Abstract
This article investigates a novel fuzzy-approximation-based nonaffine control strategy for a flexible air-breathing hypersonic vehicle (FHV). Firstly, the nonaffine models are decomposed into an altitude subsystem and a velocity subsystem, and the nonaffine dynamics of the subsystems are processed by using low-pass filters. For the unknown functions and uncertainties in each subsystem, fuzzy approximators are used to approximate the total uncertainties, and norm estimation approach is introduced to reduce the computational complexity of the algorithm. Aiming at the saturation problem of actuator, a saturation auxiliary system is designed to transform the original control problem with input constraints into a new control problem without input constraints. Finally, the superiority of the proposed method is verified by simulation.
1. Introduction
Air-breathing hypersonic vehicle (AHV) is a kind of high-velocity aircraft flying at more than 5 times the speed of sound and cruising at an altitude in the near space. With the increasing extension of human activities to space, high-velocity and high-altitude vehicles represented by air-breathing hypersonic vehicles will play an important role in the field of aerospace. The control system is the “nerve center” of an aircraft, and the design of the AHV control system has become one of the frontier issues in the field of control science [1].
Compared with traditional aircrafts, the flight environment of flexible air-breathing hypersonic vehicles is more complex, the flight envelope span is larger, and the aerodynamic characteristic change is more complex. These problems bring difficulties to the design of the FHV control system. At the same time, in the control theory, FHV has the characteristics of strong nonlinearity, strong coupling, fast time-varying, and uncertainty, which make FHV extremely sensitive to flight attitude in dynamic characteristics, and the design of the FHV control system is also subject to many constraints [2–4].
For the control of FHV, previous research has achieved fruitful results, for example, sliding mode variable structure control [5–8], neural network control [9–11], prescribed performance control [12–14], and fuzzy control [15, 16]. Hu et al. designed the FHV robust controller and the observers by means of sliding mode control in [6]. But in practical application, chattering problem exists in sliding mode control. In [7], the method of multimode switching between ordinary sliding surface and integral sliding surface is proposed to solve the chattering problem, but the switching between sliding surfaces is still relatively complex. In [8], Song et al. studied the double-loop sliding mode control method of HFV, unknown disturbances are observed by sliding mode observers, which enhanced the robustness of the control law. Bu et al. proposed an adaptive neural control method in [10]. The simulation results show that the control method can still guarantee the stable tracking of flight velocity and altitude to their respective reference inputs. In [11], the altitude subsystem of HFV is rewritten to a pure feedback form, and an adaptive neural control law is designed to ensure the realizability of the control law. By constructing the prescribed performance functions in [14], the velocity tracking error and altitude tracking error of FHV are guaranteed, and the prescribed transient performance and steady-state accuracy of the system are satisfied. In [16], unknown functions of FHV models are approximated online by using fuzzy systems, but the algorithm requires that the unknown functions be strictly positive and bounded. There are a lot of unknown functions in each subsystem, and computational costs are heavy.
Most of the above studies focus on the affine models of FHV, which require simplifying the nonaffine models of FHV to the complete affine forms of the control input under certain assumptions, and then conducting research [17]. However, in fact, many of the dynamic performances of FHV are nonlinear functions of rudder deflection, angle of attack, velocity, and altitude. Particularly in the hypersonic flight process of FHV, these nonaffine dynamic characteristics cannot be affine, so it is necessary to strengthen the study of nonaffine models of FHV. [18] designed a control law of FHV based on the Mean Value Theorem [19], which has achieved certain effects. In [20], for a class of high-order nonaffine nonlinear systems with completely unknown dynamics, Meng et al. developed a novel system transformation to transform the nonaffine system into an affine system. According to the idea expressed in [20, 21] combined the system transformation with prescribed performance control scheme, the simulation results verified the effectiveness of the method. In [22], the novel system transformation with a low-pass filter is applied to constrained systems.
Based on above research, a fuzzy-approximation-based novel back-stepping control scheme of FHV with nonaffine models is proposed in this article. The FHV’s nonaffine dynamics is decomposed into velocity subsystem and altitude subsystem to be controlled separately, and then nonaffine dynamics of FHV is processed by low-pass filters. A fuzzy approximator is introduced to approximate unknown function in each subsystem, and norm estimation approach [23] is introduced in the approximation of unknown functions. A novel back-stepping control method without virtual control law is designed. Considering the constraints of actuators, a new auxiliary system is designed to transform the control constraints into unconstrained ones. Finally, based on the simulation experiments, the performance of the control method is verified. The special contributions of this article are summarized as follows: (1)The traditional affine models need to simplify the nonaffine dynamics into fully affine forms. In this article, low-pass filters are used to solve the nonaffine dynamics of FHV(2)For each subsystem of FHV, only a fuzzy approximator is introduced to approximate the total uncertainties in the subsystem, which reduces the computational complexity of the system on the basis of ensuring the robustness of the controller. At the same time, the norm estimation approach is introduced in this article to ensure the computational speed compared with the previous fuzzy control method(3)Compared with the traditional back-stepping control method, all the virtual controllers proposed in this article are only used as intermediate variables for the stability analysis of the system and do not participate in the final solution and execution(4)An auxiliary system is introduced to transform the original control problem with input constraints into a new control problem without input constraints to solve the transient saturation problem in the control input
2. FHV Model and Preliminaries
2.1. FHV Model
Taking X-43A as the modeling objective, the longitudinal integration analytical model of FHV was established by Bolender and Doman [24]; Parker et al. ignored the weak coupling in the dynamic model and established the parameter fitting model of FHV [17]. where where and are structural mode shapes; other variables and functions are given in [24].
The FHV models contain five rigid body states and two elastic modes. The five rigid body states are flight velocity , flight altitude , track angle , pitching angle , and pitch rate ; and represent the two elastic modes of FHV; the distance between the vehicle and the earth’s center is expressed by ; is the mass of FHV; moment of inertia is ; define the flight angle of attack . and represent the damping ratio, and represent the natural frequency for elastic modes, respectively; and are generalized forces; , , , and represent the thrust, resistance, lift, and pitching moment of FHV, respectively; where the approximations of , , , , , and are designed as follows [17]: with
The control inputs are fuel-air ratio and elevator deflection angle ; the average aerodynamic chord length of FHV is ; the reference area is ; represents the dynamic pressure of FHV; the air density at is ; is the coupling coefficient of thrust and moment; represents rudder deflection coefficient; represents nominal altitude; the air density at nominal altitude is ; is the decay rate of air density.
Remark 1. In the following controller design, the rigid body states can be measured, and the elastic modes are usually regarded as the unknown dynamics, which are treated as the robustness of the controller [10, 25].
Remark 2. It can be seen from equation (10) that the FHV models are nonaffine for the control inputs, because the parameter fitting form of resistance contains [17]. Therefore, in the subsequent control system design, based on the back-stepping control framework, in the case of uncertainties and nonaffine dynamics, fuzzy-approximation-based nonlinear tracking control laws and are designed to achieve robust tracking of velocity and altitude commands and .
Remark 3. Because the fuel-air ratio is related to thrust , so the velocity is mainly controlled by . The elevator angle controls the change of altitude by affecting the pitch rate , pitch angle , and track angle . Therefore, the FHV models can be decomposed into velocity subsystem (equation (1)) and altitude subsystem (equations (2)-(5)), and then the control laws are designed separately [26].
2.2. Theory of Fuzzy Approximation
In recent years, the universal approximation characteristics of the fuzzy system have been gradually discovered, proposed, and applied. Under sufficient rules of the fuzzy system, the fuzzy system can approximate the given nonlinear continuous function with arbitrary precision [27, 28]. The complete fuzzy system includes inference engine, knowledge base, fuzzification interface, and defuzzification interface. Compared with neural network and polynomial function approximation, the advantage of the fuzzy system is that it can effectively control based on fuzzy logic. Where the input of the system , the output is .
In the fuzzy system, represents the parameter vector of ideal weight coefficient, fuzzy basis function . Fuzzy basis functions are expressed as
In equation (12), is the th fuzzy rule corresponding to the th input point, is the number of fuzzy rules, is the membership function, . In this article, the Gauss basis function is chosen as the membership function.
Lemma 1 [27]. Suppose that the real continuous function is an arbitrary function defined on compact set . For any , the fuzzy system can be obtained, and the following conditions are guaranteed:
2.3. Saturation Auxiliary System
In order to maintain the normal working mode of scramjet and the physical constraint of FHV by deflection angle, the actuators of control inputs and are limited to some extent. The saturation problem of control input is described as follows: where denotes the fuel-air ratio in the case of input saturation, denotes the elevator deflection in the case of input saturation, and are the lower and upper bounds of , and and are the lower and upper bounds of . In this article, a saturation auxiliary system is designed under the condition of input saturation. The input constraints of the control system are transformed into unconstrained ones.
When the inputs of the saturated auxiliary system are and , the sigmoid functions are used to approximate the outputs and of the auxiliary system.
According to equation (16) and the characteristic of the sigmoid function, whether and take any value, the saturation auxiliary system can make and meet the constraints of equation (15).
Remark 4. In the controller design, the original control objective can be converted to design unconstrained and such that velocity and altitude track their reference commands and .
2.4. Controller Design
In the following section, the altitude subsystem and velocity subsystem of the controller are designed with the new fuzzy-approximation-based control scheme in this article.
2.4.1. Controller Design of the Altitude Subsystem
In the altitude subsystem, we design an altitude controller based on nonaffine dynamics and a fuzzy approximator to achieve stable tracking of reference altitude .
Define altitude tracking error , the reference command of is chosen as where and are design parameters.
If the design parameters , and , then can be regulated to zero exponentially [29]. In this way, the control objective of the altitude subsystem can be transformed to by designing the appropriate control law .
According to the idea expressed in [20, 21], we define a set of newly state variables; the control input is also viewed as the newly defined state to deal with the nonaffine features. Define new state variables
Original nonaffine systems (3)-(5) can be rewritten to the following equivalent general nonaffine form according to equation (18). where and are continuous differentiable nonlinear functions; is an introduced variable, and it is regarded as the virtual control input of (19); the control inputs are generated by a low-pass filter “” driven by . “” is input-to-state stable. is the output of (19); is the design parameter.
From equations (3)-(5) together with previous studies [17], we have
Next, transform the nonaffine models.
Define , , combined with equation (19), the time derivative of is derived as
It can be noted that
We further define . Taking time derivative of and substituting (19)
Finally, define . Invoking (19), the time derivative of is derived as where
Through model transformation, the nonaffine system (19) can be converted into affine form.
The fuzzy-approximation-based altitude subsystem controller is designed by using the transformed affine form.
Remark 5. According to the theory of input-to-state stability, when the input is bounded, the state of the system is also bounded. So we need to prove the boundedness of .
Theorem 1. If the control inputs are bounded, there must be fixed values , such that
Proof. Define the Lyapunov function
Taking time derivative along (28).
If , will ultimately converge to the compact , where . That is, is bounded and the low-pass filter “” is stable when is bounded.
Define tracking error as
Taking time derivative along (30) and employing (26), it leads to
Define intermediate variable as
where is the design parameter.
Then, define tracking error as
Applying (26), the time derivative of is derived as
Define a new intermediate state variable as
where is the design parameter.
Define tracking error as
Differentiating along (36) with respect to time and applying (26), we get
Then, the intermediate variable is designed as
where is the design parameter.
Finally, define tracking error as
Taking the time derivative of and invoking (26) lead to
Define unknown function as
where . Since the function is an unknown function, the fuzzy function approximator designed in the above part is used to approximate it. The estimated value of the nonlinear function can be expressed as
where is the input vector to the unknown function, fuzzy basis function , is the ideal weight coefficient parameter vector, and the unknown function can be expressed as
where represents the estimated error, and ; is the upper bound of the estimated error.
Finally, the control law is devised as
where is the design parameter. denotes the estimation of ; its adaptive law is selected as
where is the design parameter.
Substitute equation (32) into equation (33) to obtain
Taking time derivative along (46),
By combining equations (26) and (35), equation (36) can be obtained.
Substituting equations (46) and (47) into equation (48), it yields
Taking time derivative along (49), we obtain
Substituting (26) and (38) into (39), is formulated as
Combining equations (46), (49), and (50), we can obtain
Substituting (49) and (52) into (44) leads to
Remark 6. Unlike the back-stepping control method proposed by Bu et al. [30], the presented altitude controller is exploited using an improved back-stepping framework, in which the virtual controller is transformed into intermediate variables and only the final actual control laws are needed to be implemented, which avoids repeated calculation and greatly reduces the computational pressure of the system.
Remark 7. Compared with the method in [16, 31, 32], each weight update requires a lot of calculation. In this article, only one online learning parameter is needed by using the norm estimation approach, which makes the online calculation of the system significantly reduced.
Theorem 2. Considering the closed-loop subsystem consisting of plant (26), with controller (53), and adaptive law (45). Then all the signals involved are semiglobally uniformly ultimately bounded.
Proof. Consider the following Lyapunov function:
with .
Applying (31), (34), (37), (40), (41), and (45), is derived as
Considering (32), (35), and (38), becomes
Substituting (44) into (56) yields, we can get
From the fact that
we obtain
Thereby, equation (57) becomes
In this article, since , and applying (12), we know that . There exists a constant, such that . Hence, we get
Then, (60) becomes
Let , , , , and . And define the following compact sets
If or or or or , will be negative. It is clear that , , , and are semiglobally uniformly ultimately bounded. This completes the proof.
2.5. Controller Design of Velocity Subsystem
For the velocity subsystem, its tracking target is to achieve the stable tracking of velocity to reference velocity .
Define new state variables and. In order to ensure the control effect, the velocity subsystem should be changed into an equivalent general nonaffine form with and . where the transformation consists of a low-pass filter “” and 2 newly defined state variables. is an unknown continuous differentiable nonlinear function, is the input of (64), and is the output of (64); is the design parameter.
Then, define state variables and . Where , , we can get Where nonlinear continuous functions and are expressed as
Remark 8. Noting that , so we can get the definitions of and as and , respectively. And we can calculate that .
According to the above analysis, we can express the nonaffine model in the form of the affine model, as shown in equation (67).
Next, we need to prove the boundedness of .
Theorem 3. If the control input is bounded, there must be a fixed value , such that
Proof. Construct the Lyapunov function
Taking time derivative along (69)
If will ultimately converge to the compact . Where . As a result, the boundedness of leads to the newly introduced system “” which is stable.
Define tracking error as
Taking time derivative along (71) and employing (67), it leads to
Define intermediate variable as
where is the design parameter.
Define tracking error as
Utilizing (67) and (73), the time derivative of is derived as
where . Since the function is an unknown function, the fuzzy function approximator is used to approximate it. The estimated value of the nonlinear function can be expressed as
is the input vector of the unknown function; is the parameter vector of ideal weight coefficient, fuzzy basis function . The unknown function can be expressed as
where stands for estimation error, ; represents the upper bound of estimation error.
The velocity control law is designed as
where , and its adaptive law is chosen to be
where is the design parameter.
Substituting equations (67) and (73) into equation (74), we can obtain
Substitute equation (80) into equation (78) to get
Theorem 4. Considering the closed-loop subsystem consisting of plant (67), with controller (81) and adaptive law (79). Then, all the signals involved are semiglobally uniformly ultimately bounded.
Proof. Consider the following Lyapunov function candidate:
with .
Applying (73), (74) into (72), is expressed as
Considering (75), (77), and (79), becomes
To simplify equation (84) and substituting (78) into (84) yields, we can get
From the fact that
we obtain
Thereby, equation (85) becomes
In this article, since, and applying (12), we know that .There exists a constant, such that . Hence, we get
Then, (88) becomes
Let , , . And define the following compact sets:
If or or , will be negative. Thus, , , and are semiglobally uniformly ultimately bounded. This is the end of the proof.
3. Simulation Results
To verify the effectiveness of the new control strategy proposed in this article, the proposed controller is tested in the simulation adopting the nonlinear models (1)-(7) implemented in MATLAB/SIMULINKs. The simulation step size is fixed, the step size is 0.01 s, and the simulation period is 100 s. The initial trim conditions of the FHV are shown in Table 1. The velocity step , altitude step . Both the velocity and altitude reference inputs are given by a second-order reference model with a damping ratio of 0.95 and a natural frequency of . All the aerodynamic coefficients and parameters are the same as [17, 23].
Design parameters are chosen as, , , , , , , , , , , and . In order to verify the robustness of the system, it is assumed that the aerodynamic coefficients of the FHV models have a perturbation of . where is the value of uncertain coefficient; denotes the nominal value of .
The fuzzy function approximators choose the Gauss basis functions as the membership functions, set the fuzzy set of each variable to 100. The input vectors are , . Where the fuzzy centers , , , , and of each fuzzy set are uniformly distributed on , , , , , and , respectively. The membership functions of the above variables are shown in Table 2. To show the superiority of the proposed method, we do the simulation study compared with recent research methods of others. The following two different cases are considered.
Case 1. The method proposed in this paper (controller 1) is compared with the novel back-stepping control method (controller 2) in [33] without adding the saturation auxiliary system.
The obtained simulation results are shown in Figures 1–7. As can be seen from Figures 1 and 2, both of the two methods can achieve the desired tracking effects, but the tracking accuracy of controller 1 is higher than that of controller 2 in both altitude tracking effects and velocity tracking effects. According to Figure 3, the attitude angle changes of the two methods are roughly the same, but the change amplitude of controller 1 is less than that of controller 2. Figure 4 shows that the trend of and are similar, the lines are relatively smooth, there is no high-frequency chattering phenomenon, and the amplitude of controller 1 is small. Figure 5 shows that and are bounded. Finally, variation curves of flexible states and the estimations of and are presented in Figures 6 and 7.
Case 2. Assuming that the actuators are subject to theoretical constraints:, . The closed-loop system simulation of FHV is carried out by using the method presented in this paper (controller 1) and the method of adaptive neural control with nonaffine dynamics (controller 2) in [34].
When the actuators are constrained, the simulation results using the two methods are shown in Figures 8–14. From Figures 8 and 9, it can be seen that both two methods can achieve stable tracking of reference altitude and velocity, but the altitude tracking error and velocity tracking error in controller 2 are relatively large, the maximum altitude tracking error is more than , and the maximum velocity tracking error is close to . At the same time, it can be seen from Figure 10 that the attitude angle changes of controller 1 are obviously smoother than that of controller 2. In Figure 11, although controller 2 can maintain the robustness of the system, the actuators are saturated for a long time. When controller 1 is saturated, it can also rely on the auxiliary system to maintain the stability of the system. In Figure 12, and are bounded. Variation curves of flexible states and the estimations of and are presented in Figures 13 and 14. To sum up, in both cases, the superiority of the proposed method can be demonstrated.
4. Conclusions
(1)In this paper, a fuzzy-approximation-based novel back-stepping control method is proposed. The method does not need virtual control laws, but only needs a final actual control law, and the control algorithm is more simplified(2)For each subsystem of FHV, only a fuzzy approximator is needed to approximate the total uncertainties in the subsystem. Meanwhile, the norm estimation approach is introduced to directly estimate the norm of the weight vector in the fuzzy approximator, instead of the element itself, which reduces the computational complexity of the control algorithm and guarantees the real-time performance of the algorithm(3)The control method in this paper is based on nonaffine models and uses low-pass filters to deal with nonaffine dynamics, avoiding the loss of dynamic characteristics in affine models(4)The saturation problem of actuators in nonaffine models is discussed. By designing an auxiliary system, the original control problem with input constraints is transformed into a new control problem without input constraints(5)In two cases, the simulation results are compared with the recent methods to verify the advantages of the proposed method
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Aeronautical Science Foundation of China (Grant No. 20175896023) and the National Natural Science Foundation of China (Grant Nos. 61603410, 61703424).