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The Transactions of the Korean Institute of Electrical Engineers

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The Transactions of the Korean Institute of Electrical Engineers

KIEE Vol. 68, No. 6, p.775-786

ISSN (print) :

1975-8359

ISSN (online) :

2287-4364

Received : 21 December 2018Revised : Accepted : 7 May 2019

DOI :

http://doi.org/10.5370/KIEE.2019.68.6.775

불확실 일반 선형 시스템의 레규레이션 제어를 위한 사전 제어 성능을 갖는 개선된 연속 변환된 적분 슬라이딩 모드 제어

An Improved Continuous Transformed Integral Sliding Mode Control(ICTISMC) with Prescribed Control Performance for Regulation Controls of Uncertain General Linear Systems

이정훈 (Jung-Hoon Lee) ¹^†

(ERI, Dept. of Control and Instrumentation Engineering, Gyeongsang National Univerity, Korea.)

^†Corresponding author, E-mail: wangwang7@naver.com

License :

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.(www.kiee.or.kr).

Abstract

In this paper, an improved continuous transformed integral sliding mode control(ICTISMC) with the prescribed control performance is designed for simple regulation controls of uncertain general linear systems, as an alternative approach of [51]^[51] with the same performance. An transformed integral sliding surface with an integral state having a special initial condition is adopted for removing the reaching phase and predetermining the ideal sliding trajectory from a given initial state to the origin in the state space. The ideal sliding dynamics of the transformed integral sliding surface is dynamically obtained and the solution of the ideal sliding dynamics can predetermine the ideal sliding trajectory(transformed integral sliding surface) from the given initial state to the origin. Provided that the value of the transformed integral sliding surface is bounded by certain value by means of the continuous input, the norm of the state error to the ideal sliding trajectory is analyzed and obtained in Theorem 1. A corresponding discontinuous control input with the exponential stability is proposed to generate the perfect sliding mode on the every point of the pre-selected transformed integral sliding surface. The existence condition of the sliding mode together with the closed loop stability is proved in Theorem 2 for the complete formulation. For practical applications, the discontinuity of the VSS control input is approximated to be continuous based on the modified fixed boundary layer method. The bounded stability by the continuous input is investigated in Theorem 3. With combining the results of Theorem 1 and Theorem 3, as the prescribed control performance, the pre specification on the error to the ideal sliding trajectory is possible by means of the fixed boundary layer continuous input with the transformed integral sliding surface. The suggested algorithm with the continuous input can provide the effective method to increase the control accuracy within the boundary layer by means of the increase of the $G_{1}$ gain. Through an illustrative design example and simulation study, the usefulness of the main results is verified.

Key words

variable structure system, sliding mode control, sliding surface transformation, boundary layer method, prescribed performance

1. Introduction

The VSS with the SMC can provide the effective means to the control of uncertain linear dynamical systems under parameter variations and external disturbances^[1-^3]. One of its essential advantages is the robustness of the controlled system to matched parameter uncertainties and external disturbances in the sliding mode on the predetermined sliding surface^[4-^6]. However the VSS has the three problems, those are the reaching phase^[3], chattering problem^[5], and proof problem of the existence condition of the sliding mode on the predetermined sliding surface for the complete formation of the VSS design.

The reaching phase is the transient period until the controlled system first touches to the sliding surface. During this phase, the sliding mode can not be realized. So the robustness is not guaran- teed^[6]. The alleviations of this reaching phase problem are the use of the high-gain feedback^[10], adaptive change of the rotation of the sliding surface^[3], segmented sliding surface^[11], moving sliding surfaces^[12-^14]. integral sliding surface^[6,^9,^15-^17,^19,^20], nonlinear integral type sliding surface^[21-^25], etc^[7]. In the integral sliding surface^[6,^9,^15-^17,^19,^20] and the two works of the nonlinear integral type sliding surface^[24,^25], the reaching phase is completely removed, and the controlled output is predicted and predetermined by means of the solution of the ideal sliding dynamics of the integral sliding surface or nonlinear integral type sliding surface.

On the other hand, the chattering in the VSS is the discon- tinuously high frequency inherent switching of the control input according to the sign of the sliding surface in the neighborhood of the sliding surface, which is undesirable for practical real plants, may excite the unmodeled high frequency dynamics, reduces the usable life time of actuators, and results in the loss of the asymptotic stability and poor steady state tracking error^[26]. Until now, there are many approaches to attenuate the chattering pro- blems, those are the saturation function^[17,^27,^28], boundary layer method^[29-^31], observer-based approach^[32,^33], higher-order approach^[34-^36], adaptive method^[37], fuzzy SMC^[31,^38-^40], neural net SMC^[38,^41,^42], filtering technique^[43], digital sliding mode scheme^[44], fast nonsingular terminal sliding mode^[45,^46], and uncertainty and disturbance estimation technique^[47], etc^[48]. Each method has the advantages and disadvantages at the same time.

About the proof problem of the existence condition of the sliding mode on the predetermined sliding surface for the complete formulation of the VSS design, in case of single input uncertain linear plants, this proof problem coincides with the stability pro- blems of the closed loop systems. However, in case of multi input systems, the existence condition of the sliding mode is a much more strict problem than the Lyapunov stability one^[49]. Through the proof of the existence condition of the sliding mode as the complete formulation of the VSS design, the strong robustness on the every point on the pre-selected sliding surface is guaranteed and shown. Utkin proposed the invariant theorem of the two methods of the two transformation(diagonalization) techniques, i.e. the control input transformation and sliding surface transformation in [2]^[2] without the complete proof. That was reviewed in [4]^[4]. For multi input uncertain linear plants, normally the Lya-punov stbility is used instead of proving the existence condition of the sliding mode. In case of multi input uncertain linear plants, Utkin’s theorem is proved completely in [49]^[49]. For single input uncertain nonlinear plants, Utkin;s theorem is proved in [50]^[50]. Owing to the Utkin’s theorem, the proof problem of the existence condition of the sliding mode on the predetermined sliding surface becomes the much more easy job, and there are the two algorithms(approaches) of the control input transformed VSS and sliding surface transformed VSS due to the two transfor- mation methods.

In [51]^[51], to provide the prescribed control performance with an initial valued integral sliding surface and continuous transformed control input without the reaching phase, chattering, and proof problems of the existence condition of the sliding mode, an improved continuous control input transformation integral variable structure system is designed for simple regulation control of single input uncertain general linear systems.

In this paper, as an alternative approach of the control input transformed VSS in [51]^[51], an sliding surface transformation ICTISMC with the prescribed control performance is presented for simple regulation controls of uncertain general linear systems. In the proposed algorithm, the three problems of the VSS, ie. the reaching phase, chattering problem, and proof problem are addressed to by means of the transformed integral sliding surface with a special initial condition and modified fixed boundary layer methods. The reaching phase is completely removed, the chattering is dramatically improved, and the existence condition of the sliding mode together with the closed loop exponential stability is proved clearly for the complete formation. The ideal sliding dynamics of the transformed integral sliding surface is dynamically obtained from a given initial condition to the origin in advance after the state transformation. By using the solution of the ideal sliding dynamics, the controlled output from a given initial condition to the origin is predictable and predetermined. The norm of the error of tracking to the transformed integral sliding surface is analyzed analytically as a specification on tracking to the trans- formed integral sliding surface. Theoretically a discontinuous input with the exponential stability is proposed and practically by means of the modified fixed boundary layer method, a continuous input is suggested with providing the tool of the increase of the tracking accuracy to the transformed integral sliding surface even better tracking accuracy than that of the discontinuous input. With the continuous input, the exponential stability is lost and the bounded stability is obtained but the tracking accuracy is even improved with the prescribed control performance. A design example and simulation study shows the usefulness of the main results.

2. An Continuous Integral Variable Structure Systems

2.1 Descriptions of plants

An n-th order uncertain general linear plant is described by

(1)

$\dot{z}=\left(A_{0}+\Delta A\right) \cdot z(t)+\left(B_{0}+\Delta B\right) \cdot u(t)+D f(t) \qquad z(0)$

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

(2)

$R(\Delta A) \subset R\left(B_{0}\right)$

$R(\Delta B) \subset R\left(B_{0}\right)$

$R(D) \subset R\left(B_{0}\right)$

Moreover the assumption on $\Delta B$ is made.

AssumptionA1:It is assumed the following equation is satisfied for a non zero element coefficient vector $C_{z 1} \in R^{1 \times n}$

(3)

$\left|\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \Delta B\right|=|\Delta I| ≤ \eta < 1$

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

(4)

$\dot{x}=\Lambda x(t)+\Gamma u(t)+\Gamma d(t), \qquad x(0)$

where

(5)

$\Lambda=P A_{0} P^{-1}=\left[\begin{array}{ccccc}{0} & {1} & {0} & {\dots} & {0} \\ {0} & {0} & {1} & {\dots} & {0} \\ {\vdots} & {\vdots} & {\vdots} & {\ldots} & {\vdots} \\ {-a_{1}} & {-a_{2}} & {-a_{3}} & {\dots} & {-a_{n}}\end{array}\right]$ and

$\Gamma=P B_{0}=\left[\begin{array}{c}{0} \\ {\vdots} \\ {0} \\ {b}\end{array}\right]$

where $x(0)$ is the initial condition transformed from $z(0)$ and $d(t)$ is the lumped uncertainty in the transformed system as

(6)

$d(t)=\Delta A^{\prime} P^{-1} x(t)+\Delta B u(t)+D^{\prime} f(t)$

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

(7)

$\begin{aligned} s(z, t) & =\left(C_{z 1} B_{0}\right)^{-1} C_{z 0} \cdot\left[\int_{0}^{t} z d t+\int_{-\infty}^{0} z d t\right]+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \cdot z \\ & =\left(C_{z 1} B_{0}\right)^{-1} C_{z 0} \cdot z_{0}+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \cdot z \end{aligned}$

(8)

$\begin{aligned} s(x, t) & =\left(C_{x 1} \Gamma\right)^{-1} C_{x 0} \cdot\left[\int_{0}^{t} x d t+\int_{-\infty}^{0} x d t\right]+\left(C_{x 1} \Gamma\right)^{-1} C_{x 1} \cdot x \\ & =\left(C_{x 1} \Gamma\right)^{-1} C_{x 0} \cdot x_{0}+\left(C_{x 1} \Gamma\right)^{-1} C_{x 1} \cdot x \end{aligned}$

where the coefficient matrices and the initial conditions for the integral states are expressed as shown

(9)

$C_{x 1}=\left[\begin{array}{lllll}{c_{1}} & {c_{2}} & {\dots} & {c_{n}}\end{array}\right] \in R^{1 \times n}, \quad c_{n}=1$

$C_{a 1} B_{0}=C_{x 1} \Gamma, \quad C_{z 0}=C_{x 0} P, \quad C_{z 1}=C_{x 1} P \in R^{1 \times n}$

(10a)

$\int_{-\infty}^{0} x_{i} d t=-c_{i} x_{i}(0) / C_{x 0 i}, \quad i=1,2, \ldots, n$

(10b)

$\int_{-\infty}^{0} z_{i} d t=-c_{z 1 i} z_{i}(0) / C_{z 0 i}, \quad i=1,2, \ldots, n$

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

(11)

$\dot{s}(x, t)=\left(C_{x 1} \Gamma\right)^{-1} C_{x 0} \cdot x+\left(C_{x 1} \Gamma\right)^{-1} C_{x 1} \cdot \dot{x}=0$

the differential equation for $x_{n}$ is obtained as

(12)

$\dot{x_{n}}(t) =-C_{x 0} \cdot x-\left[0 c_{1} c_{2} \ldots c_{n-1}\right] \cdot x =-C_{x} \cdot x$

where

(13)

$C_{x}=\left[c_{x 1} \quad c_{x 2} \quad \ldots \quad c_{x n}\right]=C_{x 0}+\left[0 \quad c_{1} \quad c_{2} \quad \ldots \quad c_{n-1}\right]$

Combing (12) with the first n-1 differential equation in the system (4) leads to the ideal sliding dynamics

(14)

$\dot{x_{s}}^{*}=\Lambda_{c} x_{s}^{*} \qquad x_{s}^{*}(0)=x(0)$

and

(15)

$\dot{z}_{s}^{*}=P^{-1} \Lambda_{c} P z_{s}^{*} \qquad z_{s}^{*}(0)=z(0)$

where

(16)

$\Lambda_{c}=\left[\begin{array}{c}{O^{(n-1) \times 1}} & {I^{(n-1) \times(n-1)}} \\ {-C_{x}}\end{array}\right]$

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)

(17)

$\dot{x_{s}}^{*}=\Lambda \cdot x_{s}^{*}+\Gamma u_{s}\left(x_{s}^{*}, t\right)$

$u_{s}\left(x_{s}^{*}, t\right)=-G x_{s}^{*}(t)$

where

(18)

$\Lambda_{c}=\Lambda-\Gamma G$

and expressed with the original state as

(19)

$\dot{z}_{s}^{*}=A_{0} z_{s}^{*}+B_{0} u_{s}\left(z_{s}^{*}, t\right)$

$u_{s}\left(z_{s}^{*}, t\right)=-G P z_{s}^{*}(t)=-K{z}_{s}^{*}(t)$

where

(20)

$P^{-1} \Lambda_{c} P=A_{0}-B_{0} K$

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

(21)

$\begin{aligned} C_{x} & =\left[c_{x 1} \quad c_{x 2} \quad \ldots \quad c_{x n}\right]=C_{x 0}+\left[0 \quad c_{1} \quad c_{2} \quad \ldots \quad c_{n-1}\right] \\ & =\left[a_{1} \quad a_{2} \quad \ldots \quad a_{n}\right]+b G \\ & =\left[a_{1} \quad a_{2} \quad \cdots \quad a_{n}\right]+b K P^{-1} \\ c_{x 1} & =c_{x 01} \end{aligned}$

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

(22)

$\left\|e^{\Lambda_{c} t}\right\| \leqq K_{1} \cdot e^{-\kappa t}$

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

(23)

$\overline{E_{0}}(t)=\left[\begin{array}{cccc}{e_{0}} & {e_{1}} & {e_{2}} & {\dots} & {e_{n-1}}\end{array}\right]^{T}$

$\overline{E_{1}}(t)=\dot{\overline{E}}_{0}(t)=\left[\begin{array}{ccc} {e_{1}} & {e_{2}} & {\dots} & {e_{n-1}}\end{array}\right]^{T}$

where

(24)

$e_{0}(t)=\int_{0}^{t} x_{1}-x_{s 1}^{*} d t+e_{0}(0)$

$e_{i}(t)=x_{i}-x_{s i}^{*}, \quad i=1,2, \ldots, n$

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

(25)

$\left\|\overline{E_{0}}(t)\right\| \leqq \epsilon_{1}$

$\left\|\overline{E_{1}}(t)\right\| \leqq \epsilon_{2}$

is satisfied for all $t \geqq 0$ where $\epsilon_{1}$ and $\epsilon_{2}$ are the positive con- stants defined as follows:

(26)

$\epsilon_{1}=\frac{K}{\kappa}\left|C_{x 1} \Gamma \right| \cdot \gamma, \quad \epsilon_{2}=\gamma \cdot\left|C_{x 1} \Gamma \right| [ 1+\left\|\Lambda_{c}\right\| \cdot \frac{K_{1}}{\kappa} ]$

Proof: The transformed integral sliding surface can be re-written as

(27)

$\begin{aligned} s(x, t)\left|C_{x 1} \Gamma \right| & =C_{x 0} \cdot x_{0}+C_{x 1} \cdot x-\left\{C_{x 0} \cdot x_{0 s}^{*}+C_{x 1} \cdot x_{s}^{*}\right\} \\ & =C_{x 0} \cdot\left[x_{0}-x_{0 s}^{*}\right]+C_{x 1} \cdot\left[x-x_{s}^{*}\right] \\ & =C_{x 0} \cdot \overline{E_{0}}+C_{x 1} \cdot \overline{E_{1}} \end{aligned}$

and can be re-expressed in a differential matrix from as

(28)

$\dot{\vec{E}_{0}}=\Lambda_{c} \cdot \overline{E_{0}}+\left[\begin{array}{c}{0} \\ {\vdots} \\ {0} \\ {C_{x 1} \Gamma}\end{array}\right] \cdot s(x, t)$

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

(29)

$\overline{E_{0}}(t)=e^{\Lambda_{c} t} \cdot \overline{E_{0}}(0)+\int_{0}^{t}\left\{e^{\Lambda_{c} t} \cdot\left[\begin{array}{c}{0} \\ {\vdots} \\ {0} \\ {C_{x 1} \Gamma}\end{array}\right] \cdot s(x, t-\tau)\right\} d \tau$

From the boundness of the transformed sliding surface and (22), the Euclidean norm of the vector $\overline{E_{0}}$ becomes

(30)

$\left\|\overline{E_{0}}(t)\right\|=\left\|e^{\Lambda_{c} t}\right\| \cdot\left\|\overline{E_{0}}(0)\right\|+ \int_{0}^{t}\left\|\left\{e^{\Lambda_{c} t} \cdot\left[\begin{array}{c}{0} \\ {\vdots} \\ {0} \\ {C_{x 1} \Gamma}\end{array}\right] \cdot s(x, t-\tau)\right\}\right\| d \tau$

$\leqq K_{1} \cdot e^{-\kappa t} \cdot\left\|\overline{E_{0}}(0)\right\| + \int_{0}^{t}\left\|K_{1} \cdot e^{-\kappa t \|} \cdot\right\|\left[\begin{array}{c}{0} \\ {\vdots} \\ {0} \\ {C_{x 1} \Gamma}\end{array}\right]\|\cdot\| s(x, t-\tau) \| d \tau$

$\leqq \frac{K_{1}}{\kappa}\left|C_{x 1} \Gamma\right| \cdot \gamma+\left(\left\|\overline{E_{0}}(0)\right\|-\frac{\gamma}{\kappa} | C_{x 1} \Gamma\right|) \cdot K_{1} \cdot e^{-\kappa t}$

$\leqq \frac{K_{1}}{\kappa}\left|C_{x 1} \Gamma \right| \cdot \gamma$

$=\epsilon_{1}$

for all time, $t \geqq 0$ . From (28), the following equation is obtained

(31)

$\begin{aligned}\left\|\overline{E_{1}}\right\| & =\left\|\Lambda_{c \|} \cdot\right\| \overline{E_{0}}\|+\|\left[\begin{array}{c}{0} \\ {\vdots} \\ {0} \\ {C_{x 1} \Gamma}\end{array}\right] \|\cdot\| s(x, t) \| \\ & \leqq \epsilon_{2} \end{aligned}$

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

(32)

$u(t)=-\left\{K_{z} \cdot z+G_{1} \cdot s\right\}-\left\{\Delta K_{z} \cdot z+G_{2} \operatorname{sign}(s)\right\}$

where

(33)

$K_{z}=\left(C_{z 1} B_{0}\right)^{-1}\left(C_{z 0}+C_{z 1} A_{0}\right)$

(34)

$G_{1} > 0$

(35)

$\Delta k _{zi}= \left\{\begin{array}{l} { \geq {\frac{\max \left\{\left(C_{x 1} B_{0}\right)^{-1} C_{z 1} \Delta A-\Delta I K_{z}\right\}_{i}}{\min \{I+\Delta I\}_{i}}} {\operatorname{sign}\left(s z_{i}\right) > 0}} \\ { \leq {\frac{\min \left\{\left(C_{x 1} B_{0}\right)^{-1} C_{z 1} \Delta A-\Delta I K_{z}\right\}_{i}}{\min \{I+\Delta I\}_{i}}} {\operatorname{sign}\left(s z_{i}\right) < 0}} \end{array}\right.$

$i=1,2, \dots, n$

(36)

$G _{2}= \left\{\begin{array}{l} { \geq {\frac{\max \left\{\left(C_{x 1} B_{0}\right)^{-1} C_{z 1} D f(t)\right\}}{\min \{I+\Delta I\}}} {\operatorname{sign}\left(s \right) > 0}} \\ { \leq {\frac{\min \left\{\left(C_{x 1} B_{0}\right)^{-1} C_{z 1} D f(t)\right\}}{\min \{I+\Delta I\}}} {\operatorname{sign}\left(s \right) < 0}} \end{array}\right.$

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

(37)

$\dot{s}(z, t)=\left(C_{z 1} B_{0}\right)^{-1} C_{z 0} z+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \dot{z}$

$=\left(C_{z 1} B_{0}\right)^{-1} C_{z 0} z+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1}\left(A_{0}+\Delta A\right) z +\left(C_{z 1} B_{0}\right)^{-1} C_{z1}\left(B_{0}+\Delta B\right) u+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} D f(t)$

$=\left(C_{z 1} B_{0}\right)^{-1}\left(C_{z 0}+C_{z 1} A_{0}\right) z+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \Delta A z$

$-\left(C_{z 1} B_{0}\right)^{-1} C_{z 1}\left(B_{0}+\Delta B\right)\left(K_{z} z+G_{1} s+\Delta K z+G_{2} \operatorname{sign}(s)\right) +\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} D f(t)$

$=\left(C_{z 1} B_{0}\right)^{-1}\left(C_{z 0}+C_{z 1} A_{0}\right) z-K_{z} z+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \Delta A z$

$-\Delta I K_{z} z-(I+\Delta I) \Delta K z-(I+\Delta I) G_{1} s+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} D f(t) -(I+\Delta I) G_{2} \operatorname{sign}(s)$

$=\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \Delta A z-\Delta I K_{z} z-(I+\Delta I) \Delta K z -(I+\Delta I) G_{1} s+\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} D f(t) -(I+\Delta I) G_{2} \operatorname{sign}(s)$

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

(38)

$V(t)=\frac{1}{2} s^{2}(z, t)$

Differentiating (38) with time leads to

(39)

$\dot{V}(t)=s(z, t) \cdot \dot{s}(z, t)$

Substituting (37) into (39) and by (33)-(36), one can obtain the following equation

(40)

$\begin{aligned} \dot{V}(t)=s(z, t) \cdot \dot{s}(z, t) & < -(1-\eta) G_{1} s^{2}(z, t) \\ & =-2(1-\eta) G_{1} V(t) \end{aligned}$

From (40), the following equation is obtained as

(41)

$\dot{V}(t)+2(1-\eta) G_{1} V(t) \leqq 0$

$V(t) \leqq V(0) e^{-2(1-\eta) G_{1} t}$

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

(42)

$\begin{aligned} u_{c}(t)= & -\left\{K_{z} \cdot z+G_{1} \cdot s\right\} \\ & -\left\{\Delta K_{z} \cdot z+G_{2} \operatorname{sign}(s)\right\} \cdot M B L F(s) \end{aligned}$

where $MBLF(s)$ is defined as a modified fixed boundary layer function as follows:

(43)

$\operatorname{MBLF}(s)=\left\{\begin{array}{ll}{1} & {\text { for } \quad s \geqq l_{+}} \\ {s / l_{+}} & {\text { for } \quad 0 \leqq s < l_{+}} \\ {|s| / l_{-}} & {\text { for } \quad-l_{-} < s \leqq 0} \\ {1} & {\text { for } \quad s \leqq-l_{-}}\end{array}\right.$

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

(44)

$V(t)=\frac{1}{2} s^{2}(z, t)$

From the proof of Theorem 2, we can obtain the following equation

(45)

$\begin{aligned} \dot{V}(t)=s(z, t) \cdot \dot{s}(z, t) & < -(1-\eta) G_{1} s^{2}(z, t) \\ & =-2(1-\eta) G_{1} V(t) \end{aligned}$

as long as $\left.\right|_{S}(z, t) | \geqq l=\max \left(l_{+}, l_{-}\right)$ . From (45), the following equation is obtained as

(46)

$\dot{V}(t)+2(1-\eta) G_{1} V(t) \leqq 0$

$V(t) \leqq V(0) e^{-2(1-\eta) G_{1} t}$

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.

3. Design Examples and Simulation Studies

Consider a following plant with uncertainties and disturbances^[9]

(47)

$\dot{z}_{1}=\left(-2+\Delta a_{1}\right) z_{1}(t)+\left(2+\Delta b_{1}\right) u(t)+f(t)$

$\dot{z}_{2}=\Delta a_{1} z_{1}(t)-3 z_{2}(t)+\left(2+\Delta b_{2}\right) u(t)+f(t)$

where

(48)

$A_{0}=\left[\begin{array}{cc}{-2} & {0} \\ {0} & {-3}\end{array}\right], \quad B_{0}=\left[\begin{array}{l}{2} \\ {2}\end{array}\right]$

$\Delta a_{1}=0.01 \sin (3 t), \quad \Delta b_{1}=\Delta b_{2}=0.3 \sin (5 t), \quad f(t)=0.5 \cos (8 t)$

$\left|\Delta a_{1}\right| \leq 0.01, \quad\left|\Delta b_{1}\right|=\left|\Delta b_{2}\right| \leq 0.3, \quad|f(t)| \leq 0.5$

The ICTISMC controller aims to drive the output of the plant (47) to the ideal transformed sliding surface from any given initial state to the origin. The transformation matrix to a control- lable weak canonical form and the resultant transformed system matrices are

(49)

$P=\left[\begin{array}{cc}{1} & {-1} \\ {-2} & {3}\end{array}\right]$ , $\Lambda=\left[\begin{array}{cc}{0} & {1} \\ {-6} & {-5}\end{array}\right]$ , $\quad$ $\Gamma=\left[\begin{array}{l}{0} \\ {2}\end{array}\right]$

By means of Ackermanns formula, the continuous static gain is obtained

(50)

$K=\left[\begin{array}{cc}{0.5} & {0}\end{array}\right]$ and

$G=\left[\begin{array}{cc}{1.5} & {0.5}\end{array}\right]$

so that the closed loop double eigenvalues of $\Lambda_{c}$ are located at $-3$ . Hence, the $\Lambda_{c}$ in (14) and (18) and $P^{-1} \Lambda_{c} P$ in (15) and (20) become

(51)

$\Lambda_{c}=\left[\begin{array}{cc}{0} & {1} \\ {-9} & {-6}\end{array}\right]$ , $\quad \quad$ $P^{-1} \Lambda_{c} P=\left[\begin{array}{cc}{-3} & {0} \\ {-1} & {-3}\end{array}\right]$

By using the solution (14) or (15), the regulated output from a given initial state to the origin can be predicted and predetermined. By the relationship (21), the coefficient matrices of the integral sliding surface directly becomes

(52)

$C_{x}=\left[\begin{array}{cc}{\alpha_{1}} & {\alpha_{2}}\end{array}\right]+b G=\left[\begin{array}{cc}{6} & {5}\end{array}\right]+2 \left[\begin{array}{cc}{1.5} & {0.5}\end{array}\right]=\left[\begin{array}{cc}{9} & {6}\end{array}\right]$

$C_{x 0}=\left[\begin{array}{cc}{9} & {6}\end{array}\right], \quad C_{x 1}=\left[\begin{array}{cc}{6} & {1}\end{array}\right]$

(53)

$C_{z 0}=C_{x 0} P=\left[\begin{array}{cc}{9} & {-9}\end{array}\right], \quad C_{z 1}=C_{x 1} P=\left[\begin{array}{cc}{4} & {-3}\end{array}\right], C_{z 1} B_{0}=C_{x 1} \Gamma=2$

As a result, the transformed integral sliding surface becomes

(54)

$\begin{aligned} s(z, t)=9 / 2 & \left\{\int_{0}^{t} z_{1} d \tau-4 / 9 z_{1}(0)\right\} \\ & -9 / 2\left\{\int_{0}^{t} z_{2} d \tau-3 / 9 z_{2}(0)\right\}+4 z_{1} / 2-3 z_{2} / 2 \end{aligned}$

The constants $K_{1}$ and $k$ in the equation (22) are selected as $K_{1}=3.8$ and $k=1.0$ , hence the constants $\epsilon_{1}$ and $\epsilon_{2}$ in (25) and (26) are determined as $\epsilon_{1}=7.6 \gamma$ and $\epsilon_{2}=84.56 \gamma$ . The specification on the norms of the error vector to the ideal sliding surface and the modified error vector, $\epsilon_{2}$ and $\epsilon_{1}$ are given as $\epsilon_{2}=2$ and $\epsilon_{1}=0.1798$ for an example. Then the $\gamma$ is determined as $\gamma =0.0237$ . The discontinuous input automatically and theoretically satisfy that the norm value of the integral sliding surface is bounded by $\gamma = 0.0237$ . For practical applications, the continuous input essen- tially adapted with little performance degradation as expected in the design stage. Therefore, $l$ is determined less than $\gamma = 0.0237$ that is $l=l_{+}=l_{-}=0.02$ . For the second design phase of the ICTISMC, the equation (3) in the Assumption A1 is calculated

(55)

$\left(C_{z 1} B_{0}\right)^{-1} C_{z 1} \Delta B=\Delta I \leq 0.15=\eta < 1$

Thus the Assumption A1 is satisfied in this design. The $K_{z}$ of (33) becomes

(56)

$K_{z}=\left(C_{z 1} B_{0}\right)^{-1}\left(C_{z 0}+C_{z 1} A_{0}\right)=\left[\begin{array}{cc}{0.5} & {0}\end{array}\right]$

The inequalities for the switching gains in discontinuous input terms, (34)-(36) become

(57)

$G_{1} > 0$

$k_{z 1}=\left\{\begin{array}{l}{ > \quad \frac{0.08}{0.85}=0.0941 \quad \text { for }\left(s z_{1}\right) > 0} \\ { < -\frac{0.08}{0.85}=-0.0941 \quad \text { for }\left(s z_{1}\right) < 0}\end{array}\right.$

$k_{z 2}=\left\{\begin{array}{l}{ > 0 \text { for }\left(s z_{2}\right) > 0} \\ { < 0 \text { for }\left(s z_{2}\right) < 0}\end{array}\right.$

$G_{2} > \frac{0.25}{0.85}=0.2941$

Finally the selected control gains are

(58)

$G_{1}=10000$

$k_{z 1}= \left\{\begin{array}{l} { 20.5 \text { for }\left(s z_{1}\right) > 0} \\ { -20.5 \text { for }\left(s z_{1}\right) < 0} \end{array}\right.$

$k_{z 2}= \left\{\begin{array}{l} { 24.5 \text { for }\left(s z_{2}\right) > 0} \\ { -24.5 \text { for }\left(s z_{2}\right) < 0} \end{array}\right.$

$G_{2}=10.0$

The simulation is carried out using a Fortran software under 0.1[msec] sampling time and with $z(0)=[2-1.5]^{T}$ initial condi- tion. Fig. 1 shows the control results of the designed ICTISMC by the proposed discontinuous control input (32) with the trans- formed integral sliding surface (54) in the upper figure the two output responses, $z_{1}$ and $z_{2}$ for the three cases (i) the ideal sliding outputs that is the solution of (15), (ii) the outputs without the uncertainty and disturbance, and (iii) the outputs with the uncertainty and disturbance and in the bottom figure, the three case phase trajectories. As can be seen in the upper figure, the three case outputs are almost equal, which means that Fig. 1 shows the strong and complete robustness against uncertainty and disturbance because of removing the reaching phase, prediction of the output by using the solution of (15), and predetermination of the output directly according to the pre-chosen of the integral sliding surface, as the attractive features in the theoretical point of view. Those are the same performances as those of [51]^[51]. Fig. 2 shows the discontinuous sliding surface with the uncertainty and disturbance in the upper figure and in the bottom figure the discontinuous control input with the uncertainty and disturbance. As can be seen in the upper and bottom figures, the controlled system chatters and slides from $t=0$ without the reaching phase. Since the integral sliding surface is naturally defined from any given initial condition to the origin, there is no need of consideration of the reaching mode. The value of the integral sliding surface is no more decreased as increase of the switching gains and $G_{1}$ of the input because of the finite sampling frequency, discontinuous chattering of the switching input, and digital imple- mentation of the VSS. The transformed integral sliding surface and the control input (32) is discontinuous because of the swit- ching of the sign function in the control input (32), which is undesirable for practical applications. Therefore, the continuous approximation of the discontinuous input is essentially necessary. Based on the modified boundary layer function (43), the control input is continuously implemented as (42). The positive(negative) thickness of the boundary layer is not smaller than the positive (negative) maximum magnitude of the chattering of the integral sliding surface in the upper figure of Fig. 2. Thus the positive (negative) maximum magnitude of the chattering of the transformed integral sliding surface must be smaller than $l_{+}\left(l_{-}\right)$ . If not, re- design with larger $\epsilon_{2}$ . Fig. 3 shows the control results of the designed ICTISMC by the proposed continuous control input (42) with the transformed integral sliding surface (54) in the upper figure the two output responses, $z_{1}$ and $z_{2}$ for the three cases (i) the ideal sliding outputs that is the solution of (15), (ii) the outputs without the uncertainty and disturbance, and (iii) the outputs with the uncertainty and disturbance and in the bottom figure the three case phase trajectories. As can be seen in Fig. 3, the three outputs and phase trajectories are almost identical to each other by the continuous input with the better performance than that of the discontinuous input in Fig.1, which is the same performance as that of [51]^[51]. Fig. 4 shows the continuous trans- formed sliding surface with the uncertainty and disturbance in the upper figure and in the bottom figure the continuous control input with the uncertainty and disturbance. The transformed integral sliding surface is continuous, is bounded by $l=0.02$ , and much smaller than that of the discontinuous input because of the large $G_{1}$ . The control input in the bottom figure is dramatically improved from the bottom figure of Fig. 2. There exists the tool to increase the tracking accuracy and steady state performance by means of increase of $G_{1}$ gain. But, the increase over $G_{1}=14650.0$ makes the chattering and more increase does unstable in the closed loop system due to the high gain effect. Fig. 5 shows the norms of the tracking error vector to the transformed integral sliding surface (i) for the discontinuous input with uncertainty and disturbance and (ii) for the continuous input with uncertainty and disturbance. Both the norms of the tracking vectors to the transfor- med integral sliding surface are smaller than $\epsilon_{2}^{\prime}=\left\|P^{-1}\right\| \epsilon_{2}=7.746$ , which means that the specification on the tracking error to the transformed integral sliding surface is satisfied. In Fig. 5 at 2 second, the norms of tracking vectors of the discontinuous input and continuous input is $0.0114$ and $0.0089$ , respectively. By com-paring the simulation figures of the discontinuous and continuous inputs, it is concluded that the performance of the continuous input is better than that of the discontinuous input in view of the tracking to the ideal transformed integral sliding surface, the accuracy of the transformed integral sliding surface, and the continuity and magnitude of the control input. While in the theoretical point of view, one can use the discontinuous input directly, in practical aspects, one can use the continuous input based on the modified fixed boundary layer method with the prescribed and better control performance.

그림. 1. 제안된 불연속 제어입력 (32)에 의한 설계된 ICTISMC의 제어결과

Fig. 1. Control results of the designed ICTISMC by the proposed discontinuous control input (32) with the transformed integral sliding surface (54) in the upper figure the two output responses, $z_{1}$ and $z_{2}$ for the three cases (i) the ideal sliding output that is the solution of (15), (ii) the output without the uncertainty and disturbance, and (iii) the output with the uncertainty and disturbance and in the bottom figure the three case phase trajectories

그림. 2. 불연속 슬라이딩 면과 불연속 제어입력

Fig. 2. Control results of the designed ICTISMC by the proposed discontinuous control input (32) with the transformed integral sliding surface (54), in the upper figure the discontinuous sliding surface with the uncertainty and disturbance and in the bottom figure the discontinuous control input with the uncertainty and disturbance

그림. 3. 제안된 연속 제어입력 (42)에 의한 설계된 ICTISMC의 결과

Fig. 3. Control results of the designed ICTISMC by the proposed continuous control input (42) with the transformed integral sliding surface (54) in the upper figure the two output responses, $z_{1}$ and $z_{2}$ for the three cases (i) the ideal sliding output that is the solution of (15), (ii) the output without the uncertainty and disturbance, and (iii) the outputs with the uncertainty and disturbance and disturbance, and (iii) the output with the uncertainty and disturbance and in the bottom figure the three case phase trajectories

그림. 4. 연속 슬라이딩 면과 연속 제어입력

Fig. 4. Control results of the designed ICTISMC by the proposed continuous control input (42) with the transformed integral sliding surface (54) in the upper figure the continuous sliding surface with the uncertainty and disturbance and in the bottom figure the continuous control input with the uncertainty and disturbance

그림. 5. 추적 오차의 노옴

Fig. 5. Norms of the tracking error vector to the transformed integral sliding surface (i) for the discontinuous input with uncertainty and disturbance and (ii) for the continuous input with uncertainty and disturbance.

4. Conclusions

Due to Utkin’s theorem, there are the two algorithms(approaches) in the design of the VSS for the complete formation, those are the control input transformation VSS and sliding surface trans- formation VSS. The control input transformed VSS with the prescribed control performance was proposed in [51]^[51]. In this paper, the sliding surface transformed VSS is designed as an alternative approach of[51] with the same performance. With the sliding surface transformation, the simple regulation control of uncertain general linear systems is handled by means of a discontinuous and continuous improved integral sliding mode control with the prescribed control performance. To remove the reaching phase, a transformed integral sliding surface with an integral state having a special initial condition is defined from a given initial state to the origin. The ideal sliding dynamics of the transformed integral sliding surface is obtained in the dynamic form. The solution of the ideal sliding dynamics coincides with the transformed integral sliding surface from a given initial condition to the origin. Also by using the solution of the ideal sliding dynamics of the transformed integral sliding surface, the controlled output can be predicted and predetermined in advance as an attractive property in the theoretical aspect. The relationship between the norm of the error vector to the ideal transformed integral sliding surface and the non-zero value of the transformed sliding surface due to the continuous control input is analyzed and obtained analytically in Theorem 1, provided that the value of the transformed integral sliding surface is bounded by $\gamma$ for all $t$ . In the theoretical aspect, a corresponding discontinuous input with a feedback of the transformed sliding surface itself is proposed to generate the sliding mode on the every point of the transformed integral sliding surface from g given initial condition to the origin. The exponential stability to the transformed integral sliding surface including the origin together with the existence condition of the sliding mode is investigated in Theorem 2. For the high potential of practical applications, the continuous modification of the discontinuous input is made based on the modified fixed boundary layer method. The bounded stability of the continuous input is studied in Theorem 3. Outside the boun- dary layer, the exponential stability is still guaranteed. Inside the boundary layer, the $G_{1} \cdot s$ term increase the control accuracy and steady state performance. If one can design that $l$ is smaller than $\gamma$ , then it is possible that the value of the transformed integral sliding surface is bounded by $\gamma$ , and thus the norm of the error vector to the ideal transformed sliding surface is bounded by $\epsilon_{2}$ with the continuous input proposed in this paper as the prescribed control performance. The algorithm with the continuous input can provide the effective mean to increase the tracking accuracy to the transformed integral sliding surface from a given initial state to the origin and the steady state performance by means of the increase of $G_{1}$ . In fact, because of the large $G_{1}$ , the performance of the continuous input is better than that of the discontinuous input, while the performance of the discontinuous input is no more improved as the increase of the control gains because of the finite sampling frequency, chattering of the input, and digital implementation of the VSS. The continuity of the input is dramatically improved based on the modified fixed boundary layer method. Through an illustrative example and simulation study, the effectiveness of the proposed main results is verified. In the simulation study, it is shown that the same performance as that of [51]^[51] is obtained. In the theoretical point of view, one can use the discontinuous input for the attractive performance of output prediction and predetermination and ex- ponential stability to the transformed integral sliding surface including the origin, however in the aspect of practical applications, one can use the proposed continuous input with not the performance degradation but the better performance.

References

V. I. Utkin, 1977, Variable Structure Systems with Sliding Mode, IEEE Trans. on Automatic Control, Vol. 22, No. , pp. no.2 212-222

V. I. Utkin, 1978, Sliding Modes and Their Application in Variable Structure Systems, Moscow

U. Itkis, 1976, Control Systems of Variable Structure, New York; John Wiley & Sons

R. A. Decarlo, S. H. Zak, G. P. Mattews, 1988, Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial, Proc. IEEE, Vol. 76, No. , pp. 212-232

K. D. Young, V. I. Utkin, U. Ozguner, 1996, A Control Engineer’s Guide to Sliding Mode Control, 1996 IEEE Workshop on Variable Structure Systems, Vol. , No. , pp. 1-14

J. H. Lee, 1995, Design of Integral-Augmented Optimal Variable Structure Controllers, Ph. D. Dissertation, KAIST

H. H. Choi, 2007, LMI-Based Sliding Surface Design for Integral Sliding Mode Control of Mismatched Uncertain Systems, IEEE T. Automatic Control, Vol. 52, No. 2, pp. 736-742

T. L. Chern, Y. C. Wu, 1992, An Optimal Variable Structure Control with Integral Compensation for Electrohydraulic postion Servo Control Systems, IEEE T. Industrial Electronics, Vol. 39, No. 5, pp. 460-463

J. H. Lee, 2014, A Variable Structure Point-to-Point Regulation Controller for Uncertain General Systems, KIEE(The Korean Institute of Electrical Engineers), Vol. 63, No. 4, pp. 519-525

K. K. D. Young, p. V. Kokotovic, V. I. Utkin, 1977, A Singular Perturbation Analysis of High-Gain Feedback Systems, IEEE Trans. Autom. Contr., Vol. AC-22, No. 6, pp. 931-938

F. Harashima, H. Hashimoto, S. Kondo, 1985, MOSFET Converter-Fed Position Servo System with Sliding Mode Control, IEEE Trans. Ind. Electron, Vol. I.E.-32, No. 3

S. B. Choi, C. C. Cheong, and D. W. Park, 1993, Moving Switching Surface for Robust Control of Second-Order Variable Structure System, Int. J. Control, Vol. 58, No. 1, pp. 229-245

R. Roy, N. Olgac, 1997, Robust Nonlinear Control via Moving Sliding Surfaces-n-th Order Case, Proceedings of 36th Conference on Decision and Control, San Diego, CA, USA, pp. 943-948

P. Husek, 2016, Adaptive Sliding Mode Control with Moving Sliding Surface, Applied Soft Computing, Vol. 42, No. , pp. 178-183

J. H. Lee, M. J. Youn, 1994, An Integral-Augmented Optimal Variable Structure control for Uncertain dynamical SISO System, KIEE(The Korean Institute of Electrical Engineers), Vol. 43, No. 8, pp. 1333-1351

J. H. Lee, W. B. Shin, 1996, Continuous Variable Structure System for Tracking Control of BLDDSM, IEEE Int. Workshop on Variable Structure Systems, Vol. VSS’96, No. Tokyo, pp. 84-90

J. H. Lee, M. J. Youn, 2004, A New Improved Continuous Variable Structure Controller for Accurately Prescribed Tracking Control of BLDD Servo Motors, Automatica, Vol. 40, No. , pp. 2069-2074

J. H. Lee, 2006, Highly Robust Position Control of BLDDSM Using an Improved Integral Variable Structure System, Automatica, Vol. 42, No. , pp. 929-935

J. H. Youn, M. J. Youn, 1999, A New Integral Variable Structure Regulation controller for Robot Manipulators with Accurately Predetermined Output, Proceedings of the IEEE Int. Symposium on Industrial Electronics(ISIE’99), Vol. 1, pp. 336-341

J. H. Lee, 2011, New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems, Chapter 10 in Recent Advances in Robust Control-Novel Approaches and Design Methods Edited by Andreas Mueller Intech Open Access Pub. www,intechopen.com

V. I. Utkin, J. Shi, Dec. 1996, Integral Sliding Mode in Systems Operating under Uncertainty Conditions, Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 4591-4596

W. J. Cao, J. X. Xu, 2004, Nonlinear Integral-Type Sliding Surface for Both Matched and Unmatched Uncertain Systems, IEEE T. Automatic Control, Vol. 49, No. 83, pp. 1355-1360

J. H. Zhang, X. W. Liu, Y. Q. Xia, Z. Q. Zuo, Y. J. Wang, 2016, Disturbance Observer-Based Integral Siding-Mode Conrtol for Systems with Mismatched Disturbance, IEEE Trans. Industr. Electron, Vol. 63, No. 11, pp. 7040-7048

J. H. Lee, 2010, A New Robust Variable Structure Controller with Nonlinear Integral-Type Sliding Surface for Uncertain Systems with Mismatched Uncertainties and Disturbance, KIEE, Vol. 59, No. 3, pp. 623-629

J. H. Lee, 2010, A New Robust Variable Structure Controller with Nonlinear Integral-Type Sliding Surface for Uncertain More Affine Nonlinear Systems with Mismatched Uncertainties and Disturbance, KIEE, Vol. 59, No. 7, pp. 1295-1301

I. Boiko, L. Fridman, A. Pisano, E. Usai, 2007, Analysis of Chattering in systems with Second-Order Sliding Mode, IEEE Trans. on Automatic Conrtol, Vol. 52, No. 11, pp. 2085-2102

S. M. Chen, Y. R. Hwang, M. Tomizuka, 2002, A State-Dependent Boundary Layer Design for Sliding Mode Conrtol, IEEE Trans. on Automatic Conrtol, Vol. 47, No. 10, pp. 1677-1681

J. H. Lee, 2115, A Continuous Sliding Surface Transformed VSS by Saturation Function for MIMO Uncertain Linear Plants, Journal of The Institute of Electronics and Information Engineers(IEIE), Vol. 52, No. 7, pp. 127-134

J. J. Slotine, S. S. Sastry, 1983, Tracking Control of Non-linear Systems Using Sliding Surfaces with Aapplication to Robot Manipulators, Int. J. Control, Vol. 38, pp. 465-492

J. J. Slotine, W. Li, 1991, Applied Nonliear Control, Englewood Cliffs, NJ:Prentice-Hall

H. J. Lee, E. T. Kim, H. J. Kang, M. N. Park, 2001, A New Slding Mode Control with Fuzzy Boundary Layer, Fuzzy Sets and Systems, Vol. 120, pp. 135-143

A. G. Bondarev, S. A. Bondarev, N. E. Kostyleva, V. I. Utkin, 1985, Sliding Modes in Systems with Asymptotic State Observers, Automat. Rem. Control, Vol. 46, No. 5, pp. 679-684

V. I. Utkin, J. Guldner, J. Shi, 1999, Sliding Modes in Elec-tromechanical Systems, London, U.K.: Taylor & Francis

G. Bartolini, P. Pydynowski, 1996, An Improved, Chattering Free, V.S.C. Scheme for Uncertain Dynamical Systems, IEEE T. Automatic Control, Vol. 41, No. 8, pp. 1220-1226

G. Bartolini, A. Ferrara, E. Usai, 1998, Chattering Avoi-dance by Second-Order Sliding Mode Control, IEEE T. Automatic Control, Vol. 43, No. 2, pp. 241-246

G. Bartolini, A. Ferrara, E. Usai, V. I. Utkin, 2000, On Multi-Input Chattering Free Second-Order Sliding Mode Control, IEEE T. Automatic Control, Vol. 45, No. 9, pp. 1711-1717

X. Zhang, X. Liu, 2014, Chattering Free Adaptive Sliding Mode Control for Attitude Tracking pf Spacecraft with External disturbance, Proceedings of the 33rd Chinese Control Conference, Nanjing China, pp. 2224-2228

O. Kaynak, K. Erbatur, M. Ertugrul, 2001, The Fusion of Computationally Intelligent Methodologies and Sliding Mode Control-A Survey, IEEE Trans. on Industrial Electronics, Vol. 48, No. 1, pp. 4-17

M. H. Khooban, T. Niknam, F. Blaabjerg, M. Dehgni, 2016, Free Chattering Hybrid Sliding Mode Control of a Class of Non-Linear Systems:Electric Vehicles as a Case Study, IET Science, Measurement & Technology, Vol. 10, No. 7, pp. 776-785

Y. Yang, Y. Yan, Z. Zhu, X. Huang, 2017, Chattering-Free Sliding Mode Control for Strict-Feedback Nonlinear System Using Back Stepping Technique, 2017 Int. Conference on Mechanical, System and Control Engineering, pp. 53-57

H. Morioka, K. Wada, A. Sabanovic, K. Jezernik, 1995, Neural Network Based Chattering Free Sliding Mode Control, Proceedings of the 34th SICE’95 Annual Conference: Int. Session papers, pp. 1303-1308

M. Yasser, M. Arifin, T. Yahagi, 2011, Sliding Mode Control Using Neural Networks, Chapter 17 in Sliding Mode Control Edited by Andrzej Bantoszewicz, Vol. Intech Open Access Pub. www,intechopen.com, No.

H. J. Shieh, P. K. Huang, 2007, Precise Tracking of a Piezoelectric Positioning Stage via Filtering-Type Sliding Surface Control with Chattering Alleviation, IET Conrtol Theory Applications, Vol. 1, No. 3, pp. 586-594

V. Acary, B. Brogliato, Y. Orlov, 2012, Chattering-Free Digital Sliding Mode Control with State Observer and Disturbance Rejection, IEEE Trans. on Automatic Conrtol, Vol. 57, No. 5, pp. 1087-1101

J. Zheng, H. Wang, Z. Man, J. Jin, M. Fu, 2015, Robust Motion Control of a Linear Motor Positioner Using Fast Nonsingular Terminal Sliding Mode, IEEE/ASME Trans. Mechatronics, Vol. 20, No. 4, pp. 1743-1752

S. Lyu, Z. H. Zhu, S. Tang, X. Yan, 2016, Fast Nonsingular Terminal Sliding Mode to Attenuate the Chattering for Missile Interception with Finite Time Convergence, IFAC-Papers-OnLine, Vol. 49, No. 17, pp. 34-39

Q. S. Xu, 2017, Precision Motion Control of Piezoelectric Nano-positioning Stage with Chattering_Free Adaptive Sliding Mode Control, IEEE Trans. on Automation Science and Engineering, Vol. 14, No. 1, pp. 238-248

Y. J. Xu, 2008, Chattering Free Robust Control of Nonlinear Systems, IEEE Trans. on Control Systems Technology, Vol. 16, No. 6, pp. 1352-1359

J. H. Lee, 2010, A Proof of Utkin’s Theorem for a MI Uncertain Linear Case, KIEE, Vol. 59, No. 9, pp. 1680-1685

J. H. Lee, 2017, A Proof of Utkin’s Theorem for Uncertain Nonlinear Systems, KIEE, Vol. 66, No. 11, pp. 1612-1619

J. H. Lee, 2017, An Improved Continuous Integral Variable Structure Systems with Prescribed Control Performance for Regulation Controls of Uncertain General Linear Systems, KIEE, Vol. 66, No. 12, pp. 1759-1771

저자소개

이정훈(李政勳, Jung-Hoon Lee)

1966년 2월 1일생

1988년 경북대학교 전자공학과 졸업(공학사)

1990년 한국과학기술원 전기 및 전자공학과 졸업(석사)

1995년 한국과학기술원 전기 및 전자공학과 졸업(공박)

2005년 3월~현재 경상대학교 공과대학 제어계측공학과 교수

경상대학교 공대 공학연구원 연구원

1997-1999 경상대학교 제어계측공학과 학과장

마르퀘스사의 Who’s Who in the world 2000년 판에 등재

American Biograhpical Institute(ABI)의 500 Leaders of Influence에 선정

Tel:+82-55-772-1742

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E-mail : wangwang7@naver.com

KIEEThe Transactions ofthe Korean Institute of Electrical Engineers

The Transactions of the Korean Institute of Electrical Engineers

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An Improved Continuous Transformed Integral Sliding Mode Control(ICTISMC) with Prescribed Control Performance for Regulation Controls of Uncertain General Linear Systems

Abstract

Key words

1. Introduction

2. An Continuous Integral Variable Structure Systems

2.1 Descriptions of plants

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2.2 Design of Transformed Integral sliding Surfaces

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2.3 Transformed Discontinuous and Continuous Control Inputs

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3. Design Examples and Simulation Studies

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4. Conclusions

References

저자소개

이정훈(李 政 勳, Jung-Hoon Lee)

Article Information (continued)

Key words

KIEEThe Transactions of
the Korean Institute of Electrical Engineers

이정훈(李政勳, Jung-Hoon Lee)