Research Article | Open Access
Second-Order Sliding Mode Disturbance Observer-Based Adaptive Fuzzy Tracking Control for Near-Space Vehicles with Prescribed Tracking Performance
An adaptive fuzzy fault-tolerant tracking controller is developed for Near-Space Vehicles (NSVs) suffering from quickly varying uncertainties and actuator faults. For the purpose of estimating and compensating the mismatched external disturbances and modeling errors, a second-order sliding mode disturbance observer (SOSMDO) is constructed. By introducing the norm estimation approach, the negative effects of the quickly varying multiple matched disturbances can be handled. Meanwhile, a hierarchical fuzzy system (HFS) is employed to approximate and compensate the unknown nonlinearities. Several performance functions are introduced and the original system is transformed into one incorporating the desired performance criteria. Then, an adaptive fuzzy tracking control structure is established for the transformed system, and the predefined transient tracking performance can be guaranteed. The rigorous stability of the closed-loop system is proved by using the Lyapunov method. Finally, simulation results are presented to illustrate the effectiveness of the proposed control scheme.
As is well known, disturbances and uncertainties widely exist in industrial systems and may induce negative effects on control performance or even stability of practical control systems . Therefore, a wealth of disturbance and uncertainty estimation and rejection methods, such as unknown input observer (UIO) [2, 3], perturbation observer , uncertainty and disturbance estimator , and disturbance observer (DO) , have been proposed. Meanwhile, a disturbance observer in frequency domain was developed in . For the purpose of suppressing the periodic-disturbances, an adaptive periodic-disturbance observer is designed in . In , a nonlinear disturbance observer is utilized in the robust control for spacecraft formation flying. In , a disturbance observer-based controller is developed for a magnetically suspended wheel with synchronous noise. In , a disturbance observer-based robust approaching control law is proposed for a tethered space robot. In [12, 13], several adaptive composite antidisturbance control methods have been investigated. In , a practical control method which can estimate the lumped disturbances consisting of both unknown uncertainties and external disturbances, called active disturbance rejection control (ADRC), was developed by Han and his collogues. Moreover, by using the equivalent-control-principle of the sliding mode control technique to estimate the lumped disturbances, sliding mode disturbance observer (SMDO) has been constructed . In , to address the problem of disturbance rejection control for Markovian jump linear systems with matched and mismatched disturbances, an extended sliding mode observer based control law has been developed. In , sliding mode observers are utilized for state and disturbance estimation in electrohydraulic systems. For the vibration control of a train-car suspension with magnetorheological dampers, a SMDO-based controller has been investigated in . For the air-breathing hypersonic flight vehicles subject to external disturbances and actuator saturations, a sliding mode exact disturbance observer (SMEDO) is exploited to exactly estimate the lumped disturbances [19, 20]. As is stressed in , SMDO possess strong robustness and accuracy to estimate a lumped disturbance including unknown external disturbances and parametric uncertainties. In , a sliding mode disturbance observer with switching-gain adaptation was proposed. In [23, 24], the second-order sliding mode disturbance observer, which can produce continuous estimation signals and possess strong robustness, had been proposed and applied to industrial systems.
On the other hand, in the practical control systems, it is of significant importance to consider the transient control performances including the overshoot, undershoot, and convergence rate. However, there are few results focusing on this issue. Usually, the traditional adaptive control systems can only steer the tracking errors to converge to a residual set whose size resting with the control gains and the disturbances ranges [25, 26]. Recently, by utilizing appropriately defined functions to transform the original system into one that incorporates the desired performance criteria, an effective control scheme which can guarantee the prescribed transient performances has been proposed by Bechlioulis . In , a fault-tolerant controller guaranteeing prescribed tracking performance has been proposed for a class of nonlinear uncertain systems. Moreover, the prescribed performance controllers for the cascade systems  and the nonaffine nonlinear large scale systems  have also been developed.
Since NSVs are not constrained by orbital mechanics and fuel consumption, they can offer significant advantages to Low Earth Orbit (LEO) satellites and airplanes . In general, the NSV has larger flight envelope, rapid flight speed, and time-varying aerodynamic characteristics; the controller design becomes challenging and it is necessary to develop effective control methods to guarantee the safety and reliability . In the past decades, a number of advanced control approaches have been developed for the NSVs. In , by introducing the neural networks into an adaptive backstepping controller, an effective attitude controller has been proposed. For the NSV suffering from the dynamical uncertainties, an adaptive functional link network control structure has been constructed in . In , a robust attitude controller has been designed for NSVs subjected to time-varying disturbances. In , a globally convergent Levenberg–Marquardt (LM) algorithm based on Takagi–Sugeno fuzzy training has been proposed for the NSV. In , a terminal sliding mode controller combined with dynamic sliding mode was designed based on nonlinear disturbance observer.
In spite of the progress, the control results for the NSVs with guaranteed transient performances have rarely been reported. Moreover, in practice, the aerodynamic parameters perturbations, measurement errors, actuator faults, and wind effects may induce significant uncertainties in the flight control systems of NSVs. To the best of the authors’ knowledge, the existing controllers are commonly designed for the NSVs suffering from only constant disturbances, and the aforementioned quickly varying multiple uncertainties cannot be handled. Therefore, in this paper, we are dedicated to designing an adaptive tracking control structure which can guarantee the prescribed transient tracking performance and possess satisfactory disturbance attention ability. By introducing a second-order sliding mode disturbance observer, the time-varying mismatched uncertainties can be estimated and compensated. With the aid of the hierarchical fuzzy approximator and a novel nonlinear function, the negative effects of the quickly varying multiple matched disturbances can be handled. Based on the performance functions and error transformation, the robustness and predefined tracking performance can be ensured. The adverse actuator derivations are also overcome by the proposed controller. Compared with the previous works, the contributions of this paper are summarized as follows.(1)An effective adaptive fuzzy control algorithm, which can guarantee the prescribed transient tracking performances including the convergence rate, the tracking error, and the overshoot, is developed for the NSVs.(2)The NSV model suffering from the multisource quickly varying uncertainties, including the actuator faults, the wind effects, aerodynamic uncertainties, and measurement errors, is established.(3)The adverse effects of the quickly varying uncertainties can be circumvented. The theoretic developments of this paper are valuable for handling the time-varying disturbances.
2. Problem Formulation and Preliminaries
2.1. NSV Model
According to [38, 39], the attitude dynamic model of NSVs can be described bywhere , and are the angle of attack, sideslip angle, and bank angle of the NSV, respectively. is the flight path angle. and represent lift and lateral force. is the velocity of the vehicle. , and are the roll, pitch, and yaw angular rates of the vehicle, respectively. The inertial parameters can be formulated byThe aerodynamic forces can be described aswhereThe aerodynamic moment are as follows: where denote the aerodynamic coefficients when the deflection angles are zero. represent the coefficients of aerodynamic damping torque. is the aerodynamic coefficient of the lift force; is the lateral force coefficient. The aerodynamic partial derivatives are represented by . are the deflection angles of control surfaces. is the reference area. is the reference length. is dynamic pressure. are calculated as
It should be highlighted that multiple uncertainties including the aerodynamic uncertainties, measurement errors, and unmodeled dynamics are often encountered in the NSV. Meanwhile, the NSVs often experience a complex flight environment, and the undesired stochastic winds have to be considered. Let and represent the additional angle of attack and the additional sideslip angle, respectively. Then the disturbance forces and torques caused by the unknown winds can be described as Moreover, the aerodynamic-perturbation-caused disturbance forces and torques are defined as .
In general, the NSVs are commonly equipped with the electromechanical aerodynamic rudders. However, there often exist multifarious faults in the aerodynamic rudders. As is revealed in [3, 4], actuator faults constitute the main reason for the undesired control performance and instability of the flight control system. Define . The faulty actuator model can be provided as where and denote the input and output of the actuator, respectively. denotes the deviation caused by the actuator faults. It is supposed that .
The neglect of the above-mentioned multiple uncertainties and actuator faults may cause worse control performance and instability of the flight control system. The intrinsic high dynamic characteristics and the strong coupled properties make it a challenge to establish an effective control scheme. Define and . Taking the multiple uncertainties into consideration, we can get the following NSV model:where and are the unknown nonlinearities caused by the model simplification. denotes the disturbances caused by inaccurate measurement information. represents the uncertainty part of the control distribution matrix, which is given as
In practice, the transient performance including the convergence rate, the tracking error, and the predefined maximum overshoot should be guaranteed. Therefore, the design objective of this work is to establish an effective control structure to force the angle of attack , the sideslip angle , and the bank angle to track the reference trajectories with predefined transient performance in the simultaneous presence of the actuator faults and multiple uncertainties.
2.2. Assumptions and Supporting Lemmas
The following assumptions and lemmas are necessary in this work.
Assumption 1. The actuator deviations are supposed to satisfy .
Assumption 2. The disturbances existing in the NSV system are assumed to be bounded; i.e., .
Lemma 3. Consider the systemwhere is continuous on an open neighborhood and the origin is . Suppose there is a continuous function defined on with the origin such that the following conditions hold:
is positive definite on .
There exist real numbers and , such that , and then, system (12) is locally finite-time stable. The settling time, depending on the initial state , satisfiesfor all in some open neighborhood of the origin. If and is also unbounded, system (12) is globally finite-time stable.
Lemma 4 (see ). Given any and , then one has
2.3. Hierarchical Fuzzy Systems
For the purpose of compensating the unknown nonlinear functions, we introduce a HFS in the controller. Commonly, the HFS contains several low dimensional fuzzy systems. The structure of the HFS is provided in Figure 1. Let represent the input of the HFS. Then the first subsystem can be established by using the following rules.
Rulel. If is and is , then is .
is the total number of fuzzy rules for the first fuzzy subsystem; are the corresponding fuzzy sets , and represent the fuzzy functions. A subsystem includes a singleton fuzzifier, a center-average defuzzifier, and a product inference engine. The following equation provides the output of a fuzzy subsystem:where . Then, we can establish the i-th ( ) fuzzy subsystem with the aid of the following rules.
Rulel. If is and is , then is .
and are the inputs, and is the output. Hence, , which is the output of the i-th fuzzy subsystem, is calculated as Finally, a hierarchical fuzzy system can be established and the final output can be given asIt is clear that the total number of fuzzy rules is .
Lemma 5 (see ). Define . Consider a nonlinear function , which is continuously differentiable and defined on . Then we can construct a hierarchical fuzzy system within (15) ~ (18) such thatwhere .
Lemma 6 (see ). Define . Consider any function which is continuously differentiable and defined on . Then one can construct a hierarchical fuzzy system within (15) ~ (18) such that , where is an arbitrarily small positive constant.
3. Main Results
3.1. Performance Functions
Define and . Then, for the purpose of guaranteeing the predefined control performance, we introduce the performance functions in the following text. Define the following exponentially decaying functions as the performance functions:where and . Define . If the inequality holds for , then the transient performance is ensured. To make this point more clear, we make the following explanations. and restrict the range of the overshoot and the lower bound of the undershoot of . The convergence speed of is restrained by the decreasing speed of . The steady error is constrained by .
Consider the following strictly increasing function:Clearly, function (21) satisfiesIt is not difficult to verify that the inverse function is well-defined and strictly increasing as well. Obviously, if is bounded, remains within a compact subset of . As a result, the control issue forces to be bounded.
3.2. Second-Order Sliding Mode Disturbance Observer
Define and . Consider the NSV model given in (9); we can get the following dynamic equation: Using the strictly increasing function provided in (21), it can be proven that where Consider the mismatched uncertainties existing in system, a second-order sliding mode disturbance observer is introduced for estimation and compensation. The SOSMDO is designed as In this work, we divide the mismatched uncertainties into two parts. One is the unknown disturbance related to the system states; the other is the time-varying unknown uncertainty. In other words, Without loss of generality, it is supposed that there exist constants such that
Theorem 7. Consider system (23). If one selects such thatwhere then the estimation error of the mismatched uncertainties can be forced to zero in finite time.
Proof. From (23) and (26), we can get the following closed-loop dynamic equation: Consider the following Lyapunov function:where . Taking the derivatives of (32) yields Substituting (31) into (33) yields Furthermore, By using (28) we can get that Denote ; then we can rewrite (36) aswhereClearly, by using condition (29),we can easily get that and . Then, it follows from Rayleigh’s inequality thatBy defining , we can rewrite (39) asFurthermore, Since , it can be proven that According to Lemma 3, it can be observed that the equilibrium point can be reached in finite time. Then by substituting into (31), we know . As a result, is achieved in finite time. Define the estimation error of the mismatched uncertainty as . Then from (26), it can be proven that . Therefore, it can be concluded that is forced to zero in finite time. The proof is completed.
3.3. Hierarchical Fuzzy Approximation-Based Adaptive Fault-Tolerant Tracking Control
Considering the following transformed dynamic equation we design the virtual control signal as follows: where . To avoid the excessive complexity of control design, a first-order filter is established aswhere is the parameter matrix of the filter. Denote . Then from (43) ~ (45) we can infer the closed-loop dynamic equation asAccordingly, along (43), take the derivate of asIn view of the unknown nonlinearities existing in (47), a hierarchical fuzzy logic system is constructed for compensation. Ideally, . is the weight matrix of the HFLS, is the number of the rules, and is the membership function. The final control law is developed aswhere ; is the estimation of . represent the estimations of , . The adaptive laws are given by are design constants. are the gain matrixes of the adaptive laws. The structure of the proposed control method is provided in Figure 2.
Theorem 8. Consider system (43). Suppose the estimation error of the mismatched uncertainty is bounded. Then, by using control laws (44) and (48) and adaptive laws (49), it can be ensured that all signals of the overall closed-loop system are globally uniformly bounded. As a result, the predefined control performance bounds can be guaranteed.
Proof. Consider the following Lyapunov function candidate:Then, along (46), the derivative of can be taken asResorting to Young’s inequality, it can be proven that where . Therefore, we can rewrite (51) aswhere From (47) and (48), it can be obtained that where , . Take the derivative of along (55) asBy using Lemma 4, we can get that FurtherCombining (56) ~ (58) yields By using the vector trace identity we can obtain that Subsisting (49) into (60), we come toThen it follows from (53) and (61) that Clearly, Therefore, it can be checked that