An Adaptive Optimal Multi-Surface Back-Stepping Sliding Mode Control Design for the Takagi-Sugeno Fuzzy Model of Uncertain Nonlinear System with External Disturbance

This paper proposes a novel scheme of the multi-objective robust control design for a class of uncertain nonlinear systems in strict-feedback-form based on Takagi–Sugeno fuzzy model (TSFM). The nonlinear system contains both the matched and the unmatched uncertainties and also subjected to the external disturbances. The TSFM provides the generalization of the linear systems concepts to the nonlinear systems field in a convex framework. First, a new sliding surface is defined using a convex combination of the surfaces which are defined for each fuzzy rule consequence local linear subsystems. Then, their gains are designed optimally via a generalized eigenvalue problem (GEVP). Also, the upper-bounds of the matched and unmatched uncertainties are estimated using the adaptive update laws. The multi-objective control aims not only to satisfy the H2-optimization performance, but also, α-stability region is formulated to improve the transient response performance. The H2-optimization characterization and α-stability design conditions are derived in terms of new linear matrix inequalities (LMIs) conditions. Finally, the effectiveness of the proposed approach is demonstrated by considering a comparative practical example.


I. INTRODUCTION
Takagi-Sugeno (T-S) fuzzy model with sector nonlinearity approaches is commonly used to describe an exact model of a nonlinear system with uncertainties [1]. The basic idea behind the T-S fuzzy approach is to decompose a non-linear system into several local linear models. Then, each linear model aggregates in a convex structure in terms of the normalized weighting. The linear control theory can be applied to each local linear model. After that, the nonlinear control can be obtained by fuzzy blending. These advantages make the T-S fuzzy model a useful tool to model the complex nonlinear system.
A review of the literature of nonlinear control approaches reveals that the back-stepping mechanism are recognized as a useful method and has been employed in many industrial processes such as drug injection, missiles, AC motors, hybrid electric vehicles, and so on [2][3][4][5][6][7][8]. In general, it is a systematic and recursive Lyapunov-based design technique through state feedback design that can be employed to the nonlinear systems in a strict feedback form. It can be used to solve both of the stabilization and the tracking control problems [4]. The basic idea of a back-stepping design mechanism is to recursively select appropriate functions of state variables as virtual control inputs for lower dimensions subsystems of the overall system. Thus, the final control outputs can be obtained step by step via selecting a suitable Lyapunov function. Although, backstepping control is a powerful nonlinear control method [9,10], but it is not inherently robust against the inevitable undesired phenomena such as uncertainties and external disturbance. To design a robust backstepping control, some robust approaches have been synthesized with this scheme [11,12]. However, these studies were concerned just for special classes of nonlinear systems including, e.g., strict-feedback systems [13], block-strict-feedback systems [14,15] and special practical systems [16,17]. Furthermore, backstepping control does not generate a constrained control input and the previous studies that considered the constrained control are established for a strict class of the systems or special cases [18,19]. Summary of the above, a universal backstepping control in which incorporate the robustness, constraints and optimality, simultaneously for a wide class of the nonlinear systems is not presented up to now. However, the desired tracking performance for the systems with parameters/ functions uncertainty cannot be guaranteed via back-stepping control law design. The system stability also may be destroyed in some cases. In contrast, sliding mode control (SMC) is a powerful technique that make ensures robustness of the desired system performance in presence of uncertainty [20][21][22][23][24][25]. It has received a great deal of attention in study of nonlinear control field such as biological systems, fault-tolerant control systems, robotic systems, space, and so on [26][27][28][29]. In addition, SMC has been utilized to compensate the additional uncertainties of the T-S fuzzy system [30]. In [31], for a class of uncertain discrete systems with packet losses, time delays, and ambient noise, a sliding surface was designed. Their studied system can be considered as a linear system with a matched nonlinear uncertainty which instead of the external disturbance, the ambient noise is studied. As a restriction of the approach in [31], satisfying some of the conditions are necessary, which can reduce the applicability of the proposed scheme. On another side, in [31], during the sliding surface design, they neglect some of the system dynamic terms. To do this, a gain is considered in both the integral sliding mode surface and the sliding mode controller structures to attenuate the noise effects on the dynamics. The approach in [31] may lead to non-optimal sliding surface and then needs to more control effort to reach the sliding surface.. Since, utilizing SMC design requires a priori knowledge of the uncertainties, a new sliding mode back-stepping technique was proposed in [32] by combining sliding mode control design and the adaptive back-stepping technique with tuning functions to make ensure the movement of the tracking error will be along the sliding hyperplane. Besides, internal dynamic stability is a necessity for applying the SMC approach [33]; but, the system's zero dynamics stability seldom has been inspected in the literature.
Recently, to solve the robust performance control design problem, many back-stepping sliding mode controls (BSMCs) schemes through the combination of backstepping control mechanism and SMC design have been developed, while the unmatched uncertainty generally was not considered [21,22,34]. In [22], for a knee exoskeleton used in order to assist human movement, a nonlinear disturbance observer was suggested and combined with a back-stepping sliding mode control. The approach in [22] had been used to reduce the influence of the modeling uncertainty and disturbance. In [21], an observer based backstepping terminal sliding mode controller for three degrees of freedom (DOF) robot manipulator was employed. In [34], using high-order extended state observer for the six DOF quadrotor unmanned aerial vehicle system, a BSMC algorithm was proposed to overcome the problems of the external wind disturbances and etc. Since, in the abovementioned BSMC designs, the control system stability is strictly guaranteed in a step-by-step backstepping algorithm; therefore, the unstable zero dynamics effect on the system can be avoided. However, information about the parameter uncertainty and external disturbance upper bounds must be available to derive the switching gain of the BSMC; but, in most cases, such information cannot be obtained and if one uses a conservative method to make ensure the system stability by selecting a large switching gain, severe chattering may be occurred [35]. In other sides, 2 -characterization can be used in the SMC design to optimize the desired performance [36,37]. In [36], a combination of the 2 -characterization and partial eigenstructure assignment was proposed to obtain the optimal gain of the conventional sliding surface for the linear system with matched uncertainty and external disturbances. In order to improve the buck converter robustness, a different hybrid 2 model following control based on the digital redesign SMC approach was proposed in [37]. In addition, to improve the transient time performance, pole placement has been combined with 2 -characterization problem [38,39]. In [38], an optimal control with 2 -characterization and regional pole constraint had been formulated using static state feedback. The design conditions were derived in terms of the LMIs and was employed on a power system model. In [39], for a small power system to be secured against interarea modes, its controllers were designed with a mixed 2 / ∞ optimization with regional pole placement.
The literature reviews show that the existing works tolerates from some practical issues in their design methods, i.e., poor transient response, practical aspects such as uncertainties and disturbances, and control effort. Thanks to fruitful TSFM representations of complex nonlinear systems, allows generalizing the well-known techniques in linear systems theory to nonlinear systems analysis and design. Transient response improvement for SMC-based nonlinear controller design is an interesting and challenging topic. Convex structure of TSFM representation of nonlinear systems provides us to design linear controller for the consequence subsystems optimally and then aggregate the controllers for the original nonlinear system. Moreover, elaborations of uncertainties and external disturbances, simultaneously, are another challenge topic for considering practical considerations in controller design.
In this paper, TSFM is used to describe a class of nonlinear systems with both the matched and the unmatched uncertainties. In this work, we suppose that the studied TSFM has an unknown bound of uncertainty and is exposed to an external disturbance. Therefore, as a well-known approach in robust control of nonlinear systems with bounded norm uncertainty, SMC has been utilized in this paper. Besides, due to the weakness of the SMC attenuation of the unmatched uncertainty as well as the matched uncertainty, by proposing new sliding surfaces to deal with both of the matched and the unmatched uncertainties, a new scheme of the SMC design based on the back-stepping procedure is proposed. Also, some adaptive laws are derived to estimate the bound of uncertainties. This paper aims to reduce the undeniable chattering phenomena in high frequency and tries to attenuate the external disturbance. In addition, in contrast to the previous works, we design an optimal sliding gain by considering all of the system specification, system limitation, and the required control effort to reach the desired transient time performance. To do this, first, the sliding surface and the optimal control input are defined such that they depend on the desired optimal state feedback gains. The optimal state feedback gains are designed such that the 2 -performance specification and α-stability condition are satisfied. Hence, in this paper to overcome the SMC weakness, a new robust control scheme is proposed to design an optimal dynamic BSMC for the nonlinear system described by TSFM. The main advantage and contribution of our proposed approach can be recounted as following: 1-An optimal and adaptive BSMC scheme is designed for a nonlinear system with all kinds of uncertainties and subjected to the external disturbance. To do this, we consider the 2 -performance problem specification and α-stability constraint during the sliding surface design. In addition, we propose a novel virtual control law to improve the robust performance of the nonlinear system. 2-The SMC design approaches usually suffer from knowing a prior knowledge about the bounds of the uncertainties. These bounds of the uncertainties are not available in many practical systems; therefore, to solve this, we derived some adaptive laws to estimate the unknown bounds of the uncertainties. 3-In this paper, we use the hyperbolic function instead of the signum function to reduce the chattering phenomena; i.e., ℎ(•). This paper is organized as follows: In Section II, a class of nonlinear system with the matched and the unmatched uncertainties based on TSFM description is introduced. In Section III, a novel sliding mode controller is proposed to be designed based on the back-stepping approach. In Section IV, first, new LMIs characterization for the 2 -problem and the -stability constraint will be introduced to find optimal state feedback gain. Then, in the framework of the convex optimization problem, the optimal matrix gains of the sliding surface are designed. In Section V, to show the effectiveness of the proposed method, a two-link robot manipulator system is considered in a comparative example and the paper conclusion and future works are drawn in Section VI.
In this paper to handle both the matched and the unmatched uncertainties, first, the system states (1) are decoupled into the two subsystems as follows: (3), by considering the state variable 2 ( ) as a virtual control input, the matched uncertainty 1 ( ) can be handled by SMC approaches. Using system decomposition, the second subsystem is presented as follows: where the matched uncertainty 2 ( ) can be handled by selecting an appropriate control input. We will aim to design a BSMC for dynamics (3)-(4) to achieve transient response improvement optimally and providing acceptable attenuation level of the effects of the uncertainties and external disturbances. Moreover, practical issues are considered in the control effort design. Therefore, a multi-objective controller design is designed based on H 2 -performance characterization and α-stability methods in the BSMC framework.

III. BACKSTEPPTING-BASED SLIDING MODE CONTROLLER DESIGN
In this section, a new approach to design SMC for the nonlinear system (1) is proposed. Since ( ) is the measured output vector, consider ( ) as the desired output vector, the output tracking error can be defined as 1 ( ) = 1 ( ) − ( ). Hence, the first and the second time-derivatives of 1 ( ) are obtained as follows: Define 2 : =̇1, the tracking error dynamics (5)-(6) can be rewritten as ̇1 = 2 (7) ̇2 = ∑ ℎ ( ) =1 { , ( ) + 1 , ( ( ))} + 2 ( ) −̈ (8) Now, a sliding mode controller for the tracking error dynamic (7)-(8) will be designed based on the back-stepping approach in two steps. In the first step, a virtual controller is designed for (7). After that, in the second step, a controller is designed for (8), and consequently, for the overall output tracking error system.

IV. OPTIMAL SLIDING MODE CONTROLLER DESIGN
In this section, for the nonlinear system (1) which can be exposed to an external disturbance, an optimal controller is designed. To do this, first, we consider the TSFM (1) subjected to the external disturbance ( ). Therefore, we design a multi-objective state feedback control by incorporating of (i) the regional pole placement constraint, and (ii) the 2 -characterization problem both for the consequence linear local subsystems. The 2 optimization provides the minimization of the effect of the disturbance upon the 2 -performance output vector , while the additional uncertainties are neglected, i.e., 1 ( )=0 and 2 ( ) = 0. Therefore, for the TSFM (1) subjected to the external disturbance, we design the adaptive and optimal BSMC gains in the control law (28).
The T-S fuzzy model (1), is rewritten as ], and ∈ ℝ denotes the external disturbance input, while its input matrix gain 2 has an appropriate dimension, and To design the multi-objective optimal BSMC control, consider the TSFM (29) with no additional uncertainty as In lemma 2, for the TSFM (30), using LMI characterization, an 2 performance from the disturbance input w to the output is considered.
Lemma 2: For the nonlinear system (30), the following statements are equivalent: a) ∃ , = 1, … , , such that + 1, is stable and the 2 performance from the disturbance w to the output is less than .
∎ To design the multi-objective state feedback control and to achieve the transient performance improvement and disturbance attenuation objects concurrently, the following generalized eigenvalue problem (GEVP) is proposed: (38), (39) (50) (30).