RLV integrated guidance and control based on adaptive high-order sliding mode

In this paper, a novel integrated guidance and control algorithm based on adaptive high-order sliding mode is proposed for reusable launch vehicle subject to unknown disturbances and actuator faults. We propose a time-varying barrier function-based adaptive control law to offset the effects of uncertainties. The remarkable feature of the developed algorithm is its ability to track the reference commands in finite time despite unknown disturbances and actuator faults, without designing the guidance law and attitude controller separately. Finally, the effectiveness of the proposed algorithm is confirmed by the simulation results.

sufficient robustness is essential to ensure stable flight. Traditionally, the G&C system is structured into of two parts, with the guidance subsystem and the attitude control subsystem being designed separately. In general, there are two strategies for creating guidance commands. The first strategy involves storing the reentry trajectory calculated offline and comparing it with the actual flight data in advance to obtain the feedback guidance law [1][2][3][4]. The second strategy, called the predictor-corrector guidance method, does not rely on a reference trajectory but constantly predicts the end of the flight and adjusts the control amount according to the deviation between the prediction and the desired target [5][6][7][8]. Sliding mode control (SMC) [9], model predictive control [10], robust fuzzy control [11], and adaptive dynamic programming [12] have been proposed for improving the guidance performance and eliminating the influence of model uncertainties.
After obtaining the guidance command, the design of attitude controller becomes the top priority. SMC is an attractive special nonlinear control method for attitude control of RLV due to its inherent insensitivity and complete robustness against external disturbances [13]. To reduce control chattering in traditional SMC, a quasi-continuous higher-order sliding mode control (HOSMC) is designed for RLV with bounded comprehensive disturbances, which could provide high accuracy in realization [14]. However, the upper bounds of disturbances are required in the controller design, which is not feasible in some practical applications. Therefore, the issue has motivated the researchers to develop adaptive sliding mode. In [15] an adaptive multivariable finite-time control algorithm based on supertwisting algorithm is proposed and applied in attitude control of RLV with unknown disturbances. Furthermore, to address the issue of overestimated adaptive gain, an adaptive multivariable control algorithm is developed in [16], which utilizes a time-varying barrier function to design the adaptive control gain. This approach ensures that the gain is as small as possible but large enough to resist disturbances.
To maintain the safety of RLV system subject to actuator fault, the fault-tolerant control (FTC) has been extensively studied. In [17], the combination of fixed-Time observer and finite-time FTC, where the observers are designed to estimate the information of actuator faults and model uncertainties. The results are further extended in [18], which allows the fault compensation to be completed in a fixed time. In [19], an anti-saturation adaptive backstepping attitude control scheme is proposed, and the convergence of the closed-loop system can be guaranteed within a predefined time. Moreover, the FTC that integrates a learning method is proposed in [20] to improve the performance of intelligent approximation. Specifically, neural learning is utilized to enhance the accuracy of the approximation results, which can be beneficial in situations where the system dynamics are uncertain. The aforementioned fault-tolerant control methods typically operate within individual attitude control loops.
Although significant efforts have been made to improve the performance of G&C system, instability may still occur in separated G&C system due to the spectral separation between the guidance and attitude subsystems. Redesign the subsystems separately can be time-consuming, and satisfactory results may not be guaranteed when the overall system performance is inadequate [21]. In contrast to the traditional separated G&C, integrated guidance and control (IGC) regards the guidance system and the control system as a single-loop system, and it is able to improve the coordination and matching degree of G&C system, reduce the amount of design iterations and calculation time. Due to the significant potential benefits of IGC systems, numerous research efforts have been directed towards IGC design, such as small-gain method [22], inverse dynamic method [23], robust model predictive control method [24], barrier Lyapunov function-based method [25] and so on. Moreover, the backstepping approach is well-studied for designing IGC systems due to their strict feedback structure. In [26], a backstepping based multiconstraints adaptive scheme is developed, where a saturation function is employed to guarantee the prescribed performance. In [27], the global stability of RLV closed-loop system is guaranteed through a finitetime IGC method. However, when using the backstepping approach, differential explosion phenomenon may occur. SMC is also a potential candidate for IGC, and HOSMC can mitigate the problem of arbitrary degree dynamic system. In [28], a HOSMC based smooth controller within the single-loop IGC structure is presented for a missile interceptor, where the upper bounds of disturbances are required in the IGC design. Nevertheless, in RLV practical applications, disturbances with unknown upper bounds are usually present. Thus, the issue has motivated the researchers to develop adaptive HOSMC. The Lyapunov based adaptive HOSMC is proposed in [29], where the presented controller gain can be adjusted in both increasing and decreasing directions rapidly. Further research were made in [30] and an adaptive controller using the barrier functions is proposed to achieve the finite-time convergence of the sliding variable without the knowledge of the upper bounds of disturbances. However, there are few studies that have applied the adaptive HOSMC to the IGC system of RLVs.

Motivation and contribution:
Inspired by the observations, we have developed a novel barrier functionbased adaptive HOSMC algorithm for the IGC system to improve its performance within the single-loop integrated and control structure. In contrast to the traditional guidance and attitude coordination control in [31], the single-loop G&C structure offers the opportunity to control the vehicle using a single feedback path, which makes the controller easier to design. Compared to the IGC method in [32], the proposed adaptive HOSMC algorithm provides the advantage of guaranteed convergence properties, even in the presence of unknown efficiency loss and external disturbances. Furthermore, the proposed algorithm ensures the finitetime convergence of solutions to a prescribed function from the initial time, making it easier to implement than the switched controller outlined in [30], which requires the use of a detection mechanism. The main contributions of (1) A novel adaptive HOSMC algorithm is proposed for IGC system of RLV, and it has proven to fulfill the tracking requirements of RLV even in the presence of unknown external disturbances and control efficiency loss. (2) The developed algorithm can guarantee the finitetime convergence of solutions to a prescribed function from the initial time, without utilizing the knowledge of upper bounds of disturbances.
The structure of the remainder of this paper is organized as follows. The problem studied in this research is formulated in Sect. 2. In Sect. 3, the adaptive HOSMC is provided in detail and the proposed algorithm is applied to IGC system design for RLV. Some representative simulation tests are provided in Sect. 4. Finally, the conclusion and future work are presented in Sect. 5.

Integrated guidance and control Model
In this paper, the unpowered RLV system is considered, whose nominal IGC model can be described by a set of translation and rotation differential equations as [33] x M x + I p,z M z + I p, pq pq + I p,qr qr, q = I q,y M y + I q, p p 2 + I q,r r 2 + I q, pr pr, r = I r,x M x + I r,z M z + I r, pq pq + I r,qr qr, (1) where h represents the flight altitude; v is velocity; γ is flight path angle; α, β, σ denote attack angle, sideslip angle and bank angle, respectively; p, q, r denote roll, pitch and yaw angular rates; g is the gravity acceleration and R E is radius of Earth, whereas L and D are the lift and drag accelerations denoted by L = ρv 2 SC L /2m and D = ρv 2 SC D /2m, where ρ is atmospheric density, S and m represent the mass and reference area of the vehicle; the aerodynamic coefficients are function of attack angle, which are given by C L = cl 0 + cl 1 α and C D = cd 0 + cd 1 α + cd 2 α 2 . I p,x , · · · , I r,qr are moments inertia coefficients whose expressions are given in the Appendix.A. The control inputs are M x roll moment, M y pitch moment and M z yaw moment. The main objective of the present work is to develop an IGC algorithm which makes altitude, velocity and sideslip angle track their desired values. The block diagram of IGC system of RLV in this paper is presented in Fig. 1.

Input/output linearization
Considering the tracking requirements of the IGC system, the sliding surface can be established as Then, to facilitate controller design, we differentiate the sliding surface such that it can be written in a form in which control moments appear explicitly. Taking the derivatives of the altitude and velocity, the following formulas are obtained where the drag and lift acceleration derivatives with respect to time can be computed as It is assumed that the atmospheric density ρ and the gravity acceleration g depend only on the altitude. And the time derivatives of the aerodynamic coefficients can be computed aṡ By differentiating the attitude angles twice with respect to time, we have the controls appearing inṗ,q With all these relationships, taking the actuator faults and external disturbances in to consideration, the entire linearized input/output model can be obtained, which has the following compact forṁ The matrices S, A, B, M are defined as where the expressions for the terms a h , a v , · · · , b β,z in A and B are given explicitly in Appendix.A, whose values depend on the states which are known at every moment if all system states are mensurable. Δ M denotes additive actuator fault and δ(t) denotes the efficiency loss function with δ m ≤ δ(t) ≤ 1; Δ h , Δ v , Δ β represent the external disturbances imposed on each channel; M x , M y and M z are the control moments needed to be determined.

Assumption 1
It is assumed that all the states in model (1) are available for feedback.
Remark 1 Taking the control moments as the control input, the relative degree of (7) is 4 + 3 + 2 = 9, equals to the order of IGC system. Thus, the nonlinear model can be linearized completely [34]. Moreover, with the states available in model (1), the system's higher-order derivatives of states could be obtained by rigorous mathematical expressions.
In next section, we will provide the adaptive HOSMC algorithm, which can ensure system (7) convergent to the origin in finite time.

Adaptive high-order sliding mode controller
In this section, in order to facilitate the design of the controller, the IGC system in (7) can be rewritten as which can be converted to the following general nthorder integral chain system.
where z 1 is the output variable, which can represent s h for n = 4, s v for n = 3 and s β for n = 2. z 2 , · · · , z n are internal variable, a z is the known system term, u represents the system input, δ(t) is the efficiency loss function, and d(t) is the unknown disturbance.
In what follows, the proposed algorithm is developed base on (10) without loss of generality. With the controller defined by u =ũ − a z , (10) can be transformed intȯ Then, without loss of generality, the developed algorithm is progressed on (11). Moreover, from a practical point of view, the system states and the uncertainty are always bounded as it is derived from a finite vehicle response, so the following assumption could be made to facilitate controller design.

Assumption 2
It is assumed that the uncertain function ζ (t) is bounded with unknown upper bound and there exist constant ζ > 0 such that |ζ m | ≤ ζ holds.
We now present an adaptive time-varying high-order sliding mode controller for system (11), and the construction of the proposed algorithm relies on the following lemma: Lemma 1 [35] Consider system (11) with δ ≡ 1 and ζ ≡ 0. Suppose there exists a continuous statefeedback control lawũ =ũ 0 (z), a positive definite C 1 function V (z) : R r → R + , and real numbers c > 0 and 0 < η < 1, such that the following condition is true for every trajectory z = [z 1 , · · · , z n ] T of system (11) Then system (11) with the feedbackũ 0 (z) is globally finite time stable with respect to the origin.
Regarding our problem, an adaptive function is adopted to handle the disturbances. In the spirit of the work of Laghrouche et al [30], with the knowledge of the initial value of the system, the adaptive function is defined based on the following time-varying barrier function where z 0 is the initial value of z, and V (z) is provided by Lemma 1. f : R + → R + is a non-increasing C 1 function, with lim t→+∞ f (V (z 0 ), t) ≥ 0 and f (V (z 0 ), 0) = + V (z 0 ), where is tuning positive constant. The proposed adaptive gain K (V (z), V (z 0 ), t) can change according to the current value of the Lyapunov function V (z). When V (z) increases towards the boundary of f (V (z 0 ), t), K increases accordingly, which forces the state to converge. On the other hand, when V (z) is going to zero, the adaptive gain decreases till the value which allows to compensate the disturbance, which avoids the overestimation control gain.
Remark 2 Compared with the barrier function-based piecewise function in the work of Laghrouche et al [30], the time-varying barrier function in (13) allows V (z) to stay within f (V (z 0 ), t) from the initial instant, so the detection mechanism can be avoided.
In what follows, a theorem is provided to summarize the main results of the developed algorithm.

Theorem 1 Considering system (11) with Assumption 1, if the feedback control law is defined by
where k 1 and k 2 are positive tuning parameters, K is the time-varying adaptive function defined in (13).ũ 0 (z) is any state-feedback nominal control law that satisfies Lemma 1 and obeys the following further conditions: where V (z) is the Lyapunov function introduced in Lemma 1, which can ensure the finite-time convergence of the nominal part of system (11). Then system (11) with the feedbackũ (z, t) is globally finite time stable with respect to the origin, and function V (z) will be confined in the prescribed region f (V (z 0 ), t). Proof 1 Consider system (11) and the control law defined in (14): Considering the Lyapunov function in Lemma 1 with a new control inputũ(z, t), We can obtain the following inequality for the time derivative of V (z) along the system (11).
Notice that (12), and (∂ V /∂z n )ũ 0 ≤ 0 holds according to the condition in (15). Therefor, we havė For brevity, define the following function and taking into account the definition of K in (13), the solution of F(V (z)) = 0 yields a unique solution Arguing by contradiction, one gets that V (z) > V * (t) for all non-negative time, and hence leads toV < −cV η . This yields the system convergence to the origin in finite-time, which is a contradiction. Then it can be deduced that system (11) is finite-time stable with respect to the origin for all V (z) > V * (t), and V (z) is deceasing accordingly, which leads to V (z)) < V * (t) < f (V (z 0 ), t). On the other hand, once V (z) increases and crosses the boundary of V * (t),V < 0 and V (z) decreases. Additionally, it follows from the properties of barrier function that when V (z) tends to the boundary of f (V (z 0 ), t),V tends to negative infinity, which implies that V (z) will always be confined in the prescribed In what follows, the algorithm in the work of Hong [36] will be utilized to develop the controller lawũ 0 (z). To show that, with x λ denotes |x| λ sign(x), the controller is defined as follows: Let κ < 0 and l 1 , · · · , l n be positive real numbers. For z = (z 1 , · · · , z n ), the controllerũ 0 (z) = ν n can be defined as where m i = 1 + (i − 1)κ, λ 0 = m 2 , (λ i + 1)m i+1 = λ 0 +1 and μ i = m i+1 /m i . And the Lyapunov function V is defined as With the definition of V , the conditions in (15) can be verified by ∂ V ∂z n ν n = −l n z n λ n−1 − ν n−1 and ν n = 0 if and only if ∂ V /∂z n = 0. Finally, with the development of the feedback controller in (14), the adaptive HOSMC algorithm applied to RLV system can be summarized as the following proposition.

Proposition 1 Considering IGC input/output model (7), if the control moment vector M is designed by
with virtual control vector U designed by where the nominal controllersũ 0 (s h ),ũ 0 (s v ),ũ 0 (s β ) and Lyapunov functions V (s h ), V (s v ), V (s β ) are designed based on the algorithm in the work of Hong [36], whose explicitly forms are given in Appendix.B. Then the RLV system could be able to track the reference altitude h re f (t), velocity v re f (t) and sideslip angle β re f (t).
Proof of Proposition 1. Since model (7) is a special case of the general nth-order system (10) discussed in Theorem 1, the convergence of the IGC tracking system can be proved, similar to Proof 1, which is omitted here.

Simulation results analysis
In this section, simulation experiments are provided to demonstrate the effectiveness of the developed algorithm. The RLV characteristic parameters used in Finally, we make a comparison between the proposed IGC algorithm and the separate guidance and control (SGC) strategy developed in the work of Tian et al [15]. Figures 2 and 3 illustrate the trajectory tracking results of RLV system. From the results, it can be observed that developed IGC algorithm can effectively drive the RLV to track the desired reference command, overcoming the effects of disturbances and actuator faults. Meanwhile, to illustrate the difference in control effects between the two methods, the enlarged images are provided in Fig. 3 that show a deviation between the SGC and the reference. This suggests that actua- tor faults may cause the SGC system to fail despite its ability to handle perturbations in initial conditions as demonstrated in previous work [15]. To better illustrate the convergence characteristic of the system, Fig. 3 provides the derivatives of the tracking errors, which shows that the developed adaptive HOSMC can effectively guarantee convergence performance for all orders of system derivatives, and it more clearly demonstrates the advantages of IGC over SGC. It should be noted that the convergence time of the system is mainly related to the initial deviation and the parameters of the nominal controllersũ 0 . Finally, as the control moments illustrated in Fig. 4, the convergence characteristic of the system is guaranteed with continuous control inputs. The above results verify the effectiveness of the developed IGC scheme based on adaptive HOSMC even in complex disturbance environment.

Conclusions
A novel adaptive HOSMC based IGC algorithm is developed for RLV guidance and control system subject to unknown disturbances and actuator faults. The proposed approach provides the finite-time stability utilizing a time-varying barrier function based adaptive controller, which is able to restrict the RLV tracking error in a prescribed region from the initial instant. The remarkable advantage of the proposed method is that it can guarantee the finite-time convergence of solutions, even in the presence of unknown external disturbances and control efficiency loss, without utilizing the knowledge of upper bounds of disturbances. The developed algorithm is compared with a separate guidance and control method, and simulation results demonstrate that the proposed IGC system has better robust performance and can ensure the system state to track the reference Fig. 4 Curves of control moments trajectory in a finite time. The primary limitation of this method is its reliance on a verified nominal controller, and it is difficult to determine the exact convergence time of the system. Future work will focus on the combination of the proposed algorithm with fixed-time control method.