Adaptive fast fixed-time three-dimensional guidance law with acceleration saturation constraints

This paper investigates continuous anti-saturation three-dimensional guidance law against maneuvering target at the desired line-of-sight (LOS) angles. Firstly, in the light of fixed-time stability, fast nonsingular terminal sliding mode is designed without the limitation of odd-ratio fractional order. Then, with the limits of lateral acceleration saturation, continuous guidance law is proposed to guarantee that desired LOS angles are reached and maintained after the specified time, while adaptive law is implemented to estimate the information on maneuver accelerations of the attacking missile. Finally, numerical simulations are presented to validate the effectiveness and rationality of the designed scheme.


I. INTRODUCTION
Owing to the need of modern war, various missiles have developed rapidly. In the meantime, the theory and technology, which are related to missiles or other aircrafts, have attracted the close attention of scientific researchers. Therefore, several results are also achieved about guidance laws for intercepting coming targets. However, with requirement improvements of missile performance, it is still challenging to design guidance laws.
Among all guidance laws, the proportional navigation (PN) guidance law and its variants [1][2][3][4] have been widely investigated and implemented, because of their generality and simplicity. However, the engagement systems of intercepting maneuvering target can be essentially classified as uncertain nonlinear systems, and therefore various PN guidance laws, which do not possess the ability of suppressing disturbances, can not effectively solve this problem. In recent years, with the development of nonlinear control theory, various guidance laws have been proposed to deal with this issue, such as adaptive control [6][7], sliding mode control (SMC) [8][9][10], H  control [10], backstepping control [11]. Herein, SMC system, which has fast response and robustness against external noise and parameter perturbation, has been widely implemented and studied in the field of guidance laws.
Apart from robustness, finite-time convergence of LOS rates is important for terminal guidance of ballistic missile or air-to-air combat situation. Therefore, finite-time control schemes have been gradually proposed for guidance laws. Switching control was used to design robust guidance law for steering LOS angular rates to the neighborhood of the origin in the planar and three-dimensional (3D) environments [12], and then new finite-time guidance law [13] was developed to alleviate the cross coupling effect, which was ignored in [12]. Lateral acceleration was built to enforce nonsingular sliding mode on the switching hypersurface, and then the desired impact angle was achieved in finite time [14]. Similarly, SMC guidance law [15] was derived to guarantee that the flight-path angle can meet the desired impact angle with finite-time convergence. With regard to non-maneuvering target, continuous finite time control laws were developed to achieve high-accuracy guidance process, and furthermore one observer was united to detect target acceleration [17]. Based on the slightly complex and detailed three-dimensional engagement model, continuous finite-time guidance laws [16] were structured to intercept maneuvering target with the impact angles. These guidance laws could significantly reduce the miss distance and improve strike effect against maneuvering target.
However, the settling time of finite-time control system depends on the initial values of system state, and therefore fixed-time control algorithms, which remove this restriction, are subsequently investigated to design guidance laws. With impact angle constraints, robust guidance laws [18] were proposed to nullifying the LOS rates without decoupling cross couplings, and the convergent time can be set beforehand. Adaptive smooth fixed-time guidance law [19] was derived for intercepting maneuvering target based on fast stable system and nonsinguar terminal SMC (NTSMC) in planar homing engagement geometry. Fixedtime 3D guidance law was designed to intercept maneuvering target in the light of fast NTSM, saturation function and adaptive law [20]. By two-order consensus agreement and fixed-time SMC, distributed 3D cooperative guidance law [21] was presented to capture maneuvering target with desired impact angles. NTSMC was implemented to design fixed-time guidance law [22], while unknown target maneuver could be estimated by adaptive law. These results promote the study of guidance law to one new stage.
Saturation constraint is another important restrictive factor for practical second-order system. Until now, many researchers have studied this problem of the saturation constraint for missiles or other flight vehicles. Auxiliary system was constructed to tackle the effects of actuator magnitude constraints [23], while robust adaptive control scheme was proposed for air-breathing hypersonic vehicle by neural approximation and minimal-learning parameter technique. Integral sliding mode, adding power integrator and adaptive law are combined to solve spacecraft attitude tracking problem with actuator saturation, faults and misalignment [24]. In [25], approximate/adaptive dynamic programming was extended to broader nonlinear dynamic systems with asymmetry constraints. With actuator faults and input constraint, fixed/finite time guidance laws [26] were investigated to intercept maneuvering target, but unfortunately it was not deduced and proved in strict accordance with the concept of fixed-time stability. Backstepping and nonlinear disturbance observer are fully applied to guarantee robust tracking of altitude and velocity reference trajectories [27], and additional system was exploited to cope with actuator saturation. Inspired by the above literatures, robust 3D fixed-time guidance law is designed to hit maneuvering target in this paper. The main contributions of this article are provided as follows: (1) fast fixed-time system is analyzed, and NTSM is established to solve control problem of nonlinear systems; (2) novel fixed-time guidance law is proposed against incoming maneuvering missile, where the upper bound is estimated for the convergence time of LOS rates by Lyapunov stability theory; (3) the constraints of accelerate saturation and desired impact are considered, while adaptive law and boundary layer are utilized to eliminate the requirement of target maneuver information and chattering phenomenon, respectively.
The rest of this paper is arranged as follows. Section 2 states problem formulations and preliminaries. In Section 3, fixed-time NTSM is constructed, and then continuous guidance law is designed based NTSM technique and adaptive law, while fixed-time stabilization is deduced and analyzed. To demonstrate the effectiveness of the proposed guidance law, numerical simulation is provided in Section 4. Conclusions are finally drawn in Section 5.

A. PROBLEM FORMULATION OF 3D GUIDANCE
For simplicity, these assumptions are made: (1) the pursuer and target are considered as point masses; (2) seeker dynamics and the autopilot of the pursuer are fast enough to be neglected; (3) the speeds of the missile and target are constant, and the angle-of-attack is small enough to be neglected. The 3D homing guidance geometry is shown in Fig.1, where the relevant explanations are provided below. The interceptor is fixed at the origin of the reference frame.  According to Fig.1, the kinematic engagement equations can be derived from classical principles of dynamics cos cos cos cos cos cos sin cos sin sin sin cos tan cos sin cos tan sin cos tan cos sin cos tan sin cos where ym a and  . Therefore, in order to design guidance law and explore other rules, the above differential equations (1)-(7) are transformed into two-order engagement dynamics (8)-(9) between the accelerations and the LOS angles.
According to (8)- (9), it can be seen that there exist cross coupling terms between them, while acceleration components zm a and zt a simultaneously affect the changes of LOS angles L  and L  . Compared with engagement dynamics [12,21] of the spherical coordinate, the above model can express the relationship between lateral accelerations and LOS angles in more detail. Furthermore, this 3D guidance model can be also rewritten as  and zt a of the target are also bounded. Assumption 1 means that the relative distance r is the positive scalar. In the process of interception engagement, LOS angles constraint is reasonable for Assumption 2. Therefore, from Assumptions 1-2 and the relevant analysis, it can be concluded that Assumption 3 is also reasonable.

Consider the nonlinear system
with n x  , and : nn is an equilibrium point of nonlinear system (11). Then, some definitions and lemmas are provided as follows.
Definition 1 [30]. The equilibrium point of system (11) [29]. The equilibrium point of system (11) is fixed-time stable if the settling time is bounded and independent of initial conditions, i.e., max 0 0: Lemma 1 [28,29]. For holds, the trajectory of this system is practical fixed-time stable. The residual set of the solution for system (11) is provided by with 01 . The bounded time required to reach the residual set is estimated by

III. MAIN RESULTS
In this section, fast fixed-time TSM technique is developed, and then adaptive continuous fixed-time 3D guidance law will be proposed to nullify the LOS angular rates within the given time. The specific results are provided as follows.

A. NONSINGULAR FIXED-TIME FAST TSM
To construct the sliding mode surface of LOS angle errors, new variable is defined as and the derivative of this variable is  1 T , which is irrelevant with the initial value of the system states.
Design a Lyapunov function as Consider one new variable , and then taking its derivative yields By solving (18), the upper bound of the settling time can be computed as holds, the following result can be obtained that , it can be proved that 11 TT   is satisfied, and therefore system (10) is fixed-time stable after TSM variable is maintained on SM surface.
Recently, fast NTSM techniques have been applied to deal with guidance problem. In [19], fixed-time sliding mode variable can be expressed as Another fast fixed-time sliding mode can be seen in [20], which is provided as For these two fixed-time sliding mode algorithms, the power constants are       , and other parameters are selected as the same as the above.

Lemma 4.
If the parameter selection 11   is fulfilled, the convergent rate of (14) is faster than (21) after system trajectories reach fixed-time sliding mode surface. When the parameters satisfy 11   and 11   , the convergent rate of (14) is faster than (22).
Proof: Choose the same Lyapunov function as With regard to (22), the following inequality holds Therefore, the conditions of three parameters 11   and 11   are satisfied, the convergent rate of (14) is faster than (22). According to the similar approach, for the settling time 2 S T of (21), the corresponding result can be derived to meet 2 1 0 S TT   with 11   . That is to say, the convergent rate of (14) is faster than (21) after system trajectories reach fixed-time sliding mode surface. In addition, with the same parameters, the convergent time of (14) is less than (21) after sliding motion occurs.

Remark 2.
In [26], fixe-time NTSM was also designed to cope with guidance problem. However, the algorithm is only one special form of the proposed NTSM (12) or (14) when the circumstances 11  x , it can be deduced that the stability time of system states for (14) is less than that of [26] after sliding motion on the designed hypersurface. Remark 3. If system states are far from the equilibrium ) is fulfilled. When system states approach the equilibrium point,  

B. THE DESIGN OF FIXED-TIME GUIDANCE LAW
In this section, adaptive continuous anti-saturation guidance law is proposed to intercept maneuvering target with fixedtime convergence. Firstly, consider the limits of acceleration amplitudes, and 3D guidance system can be expressed as is the maximum amplitude of actual acceleration.

Remark 4.
The guidance law or acceleration is bounded as it will be designed and includes system states which are bounded. Therefore, additional control term s u can be considered bounded and reasonable.
The time derivative of the proposed TSM (12) can be expressed as Select 11 , Therefore, it can be concluded that guidance system is practically fixed-time stable. Through the analysis of the above three cases, it can be inferred that LOS angles and angular rates will reach the respectively.

Remark 5.
It should be noted that it is necessary to prove the fixed-time stability of the closed-loop guidance system by two steps. Because there is not the boundedness proof for the adaptive parameters in the first step, the following proof is not logically rigorous.

Remark 6.
Adaptive continuous anti-saturation guidance law (30)-(31) is in essence NTSM control algorithm, and therefore this technique consists of two stages. Firstly, NTSM surface is constructed according to fixed-time performance criterion for steering system trajectories to the equilibrium point or the neighborhood of the equilibrium point. Then, fixed-time reaching law is designed to force the system state to reach NTSM surface such that sliding mode occurs on this hypersurface. Therefore, the settling time of the closed-loop guidance can be estimated as 12 F T T T based on NTSM control principle and fixedtime stability theory.

Remark 7.
When the designed fixed-time guidance law is applied to engagement geometry, control parameters should be selected appropriately to accomplish guidance time, control energy consumption and miss distance. However, there is not one standard procedure to choose these parameters, which can be selected by trial and error until the specified indicators are satisfactorily acquired.

Remark 8.
If the initial conditions are far from the equilibrium point, it may need a big control action to guarantee the fast fixed-time convergence. So if there exits strong constraints on control action, the settling time can be extended by selecting appropriate parameters to guarantee fixed-time convergence with respect to guidance system. In the actual engagement process, the acceleration constraint, the predetermined convergent time and the miss distance should be considered at the same time in order to achieve reasonable balance.

IV. SIMULTTIONS
In this section, numerical simulations are carried out to illustrate the effectiveness of the presented guidance law to intercept one maneuvering target. Firstly, inertial reference system is fixed and centered at launch site at the instant of the launch, as shown in Fig.1. In this system, the X-axis is considered to be in the horizontal plane and points to the direction of launch with the positive Z-axis in the vertical plane, while the Y-axis is selected such that the reference frame forms a right-handed reference frame.
The missile's initial position is set at the origin of the inertial reference frame. Its initial velocity is 450 /  After applying the controller (30)-(31) to engagement geometry model (1)-(7), and the simulation results are shown in Figures 2-5. Figure 2 shows curves of engagement trajectories and relative distance under Case 1. It can be seen that the missile can precisely intercept maneuvering target with the desired LOS angles, and miss distance is 0.0364m for this case. Figure 3  S and 2 S in Figure 4(a), and continuous lateral accelerations are provided without chattering phenomenon in Figure 4(b), which initially exceeds the maximum limits until TSM variables 1 S and 2 S reach sliding surfaces, and then gradually converge to the small certain constants along with the convergence of LOS angular rates. Heading angles of missile and target are plotted in Figure 5; due to the designed guidance law, the curves of  Therein, the comparison results can be seen in Table 1 in detail.  In order to verify the performance of the proposed guidance law (30)-(31), the simulation comparison under Case 2 with the continuous NTSM guidance law [16] is carried out, and control parameters are the same as the above mentioned constants. Moreover, the maximum limits of missile acceleration remain unchanged. Figure 4 (a) and (b) describe the curves of three-dimensional engagement trajectory and the relative distance. It can be seen that guidance law can guarantee that the missile effectively hits maneuvering target with miss distance 0.4226m. The LOS angles and angular rates are shown in Figure 7  while it can be obtained that the accelerations are almost at the saturation state before 2 3.06 p T  s. Therefore, it can be inferred that energy consumption are more that of the proposed guidance law. Heading angles of missile and target are plotted in Figure 9 under Case 2. Compared with that of the proposed algorithm, it can be seen that missile heading angles change greatly. Other corresponding comparisons are shown in Table 2.   Figure 10 shows curves of engagement trajectories and relative distance under Case 2. It can be seen that the missile can also hit maneuvering target. However, guidance time has been increased, and miss distance is 0.0786m for this case. Figure 11 shows the varying profiles of LOS angles and angular rates, and it can be seen that the difference is relatively small compared with Figure 3. Fast nonsingular TSM variables 1 S and 2 S are described in Figure 12(a), continuous lateral accelerations are shown in Figure 12(b). Due to the varying maneuver, from Figure 12(b), guidance accelerations also change all the time after LOS angular rates are stabilized. Heading angles of missile and target are plotted in Figure 13 under Case 2; compared with the result of Case 1, there is little difference before 5 seconds; however, the fluctuation changes of missile heading angles occur along with periodical change of target heading angles after 5 seconds.

V. CONCLUSION
In this study, continuous fixed-time guidance law is designed with new NTSM technique and adaptive law, and prior information is not required. Furthermore, simulation results show LOS angular rates and NTSM variables can reach the neighborhood of the origin within the settling time, and the excellent performance is finally demonstrated for this algorithm. In the future, the balance between energy consumption, saturation restriction, guidance time and performance will be deeply investigated, while other guidance laws will be constructed to avert violation with respect to the constraint of LOS angular rates.