Fuzzy optimal tracking control of hypersonic flight vehicles via single-network adaptive critic design

—Optimal performance is extremely important for hypersonic flight control. Different from the most existing methodologies which only consider basic control performance including stability, robustness and transient performance, this article deals with the design of nearly optimal tracking controllers for hypersonic flight vehicles (HFVs). Firstly, main controllers are developed for the velocity subsystem and the altitude subsystem of HFVs via concise fuzzy approximations. Then, optimal controllers are nearly implemented utilizing single-network adaptive critic design. Moreover, the stability of closed-loop systems and the convergence of optimal controllers are theoretically proved. Finally, compared simulation results are given to verify the superiority. The special contribution is the application of low-complex control structure owing to the critic-only network and advanced learning laws developed for fuzzy approximations, which is expected to guarantee satisfied real-time performance.


I. INTRODUCTION
YPERSONIC flight vehicles (HFVs) are potential to serve as long-range transports and carriers of rapid and accurate attack weapons [1]- [5]. The design of control systems for HFVs continues to be a topic of important research interest, and it is inherently difficult due to the fact that the vehicle dynamics is nonlinear, coupled and uncertain. Furthermore, hypersonic flight controllers must handle narrow flight envelopes, rapidly time-varying flight circumstances and notable flexible effects. In addition to ensuring control authority over the entire flight envelope, optimal index is also necessary for HFVs' control systems to accomplish miscellaneous missions Hypersonic flight control has received worldwide attentions in recent years because of the interesting and hard-to-handle flight conditions connected with high Mach numbers. Thereby, considerable efforts have been made by researchers to exploit advanced controllers for HFVs for the purpose of attaining stable tracking of reference trajectories [6]- [9]. In the literature, stability performance of the closed-loop control system is a primarily considered index for HFVs by incorporating baseline controllers with additional terms [10]- [13]. In [12], a robust control method is studied for HFVs to maintain control stability and reject parametric perturbations. Firstly, dynamic inversion is combined with back-stepping to develop baseline controllers for the velocity subsystem and the altitude subsystem. And then, sliding mode switching terms together with neural estimators are applied to increase the tolerance of closed-loop control systems to external disturbances and model uncertainties. An alternative method which is available for resisting system uncertainties and external disturbances of HFVs is the active disturbance rejection control (ADRC) approach [14]. The difference from [12] is that it [14] only considers the attitude control issue. A common problem which arises in the hypersonic control domain is actuator faults/saturations [15]- [17]. This may result in tracking performance reducing even instability. A possible solution to this problem is to add auxiliary systems to baseline controllers. The compensation signals generated by auxiliary systems can stabilize hypersonic flight control systems in the presence of actuator faults/ saturations. Unlike the above disturbance-compensation methodologies, a new offset-free control approach is proposed in [18] to make control system resistant to disturbances and unknown dynamics. Besides, other estimators such as disturbance observers [8] and intelligent approximations [19]- [21] also are usually used to resist disturbances. On the other hand, except for stabilization, transient performance is widely considered to be a very significant index for hypersonic flight control systems. Prescribed performance control (PPC) [9], [22] has been shown to efficiently guarantee transient performance.
The key point of PPC is to devise performance functions which are used to impose funnel constraints on tracking errors. And then, the desired prescribed performance is realized owing to the boundedness of transformed errors [9], [22].
Most of the methodologies mentioned above however are aimed at guaranteeing steady-state performance and transient performance for hypersonic flight control systems under different actual conditions including uncertainties, disturbances and actuator saturations/faults. Unfortunately, only such basic performance isn't enough for hypersonic flight control and instead we must further seek some optimal performance indexes [23]- [27]. Adaptive critic design (ACD) is a newly emerging methodology to solve optimal control problems, and 3 4 5 6 7 8 the unknown term f  still leads to a problem related to the implementations of control laws, which is usually tackled using the adaptive strategy by online updating f  . This also results in high computational costs owing to too many elements of f  .
For an arbitrarily unknown function, we can obtain a lowcomputational approximation by directly turning 2 || || f  instead of its elements. This is very significant for HFVs to guarantee their control systems with excellent real-time performance.
Based on SACD, function approximations are necessary for critic network designs. Unfortunately, the above fuzzy system (8) isn't suitable for HFVs to devise critic networks. According to the gradient descend method, we must adjust all the elements of f  via developing adaptive laws based on Lyapunov theory.
Undoubtedly, the control real-time performance cannot meet the requirement of hypersonic flight control if we use fuzzy formulation (8) to develop critic networks. For this reason, we give another form of fuzzy approximations, called Fuzzy Hyperbolic Model (FHM) [37], to construct the critic network subsequently. FHM can be utilized to reconstruct an arbitrary dynamic system is a system state vector and are Hyperbolic basis function vectors.

Remark 2.
In what follows, we use FHM to design critic networks, which helps to reduce computation loads and ensure real-time performance because x φ and x A φ W degenerate into scalars in the design process of each critic network.

A. Vehicle controller design
This subsystem presents the design process of a fuzzy optimal controller for velocity subsystem (1) to make V→V ref and minimize cost function (20).
We represent velocity subsystem (1) as  is a main controller and *  is an optimal controller which is used to optimize performance index (20).
unknown but continuous function. Note that V a is a function of states and control inputs, and control inputs are functions of states since they are computed based on state-feedback controllers. Hence, we can use fuzzy system (8) is a weight vector and is an input vector. The estimation . We devise the following learning Utilizing (11) and (13) Because of The existing work [8] shows that 1 V  can be stabilized. For 2 V  , we will develop an optimal controller based on SACD.
We give the following dynamics constants V q and V r . We further obtain Hamilton-Jacobi-Bellman (HJB) equation Since V J  is an unknown term, we use FHM to construct the critic network via adaptively estimating V J .
tanh ( ) where Substituting (24) into (21), we have Then, the optimal control *  becomes   The HJB equation becomes . We develop the following regulation Chose Lyapunov function candidate Combining (16) and (33) It can be seen that V  and V A   are convergent, that is,

Remark 3. The sustained incentive condition leads to that
From (22) and

B. Altitude controller design
In this subsystem, we will develop a fuzzy optimal controller for altitude subsystem (2)-(5) via back-stepping such that h→href and makes cost function (64) minimal.
Similarly, we use fuzzy system (8) to approximate unknown functions a  and Q a . where + l    is a design parameter.
Define d x  as the estimate of  , and introduce the following Based on (39), e   is given by where Q e and Q s will be defined subsequently. s Q is a main virtual controller and * Q is an optimal virtual controller.
Define s Q as To obtain the time derivative of s Q , we give the following filter Step .5 2 with the constant According to (45), (49) and (54), we get The previous study [8] proves that there exist We further define Based on (58)-(60), we further obtain  In what follows, we will devise an optimal controller to handle the last term in (62).
The cost function is given by The total Lyapunov function is formulated as The time derivative of h L is described as      The proposed controller is compared with a neuralapproximation-based back-stepping control (NBC) [38] to validate its effectiveness and superiority. The values of model coefficients and parameters adopted in the simulation are referenced from [35]. The initial trim condition for HFVs is chosen as : V=7700 ft/s, h=85000 ft, γ=0 deg, θ=1.62

V. CONCLUSIONS
This article investigates an optimal tracking control problem of HFVs subject to unknown dynamics. The vehicle dynamics is consisted of the velocity subsystem and the altitude subsystem. Fuzzy approximations are applied to develop a robust tracking controller for the velocity subsystem via SACD. Furthermore, a back-stepping-based nearly optimal controller is devised for altitude dynamics. Advanced regulation algorithms are exploited for fuzzy approximations to construct a low-complexity control framework. Finally, the effectiveness and advantage of the addressed method are verified by simulation results.