Full-State-Constrained Adaptive Control for a Class of UAVs Suffering from Coupled Uncertainties Using the HOBLF

Chang, Xiaofei; Wang, Kexuan; Chen, Kang; Fu, Wenxing

doi:https://doi.org/10.1155/2021/9963426

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9963426 | https://doi.org/10.1155/2021/9963426

Full-State-Constrained Adaptive Control for a Class of UAVs Suffering from Coupled Uncertainties Using the HOBLF

Xiaofei Chang,¹Kexuan Wang,²Kang Chen,¹and Wenxing Fu¹

Academic Editor: Jose Vicente Salcedo

Received09 Mar 2021

Revised03 Sept 2021

Accepted11 Sept 2021

Published11 Oct 2021

Abstract

Nowadays, the practical tasks of UAVs are becoming more and more complicated and diversified. In the practical flight process, the large-scale changes of the flight environment, the modeling errors, and the external disturbances may induce the instability of the UAV flight system. Meanwhile, the constraints of the UAV attitudes also have to be guaranteed during the flight process. However, most existing control methods still have limitations in handling the constraints and the multisource disturbances simultaneously. To address this problem, in this paper, we focus on the actual output tracking control for the UAV systems with full-state constraints and multisource disturbances. Firstly, a high-order tan-type barrier Lyapunov function (HOBLF) has been constructed for the UAV to maintain the full-state constraints. Secondly, by combining the adaptive backstepping technique and the fuzzy logic systems, the modeling errors and the unknown nonlinearities of the UAV attitude control system can be handled. Moreover, by properly constructing several adaptive laws, the time-varying disturbances existing in the UAV attitude control system can be suppressed. Finally, the full-state-constrained antidisturbance controller is formed, ensuring that the tracking error approaches arbitrarily to small neighborhood and does not violate the given constraints. The simulation results illustrate the feasibility and the advantages of the proposed method.

1. Introduction

Unmanned aerial vehicles (UAVs) have attracted research interest on account of their simple structures, low price, high mobility, and high flexibility. Due to their significant advantages, they have broad application prospects in military [1] and civilian [2]. However, the nonlinear characteristics and strong coupling of UAVs increase the difficulty of UAV control [3]. The aerodynamic parameter perturbations, modeling errors, and external interference caused by changes in flight conditions may cause instability of the flight system or even failure [4].

In order to tackle the control problem of UAVs [5], several approaches have been proposed, such as robust control [6], adaptive control [7], neural networks [8], fuzzy control [9], sliding mode control [10–12], genetic algorithms [13], and linear parameter-varying (LPV) method [14]. Recently, various advanced control methods have been presented for the attitude control of the UAVs. In [15], a nonlinear model predictive control (NMPC) is used to design a high-level controller for a fixed-wing unmanned aerial vehicle. In [16], this method has been improved, and it improved the optimality of the MPC controllers. In [17], H-loop-shaping method is used to provide the stabilizer mode while successfully improving the robustness of the UAV. The LPV model reference method [14] has been further developed in [18] to achieve agile and high-performance tracking objectives of UAVs. Chen et al. [19] used a disturbance observer-based control (DOBC) scheme with transient performance design for UAVs to tackle the mismatched disturbance problem. Besides the disturbance observer, another effective way to account for the unknown dynamics in the UAV is employing active disturbance rejection control (ADRC) [20]. Moreover, by a combination of backstepping and ADRC, an integrated controller is devised for the quadrotor UAV with multiple uncertainties in [21]. Based on filtered dynamic inversion, the controller in [22] enabled the fixed-wing UAV to achieve effective altitude command following in the presence of turbulent wind. In [23], an adaptive second-order sliding mode control based on an extended observer is proposed for UAVs to improve performance under different operating conditions. To achieve an ideal convergence performance of the UAV, prescribed performance controllers [24] and finite-time stable controllers [25–27] were designed.

Due to the limitations of actual physical devices, system performance requirements, and safety requirements, most practical systems will have constraints on the input, state, and output, such as nonuniform gantry crane [28], flexible crane systems [29], and robotic manipulator systems [30]. A series of constraint control methods have been proposed, including reference governors [31], model predictive control [32], extremum-seeking control [33], and the notion of set invariance [34, 35]. In recent years, BLF-based constraint control methods have been widely used to deal with state constraints and output constraints because they do not require precise solutions of the system. The synthesis of the barrier Lyapunov function (BLF) with the constraint function is proposed firstly in dealing with the Brunovsky constraint system [36]. Since then, many scholars have focused on studying the adaptive control for uncertain nonlinear systems based on the barrier Lyapunov function. In [37], by the combination of the backstepping method and BLF, an adaptive constrained controller was proposed for partial-state-constrained systems. In [38, 39], nonlinear pure-feedback systems and stochastic systems with full-state constraints have been investigated by constructing barrier Lyapunov functions (BLFs). For the MIMO full-state-constrained nonlinear systems, the opportune integral BLFs were established to prevent violation of the state constraints in [40]. Tee et al. [41] extended the constant constraint in [41] to the time-varying constraint. Furthermore, BLF has also been successfully applied in practical systems, such as electrostatic microactuator [42] and HSV [43, 44]. In [45], a higher-order barrier Lyapunov function (HOBLF) has been firstly proposed while achieving constraint satisfaction without solving the displayed solution of the system, especially for the higher-order uncertain nonlinear system. Given the influence of the external aerodynamic change and strong interference on the UAV in the flight environment, the attitude control of the UAV becomes an important content related to its performance. Therefore, certain state constraints should be applied to its three attitude angles to avoid the occurrence of large deviations. The high-order barrier Lyapunov function can just solve the problem of UAV attitude constraints. Therefore, this paper introduces [45] to solve the problem of UAV state constant constraints. And as far as the authors know, there is no relevant literature that has studied the state constraints of UAVs.

In spite of the process, the HOBLF-based full-state-constrained antidisturbance control has never been investigated for the UAVs. According to Sun et al. [45], using the HOBLF can achieve practical output tracking control for a category of high-order uncertain nonlinear systems with full-state constraints. Although Tan and Guo [24] considered the full-state-constrained control for the UAVs, the HOBLF has never been utilized. Moreover, the influence of external aerodynamic changes and strong disturbances of UAVs has to be taken into consideration.

Based on the above observations, for UAV systems with full-state constraints,(i)This paper introduces a high-order tan-type BLF and applies it to the UAV system, whose three attitude angles have state constraints. In addition, when constraints disappear or the high-order system is reduced to a traditional nonlinear system in the form of strict feedback, this BLF is still effective.(ii)For the time-varying interference that the UAV may encounter during the flight, an adaptive law is constructed to enable the UAV to track the desired signal stably even under interference and has a strong adaptability.(iii)Compared with the traditional adaptive, DOBC, and general BLF [36], computational burden has been decreased, and the proposed control method has a better tracking effect on account of the HOBLF and fuzzy system.

This paper is organized as follows. In Section 2, the problem formulation and related knowledge are given, while in Section 3, the high-order tan-type BLF, the design procedures, the controllers, and the adaptive law are designed. Section 4 presents the simulation results and comparison, and Section 5 concludes this paper.

2.1. System Model

In this paper, we use the nonlinear dynamics of the UAV built in [46]; the details are shown as follows:where , respectively, represent the pitching, yaw, and rolling attitude angle of the UAV relative to the inertial coordinate system. represent the angular velocity of each degree of freedom, respectively, where . is the rotational inertia matrix of the UAV. are unknown unmodeled dynamics of the plant. The control scheme is depicted in Figure 1, and the details and expressions of and are shown as follows: is the product of inertia in both directions, is the dynamic pressure of air, is the total airspeed of the UAV, is the air density, is the total area of the wing, is the wingspan, and is the average chord length. The rolling, pitching, and yaw moments of the UAV are mainly controlled by , which can be given bywhere are the right and left aileron rudder deviation. are the deviation of the left and right lifter rudder; is the direction of the wing rudder deviation; is the drag coefficient with being the coefficient of to ; is the drag coefficient with being the coefficient of to ; is the drag coefficient with being the coefficient of to ; are the angle of attack and sideslip angle.

In order to build a model of the UAV attitude system, define , and . Then, the UAV attitude dynamic equation with time-varying interference can be expressed aswhere . They are denoted by

All states are bound to the compact setwhere is a known positive constant.

Remark 1. In fact, it is difficult to develop an exact model for the UAV since we cannot reproduce the complex flight environment of the UAV in a wind tunnel. So, the unmodeled dynamics and uncertain parameters are inevitable.

2.2. Assumptions and Lemmas

The purpose of this paper is to design a controller : (1) for any initial state, there is a finite time so that the output tracking error meets , where is the desired reference trajectory; (2) all closed-loop signals are bounded; (3) state constraint requirements are not violated.

In order to design the controller, analyze the stability of the system, and achieve the above control objectives, the following assumption and lemmas are introduced.

Assumption 1. For a continuously differentiable expected trajectory , there exists known positive such that , and .

Lemma 1. According to Young’s inequality, given any and ,where is arbitrarily small and is arbitrarily large, and .

Lemma 2. According to Young’s inequality, for given positive integers and for each , the following inequality holds:where and can be any constant or real-valued function.

Lemma 3. For any given constant and , there iswhere is a constant that satisfies ; for instance, .

2.3. Fuzzy Logic Theory

The specific form of the fuzzy logic system is as follows, which is composed of single point fuzzy set, product inference rule, center average weighted nonfuzzy set and Gaussian membership function:

Let

and . Then, the fuzzy system is

Lemma 4. There is a fuzzy system ; for any given constant , there iswhere is a continuous function defined on the compact set .

3. Main Results

Next, introduce the following transformations:where and , respectively, represent virtual tracking errors and virtual control signal. Furthermore, is constructed, where is a constant. The following high-order tan-type BLFs are introduced to solve the problem of system state constraints of the fixed-wing UAV system:

For each , define , where are odd positive integers and , is the order of the system, is the dimension of the system, and is a constant that will be given later. For the second-order and three-dimensional UAV system in this article, .

Remark 2. Tan-type BLFs (15) are proposed to handle full-state constraints for high-order systems, where . Because means when . From this, the formula can be further inferred as follows:Therefore, the high-order tan BLF introduced in this paper will be transformed into a general Lyapunov function, and the design method is also effective for unconstrained high-order systems.

The parameters used in the subsequent control design are defined as follows:where and are adjustment parameters.where .

Using the fuzzy system , it is easy to getwhere are the approximation errors. This comes together with Lemma 2:where are arbitrary constants.

Substitute (20) into (18) to getwhere . Define

Then, according to Lemma 3,

is designed as

In this way,

Substituting (25) into (21), we have

The adaptive law is as follows:

So, it can be deduced that

Given , (28) can be rewritten aswhere . Then, the asymptotic bounded convergence of the system is proved.

Remark 3. The negative term designed in (29) supplies an adequate stable domain to stabilize the positive terms containing which will appear in the next design step.

Next, consider the design of .

By substituting (29) into (30), it can be obtained that

In accordance with Lemma 1,where .

Substituting (32) into (31),

Further deformation of the above equation can be obtained:where .

Use the fuzzy system to getwhere is the approximation error; through Lemma 2, we getwhere , , and , being adjustment parameters. Substituting (36) into (34) results inwhere .

Same as above, according to Lemma 3, define so that

Design as

Then,

This together with (37) yields

Design the adaptive law as

So, it can be deduced that

In the same way, . Therefore, (43) can be rewritten as

Then, the asymptotic bounded convergence of the system can be proved.

Because the system is a second-order three-dimensional system, , are asymptotically bounded convergences, and , . So, the system is also asymptotically bounded. According to , we know that . Then,where ,.

Proof. If the system with full-state constraints meets the assumptions, the trace controller can accomplish the following:(i)All states of a closed-loop system are bounded(ii)It will not violate the complete state constraints(iii)The tracking errors converge to any small neighborhood of zero(iv)The system is stable under time-varying disturbanceConsidering , according to formula (45), we can get . Therefore, we can infer that . Hence, is constrained; it also represents that is bounded. Therefore, . According to the Assumption 1, we can know that . On the contrary, it can be inferred from the continuity of and the boundedness of that is also constrained. Hence, there is a finite constant that makes . From the boundedness and error transformation of and , the conclusion of can be drawn. As a result, all states are bounded and meet the constraints.

Remark 4. For the possible disturbance , it is replaced by , which greatly reduces the computation. If the adaptive parameters are used to cancel the disturbance, the adaptive parameters should be designed for , respectively, so that the computation burden will be increased.

4. Simulation Experiment

4.1. Initial Value Selection

In order to verify the effectiveness and superiority of the HOBLF method proposed in this paper under the influence of time-varying interference, this section studies the simulation effect under the influence of strong time-varying interference and weak time-varying interference, respectively, to verify its validity. Under the same conditions, the simulation effects of the proposed method and DOBC, general adaptive control, and low-order general BLF were compared to verify its superiority.

Now, suppose that the UAV rises at a fixed attitude angle in the air, and the UAV needs to change from the rising stage to the flat flying stage. The square wave signal filtered by the second-order link is selected as the expected signal with an amplitude of 1 and a period of 20 s. Related parameters have been listed in Tables 1 and 2.

4.2. Simulation Comparison

The effectiveness of the HOBLF is verified by performing the simulation under three cases.

Case 1. First, consider the case of using the HOBLF for control under weak time-varying interference. For pitch angle , yaw angle , and roll angle , the time-varying interferences are , and , respectively.

Case 2. Then, consider the case of controlling with the HOBLF under strong time-varying interference. For pitch angle , yaw angle , and roll angle , the time-varying interferences are , , and , respectively.

The simulation results are shown in Figures 2–5. It can be analyzed from Figures 2–5 that, under the influence of strong or weak time-varying interference, the controlled states , and can track the expected signal , and smoothly without violating the state constraint. Therefore, the effectiveness and superiority of the HOBLF can be verified.

Case 3. high-order tan-type BLF is selected. The tangent Lyapunov function can not only deal with the adaptive control of nonlinear systems with state constraints but also can be regarded as a quadratic Lyapunov function without constraints. Therefore, the tangential Lyapunov function can deal with constrained and unconstrained systems. The state constraint on is 1.5.
The design of inside loop controller is as follows:The design of outside loop controller is as follows:Other relevant parameters are set as follows: .

To show the advantages, the TAC (traditional adaptive control), DOBC, and general BLF are applied to the simulation.

Using the traditional adaptive method, it first applies backstepping design technology to constructively give the design steps of the state feedback adaptive controller and then combines Barbalat’s lemma to analyze the stability of the closed-loop system, making the closed-loop system state globally asymptotically stable. The remaining closed-loop signals are uniformly bounded. The selected Lyapunov function is . The relevant parameters are designed as diagdiag .

DOBC (disturbance-observer-based control): by designing a disturbance observer to cancel the influence of disturbance on the servo system, the disturbance is regarded as a part of the system state, and the reduced-order state observer is used to estimate and cancel the disturbance. The design of the disturbance observer is which is an auxiliary variable, and the relevant parameters are designed as diag diag .

Using the general BLF method, the symmetric positive definite log-type BLF is as follows:

It can guarantee that every step in the backstepping method does not violate the state constraint and guarantee the stability of the system, where and are positive design parameters, , and is an adaptive parameter. The relevant parameters are designed as .

HOBLF, DOBC, general adaptive, and general BLF were, respectively, used for simulation. The expected signal tracked was the square wave signal after second-order filtering, with an amplitude of 1 and a period of 20 s.

The simulation results of the four methods are shown in Figures 6–11, where Figures 6 and 9 show the tracking trajectory and error curves of the pitch angle, respectively. Figures 7 and 10 show the yaw angle tracking track and error curve, respectively. Figures 8 and 11, respectively, show the tracking track and error curve of the roll angle. According to the analysis in Figures 6–8, the states in the traditional adaptive method and DOBC will exceed the state constraint and have a large deviation. The tracking trajectory of the high-order BLF and general BLF in three directions is always between the upper and lower limits of the state constraint. However, for the tracking with a constant expected signal, high-order BLF is more stable and has less error. It can be easily known from Figures 9–11 that compared with the high-order BLF method, the traditional adaptive method, DOBC, and general BLF have a larger tracking trajectory and error.

5. Conclusion

In this work, an adaptive fuzzy controller has been proposed for a fixed-wing UAV control system of full-state constraints and unknown external disturbances. The high-order BLF (HOBLF) is firstly introduced for the UAV systems to realize the state constraints. Meanwhile, the unknown nonlinear functions have been compensated by the fuzzy system and backstepping control method to ensure the stability of the system. According to simulation results and comparison, we have made the conclusion that the proposed control is able to track a desired trajectory with an excellent performance without violating the state constraints. Moreover, further research based on the current results of this paper is how to verify the control feasibility of the UAV system with existing time-varying state constraint issues.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Basic Research Program of Natural Science of Shaanxi Province (Grant no. 2019ZY-CXPT-03-02).

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Mathematical Problems in Engineering

Full-State-Constrained Adaptive Control for a Class of UAVs Suffering from Coupled Uncertainties Using the HOBLF

Abstract

1. Introduction

2. Problem Formulation and Related Knowledge

2.1. System Model

2.2. Assumptions and Lemmas

2.3. Fuzzy Logic Theory

3. Main Results

4. Simulation Experiment

4.1. Initial Value Selection

4.2. Simulation Comparison

5. Conclusion

Data Availability

Conflicts of Interest

Acknowledgments

References

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