Mobile QR Code QR CODE : The Transactions of the Korean Institute of Electrical Engineers

  1. (ERI, Dept. of Control & Instrum. Eng. Gyeongsang Nat'l University, Korea)



Variable structure system, Sliding mode control, Proof of Ukin's Theorem, Nonlinear system transformation methods

1. Introduction

The VSS with the SMC can provide the effective means to the control of uncertain nonlinear dynamical systems under parameter variations and external disturbances[1-4]. One of its essential advantages is the robustness of the controlled system against matched parameter uncertainties and external disturbances in the sliding mode on the predetermined sliding surface, s=0[5-7]. To take the advantages of the sliding mode on the predetermined sliding surface, the precise existence condition of the sliding mode, $s \cdot \dot{s} < 0$ for the SI case as well as $s_{i} \cdot \dot{s}_{i} < 0, \quad i=1,2, \ldots m$ for the MI(Multi Input) case should be satisfied[824]. Therefore the precise existence condition of the sliding mode must be proved completely for linear plants moreover for nonlinear plants. Utkin in [4][4] presented the two nonlinear methodologies without the complete proofs in order to prove the precise existence condition of the sliding mode on the pre-selected sliding surface. It is so called the invariance theorem, that is the equation of the sliding mode is invariant with respect to the two nonlinear transformations. Those are the control input transformation and sliding surface transformation, so called the two diagonalization methods. The essential feature of both nonlinear transformation methods is the conversion of a multi-input design problem into m single-input design problems[4,24]. Those were only reviewed in [5][5]. DeCarlo, Zak, and Matthews tried to prove Utkin's invariance theorem. But, the proof also is incomplete, and only the same as that of Utkin. In [9][9], Su, Drakunov, and Ozguner mentioned the sliding surface transformation, which would diagonalize the control coefficient matrix to the dynamics for the sliding surface $s$. But they did not completely prove the precise existence condition of the sliding mode on the predetermined sliding surface. Even for a SI linear case, that proof is hardly reported. For MI uncertain linear plants, instead of proving the precise existence condition of the sliding mode, some design methods were studied, those are, including the two diagonalization methods [4][5], the hierarchical control methodology[4,6], simplex algorithm[14], Lyapunov approach[1,9,18,22], and so on. Until now in MIMO(Multi input multi output) VSSs, it is difficult to prove the precise existence condition of the sliding mode on the predetermined sliding surface theoretically, but in [9][9], [18][18], [22][22], only the result that the derivative of the Lyapunov candidate function is negative, i,e. $\dot{V} < 0$ is obtained when the Lyapunov candidate function is taken as $V=1 / 2 s^{T} s$. In SI systems, both the VSS existence condition of the sliding mode and the Lyapunov stability are the same when the Lyapunov candidate function is taken as $V=1 / 2 s^{T} s$. However, the VSS existence condition of the sliding mode in multi input systems is the more strict condition than the Lyapunov stability because if the VSS existence condition of the sliding mode is satisfied, then the Lyapunov stability is did but the reverse argument does not hold generally[24]. In [23][23], for MI uncertain linear plants, the proof of Utkin’s theorem is given comparatively, the precise existence condition of the sliding mode is proved completely, and the complete formulation of the multivariable VSS is possible from the design of the sliding surface to the proof of the precise existence of the sliding mode and the proof of the stability of the closed loop system. For SI uncertain nonlinear plants, the proof of Utkin’s theorem is given in [24][24]. Thus, there are the two methodologies, i.e., sliding surface transformation and control input transformation to prove the existence condition of the sliding mode in the SMC for uncertain nonlinear systems, but the controlled output is not predictable.

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

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

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

2. Main Results of Proof of Utkin's Theorem

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

Theorem 1: The equation of the sliding mode is invariant with respect to the two nonlinear transformations, i.e. the control input transformation and sliding surface trans- formation:

(1)
$\begin{aligned} u^{*}(X) & =H_{u}(X, t) \cdot u(X) \\ s^{*}(X) & =H_{s}(X, t) \cdot s(X) \end{aligned}$

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

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

(2)
$\dot{X}=f(X, t)+g(X, t) u+d^{\prime}(X, t) \quad X(0)$

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

Assumption

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

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

(3)
$\dot{X}=\left[f_{0}(X, t)+\Delta f_{1}(X, t)\right] X+\triangle f_{2}(X, t) +\left[g_{0}(X, t)+\triangle g(X, t)\right] u+d^{\prime}(X, t) $

$=f_{0}(X, t) X+g_{0}(X, t) u+d(X, t)$

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

(4a)
$f(X, t)=\left[f_{0}(X, t)+\Delta f_{1}(X, t)\right] X+\triangle f_{2}(X, t)$

(4b)
$g(X . t)=\left[g_{0}(X, t)+\Delta g(X, t)\right]$

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

Assumption:

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

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

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

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

(5)
$X_{0}(t)=\int_{0}^{t} X(\tau) d \tau+\int_{-\infty}^{0} X(\tau) d \tau =\int_{0}^{t} X(\tau) d \tau+X_{0}(0)$

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

(6)
$X_{0_{i}}(0)=-\left(c_{i} / c_{0_{i}}\right) X_{i}(0), \quad i=1,2, \dots n$

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

(7)
$s=C^{T} \cdot X+C_{0}^{T} \cdot X_{0} =\sum_{i=1}^{n} c_{i} X_{i}+\sum_{i=1}^{n} c_{0_{i}} X_{0_{i}}(=0), \quad c_{n}=1$

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

(8)
$\dot{X}_{n}=-\sum_{i=1}^{n-1} c_{i} \dot{X}_{i}-\sum_{i=1}^{n} c_{0_{i}} X_{i}$

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

(9)
$u_{1}=-K(X) \cdot X-\Delta K \cdot X-G \cdot \operatorname{sign}(s)$

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

2.1 Sliding surface transformation[4,5,9]

(10)
$s^{*}=\left[C^{T} g_{0}(X, t)\right]^{-1} \cdot s, H_{s}(X, t)=\left[C^{T} g_{0}(X, t)\right]^{-1}$

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

(11)
$u_{2}=-K(X) \cdot X-\Delta K \cdot X-G \cdot \operatorname{sign}\left(s^{*}\right)$

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

(12)
$\dot {s^{*}}=\left[\dot {C^{T} g_{0}(X, t)}\right]^{-1} s+\left[C^{T} g_{0}(X, t)\right]^{-1} \dot {s} $

$=\frac{\partial}{\partial X}\left[C^{T} g_{0}(X, t)\right]^{-1} \dot{X}\left(C^{T} X+C_{0}^{T} X_{0}\right) +\left[C^{T} g_{0}(X, t)\right]^{-1}\left(C^{T} \dot{X}+C_{0}^{T} X\right)$

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

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

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

(13)
$s^{*}=\left[C^{T} B_{0}\right]^{-1}\left(C^{T} \dot{X}+C_{0}^{T} X\right)$

$=\left[C^{T} B_{0}\right]^{-1}\left(C_{0}^{T} X\right)$ $+\left[C^{T} B_{0}\right]^{-1} C^{T}\left(f_{0}(X, t)+\Delta f_{1}(X, t)\right) X$ $+\left[C^{T} B_{0}\right]^{-1} C^{T} \Delta f_{2}(X, t)$ $+\left[C^{T} B_{0}\right]^{-1} C^{T}\left(B_{0}+\Delta g(X, t)\right) u_{2}$ $+\left[C^{T} B_{0}\right]^{-1} C^{T} d^{\prime}(X, t)$

$=\left[C^{T} B_{0}\right]^{-1} C_{0}^{T} X$ $+\left[C^{T} B_{0}\right]^{-1} C^{T}\left(f_{0}(X, t)+\Delta f_{1}(X, t)\right) X$ $+\left[C^{T} B_{0}\right]^{-1} C^{T} \triangle f_{2}(X, t)$ $\quad+(\dot {I}+\Delta t) u_{2}+\left[C^{T} B_{0}\right]^{-1} C^{T} d^{\prime}(X, t)$

$=\left[C^{T} B_{0}\right]^{-1} C_{0}^{T} X$ $+\left[C^{T} B_{0}\right]^{-1} C^{T}\left(f_{0}(X, t)+\Delta f_{1}(X, t)\right)_{X}$ $+\left[C^{T} B_{0}\right]^{-1} C^{T} \triangle f_{2}(X, t)$ $\quad+(I+\Delta t)\left(-K(X) X-\Delta K X-G \operatorname{sign}\left(s^{*}\right)\right)$ $\quad+\left[C^{T} B_{0}\right]^{-1} C^{T} d^{\prime}(X, t)$

By letting the static nonlinear feedback gain as

(14)
$K(X)=\left[C^{T} B_{0}\right]^{-1}\left(C^{T} f_{0}(X, t)+C_{0}^{T}\right)$

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

(15)
$\dot {s^{*}}=\left[\left(C^{T} B_{0}\right)^{-1} C^{T} \Delta f_{1}(X, t)-\Delta \dot {I} K(X)\right] X$ $-(I+\Delta \dot {I}) \Delta K X$

$+\left(C^{T} B_{0}\right)^{-1} C^{T} \triangle f_{2}(X, t)+\left(C^{T} B_{0}\right)^{-1} C^{T} d^{\prime}(X, t)$ $-(I+\Delta \dot {I}) G \operatorname{sign}\left(s^{*}\right)$

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

(16)
$\triangle k_{j}=\left\{\begin{array}{l} { \geq \frac{\max \left\{\left(C^{T} B_{0}\right)^{-1} C^{T}\left(\Delta f_{1}(X, t)+ \Delta \dot {I} K(X) \right)\right\}}{\min \{I+\Delta \dot {I}\} _ {j}} \quad \quad \text { for sign }\left(s^{*} X_ {j}\right) > 0} \\ { \leq \frac{\min \left\{\left(C^{T} B_{0}\right)^{-1} C^{T}\left(\Delta f_{1}(X, t)+ \Delta \dot {I} K(X) \right)\right\}}{\min \{I+\Delta \dot {I}\} _ {j}} \quad \quad \text { for sign }\left(s^{*} X_ {j}\right) < 0}\end{array}\right.$

$j=1,2, \ldots, n$

(17)
$G=\left\{\begin{array}{l}{ \geq \frac{\max \left\{\left(C^{T} B_{0}\right)^{-1} C^{T}\left(\Delta f_{2}(X, t)+d^{\prime}(X, t)\right)\right\}}{\min \{I+\Delta \dot {I}\}} \quad \quad \text { for sign }\left(s^{*}\right) > 0} \\ { \leq \frac{\min \left\{\left(C^{T} B_{0}\right)^{-1} C^{T}\left(\Delta f_{2}(X, t)+d^{\prime}(X, t)\right)\right\}}{\min \{I+\Delta \dot {I}\}} \quad \quad \text { for sign }\left(s^{*}\right) < 0}\end{array}\right.$

then

(18)
$s^{*} \cdot \dot{s}^{*} < 0$

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

2.2 Control input transformation[4,5]

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

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

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

(19)
$\begin{aligned} u^{*} & =\left[C^{T} g_{0}(X, t)\right]^{-1} u_{1}, \quad H_{u}=\left[C^{T} g_{0}(X, t)\right]^{-1} \\ & =\left[C^{T} g_{0}(X, t)\right]^{-1}[-K(X) X-\Delta K X-G \operatorname{sign}(s)] \end{aligned}$

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

(20)
$\dot{s}=C^{T} \dot{X}+C_{0}^{T} X$

$= C_{0}^{T} X+C^{T}\left(f_{0}(X, t)+\Delta f_{1}(X, t)\right) X+C^{T} \Delta f_{2}(X, t) +C^{T}\left(g_{0}(X, t)+\Delta g(X, t)\right) u^{*}+C^{T} d^{\prime}(X, t) $

$=C_{0}^{T} X+C^{T}\left(f_{0}(X, t)+\Delta f_{1}(X, t)\right) X+C^{T} \Delta f_{2}(X, t) +(I+\Delta I) u_{1}+C^{T} d^{\prime}(X, t)$

$=C_{0}^{T} X+C^{T}\left(f_{0}(X, t)+\Delta f_{1}(X, t)\right) X+C^{T} \Delta f_{2}(X, t) +(I+\Delta I)(-K(X) X-\Delta K X-G s i g n(s))+C^{T} d^{\prime}(X, t)$

$=C_{0}^{T} X+C^{T} f_{0}(X, t) X-IK(X) X+C^{T} \Delta f_{1}(X, t) X$

$-\Delta I K(X) X-(I+\Delta I) \Delta K+C^{T} \triangle f_{2}(X, t) +C^{T} d^{\prime}(X, t)-(I+\Delta I) G \operatorname{sign}(s)$

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

(21)
$K(X)=C^{T} f_{0}(X, t)+C_{0}^{T}$

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

(22)
$\dot{s}=C^{T} \Delta f_{1}(X, t) X-\Delta I K(X) X-(I+\Delta I) \Delta K X$

$+C^{T} \triangle f_{2}(X, t)+C^{T} d^{\prime}(X, t) -(I+\Delta I) G \operatorname{sign}(s)$

If one takes the switching gains as the design parameters

(23)
$\triangle k_{j}=\left\{\begin{array}{l} { \geq \frac{\max \left\{C^{T} \triangle f_{1}(X, t)-\triangle I K(X)\right\}_{j}}{\min \{I+\triangle I\}_{j}} \quad \quad \text { for sign }\left(s X_ {j}\right) > 0} \\ { \leq \frac{\min \left\{C^{T} \triangle f_{1}(X, t)-\triangle I K(X)\right\}_{j}}{\min \{I+\triangle I\}_{j}} \quad \quad \text { for sign }\left(s X_ {j}\right) < 0}\end{array}\right.$

$j=1,2, \ldots, n$

(24)
$G=\left\{\begin{array}{l} { \geq \frac{\max \left\{C^{T} \Delta f_{2}(X, t)+C^{T} d^{\prime}(X, t)\right\}}{\min \{I+\Delta I\}} \quad \quad \text { for sign }\left(s\right) > 0} \\ { \leq \frac{\min \left\{C^{T} \triangle f_{2}(X, t)+C^{T} d^{\prime}(X, t)\right\}}{\min \{I+\Delta I\}} \quad \quad \text { for sign }\left(s\right) < 0}\end{array}\right.$

then one can obtain the following equation

(25)
$s \cdot \dot{s} < 0$

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

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

3. Design Examples and Simulation Studies

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

(26)
$\dot{X}_{1}=-X_{1}-X_{1}^{3}+X_{2}$

$\dot{X}_{2}=0.7 \sin \left(X_{1}\right)+X_{2}-0.8 \sin \left(X_{2}\right)+0.2\left(X_{1}^{2}+X_{2}^{2}\right) +X_{2} \sin ^{2}\left(X_{2}\right)+(2+0.3 \sin (2 t)) u+2 \sin (5 t)$

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

(27)
$\left[ \begin{array}{c}{\dot{X}_{1}} \\ {\dot{X}_{2}}\end{array}\right]=\left[ \begin{array}{cc}{-1-X_{1}^{2}} & {1} \\ {0} & {1+\sin ^{2}\left(X_{2}\right)}\end{array}\right] X +\left[ \begin{array}{c}{0} \\ {0.7 \sin \left(X_{1}\right)-0.8 \sin \left(X_{2}\right)+0.2\left(X_{1}^{2}+X_{2}^{2}\right)}\end{array}\right]$

$+\left[ \begin{array}{c}{0} \\ {2+0.3 \sin (2 t)}\end{array}\right] u+\left[ \begin{array}{c}{0} \\ {2 \sin (5 t)}\end{array}\right]$

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

(28)
$f_{0}(X, t)=\left[ \begin{array}{cc}{-1-X_{1}^{2}} & {1} \\ {0} & {1}\end{array}\right]$,

$g_{0}(X, t)=B_{0}=\left[ \begin{array}{l}{0} \\ {2}\end{array}\right]$,

$\Delta f_{1}(X, t)=\left[ \begin{array}{cc}{0} & {0} \\ {0} & {\sin ^{2}\left(X_{2}\right)}\end{array}\right]$,

$\triangle f_{2}(X, t)=\left[ \begin{array}{c}{0} \\ {0.7 \sin \left(X_{1}\right)-0.8 \sin \left(X_{2}\right)+0.2\left(X_{1}^{2}+X_{2}^{2}\right)}\end{array}\right]$,

$\triangle g(X, t)=\left[ \begin{array}{c}{0} \\ {0.3 \sin (2 t)}\end{array}\right]$,

$d^{\prime}(X, t)=\left[ \begin{array}{c}{0} \\ {2 \sin (5 t)}\end{array}\right]$

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

(29)
$X_{0_{1}}(t)=\int_{0}^{t} X_{1}(\tau) d \tau+X_{0_{1}}(0)$

(30)
$X_{0_{2}}(t)=\int_{0}^{t} X_{2}(\tau) d \tau+X_{0_{2}}(0)$

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

3.1 Sliding surface transformation

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

(31)
$K(X)=\left[5.5+0.5 X_{1}^{2} \quad 5\right]$

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

(32)
$C^{T}=\left[ \begin{array}{cc}{-1} & {1}\end{array}\right]$ and $C_{0}^{T}=\left[ \begin{array}{ll}{11} & {10}\end{array}\right]$

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

(33)
$\dot{X}_{2}=\dot{X}_{1}-11 X_{1}-10 X_{2} =-12 X_{1}-X_{1}^{3}-9 X_{2}$

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

(34)
$\dot{X}_{s}=\left[ \begin{array}{cc}{-1-X_{1}^{2}} & {1} \\ {-12-X_{1}^{2}} & {-9}\end{array}\right] X_{s}, \quad X_{s}(0)=X(0)$

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

(35)
$s^{*}=\left(C^{T} B_{0}\right)^{-1} \cdot s, H_{s}(X, t)=\left(C^{T} B_{0}\right)^{-1}=2^{-1}$

Now, the IVSS control input is taken as follows:

(36)
$u_{2}=-K(X) \cdot X-\Delta K \cdot X-G \cdot \operatorname{sign}\left(s^{*}\right)$

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

(37)
$s^{*}=\left[\left(C^{T} B_{0}\right)^{-1} C^{T} \Delta f_{1}(X, t)-\Delta \dot {I} K_{\mathrm{K}}(X)\right] X -(I+\Delta \dot {I}) \Delta K X$

$+\left(C^{T} B_{0}\right)^{-1} C^{T} \triangle f_{2}(X, t)+\left(C^{T} B_{0}\right)^{-1} C^{T} d^{\prime}(X, t) -(I+\Delta \dot {I}) G \operatorname{sign}\left(s^{*}\right)$

where

(38)
$\triangle \dot {I}=\left[C^{T} B_{0}\right]^{-1} C^{T} \triangle g(X, t) =2^{-1} \left[ \begin{array}{cc}{-1} & {1}\end{array}\right] \left[ \begin{array}{c}{0} \\ {0.3 \sin (2 t)}\end{array}\right] =0.15 \sin (2 t) \leq 0.15 < 1$

If one takes the switching gains as follows:

(39)
$\Delta k_{1}=\left\{\begin{array}{cc}{2.35+0.1 X_{1}^{2}} & {\text { if } s^{*} X_{1} > 0} \\ {-2.35-0.1 X_{1}^{2}} & {\text { if } s^{*} X_{1} < 0} \\ {0} & {\text { if } s^{*} X_{1}=0}\end{array}\right.$

$\Delta k_{2}=\left\{\begin{array}{cc}{3.2} & { \text { if } s^{*} X_{2} > 0} \\ {-3.2} & { \text { if } s^{*} X_{2} < 0} \\ {0} & {\text { if } s^{*} X_{2}=0}\end{array}\right.$

$G=\left\{\begin{array}{cc}{2.35+0.635\left(X_{1}^{2}+X_{2}^{2}\right)} & { \text { if } s^{*} > 0} \\ {-2.35-0.635\left(X_{1}^{2}+X_{2}^{2}\right)} & { \text { if } s^{*} < 0} \\ {0} & { \text { if } s=0}\end{array}\right.$

then

(40)
$s^{*} \cdot \dot{s}^{*} < 0$

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

(41a)
$X_{0_{1}}(0)=-\left(c_{1} / c_{0_{1}}\right) X_{1}(0)=-(-1 / 11) * 10=0.9091$

(41b)
$X_{0_{2}}(0)=-\left(c_{2} / c_{0_{2}}\right) X_{2}(0)=-(1 / 10) *(2)=-0.2$

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

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

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

../../Resources/kiee/KIEE.2019.68.3.460/fig1.png

그림. 2. 실제 상 궤적과 이상 슬라이딩 궤적

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

../../Resources/kiee/KIEE.2019.68.3.460/fig2.png

그림. 3. 슬라이딩 면 $s^{*}(t)$

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

../../Resources/kiee/KIEE.2019.68.3.460/fig3.png

그림. 4. 제어입력 $u_{2}(t)$

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

../../Resources/kiee/KIEE.2019.68.3.460/fig4.png

3.2 Control input transformation

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

(42)
$K(X)=\left[11+X_{1}^{2} \quad 10\right]$

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

(43)
$C^{T}=\left[ \begin{array}{cc}{-1} & {1}\end{array}\right]$ and $C_{0}^{T}=\left[ \begin{array}{ll}{11} & {10}\end{array}\right]$

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

(44a)
$X_{0_{1}}(0)=-\left(c_{1} / c_{0_{1}}\right) X_{1}(0)=-(-1 / 11) * 10=0.9091$

(44b)
$X_{0_{2}}(0)=-\left(c_{2} / c_{0_{2}}\right) X_{2}(0)=-(1 / 10) *(2)=-0.2$

The control input transformation is designed as

(45)
$u^{*}=\left(C^{T} g_{0}(X, t)\right)^{-1} u_{1}$,

$H_{u}=\left(C^{T} g_{0}(X, t)\right)^{-1}=[-1 \quad 1] \left[ \begin{array}{l}{0} \\ {2}\end{array}\right]=2^{-1} =2^{-1}[K(X) X-\Delta K X-G \operatorname{sign}(s)]$

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

(46)
$\dot{s}=C^{T} \Delta f_{1}(X, t) X-\Delta IK(X) X-(I+\Delta I) \Delta K X +C^{T} \triangle f_{2}(X, t)+C^{T} d^{\prime}(X, t)-(I+\Delta I) G \operatorname{sign}(s)$

where

(47)
$\triangle I=C^{T} \triangle g(X, t)\left[C^{T} g_{0}(X, t)\right]^{-1} =\left[ \begin{array}{ll}{-1} & {1}\end{array}\right] \left[ \begin{array}{c}{0} \\ {0.3 \sin (2 t)}\end{array}\right] 2^{-1} =0.15 \sin (2 t) \leq 0.15 < 1$

If one takes the switching gain as design parameters

(48)
$\Delta k_{1}=\left\{\begin{array}{cc}{4.7+0.2 X_{1}^{2}} & {\text { if } s X_{1} > 0} \\ {-4.7-0.2 X_{1}^{2}} & {\text { if } s X_{1} < 0} \\ {0} & {\text { if } s X_{1}=0}\end{array}\right.$

$\Delta k_{2}=\left\{\begin{array}{cc}{6.4} & {\text { if } s X_{2} > 0} \\ {-6.4 \text { if } s X_{2} < 0} \\ {0} & {\text { if } s X_{2}=0}\end{array}\right.$

$G=\left\{\begin{array}{cc}{4.7+1.27\left(X_{1}^{2}+X_{2}^{2}\right)} & {\text { if } s > 0} \\ {-4.7-1.27\left(X_{1}^{2}+X_{2}^{2}\right)} & {\text { if } s < 0} \\ {0} & {\text { if } s=0}\end{array}\right.$

then one can obtain the following equation

(49)
$s \cdot \dot{s} < 0$

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

그림. 5. 제어입력 변환에 의한 두 출력 응답

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

../../Resources/kiee/KIEE.2019.68.3.460/fig5.png

그림. 6. 실제 상 궤적과 이상 슬라이딩 궤적

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

../../Resources/kiee/KIEE.2019.68.3.460/fig6.png

그림. 7. 슬라이딩 면 $s(t)$

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

../../Resources/kiee/KIEE.2019.68.3.460/fig7.png

그림. 8. 제어입력 $u^{*}(t)$

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

../../Resources/kiee/KIEE.2019.68.3.460/fig8.png

4. Conclusions

In this note, the invariant theorem of Utkin is rigorously, precisely, and completely proved for SI uncertain integral nonlinear VSS. During the proof, the precise existence condition of the sliding mode on the pre-selected integral sliding surface is completely proved for the IVSS of SI uncertain nonlinear plants. The invariance theorem of the two diagonal(transformation) methods i.e., the control input transformation and sliding surface transformation is proved clearly, comparatively, and completely. Therefore, the equation of the sliding mode, i.e., the integral sliding surface is invariant with respect to the two diagonalization methods. These two methods diagonalize the input system of the real dynamics of the sliding surface $S$ or $S^{*}$ so that the existence condition of the sliding mode on the predetermined integral sliding surface is easily proved. During the proof of Utkin's theorem, the integral sliding surface is adopted to remove the reaching phase and the guideline to obtain the ideal sliding dynamics of the integral sliding surface is presented. And by using the solution of the ideal sliding mode dynamics, the output is predictable under the matched uncertainties and disturbances which is shown in the design example and simulation study. The gain design rules for the two control inputs are proposed. Through an illustrative example and simulation study, the effectiveness of the proposed main results is verified. The same results in the outputs by the two diagonalization methods are obtained. The equation of the sliding mode, i.e., the integral sliding surface is invariant with respect to the two diagonalization methods. If the control input matrix $g_{0}(X, t)$ in the model of nonlinear plants is constant(not function of $X$ or $t$) i.e. $g_{0}(X, t)=B_{0}$, then both transformation (diagonalization) methods of Utkin’s theorem can be used to, otherwise, only the control input transformation can be basically applied to the proof of the existence condition of the sliding mode on the predetermined integral sliding surface in the nonlinear IVSS. It is possible to formulate completely the equation of the IVSS controller design for SI uncertain nonlinear systems from the formulation of the objective plants to the complete proof of the existence condition of the sliding mode and the proof of the asymptotic stability of the closed loop system.

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저자소개

이 정 훈 (Jung-Hoon Lee)
../../Resources/kiee/KIEE.2019.68.3.460/au1.png

1966년 2월 1일생

1988년 경북대학교 전자공학과 졸업(공학사)

1990년 한국과학기술원 전기 및 전자공학과 졸업(석사)

1995년 한국과학기술원 전기 및 전자공학과 졸업(공박)

1995.3.1.~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 : jhleew@gnu.ac.kr