1. Introduction

Because the VSS or SMC can provide an effective means for the discontinuous control of MIMO uncertain dynamical linear systems under matched parameter variations and external disturbances^(1,2) due to the sliding mode on a preselected sliding surface, recently, the VSS or SMC has been successfully applied to the control of a wide range of uncertain plants^(3-5). 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, ^(6,7). The VSS or SMC has the three shortcomings, i.,e., the reaching phase problem, chattering problem, and the proof issue of the MIMO existence condition of the sliding mode. To remove the reaching phase problems, the two conditions must be satisfied, those are that the sliding surface is defined from any initial condition to the origin and the MIMO eixstence condition of the sliding mode should be satisfied and proved for the whole trajectory⁽³³⁾. For single input single output(SISO) uncertain plants, the integral state with the special initial condition is introduced to the VSS for the first time in ⁽⁸⁾. The idea of ⁽⁸⁾ is applied to tracking controls of motors in ⁽⁹⁾ and ⁽¹⁰⁾ and to simple regulation controls of motors in ⁽¹¹⁾. The idea of ⁽⁸⁾ is also applied to set-point regulation controls of robot manipulators in ⁽¹²⁾, to simple regulation controls of nonlinear systems in ⁽¹³⁾, and to point-to-point regulation controls of uncertain general linear systems in ⁽¹⁴⁾. In those papers, the reaching phase problems are completely removed. The performance of the output prediction and predeter- mination is obtained by using the solution of the ideal sliding dynamics of the integral sliding surface. A modification of ⁽⁸⁾ so called a nonlinear integral-type sliding surface is proposed in ⁽¹⁵⁾ by Utkin and Shi. Similar results to ⁽¹⁵⁾ are obtained in ⁽¹⁶⁾ and ⁽¹⁷⁾. However, the algorithms of ⁽¹⁵⁾, ⁽¹⁶⁾, and ⁽¹⁷⁾ have the disadvantages of needing the information of the nominal input $u_{0}$ to construct the nonlinear integral-type sliding surface and proving the MIMO existence condition of the sliding mode. The resulting output can not be predicted. These demerits are overcome in ⁽¹⁸⁾ and ⁽¹⁹⁾ by means of introducing the closed-loop dynamics to the integrand in the nonlinear integral-type sliding manifolds instead of using $u_{0}$. Other versions of the integral sliding surface are proposed in ⁽²⁰⁾ in order to adopt the integral of the sliding surface itself to the conventional sliding surface. The reaching phase is also removed, but the advantages of removing the reaching phase are not discussed and the output can not be predictable. The time-varying sliding surface is surveyed in ⁽²¹⁾ and proposed for second order systems in ⁽²²⁾. However, the application plants are limited to second order systems, the optimal para- meter of the change of the time-varying sliding surface is the problem, and the output can not be predicted.

In contrast with a lot of applications of the VSS to the control of real uncertain plants, in theoretical aspect, the research on the MIMO VSS is not sufficient to predict the real output itself. For the MIMO research of the VSS control, some design methods were studied, those are the hierarchical control methodology^(1,2,23), transformation(diagonalization) methods^{(1,6,24,32-34)}, simplex algorithm⁽²⁵⁾, Lyapunov approach^(26-31), and so on. In most present MIMO SMCs except the transformation methods^{(1,6,24,32-34)}, it is difficult to prove the precise MIMO existence condition of the sliding mode on the predetermined sliding surface theoretically, but in ^(26-31), only the results that the derivative of the Lyapunov candidate function is negative, i.e. $\dot V\le 0$ is obtained when $V=1/2s^{T}s$, instead of exactly proving the MIMO existence condition of the sliding mode, which is stricter condition than the Lyapunov stability. Without the complete proofs, the two methodologies are presented by Utkin to prove the MIMO existence condition of the sliding mode on the sliding surface⁽¹⁾. It is so called the invariance theorem, that is the equation of the sliding mode is invariant with respect to the two nonlinear transformation(diagonalization). Those are the control input transformation(diagonalization)⁽³³⁾ and sliding surface transformation(diagonalization)^(33,34). For MIMO uncertain linear plants, the theorem of Utkin is proved comparatively and completely⁽²⁴⁾. Owing to the Utkin’s theorem⁽¹⁾ and its proof⁽²⁴⁾ for MIMO Plant, the work of proving the precise MIMO existence condition of the sliding mode becomes the easy task, and there are the two design approaches in the SMC for the complete formulation from the description of object plants through the presentation of the sliding surface and suggestion of the corresponding control input to the closed loop stability analysis together with the investigation of the MIMO existence condition of the sliding mode. Those are the sliding surface transformation SMC^(32,34) and control input transformation SMC⁽³³⁾. In ⁽³²⁾ the continuous sliding surface transformation SMC as one approach of MIMO VSS design is studied for MIMO uncertain linear plants, in which the MIMO existence condition of the sliding mode is proved clearly for the complete formulation of the VSS design and by using the saturation function, the continuous approxi- mation is made. The closed loop stability of the continuous VSS is not investigated, the reaching phase problems are not dealt with, and the output can not be predicted. In ⁽³³⁾, the continuous control input transformation SMC as another approach of MIMO VSS design is presented for MIMO uncertain linear plants. The MIMO existence condition of the sliding mode is proved clearly for the complete formulation of the VSS design. By using the modified boundary layer function, the continuous approximation is successfully made and its closed loop bounded stability of the continuous VSS is investigated. But the problems of the reaching phase are not dealt with. So the output can not be predictable. In ⁽³⁴⁾, by means of removing the reaching phase problems, as the one approach, an continuous sliding surface transformation VSS with the output prediction perfor- mance is studied. The transformed integral sliding surface is suggested to remove the reaching phase completely. The two ideal sliding dynamics from any given initial condition to the origin are presented and using the solution of the ideal sliding dynamics, the real output can be predetermined, pre-designed, and predicted. The complete formulation of the VSS design for the output prediction is presented by means of removing the reaching phase and the chattering problem is dealt with by the fixed modified boundary layer function. The bounded stability together with the MIMO existence condition of the sliding mode is investigated by the continuous sliding surface trans- formation VSS.

In this paper, as another approach of ⁽³⁴⁾, theoretically discontinuous and practically continuous control input transformed sliding mode controls with the output prediction and pre- determination as the alternative approach of ⁽³⁴⁾ are proposed by using the integral sliding surface for MIMO uncertain linear plants. The reaching phase is completely removed for MIMO uncertain linear plants by means of the integral sliding surface with augmentation of an special initial valued integral state by extending the idea in ⁽⁸⁾ for SISO plants to MIMO plants and proving the MIMO existence condition of the sliding mode. The one form of the ideal sliding dynamics of the integral sliding surface is obtained and using the solution of the ideal sliding dynamics, the robust real output can be pre-designed, predetermined, and predicted. Theoretically a transformed discontinuous input is suggested to satisfy the MIMO existence condition of the sliding mode on the every point on the predetermined sliding surface from a given initial condition to the origin. Closed loop exponential stability together with the MIMO existence condition of the sliding mode is investigated in Theorem 1 for the complete formulation of the VSS design for the output prediction and predetermination. The another ideal sliding dynamics is given by the other concept due to the strong robustness of the sliding mode for the entire trajectory. For the high potential of practical applications, a continuous approximation of the discontinuous VSS for the integral sliding surface is made by applying the fixed modified boundary layer function presented in ⁽³²⁾ to the discontinuous transformed input. The chattering of the control input as the inherent property of the VSS is much improved. The analysis of the closed loop bounded stability together with the MIMO existence condition of the sliding mode by the continuous control input transformation VSS is added to Theorem 2. Through a design example and comparative simulation studies, the usefulness of the proposed continuous transformed VSS controller is verified. The projection ideal sliding trajectory is given.

2. Discontinuous and Continuous Integral SMCs

Consider a MIMO uncertain linear system with m inputs:

where $X\in R^{n}$ is the state variable, $X(0)\in R^{n}$ is the initial condition for the state, $u\in R^{m}$ is the control input, $A_{0}\in R^{n\times n}$ is the nominal system matrix, $B_{0}\in R^{n\times m}$ is the nominal input matrix, $\triangle A$ and $\triangle B$ are the system matrix uncertainty and input matrix uncertainty, those are assumed to be matched and bounded, and $\Delta D(t)$ is the external disturbance which is also assumed to be matched and bounded.

The first aim of the integral SMC controller design is to maintain the integral sliding surface $s$ to be zero vector from the initial time without the reaching phase and finally to regulate the state $x$ to be zero vector from any initial condition by the transformed discontinuous input with the ideal sliding dynamics and with the output prediction performance. By the transformed continuous input for the practical applications, the second aim of the integral VSS controller design is to maintain the integral sliding surface $s$ to be bounded near to zero and finally to regulate the state $x$ to be bounded with almost output prediction by the transformed continuous input.

The intensional integral state $X_{0}\in R^{n}$ is introduced with the special initial condition for use later in the integral sliding surface as follows:

where $X_{0}(0)$ is the special initial condition of the integral state that is chosen later.

The proposed MIMO integral sliding surface $s\in R^{m}$ with defining the surface from any initial condition without any reaching phase is the linear combination of the integral state and the full state variable as

$s=C_{0}\cdot X_{0}+C\cdot X(=0)$ where $C_{0}$ is a stable coefficient matrix for the integral state and $C$ is a stable coefficient matrix for the original state. The special initial condition $X_{0}(0)$ in (2) for the integral state is determined so that the integral sliding surface (3) is the zero vector at $t=0$ for any initial condition $X(0)$ such that

The ideal of the integral sliding surface with a special initial condition stems from that in ⁽⁸⁾ in which the integral sliding surface is for SISO plants. With the initial condition (4) for the integral state, the integral sliding surface is the zero vector at the initial time $t=0$. Hence, the integral sliding surface (3) can define the surface from a given initial condition to the origin in the state space with the ideal sliding dynamics, and the controlled system slides from the initial time $t=0$ without any reaching phase. the first condition of removing reaching phase problems is satisfied⁽³³⁾. The sliding surface (3) becomes an other form⁽³⁵⁾

In (6), there is not necessary for considering the initial condition for the integral state. The integral sliding surfaces (3) and (6) have the same ideal sliding dynamics. Since the $s=\dot s =0$ is satisfied in the sliding mode, from $\dot s =0$, m the ideal sliding dynamics is obtained. From (1), n-m certain linear dynamics being independent of the input is taken. Combine both the two obtained dynamics and the following equation of n the ideal sliding dynamics of (3) is obtained as

where $x_{s}$ is the solution of (7) and becomes the ideal sliding robust output(sliding surface). The ideal sliding dynamics (7) will be given by another method and will be shown in the design example and simulation study. The ideal sliding dynamics (7) is the dynamic representation of the integral sliding surface and the solution of (7) is the same meaning of the ideal sliding surface defined by the integral sliding surface (3), by using the solution of (7), the controlled real robust output can be predesigned, predicted, and predetermined in advance from any given initial condition to the origin with the ideal sliding dynamics.

The stable design of the integral sliding surface implies to the stable design of the ideal sliding dynamics and the reverse argument also holds.

Assumption 1:

$CB_{0}$ has the full rank and has its inverse for the coefficient matrix of the integral sliding surface, $C$, i.e. $R\left\{CB_{0}\right\}=m$ and there exists $\left(CB_{0}\right)^{-1}$.

Assumption 2:

$C\triangle B(CB_{0})^{-1}=\triangle I$. $\triangle I$ is diagonal and $|\triangle I_{ii}|\le\rho_{i}<1,\:$ $i=1,\: 2,\:\cdots ,\: m$ where $\rho_{i}$ is the constant.

Now, the suggested transformed^(1,24,33) discontinuous VSS control input for the integral sliding surface is proposed as follows:

where the constant gains are taken as

and one takes the discontinuously switching gains as follows:

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

The idea of the transformed control input is stemmed from Utkin’s theorem in ⁽¹⁾ and its proof for MIMO plants in ⁽²⁴⁾, which is for easy design so as to satisfy the MIMO existence condition of the sliding mode. In ⁽³²⁾ and ⁽³⁴⁾, the control input is not transformed, but the sliding surface is transformed. In the transformed discontinuous control input (8), the sliding surface itself is one of the output feedback which makes the controlled system be closer to the ideal pre-selected sliding surface⁽⁸⁾. Then, the real dynamics of the integral sliding surface by the transformed discontinuous control input, i.e. the time derivative of $s$ becomes

From (9), the real dynamics of $s$ becomes finally

Since each gain term in the right hand side of (15) is diagonalized because of the multiplication of the $(I+\triangle I)$ term to each gain term, m the multi input design problem is changed to m the single input design problems due to the transformation of the control input in (8), which results in the easy satisfaction of the MIMO existence condition of the sliding mode for the whole trajectory. The total closed loop stability with the transformed discontinuous control input (8) and the integral sliding surface (3) together with the MIMO precise existence condition of the sliding mode will be investigated in Theorem 1.

Theorem 1: If the integral sliding surface (3) is designed to be stable, the transformed discontinuous control input (8) with the integral sliding surface (3) satisfies the MIMO existence condition of the sliding mode on the pre-determined sliding surface and closed loop exponential stability to the integral sliding surface $s=0$ including the origin.

Proof^(33,34): A Lyapunov function candidate is chosen as

Differentiating (16) with time leads to

Substituting (15) into (17) leads to

Since the uncertainty and external disturbance terms in (18) are canceled out from the chattering input by means of the switching gains in (11) and (12), the following equation remains

From (19), the following equation is obtained

The MIMO existence condition of the sliding mode on the pre-selected sliding surface by the transformed discontinuous control input is proved theoretically for the complete formulation of the integral VSS design for the output prediction performance, while the Lyapunov stability is proved instead in most VSSs which is looser condition. By only through the proof of the MIMO existence condition of the sliding mode, the strong robustness on the every point on the pre-selected sliding surface for the whole trajectory from a given initial condition finally to the origin is guaranteed and shown. Hence, the controlled robust output can be predesigned, predicted, predetermined, and exhibited as designed in the integral sliding surface. The second condition of removing the reaching phase problems is satisfied⁽³⁴⁾. From (19), the following equation is obtained.

From (21), the following equation is obtained

which completes the proof of Theorem 1.

If the constant gains and switching gains are suitably and stably designed, then the effect of the uncertainties and external disturbances is canceled out in the sliding mode due to the strong robustness of the sliding mode, and the closed loop linear dynamics with the constant feedback is only remained as

which is the same as the ideal sliding dynamics (7) of (3). Due to the proof of Theorem 1, the following is concluded by the discontinuous control input as $t arrow\infty$

$$ \begin{array}{l} \text { maintain } V(X(t)) \text { as } 0 \Rightarrow \text { maintain } s(t) \text { as } 0 \\ \qquad \Rightarrow X(t)-0 \text { with designed ideal sliding dynamics } \end{array} $$

The transformed control input (8) may have the chattering problems because of the high frequency switching of the discontinuous part of the control input (8) due to the infinitely switching of the sign function in (13) according to the value of the sliding surface which may be harmful to practical real plants^(36-38). Hence, the continuous approximation of the dis- continuous VSS is essentially needed for practical applications to real MIMO uncertain plants without a severe performance loss. Applying the idea in ⁽³³⁾ of the modified fixed boundary layer method, the transformed discontinuous input (8) has modified to the following form

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

Since the switching parts in (24) are stable itself which is shown through the proof of Theorem 1, the $mblf(s_{i})$ function can not make an effect on the closed loop stability and only can change the magnitude of the switching terms within the boundary layer in order to much improve the chattering of the discontinuous VSS input.

Theorem 2: The integral SMC with the transformed continuous input (24) and the proposed integral sliding surface (3) can exhibit the bounded stability for all the uncertainties and external disturbances.

Proof^(33,34): Take a Lyapunov candidate function as

If $s_{i}>l_{+i}$ or $s_{i}<l_{-i}$, then $mblf(s_{i})=1$ from (25) and the continuous input (24) becomes the discontinuous input (8). Therefore from the proof of Theorem 1, we can obtain the following equation

as long as $||s(X,\:t)||\ge l=\max_{i}\max(l_{+i},\:l_{-i})$. From (27), the following equation is obtained

as long as $||s(X,\:t)||\ge l$, which means that over the boundary layer, the existence condition of the sliding mode is the same as that of the discontinuous input case. From (27), the following equation is obtained as

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

The closed loop bounded stability together with the existence condition of the sliding mode by the continuous input (24) is investigated, however those are difficult in most of the continuous VSSs specially in ⁽³⁶⁾ and ⁽³⁷⁾. Due to the proof of Theorem 2, the following statement is concluded by the continuous transformed control input as $t\to\infty$

$$ \begin{aligned} V(t) & \rightarrow \text { bounded by } l^{2} / 2 \Rightarrow s(t) \rightarrow \text { bounded by } l \\ & \Rightarrow X(t) \rightarrow \text { bounded } \end{aligned} $$

3. Design Example and Comparative Simulation Studies

A fifth-order MIMO uncertain linear system with the two inputs described by the state equation is considered which is slightly changed from that in ⁽³⁹⁾

where the nominal parameter $A_{0}$ and $B_{0}$, matched uncertainties $\triangle A$ and $\triangle B$, and disturbance $\triangle D(t)$ are

To design the proposed MIMO discontinuous integral SMC with the integral sliding surface and transformed control input, first the stable coefficient matrix for the state in the suggested integral sliding surface is determined⁽³⁹⁾ as

so that the equations in Assumption 1 are calculated as

and $\triangle I$ in Assumption 2 is

The closed loop poles of the ideal sliding dynamics (7) or (23) are selected for the critical damping as

Using the MATLAB command place

$K=place(A_{0},\:B_{0},\:P)$ the constant feedback gain $K$ in (9) or (23) is designed as

The closed loop dynamics i.e. ideal sliding dynamics of (23) is calculated as

where

From (9), the coefficient matrix for the integral state in the integral sliding surface is calculated as

To find the ideal sliding dynamics of (3), the 5(n)-2(m)=3 certain linear dynamics of (30) being independent of the input is obtained as

From $\dot s =0$, the two ideal sliding dynamics is derived as

By combing both the dynamics (40) and (41), the following full ideal sliding dynamics (7) of the proposed integral sliding surface is obtained as

where

The difference of (37) and (42) is due to the truncation error and both are almost same. Using the solution of (37) or (42), the controlled robust output can be predesigned, predetermined, and predicted in advance. If one takes the switching gains as follows:

then

The MIMO existence condition of the sliding mode is satisfied for the every point on the integral sliding surface i.e. for the entire sliding trajectory from a given initial to the origin with the ideal sliding dynamics (37) or (42).

By using the Fortran software, the simulation is carried out under $0.1[m\sec]$ sampling time and with $x(0)=[2 0 0 -1.5 0]^{T}$ initial condition. For comparison, the simulation results of the previous MIMO VSS of ⁽³⁴⁾ are given from Fig. 1to Fig. 4. Fig. 1shows the five state output responses, $x_{1}$ and $x_{2}$ in a top figure, $x_{3}$ in a middle figure, $x_{4}$ and $x_{5}$ in a bottom figure by the continuous control input of the reference ⁽³⁴⁾ for comparison. The two real trajectories and two ideal trajectories are shown in Fig. 2, $x_{1}-x_{2}$ plane trajectories in a upper figure and $x_{4}-x_{5}$ plane trajectories in a lower figure. The two sliding surfaces and two continuous control inputs are depicted in Fig. 3and Fig. 4, respectively. As can be seen, the almost output prediction performance is obtained with the continuous control input.

The simulation results of the proposed MIMO integral SMC are given from Fig. 5to Fig. 12, in which from Fig. 5to Fig. 8the results by the proposed discontinuous input are shown and from Fig. 9to Fig. 12the results by the continuous input are depicted. Fig. 5shows the five real state output responses and the five ideal sliding outputs, $x_{1}$ and $x_{2}$ in a top figure, $x_{3}$ in a middle figure, $x_{4}$ and $x_{5}$ in a bottom figure by the discontinuous control input of the proposed MIMO discontinuous SMC. The ideal sliding outputs are the time solution of the ideal sliding dynamics (37) for the given initial condition. As can be seen, the ideal sliding output and real output are almost identical because the sliding output is defined by proposed (3) without the reaching phase from the given initial condition finally to the origin, the suggested discontinuous control input satisfies the MIMO existence condition of the sliding mode as (46), and the complete strong robustness of the sliding mode is guaranteed for the entire trajectory from the given initial condition to the origin. The robust output can be exhibited as designed in the integral sliding surface and the robust output can be predetermined and predicted. The two real trajectories and two ideal trajectories are shown in Fig. 6, $x_{1}-x_{2}$ plane trajectories in a upper figure and $x_{4}-x_{5}$ plane trajectories in a lower figure. The ideal sliding trajectories are depicted by the values of

\begin{align*} x_{s}(2)=-(s_{1}-c_{12}x(2))/c_{12}\\ =-(s_{1}-1.802x(2))/1.802 \end{align*}

which are obtained from $s=0$. As can be seen, therefore is no the reaching phases. The complete robustness of the output is guaranteed for the entire trajectory and the robust output can be predicted. The two sliding surfaces and two discontinuous control inputs are depicted in Fig. 7and Fig. 8, respectively. As can be seen, the controlled system slides from the initial time without the reaching phase by the discontinuous input. The large chattering of the discontinuous input is shown, which is harmful to real plants. The continuous approximation by the transformed continuous control inputs (24) with the proper layer $l_{-1}=l_{+1}=l_{-2}=l_{+2}=0.008$ are made. Fig. 9shows the five real state output responses and the five ideal sliding outputs, $x_{1}$ and $x_{2}$ in a top figure, $x_{3}$ in a middle figure, $x_{4}$ and $x_{5}$ in a bottom figure with the same performance of Fig. 1. The two real trajectories and two ideal trajectories are shown Fig. 10, $x_{1}-x_{2}$ plane trajectories in a upper figure and $x_{4}-x_{5}$ plane trajectories in a lower figure. The two sliding surfaces and two continuous control inputs are depicted in Fig. 11and Fig. 12, respectively. As can be seen, the large chattering of the two control inputs is much improved. For comparison, Fig. 13shows the five state output responses of the previous MIMO continuous VSS of ⁽³⁴⁾ and the five state output responses of the proposed MIMO continuous SMC, which shows almost the same output performance as that of the previous MIMO continuous VSS⁽³⁴⁾.

4. Conclusion

In this note, MIMO discontinuous and continuous control input transformation SMCs are proposed with the output prediction and predetermination performance for all uncertainties and disturbances as the alternative approaches of ⁽³⁴⁾ with the same performance. The integral sliding surface is suggested for removing the reaching phase and defining the ideal sliding surface from any given initial condition to the origin with the ideal sliding dynamics. The two methods of obtaining the ideal sliding dynamics from any given initial condition to the origin are given. To present the complete formulation of the SMC design for the output prediction and predetermination, the closed loop exponential stability together with the MIMO existence condition of the sliding mode on the selected sliding surface by the proposed transformed discontinuous control input is investigated in Theorem 1 for all matched uncertainties and matched disturbance. By proving the MIMO existence condition of the sliding mode, the complete strong robustness is guaran- teed for the entire trajectory. By using the solution of the ideal sliding dynamics, the real robust output can be predicted, pre-designed, and predetermined. The output(performance) designed in the sliding surface is completely guaranteed in real output for all uncertainties and disturbances as the performance robust- ness that is how can control designer decrease the difference between the design output(performance) and real output(perfor- mance), which is the problem in all the controllers. For practical application of the proposed transformed integral SMC to the continuous control of real plants, the harmful chattering of the discontinuous input is effectively much improved without severe performance loss by means of the fixed modified boundary layer function. The bounded stability together with the MIMO existence condition of the sliding mode by the continuous transformed integral SMC is investigated in Theorem 2. Through a design example and comparative simulation studies, the effectiveness of the proposed discontinuous and continuous transformed integral SMC controller is verified. Theoretically the discontinuous input is considered for the discontinuous control of simulation plants, and practically the continuous integral SMC based on the fixed modified boundary layer diagonal function method can be applied to the continuous control of real plants. The obtained projection ideal sliding trajectory is given.