 $x_{2}$

Fig. 1. Two output responses, $x_{1}$ and $x_{2}$ by previous sliding surface part transformation in ⁽³³⁾

그림. 2. ⁽³³⁾의 기존 슬라이딩 면 부분 변환에 의한 이상과 실제 플랜트의 두가지 상 궤적

Fig. 2. Two phase trajectories by previous sliding surface part transformation in ⁽³³⁾

그림. 3. ⁽³³⁾의 기존 슬라이딩 면 부분 변환에 의한 슬라이딩 면

Fig. 3. Sliding surface by previous sliding surface part transformation in ⁽³³⁾

그림. 4. ⁽³³⁾의 기존 슬라이딩 면 부분 변환에 의한 제어입력

Fig. 4. Control input by previous sliding surface part transformation in ⁽³³⁾

그림. 5. 제안된 한가지 슬라이딩 면 부분 변환에 의한 두가지 응답 $x_{1}$과 $x_{2}$

Fig. 5. Two output responses, $x_{1}$ and $x_{2}$ by proposed one of sliding surface part transformations

그림. 6. 제안된 한가지 슬라이딩 면 부분 변환에 의한 이상과 실제 플랜트의 두가지 상 궤적

Fig. 6. Two phase trajectories by proposed one of sliding surface part transformations

그림. 7. 제안된 한가지 슬라이딩 면 부분 변환에 의한 슬라이딩 면

Fig. 7. Sliding surface by proposed one of sliding surface part transformations

그림. 8. 제안된 한가지 슬라이딩 면 부분 변환에 의한 제어입력

Fig. 8. Control input by proposed one of sliding surface part transformations

그림. 9. 제안된 다른 한가지 슬라이딩 면 부분 변환에 의한 두가지 응답 $x_{1}$과 $x_{2}$

Fig. 9. Two output responses, $x_{1}$ and $x_{2}$ by proposed another of sliding surface part transformations

그림. 10. 제안된 다른 한가지 슬라이딩 면 부분 변환에 의한 이상과 실제 플랜트의 두가지 상 궤적

Fig. 10. Two phase trajectories by proposed another of sliding surface part transformations

그림. 11. 제안된 다른 한가지 슬라이딩 면 부분 변환에 의한 슬라이딩 면

Fig. 11. Sliding surface by proposed another of sliding surface part transformations

그림. 12. 제안된 다른 한가지 슬라이딩 면 부분 변환에 의한 제어입력

Fig. 12. Control input by proposed another of sliding surface part transformations

fig. 11 and 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 real outputs exhibit the same level of performance due to the identical ideal dynamics employed. Thus, the three real outputs can be predicted, predesigned, and predetermined in the same manner as designed and expected.

4. Conclusions

In this paper, we delve into the design of the two sliding surface part transformations targeting ITSMC applied to second-order SI uncertain linear systems, building upon the groundwork laid by ⁽³³⁾. These investigations stand as alternative methods to those proposed in ⁽³³⁾ and contribute to the further exploration of this domain. The ITSMC design landscape encompasses five distinct approaches: control input transformation, sliding surface transformation, and three sliding surface part transformations. While comprehensive research into the first three transformations has been presented in ⁽³³⁾, the study of the remaining two sliding surface part transformations necessitates deeper inquiry. Our paper takes the initiative to provide an intricate design analysis of these two part transformations, marking their debut in this context. It's important to note that the scope of Utkin's theorem is confined to the initial two transformations: control input transformation and sliding surface full transformation. However, the subsequent three sliding surface part transformations don't align with the applicability of Utkin's theorem. Despite this limitation, these three transformations are still valuable design methodologies for ITSMCs. The last two methods have the capability to attain identical performance levels as the first three approaches by meticulously designing the same ideal sliding dynamics of the integral terminal sliding surfaces. This is due to the precise alignment of the real output with the intended behavior defined within the integral terminal sliding surfaces. To substantiate the effectiveness of these novel approaches, we present an illustrative example alongside a comprehensive simulation study. Remarkably, the simulations yield three outputs those mirror the ideal outputs. This consistency signifies the successful elimination of the reaching phase, enabling the predetermined and predicted behavior of the real robust output. The simulation results impeccably align with the design intent of the three transformations. These proposed algorithms possess a broader scope and can be applied to higher-order SISO systems as well as MIMO plants. Moreover, they hold potential for application in motion control of various dynamical systems, contributing to an extended arena of study.