Sliding Mode Control for a Class of Uncertain MIMO Nonlinear Systems with Application to Near-Space Vehicles

Abstract

We propose a robust sliding mode control (SMC) scheme for a class of uncertain multi-input and multi-output (MIMO) nonlinear systems with the unknown external disturbance, the system uncertainty, and the backlash-like hysteresis. To tackle the continuous system uncertainty, the radial basis function (RBF) neural network is employed to approximate it. And then, combine the unknown external disturbance, and the unknown neural network approximation error with the affection caused by backlash-like hysteresis as a compounded disturbance which is estimated using the developed nonlinear disturbance observer. The robust sliding mode control based on the nonlinear disturbance observer and RBF neural network is presented to track the desired system output in the presence of the unknown system uncertainty, the external disturbance, and the backlash-like hysteresis. Finally, the designed robust sliding mode control strategy is applied to the near-space vehicle (NSV) attitude dynamics, and simulation results are given to illustrate the effectiveness of the proposed sliding mode control approach.

1. Introduction

Recently, the robust adaptive control has been extensively studied for uncertain MIMO nonlinear systems due to most control plants with multichannels, system uncertainties, and unknown external disturbances [1–10]. Sliding mode control is one of the most important approaches which is particularly suited to the deterministic control of systems with large uncertainties, nonlinearities, and bounded external disturbances [11–16]. However, it is usually difficult to directly extend the SMC design to the uncertain MIMO nonlinear system, and a few research results are directly available. In [17], dynamic SMC and higher-order sliding mode were studied for MIMO systems. Generalized SMC was proposed for multi-input nonlinear systems in [18]. In [19], robust MIMO water level control was developed for interconnected twin-tanks using second-order SMC. In general, the robust SMC needs to be further developed for the uncertain MIMO nonlinear system.

Usually, the unknown system uncertainty and the unknown external disturbance will increase the SMC design difficulty of uncertain MIMO nonlinear systems. In many existing works, universal function approximators (such as fuzzy systems and neural networks) are employed to tackle the unknown system uncertainty in nonlinear systems. On the basis of the output of universal function approximators, lots of robust adaptive control schemes were designed for the uncertain MIMO nonlinear system [20–24]. In [17], robust adaptive sliding mode control was proposed using fuzzy modelling for a class of uncertain MIMO nonlinear systems. Fuzzy adaptive sliding-mode control was presented for MIMO nonlinear systems in [25]. In [26], robust adaptive neural network control was studied for a class of uncertain MIMO nonlinear systems with input nonlinearities using the variable structure control technique. In this paper, the RBF neural network is introduced to handle the unknown system uncertainty. However, the unknown time-varying external disturbance of nonlinear systems cannot be efficiently handled via neural networks and the unknown approximation error of the neural network is hard to be tackled. To improve the antidisturbance ability of control systems, the disturbance-observer-based control strategy would provide a very promising approach where a disturbance observer is adopted to estimate unknown external disturbances.

In the recent decade, many disturbance observer design techniques have been developed to fully utilize the information of external disturbances [27, 28]. The general framework for disturbance-observer-based control (DOBC) of nonlinear systems subject to disturbances was presented in [29]. In [30], the direct adaptive neural control was developed for a class of uncertain nonaffine nonlinear systems based on disturbance observer. Nonlinear predictive control was proposed via using disturbance observers in [31]. In [32], a nonlinear disturbance observer was proposed for multivariable minimum-phase systems with arbitrary relative degrees. A nonlinear disturbance observer was developed for robotic manipulators in [33]. For uncertain structural systems [34] and nonlinear systems [35] with disturbances, disturbance-observer-based control using terminal sliding mode technique was studied. The disturbance attenuation and rejection problem was investigated for a class of MIMO nonlinear systems in the DOBC framework in [36]. Combined with disturbance observers, the robust synchronization control was proposed for uncertain chaotic systems in [37, 38]. In [39], robust DOBC was presented for time-delay uncertain systems. However, the disturbance observers need to be further developed for the robust sliding mode control of uncertain MIMO systems with backlash-like hysteresis.

Hysteresis as a special input nonlinearity exists in a wide range of practical systems, such as biology optics, mechanical actuators, electromagnetism, and electronic relay circuits [40]. The existence of the hysteresis is challenging problem for the control of uncertain nonlinear systems. For backlash hysteresis, several adaptive control schemes have recently been proposed. In [41], robust adaptive control was developed for a class of nonlinear systems with unknown backlash-like hysteresis. Decentralized adaptive stabilization was developed for the interconnected systems in the presence of unknown backlash-like hysteresis in [42]. In [43], identification method was studied for Hammerstein systems in presence of hysteresis-backlash and hysteresis-relay nonlinearities. Stable adaptive fuzzy control was proposed for nonlinear systems preceded by unknown backlash-like hysteresis in [44]. However, the SMC scheme should be further investigated for MIMO nonlinear systems with the unknown external disturbance, the system uncertainty, and the backlash-like hysteresis.

To show the effectiveness of the proposed sliding mode control approach, it is used to design the robust attitude control law for the NSV. Due to the potential military and civilian dual-use value, the near space has caused much concern around the world in recent years, and the flight control of NSVs becomes a hot research topic [45, 46]. In [47], robust attitude control was developed for NSVs with time-varying disturbances. Adaptive functional link network control was studied for the NSV with dynamical uncertainties in [48]. In [49], adaptive fault-tolerant tracking control was proposed for the NSV using Takagi-Sugeno fuzzy models. Fault tolerant control was presented for a class of nonlinear systems with application to NSVs in [50]. However, the robust sliding mode attitude control needs to be further developed for the NSV with the backlash-like hysteresis.

This work is motivated by the robust sliding mode control of uncertain MIMO nonlinear systems with the unknown external disturbance, the system uncertainty, and the backlash-like hysteresis. The organization of the paper is as follows. Section 2 details the problem formulation. The robust sliding mode control based on the disturbance observer is proposed in Section 3. Simulation results of near space vehicle attitude dynamics are presented in Section 4 to demonstrate the effectiveness of the developed sliding mode control of the uncertain MIMO nonlinear systems, followed by some concluding remarks in Section 5.

2. Problem Statement

To develop the robust sliding mode control, consider the following uncertain MIMO nonlinear system in the form of 𝑥̇=𝐹(𝑥)+𝐺(𝑥)𝑢+𝜌(𝑥)+𝑑(𝑡)𝑦=𝑥, (1) where 𝑥∈𝑅^𝑛×1 is the state vector of the uncertain nonlinear system, 𝑦∈𝑅^𝑛×1 is the system output, and 𝑢∈𝑅^𝑚×1 is the control input vector. 𝐹(𝑥)∈𝑅^𝑛×1 is the known function vector, 𝐺(𝑥)∈𝑅^𝑛×𝑚 is the known control gain matrix, 𝜌(𝑥)∈𝑅^𝑛×1 is the continuous unknown system uncertainty, and 𝑑(𝑡)∈𝑅^𝑛×1 is the unknown time-varying external disturbance.

Usually, the actuator of the practical control system has a backlash-like hysteresis nonlinearity, and 𝑢_𝑖 is the output of such hysteresis described as [41]: 𝑢_𝑖(𝑡)=𝑃𝑣_𝑖, (2) where 𝑣_𝑖, 𝑖=1,2,…,𝑚, is the input of the hysteresis for the 𝑖th actuator and 𝑃(⋅) denotes a backlash hysteresis operator.

In this paper, we consider a class of hysteresis which is expressed as the following continuous-time dynamic model [41]: 𝑑𝑢_𝑖𝑑𝑡=𝑎_𝑖𝑑𝑣_𝑖𝑏𝑑𝑡^𝑖𝑣_𝑖−𝑢_𝑖+𝑐_𝑖𝑑𝑣_𝑖,𝑑𝑡 (3) where 𝑎_𝑖, 𝑏_𝑖, and 𝑐_𝑖 are known constants with 𝑏_𝑖>𝑐_𝑖.

Considering the solution properties of the dynamic model (3), it can be solved explicitly for piecewise monotone as [41]: 𝑢_𝑖=𝑏_𝑖𝑣_𝑖+𝑑_𝑖0𝑣_𝑖, (4) where 𝑑_𝑖0(𝑣_𝑖) is defined as 𝑑_𝑖0𝑣_𝑖=𝑢_𝑖0−𝑐_𝑖𝑣_𝑖0𝑒^{−𝑎𝑖(𝑣𝑖−𝑣𝑖0̇𝑣)sgn𝑖}+𝑒^{−𝑎𝑖𝑣𝑖̇𝑣sgn𝑖}+^𝑣𝑖_𝑣𝑖0𝑐_𝑖−𝑏_𝑖𝑒^{𝑎𝑖𝜉𝑖̇𝑣(sgn𝑖)}𝑑𝜉_𝑖. (5) Considering (4), the uncertain MIMO nonlinear system (1) can be rewritten as 𝑥̇=𝐹(𝑥)+𝐺(𝑥)𝐵𝑣+𝐺(𝑥)𝐷₀(𝑡)+𝜌(𝑥)+𝑑(𝑡)𝑦=𝑥, (6) where 𝐵=diag{𝑏₁,𝑏₂,…,𝑏_𝑚}, 𝜈=[𝑣₁,𝑣₂,…,𝑣_𝑚], and 𝐷₀(𝑡)=[𝑑₁₀,𝑑₂₀,…,𝑑_𝑚0]^𝑇.

Define ℎ(𝑥)=𝐿𝜌(𝑥) with the matrix 𝐿=𝐿^𝑇>0. Then, (6) can be written as 𝑥̇=𝐹(𝑥)+𝐺(𝑥)𝐵𝑣+𝐺(𝑥)𝐷₀(𝑡)+𝐿⁻¹ℎ(𝑥)+𝑑(𝑡)𝑦=𝑥. (7) Since the unknown function vector 𝜌(𝑥), the function vector ℎ(𝑥) is also unknown. To efficiently handle the unknown ℎ(𝑥), the RBF neural network is employed to approximate it. The approximation output of the RBF neural network can be written as ℎ(𝑥)=𝑊^∗𝑇𝜙(𝑥)+𝜀^∗, (8) where 𝑊^∗∈𝑅^𝑝×𝑛 is a optimal weight value vector of the RBF neural network, 𝜙(𝑥)∈𝑅^𝑝 is the basis function and 𝜙_𝑖(𝑥)=exp[−(𝑥−𝑐_𝑟𝑖)^𝑇(𝑥−𝑐_𝑟𝑖)/𝑏²_𝑟𝑖],𝑖=1,2,…,𝑝, and 𝑐_𝑟𝑖 and 𝑏_𝑟𝑖 are the center and width of the radial basis function, respectively. 𝜀^∗ is the smallest approximation of the RBF neural network.

Substituting (8) into (7) yields 𝑥̇=𝐹(𝑥)+𝐺(𝑥)𝐵𝑣+𝐺(𝑥)𝐷₀(𝑡)+𝐿⁻¹𝑊^∗𝑇𝜙(𝑥)+𝐿⁻¹𝜀^∗+𝑑(𝑡). (9) Define 𝐷(𝑥,𝑡)=𝐺(𝑥)𝐷₀(𝑡)+𝐿⁻¹𝜀^∗+𝑑(𝑡). Then, (9) can be written as 𝑥̇=𝐹(𝑥)+𝐺(𝑥)𝐵𝑣+𝐿⁻¹𝑊^∗𝑇𝜙(𝑥)+𝐷(𝑥,𝑡). (10) Define 𝑒=𝑦−𝑦_𝑑 and 𝑦_𝑑 is the reference tracking signal of the uncertain MIMO nonlinear system. Considering (10), we obtain 𝑒̇=𝐹(𝑥)+𝐺(𝑥)𝐵𝑣+𝐿⁻¹𝑊^∗𝑇𝜙(𝑥)+𝐷(𝑥,𝑡)−𝑦̇_𝑑. (11) In this paper, the control objective is that the robust sliding mode control is designed to follow a given desired output 𝑦_𝑑 of the uncertain MIMO nonlinear system in the presence of the unknown time-varying external disturbance and the backlash-like hysteresis. For the desired system output 𝑦_𝑑, the robust sliding mode control scheme is proposed such that all closed-loop signals are uniformly asymptotically convergent. Furthermore, the developed robust sliding mode control strategy is applied to NSV attitude dynamics to illustrate its effectiveness.

To proceed the design of robust sliding mode control scheme for the uncertain MIMO nonlinear system (1), the following assumptions are required:

Assumption 1 (see [51]). For all 𝑡>0, there exist known constants Δ₀ and Δ₁ such that ‖𝑦_𝑑(𝑡)‖≤Δ₀ and ‖𝑦̇_𝑑(𝑡)‖≤Δ₁.

Assumption 2. For the time-varying unknown compounded disturbance 𝐷(𝑥,𝑡), there exists an unknown positive constant 𝛿₀ such that ‖𝐷̇(𝑥,𝑡)‖≤𝛿₀‖𝑥‖.

3. Robust Sliding Mode Control Based on Disturbance Observer

In this section, we consider the robust sliding mode control design for the uncertain MIMO nonlinear system (1). Since 𝐷(𝑥,𝑡) is unknown, it cannot be directly used to design the robust sliding mode controller. To efficiently handle it, the nonlinear disturbance observer is proposed to estimate it.

The nonlinear disturbance observer is proposed as ()̇(𝑥)(𝑥)𝐷=𝐿𝑒−𝜉𝜉=𝐹+𝐺𝐵𝑣+𝐿⁻¹𝑊^𝑇𝜙(𝑥)+𝐷−𝑦̇_𝑑, (12) where 𝐿=𝐿^𝑇>0 is a design parameter of the nonlinear disturbance observer and 𝑊 is the estimate of the optimal weight value 𝑊^∗.

Considering (11) and (12), we obtain 𝐷̇=𝐿𝑒̇−𝜉̇=𝐿𝐷(𝑡)−𝐷(𝑡)+𝑊^∗𝑇𝜙(𝑥)−𝑊^𝑇𝜙(𝑥). (13) Define 𝐷=𝐷−𝐷 and 𝑊=𝑊^∗−𝑊. Considering (13) yields ̇̇̇𝐷=̇𝐷(𝑡)−𝐷(𝑡)−𝑊𝐷=𝐷−𝐷−𝐿_∗𝑇𝜙(𝑥)−𝑊^𝑇𝜙(𝑥)=𝐷̇−𝐿𝐷−𝑊^𝑇𝜙(𝑥). (14) Invoking (14), we obtain 𝐷^𝑇̇𝐷=𝐷^𝑇𝐷̇−𝐷^𝑇𝐿𝐷−𝐷^𝑇𝑊^𝑇𝜙(𝑥). (15) Considering Assumption 2 and the following fact −2𝐷^𝑇𝑊^𝑇𝜙(𝑥)≤2𝐷𝑊𝜙(𝑥)≤𝜏𝜗²𝐷²+1𝜏𝑊² (16) yields 𝐷^𝑇̇𝐷≤0.5𝑊²+0.5𝐷̇²−𝐷^𝑇𝐿𝐷+𝜏𝜗²𝑊²+1𝜏𝑊²≤−𝐷^𝑇𝐿−0.5+𝜏𝜗²𝐼_𝑛×𝑛𝐷+𝛿‖𝑥‖²+1𝜏𝑊², (17) where ‖𝜙(𝑥)‖≤𝜗, 𝛿=0.5𝛿²₀ which is unknown due to the unknown constant 𝛿₀ and 𝜏>0 is a design parameter.

Considering Assumption 1, 𝑒=𝑦−𝑦_𝑑 and 𝑦=𝑥 yields ‖𝑥‖=𝑒+𝑦_𝑑≤‖𝑒‖+𝑦_𝑑≤‖𝑒‖+Δ₀. (18) Invoking (17) and (18), we have 𝐷^𝑇̇𝐷≤−𝐷^𝑇𝐿−0.5+𝜏𝜗²𝐼_𝑛×𝑛𝐷+𝛿‖𝑒‖+Δ₀²+1𝜏𝑊²=−𝐷^𝑇𝐿−0.5+𝜏𝜗²𝐼_𝑛×𝑛𝐷+𝛿‖𝑒‖²+2𝛿Δ₀‖𝑒‖+𝛿Δ²₀+1𝜏𝑊². (19) So far, the nonlinear disturbance observer design has been completed. In the following, the robust sliding mode control scheme will be proposed. Let the sliding surface be 𝜎=Λ𝑒, (20) where Λ=Λ^𝑇>0 is a design matrix.

The sliding condition becomes 𝜎^𝑇𝜎̇=𝜎^𝑇Λ𝑒.̇ (21) Invoking (11), we have 𝜎^𝑇𝜎̇=𝜎^𝑇Λ𝐹(𝑥)+𝐺(𝑥)𝐵𝑣+𝐿⁻¹𝑊^∗𝑇𝜙(𝑥)+𝐷(𝑥,𝑡)−𝑦̇_𝑑. (22) In accordance with the definitions of matrices 𝐵 and Λ, using the output of the designed nonlinear disturbance observer, the robust sliding control law is proposed as 𝑣=−𝐴^†Γ𝜎+Λ𝐹(𝑥)+Λ𝐿⁻¹𝑊^𝑇𝜙(𝑥)+Λ𝐷−Λ𝑦̇_𝑑+𝐾sign(𝜎)+𝑣_𝑐,𝑣_𝑐=𝛿‖𝑒‖²+2Δ₀𝛿‖𝑒‖𝜎‖𝜎‖²,‖𝜎‖‖𝜎‖>𝜖0,≤𝜖, (23) where 𝐴^†=𝐵^𝑇𝐺(𝑥)^𝑇Λ(Λ𝐺(𝑥)𝐵𝐵^𝑇𝐺(𝑥)^𝑇Λ)⁻¹, 𝐾=diag{𝑘_𝑖} is a design matrix with 𝑘_𝑖>0, sign(𝜎)=[sign(𝜎₁),…,sign(𝜎_𝑚)]^𝑇,  𝜖>0 is a small design parameter, and 𝛿 is the estimate of the unknown constant 𝛿.

Substituting (23) into (22) yields 𝜎^𝑇𝜎̇=−𝜎^𝑇Γ𝜎+𝜎^𝑇Λ𝐿⁻¹×𝑊^∗𝑇𝜙(𝑥)−𝑊^𝑇𝜙(𝑥)+𝜎^𝑇Λ𝐷(𝑥,𝑡)−𝐷(𝑥,𝑡)−𝜎^𝑇𝐾sign(𝜎)−𝛿‖𝑒‖²−2Δ₀𝛿‖𝑒‖=−𝜎^𝑇Γ𝜎+𝜎^𝑇Λ𝐿⁻¹𝑊^𝑇𝜙(𝑥)+𝜎^𝑇Λ𝐷−𝜎^𝑇𝐾sign(𝜎)−𝛿‖𝑒‖²−2Δ₀𝛿‖𝑒‖≤−𝜎^𝑇Γ−0.5𝐼_𝑛×𝑛𝜎+𝜎^𝑇Λ𝐿⁻¹𝑊^𝑇𝜙(𝑥)+0.5‖Λ‖²𝐷²−𝛿‖𝑒‖²−2Δ₀𝛿‖𝑒‖. (24) Consider the adaptive law for 𝑊 as 𝑊̇=𝑇𝜙(𝑥)𝐿⁻¹Λ𝜎+𝜆₀𝑊, (25) where 𝑇=𝑇^𝑇>0 and 𝜆₀>0.

The parameter updated law of 𝛿 is designed as ̇𝛿=𝛾‖𝑒‖²+2Δ₀‖𝑒‖+𝜆₁𝛿, (26) where 𝛾>0 and 𝜆₁>0.

The above robust sliding mode control design procedure for a class of uncertain MIMO nonlinear systems (1) can be summarized in the following theorem, which includes the result for robust sliding mode control based on the nonlinear disturbance observer of the uncertain MIMO nonlinear system (1) with the unknown external disturbance and the backlash hysteresis.

Theorem 3. Considering the uncertain MIMO nonlinear system (1) with the unknown system uncertainty, the unknown disturbance, and the backlash hysteresis, the disturbance observer is designed as (12) and the parameter updated laws are chosen as (25) and (26). Then, the robust sliding mode control law is proposed as (23). Under the developed robust sliding mode control scheme, the tracking error of the uncertain MIMO nonlinear system (1) is convergent.

Proof. Consider the Lyapunov function candidate 1𝑉=2𝜎^𝑇1𝜎+2𝐷^𝑇1𝐷+2tr𝑊^𝑇𝑇⁻¹𝑊+1𝛿2𝛾², (27) where 𝛿=𝛿−𝛿.
Invoking (19) and (24), the time derivative of 𝑉 is 𝑉̇=𝜎^𝑇𝜎̇+𝐷^𝑇𝐷̇+tr𝑊^𝑇𝑇⁻¹𝑊̇+1𝛾𝛿̇𝛿≤−𝜎^𝑇Γ−0.5𝐼_𝑛×𝑛𝜎+𝜎^𝑇Λ𝐿⁻¹𝑊^𝑇𝜙(𝑥)+0.5‖Λ‖²𝐷²−𝛿‖𝑒‖²−2Δ₀𝛿‖𝑒‖+𝛿Δ²₀−𝐷^𝑇𝐿−0.5+𝜏𝜗²𝐼_𝑛×𝑛𝐷+𝛿‖𝑒‖²+2𝛿Δ₀‖𝑒‖+1𝜏𝑊²+tr𝑊^𝑇𝑇⁻¹̇𝑊+1𝛾𝛿̇𝛿≤−𝜎^𝑇Γ−0.5𝐼_𝑛×𝑛𝜎+𝜎^𝑇Λ𝐿⁻¹𝑊^𝑇𝜙(𝑥)−𝐷^𝑇𝐿−0.5+𝜏𝜗²+0.5‖Λ‖²𝐼_𝑛×𝑛𝐷+1𝜏𝑊²+tr𝑊^𝑇𝑇⁻¹̇𝑊+‖𝑒‖²𝛿+2Δ₀‖𝑒‖1𝛿+𝛾𝛿̇𝛿+𝛿Δ²₀. (28) Since 𝑊=𝑊^∗−𝑊 and 𝛿=𝛿−𝛿, invoking the weight value adaptation law (25) and the parameter updated law (26), (28) can be written as 𝑉̇≤−𝜎^𝑇Γ−0.5𝐼_𝑛×𝑛𝜎−𝐷^𝑇𝐿−0.5+𝜏𝜗²+0.5||Λ||²𝐼_𝑛×𝑛𝐷+1𝜏𝑊²−𝜆₀tr𝑊^𝑇𝑊−𝜆₁𝛿𝛿+𝛿Δ²₀. (29) Considering the following facts 2tr𝑊^𝑇𝑊=𝑊²+𝑊²−𝑊^∗2≥𝑊²−𝑊^∗2,2𝛿𝛿=𝛿²+𝛿−𝛿²≥𝛿²−𝛿², (30) we obtain 𝑉̇≤−𝜎^𝑇Γ−0.5𝐼_𝑛×𝑛𝜎−𝐷^𝑇×𝐿−0.5+𝜏𝜗²+0.5||Λ||²𝐼_𝑛×𝑛𝐷−𝜆₀2−1𝜏𝑊²−𝜆₁2𝛿²+𝜆₀2×𝑊^∗2+𝜆₁2𝛿²+𝛿Δ²₀≤−𝜅𝑉+𝐶, (31) where 𝜅∶=min𝜆_minΓ−0.5𝐼_𝑛×𝑛,𝜆_min𝐿−0.5+𝜏𝜗²+0.5‖Λ‖²𝐼_𝑛×𝑛,2𝜆₀/2−1/𝜏𝜆_max𝑇⁻¹,2𝜆₁𝛾,𝐶∶=𝛿Δ²₀+𝜆₀2𝑊^∗2+𝜆₁2𝛿². (32) To ensure the closed-loop system stability, the corresponding design matrices Γ, 𝐿, 𝜏, Λ, and 𝜆₀ should be chosen to make Γ−0.5𝐼_𝑛×𝑛>0, 𝐿−(0.5+𝜏𝜗²)𝐼_𝑛×𝑛−0.5||Λ||²𝐼_𝑛×𝑛>0, and 𝜆₀/2−1/𝜏>0. Considering (31), it may directly show that the signals 𝜎, 𝐷, 𝑊, and 𝛿 are semiglobally uniformly bounded. According to (31), we have 𝐶0≤𝑉≤𝜅+𝑉(0)−𝐶𝜅𝑒^−𝜅𝑡. (33) From (33), we can know that 𝑉 is exponentially convergent; that is, lim_𝑡→∞𝑉=𝐶/𝜅. Hence, the sliding mode surface 𝜎 and the approximation errors 𝐷, 𝑊, and 𝛿 of the closed-loop system are bounded. From the convergence of the sliding mode surface 𝜎, we can know that the tracking errors 𝑒 is convergent. Namely, the control objective is achieved. This concludes the proof.

Remark 4. To the developed robust sliding mode control strategy, the reach condition of the sliding mode surface is always satisfied. Since the error signals 𝐷, 𝑊, and 𝑒 are convergent, 𝜎^𝑇𝜎̇<0 is held if the design parameter matrix Γ is correctly chosen on the basis of (24). On the other hand, to adjust the dynamic control performance of the developed robust sliding mode control, the design parameter 𝜖 in the control term 𝑣_𝑐 should be chosen as a small positive constant.

Remark 5. In this paper, combining the external time-varying disturbance with the effect of approximation error of backlash hysteresis and the NN approximation error is treated as a compounded disturbance which is estimated via the nonlinear disturbance observer. For the proposed nonlinear disturbance observer, we know that the estimate error with suitable approximation performance can be obtained via choosing the appropriate disturbance observer gain matrix 𝐿. For example, the estimate error could be decreased by increasing the value of 𝐿. Due to the introduction of the nonlinear disturbance observer, the disturbance rejection ability is enhanced by the closed-loop control system under the proposed robust sliding mode control approach.

4. Simulation Study

In this section, the designed robust sliding mode control strategy is applied to NSV attitude dynamics, and simulation results are given to illustrate the effectiveness of the proposed sliding mode control scheme. The considered attitude control model of the NSV is derived from the six-degree-of-freedom and twelve-state kinematic equations which can be simplified as the affine nonlinear equation as follows [46]: Ω̇=𝐹_𝑠(𝑥)+𝐺_𝑠𝜔_𝑐+Δ𝐹_𝑠𝑦_𝑠̇=Ω𝜔=𝐹_𝑓+𝐺_𝑓𝑢+Δ𝐹_𝑓(𝑡)𝑦+𝑑_𝑓=𝜔, (34) where Ω=[𝛼,𝛽,𝜇]^𝑇 is a vector of attitude angles which are angle of attack, sideslip angle, and flight-path roll angle, respectively; 𝜔=[𝑝,𝑞,𝑟]^𝑇 is the body-axis angular rates which are the fast-loop states; 𝑢=[𝛿_𝑒,𝛿_𝑎,𝛿_𝑟,𝛿_𝑥,𝛿_𝑦,𝛿_𝑧]^𝑇 is the control input vector which is the deflection vector of control surfaces. Δ𝐹_𝑠 and Δ𝐹_𝑓 are the unknown system uncertainties. 𝑑(𝑡)=[𝑑₁(𝑡),𝑑₂(𝑡),𝑑₃(𝑡)]^𝑇 is the unknown time-varying external disturbance. To the attitude dynamic, 𝐹_𝑠=[𝑓_𝛼,𝑓_𝛽,𝑓_𝜇]^𝑇 and 𝐹_𝑠=[𝑓_𝑝,𝑓_𝑞,𝑓_𝑟]^𝑇; matrices 𝐺_𝑠 and 𝐺_𝑓 can be written as [48] 𝐺_𝑠=𝐺−tan𝛽cos𝛼1−tan𝛽sin𝛼sin𝛼0−cos𝛼sec𝛽cos𝛼0sec𝛽sin𝛼_𝑓=1𝐼_𝑥0100𝐼_𝑦0100𝐼_𝑧𝑔_{𝑝,𝛿𝑒}𝑔_{𝑝,𝛿𝑎}𝑔_{𝑝,𝛿𝑟}𝑔_{𝑟,𝛿𝑥}𝑔00_{𝑞,𝛿𝑒}𝑔_{𝑞,𝛿𝑎}𝑔_{𝑞,𝛿𝑟}00𝑔_{𝑟,𝛿𝑧}𝑔_{𝑟,𝛿𝑒}𝑔_{𝑟,𝛿𝑎}𝑔_{𝑟,𝛿𝑟}0𝑔_{𝑟,𝛿𝑦}0, (35)(36) where 𝐼_𝑥, 𝐼_𝑦, and 𝐼_𝑧 are moments of inertia of the NSV. Here, the detailed expressions of 𝑓_𝛼, 𝑓_𝛽, 𝑓_𝜇, 𝑓_𝑝, 𝑓_𝑞, 𝑓_𝑟, 𝑔_𝑝,∗, 𝑔_𝑞,∗, and 𝑔_𝑟,∗ are omitted. In this simulation study, we assume that the actuator of the NSV has a backlash-like hysteresis nonlinearity and 𝑢_𝑖, 𝑖=1,2,…,6, is the output of such hysteresis described by 𝑢_𝑖(𝑡)=𝑃𝑣_𝑖, (37) where 𝑣_𝑖, 𝑖=1,2,…,6, is the input of the hysteresis for the 𝑖th actuator of NSV and 𝑃(⋅) denotes a backlash hysteresis operator. When 𝑎₁=1, 𝑐_𝑖=3.1635, 𝑏_𝑖=0.345, and 𝑣_𝑖=𝑘sin(2.3𝑡) with 𝑘=2.5,3.5,4.5,5.5,6.5, the hysteresis curves are shown in Figure 1 [41].

Figure 1 

Hysteresis curves given by (37).

Suppose that the NSV fight lies in the cruise flight phase with the velocity 2500 m/s and flight altitude 26 km. The initial attitude and attitude angular velocity conditions are arbitrarily chosen as 𝛼₀=1^∘, 𝛽₀=1^∘, 𝜇₀=0.5^∘, and 𝑝₀=𝑞₀=𝑟₀=0 rad/s. The desired flight attitude states are chosen as 𝛼_𝑑=5^∘,𝛽_𝑑=0^∘,𝜇_𝑑=0.5^∘. (38) The system uncertainties Δ𝐹_𝑠 and Δ𝐹_𝑓 are assumed as +20% and −20% variation of aerodynamic coefficients and aerodynamic moment coefficients, respectively. On the other hand, the following unknown time-varying disturbance moments are considered in the simulation [48]: 𝑑₁(𝑡)(())𝑑=150000sin6𝑡+0.15Nm₂(𝑡)(())𝑑=200000sin5𝑡+0.25Nm₃(𝑡)=250000(sin(4𝑡)+0.2)Nm. (39) In this simulation, the parameters of the backlash-like hysteresis should be chosen according to the used actuator of the near-space vehicle. All parameters of the nonlinear disturbance observer and the robust sliding mode attitude controller are chosen as Λ=diag{2}, 𝐿=diag{60}, 𝑇=diag{2}, 𝛾=2, 𝜆₀=0.15, and, 𝜆₁=0.15. The disturbance observer is designed as (12) and the parameter updated laws are designed as (25) and (26). The robust sliding mode attitude controller is designed in accordance with (23) for the NSV. The attitude control results, the control input, and the sliding mode surface are shown in Figures 2–5 under the designed robust sliding mode attitude control approach.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Tracking control results.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

Control input response.

Figure 4 

Sliding mode surface of attitude angular.

Figure 5 

Sliding mode surface of attitude angular rates.

The tracking control results of the NSV under the designed robust sliding mode control scheme are shown in Figure 2. From Figure 2, we can see that the small tracking errors of attitude angles are obtained and the expected control performances of the body-axis angular rate are guaranteed. The control inputs are presented in Figure 3 which show the convergence of the control inputs. The plots of the sliding mode surfaces are given in Figures 4 and 5 which are convergent.

In accordance with the above simulation results, the satisfactory attitude tracking control performances are obtained under the developed robust sliding mode attitude control scheme of the NSV. Thus, the developed robust sliding mode control scheme is valid for the uncertain MIMO nonlinear system.

5. Conclusion

Robust sliding mode control approach has been developed for a class of uncertain MIMO nonlinear systems in the presence of the unknown system uncertainty, the external time-varying disturbance, and backlash-like hysteresis in this paper. To enhance the robust control performance, the RBF neural network has been introduced to approximate the unknown continuous system uncertainty. At the same time, we treated the unknown external disturbance, the unknown neural network approximation error with the approximation error of backlash-like hysteresis as a compounded disturbance, and the nonlinear disturbance observer has been developed to estimate it. Then, the robust sliding mode control has been developed for the uncertain MIMO nonlinear system using the outputs of the RBF neural network and the developed nonlinear disturbance observer. Via Lyapunov analysis, the uniformly asymptotical convergence of all closed-loop signals has been guaranteed. Finally, the designed robust sliding mode control strategy has been applied to NSV attitude dynamics, and simulation results have been given to illustrate the effectiveness of the proposed sliding mode control approach.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant no.: 61174102), Program for New Century Excellent Talents in University of China (Grant no.: NCET-11-0830), Jiangsu Natural Science Foundation of China (Grant no.: SBK2011069), project-sponsored by SRF for ROCS, SEM, and project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.