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This paper generalized a new sliding mode control (SMC) without reaching phase to solve two important problems in the stability of complex interconnected systems: (1) a decentralized controller that uses only output variables directly and (2) the stability of complex interconnected systems ensured for all time. A new sliding surface is firstly designed to construct a single-phase SMC in which the desired motion is determined from the initial time instant. A new lemma is secondly established for the controller design using only output variables. The proposed single-phase SMC and the decentralized output feedback controller ensure the robust stability of complex interconnected systems from the beginning to the end. One of the key features of the single phase SMC scheme is that reaching time, which is required in most of the existing two phases of SMC approaches to stabilize the interconnected systems, is removed. Finally, a numerical example is used to demonstrate the efficacy of the method.

The theory of sliding mode control (SMC) is known to be an effective robust control technique and has been successfully applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, electrical motors, power systems, and automotive engines [

Unfortunately, the applications of two phases SMC for the stability of complex interconnected system have some drawbacks. Firstly, the system stability is not ensured for all time because the motion equation in sliding mode is determined after the system state hits the sliding surface [

In order to solve the above problems, first we develop a new SMC such that the reaching time is equal to zero and the desired motion is determined from the beginning time. Second, appropriate LMI stability conditions by the Lyapunov method are derived to guarantee the stability of the system. Third, a new lemma is established for controller design using only output variables directly. Consequently, the stability of complex interconnected systems driven by single-phase SMC law can be ensured throughout an entire response of the system starting from the initial time instance. Before demonstrating the advantages of the application of single-phase SMC to complex interconnected systems, one wants to point out some previous results about the stability analysis of uncertain systems.

The design of SMC without reaching phase can be found in [

Thus, the approaches in [

However, it is worth to point out that there are some limitations in the existing design methods of SMC in application for the stability of interconnected systems. First, the approaches proposed in [

This study therefore developed a new single-phase SMC for robust stability of a class of complex interconnected systems from beginning to end. First, a new sliding surface is designed to construct the single-phase SMC which the desired motion is determined from the initial time instant. Second, appropriate LMI stability conditions by the Lyapunov method are derived to guarantee the stability of the system. Third, a new lemma is established for controller design using only output variables. Fourth, a decentralized output feedback controller is designed to force the system states to stay on the sliding surface for all time. Unlike the existing related works such as [

Design of a new sliding surface to construct a single-phase SMC such that the desired motion is determined from the initial time instant.

Derivation of appropriate LMI stability conditions by the Lyapunov method to guarantee the stability of the system.

Establishment of a lemma for controller design using only output variables.

Development of a new approach (single-phase SMC and decentralized output feedback controller) guarantees that sliding mode exists from the initial time instant and the closed loop of the complex interconnected systems in sliding mode is asymptotically stable.

In this paper, we consider a class of complex interconnected systems with exogenous disturbance and mismatched uncertainties of each isolated subsystem and interconnection. The system is decomposed into

In order to modify the existing two phases SMC, we denote the sliding surface by

A sliding mode control is said to be a single-phase SMC, if and only if the following two conditions are satisfied:

the reaching time is equal to zero;

the order of the motion equation in sliding mode is equal to the order of the original system.

The concept of single-phase sliding mode control focusses on the robustness of the motion in the entire state space. The order of the motion equation in sliding mode is equal to the dimension of the state space. Therefore, the robustness of complex interconnected systems can be assured throughout an entire response of the system starting from the initial time instance.

In order to apply the concept of single-phase SMC for the system (

The mismatched parameter uncertainties in the state matrix of each isolated subsystem are satisfied as

The matrices

From [

The triple

From [

There exist known nonnegative constants

The mismatched interconnections are given as

Assumptions

In this section, we develop a single-phase SMC to stabilize the complex interconnected system (

Let us first design a new sliding surface without reaching phase that uses only output variables and the desired motion is determined from the initial time instant. Under Assumptions

It is obvious that

This approach concentrates on the robustness of the motion in the entire state space. The order of the motion equation in sliding mode is equal to the order of the original system. Therefore, the robustness of the system can be assured throughout an entire response of the system starting from the initial time instance.

Following design of the sliding surface, two tasks remain. First, for stability analysis, appropriate LMI stability conditions by the Lyapunov method must be derived to ensure the stability of sliding motion (

This section focuses on the former task. We begin by considering the following LMI:

We also recall the following lemmas, which will be used in proving the stability of sliding motion (

Let

Let

The linear matrix inequality:

Then, we can establish the following theorem.

Suppose that LMI (

Let us consider the following positive definition function:

Theorem

It is seen that, compared to the the recent LMI methods [

In order to design a new output feedback sliding mode control scheme for complex interconnected system (

Consider a class of interconnected systems that is decomposed into

We are now in the position to prove Lemma

It is obvious that the time function

In the last section, we proved that the sliding motion (

Now, we can establish the following theorem.

Suppose that LMI (

Now, we are going to prove Theorem

From sliding mode control theory, Theorems

Unlike the existing related work such as [

In contrast to other SMC approaches such as those presented in [

It is obvious that this approach uses the output information completely in the sliding surface and controller design. Therefore, conservatism is reduced and robustness is enhanced.

To verify the effectiveness of the proposed decentralized output feedback SMC law, we apply our single-phase SMC to a mismatched uncertain interconnected system composed of two third-order subsystems, which is modified from [

For this work, the following parameters are given as follows:

From Theorem

In the example above, the mismatched uncertainties in the state matrix of the systems (

Time responses of states

Time responses of states

Sliding surface

Sliding surface

Control input

Control input

In this paper, a single-phase SMC law is presented for decentralized robust stability of complex interconnected systems from the beginning to the end. It is proved that the proposed single-phase SMC guaranteed the robustness of complex interconnected system throughout an entire response of the system starting from the initial time instance. One of the key features of the single-phase SMC scheme is that reaching time, which is required in most of the existing two phases of SMC approaches to stabilize the complex interconnected systems, is removed. As a consequence, the proposed single-phase SMC law can be applied to complex interconnected systems, which is not always achievable in the traditional SMC design for complex interconnected systems using only output variables.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the financial support provided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3).

^{∞}control theory