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This paper proposes new methodologies for the design of adaptive integral-sliding mode control. A tensor product model transformation based adaptive integral-sliding mode control law with respect to uncertainties and perturbations is studied, while upper bounds on the perturbations and uncertainties are assumed to be unknown. The advantage of proposed controllers consists in having a dynamical adaptive control gain to establish a sliding mode right at the beginning of the process. Gain dynamics ensure a reasonable adaptive gain with respect to the uncertainties. Finally, efficacy of the proposed controller is verified by simulations on an uncertain nonlinear system model.

Sliding mode control (SMC) has advantages like ease of implementation and reduction in the order of the state equation. It also has the ability to withstand external disturbances and model uncertainties satisfying the matching condition, that is, the perturbations that enter the state equation at the same point as the control input. SMC design comprises two steps: a sliding surface is first constructed such that the system trajectories along the sliding surface meet the specified performance and then a discontinuous or continuous control law drives the states towards the sliding surface, and keeps them on the surface thereafter, regardless of disturbances or parasitic uncertainties. The resulting controller, although robust against matched perturbations, still suffers from reaching phase problem, that is, an initial period of time in which the system has not yet reached the sliding surface and it is sensitive to perturbations satisfying the matching condition.

In order to solve the reaching phase problem, an integral-sliding surface was proposed [

TP model transformation was first introduced by Baranyi et al. [

Inspired by the adaptive sliding mode controller [

The rest of the paper is organized as follows. In Section

It is known that a given bounded continuous function

All-orthogonality:

Ordering:

Consider a class of

The nominal parts

The matrix product

The actual values of

The known nominal nonlinear plant of (

The following nonlinear integral-type sliding surface [

The reaching phase is eliminated for nonlinear integral-type sliding surface defined by (

The adaptive sliding mode control law using the integral-sliding surface (

The polytopic model (

Assume that

The aforementioned control algorithm can be summarized as follows.

Consider the uncertain nonlinear system (

Assume that there exists a final gain

Briefly, the proposed TPAISMC scheme provides the following three main advantages. First, the knowledge of the upper bound of the system uncertainties is not required. Second, TP model transformation has the ability to provide a tradeoff between approximation accuracy and complexity of the resulting TP function. If an exact transformation is impossible, approximation error is bounded by discarded singular values, then this error can be incorporated into lumped uncertainty

The above adaptive method does not require knowledge of uncertainties or perturbations bound. Adaptive gain

Consider the uncertain nonlinear system (

Consider the following nonlinear system:

(a) States responses with TPASMC. (b) Sliding surface function with TPASMC.

(a) Control input with TPASMC. (b) Adaptive gain parameter.

Figure

(a) States responses with TPAISMC. (b) Sliding surface function with TPAISMC.

(a) Control input with TPASMC. (b) Adaptive gain parameter.

To avoid large adaptive gain and chattering, we apply the adaptation law (

(a) States responses with TPAISMC (

(a) Sliding surface function with TPAISMC (

In this work, an adaptive siding mode controller scheme for a class of uncertain nonlinear system is studied, and an integral-sliding surface was adopted in the designing of TPAISMC. Instead of using the traditional feedback control, TP model transformation based parallel distributed compensation controller was applied to stabilize the states. It is noted that the modeling error caused by TP model transformation can be zero if nonzero singular value is discarded. Nonzero modeling error can be modeled as lumped uncertainties, combining modeling error with perturbations and external disturbances, and can be suppressed effectively by adaptive gain controller with

This work was supported by the National Natural Science Foundation of China (no. 11101066; no. 61074044; no. 61374118; no. 61104038; no. 60834004) and the Fundamental Research Funds for the Central Universities (no. DUT13LK32), and partially supported by the Chinese National Basic Research 973 Program (2009CB320602).