Predefined-time prescribed performance guidance law design against maneuvering target under input saturation

A novel SMC-based prescribed performance guidance law combined with a base generator (TBG) is proposed. A TBG-based modified Lyapunov function is first proposed, which ensures that the LOS tracking error converges into zero within a predefined time. A novel sliding manifold with adjusting convergent time property is proposed, which brings about a smaller control magnitude and predefined time convergent rate. To ensure a satisfied transit and steady performance, some converted variables are defined based on which a novel prescribed performance guidance law is constructed. The input saturation problem is properly solved by designing a novel TBG-based one-order auxiliary system. Extensive simulation results have verified the effectiveness of the proposed method.


Introduction
With the increased maneuverability of the target and faster relative velocity in line of sight (LOS) reference, it demands that the modern guidance laws possess good robustness to disturbances and fast convergence of tracking errors.The sliding mode control (SMC) technique is one of the most popular methods for designing robust guidance law.Combined with the nonlinear observer technique, finite time, or fixed time control theory, some novel SMC-based guidance laws [1] were designed to realize a fast robust engagement against maneuvering targets.In [2], an impact-time-control guidance law was proposed for a non-maneuvering target.The authors in [3] designed a modified proportional navigation (IPN) guidance law to intercept a team of evading targets.
Given the shortness of the terminal process, the predefined time control theory, which enables stabilizing a closed system within a user-defined time, has been used in designing guidance laws recently.To realize a predefined-time consensus tracking for multiple nonlinear systems, a modified non-singular sliding mode with a desired temporal constant was proposed in [4].A robust predefined-time SMCbased control schema was proposed in [5] to solve the formation tracking issue of multiple underwater vehicles (AUVs).The works mentioned above about finite time, fixed time, or predefined time control theory have a common drawback: a huge amplitude of initial control inputs due to the big tracking errors.To deal with this issue, a novel control method combined with a time base generator (TBG) is gradually used to realize a predesigned time control, in which not only can the setting time of the closed system be set arbitrarily, but also the magnitude of the control signals can be reduced significantly.In [6], a IOP Publishing doi:10.1088/1742-6596/2764/1/012083 2 time-varying SMC-based guidance law was designed to ensure the LOS angle rates converge to zero at the interception time.The magnitude of the overload command signals has been greatly reduced.Moreover, to achieve a better formation tracking performance, a TBG-based distributed estimator was proposed in [7] to drive the estimation errors of the states of virtual leaders to converge to zero within a given time.
Apart from the requirement of terminal guidance precision, we also hope that the guidance and control system proposes a good transit performance.Prescribed performance control (PPC) has proven to be an effective method to achieve this control objective.An adaptive PPC controller for a strict feedback system was proposed by Bechlioulis and Rovithakis [8], where the transient process and the tracking error of steady states were restricted within a prescribed performance envelope.A non-singular sliding manifold combing with PPC was constructed in [9] to realize the engagement of a maneuvering target, in which the PPC was used to restrict the transit of the control process.To eliminate the chartering chattering phenomenon of SMC, a super-twisting sliding mode controller with prescribed performance was proposed in [10] to limit the overshoot and steady errors for the gyroscope control system.However, nearly none of the works mentioned above have taken the problem of input saturation into account which is inevitable in physical systems.Although there exist some research results about the anti-saturated guidance laws, the research works with the predefined time convergence, prescribed performance, and input saturation into consideration synchronously are quite limited, which is the motivation of our work.The novelties of this work are as follows: 1) A modified TBG-based Lyapunov function is constructed.Compared with the conventional predefined-time Lyapunov criteria, the proposed TBG-based function ensures that the convergent process possesses an adjustable temporal behavior with a lower risk of input saturation.
2) To ensure the tracking errors with prescribed performance, a novel non-singular TBG-based sliding manifold is proposed, in which two TBG functions are used to modify the reaching phase and sliding phase, based on which the transformed tracking variables converge into a small neighbor of zero within a given time.
3) Considering the external disturbances and input saturation, a new TBG-based compensator system with predefined time property is designed to further eliminate the control input saturation, and the upper bound of the external disturbances is estimated and compensated adaptively.

Model description
The terminal guidance process is investigated in our work.In the line of sight (LOS) coordinate frame, the dynamic in [1], [7], and [9] between an interceptor and a target in 3D space is presented as: ( Where R denotes the relative distance; L  and L  are the line of sight angles; To facilitate the prescribed performance guidance law design, we denote as the error-transformed variables, where ( ) is a prescribed performance function given by: We set and the error system in Equation ( 2) can be transformed as: The objective of this work is to design a predefined-time guidance law with prescribed performance against maneuvering target subject to input saturation and the terminal LOS angle constraints, that is:

Modified TBG-based Lyapunov function design
The conventional TBG function can only operate within a given temporal constant.To extend the application of TBG functions, a novel TBG-based lemma is proposed.Lemma 1: Considering a nonlinear system 0 ( ) ( , ( )), , (0) , we will obtain: Then ( ( ), )

V t t x
will converge into the following region Where ( ) t  is a time base generator (TBG) with a temporal constant f T , which satisfies that: Proof: We set ( ) , and Equation ( 6) can be written as: Where the relations and integrate both sides of Equation (10), which yields that: The proof has been completed.

TBG-based prescribed performance guidance law design
To satisfy the multiple terminal guidance constraints, an integrated PPC guidance strategy combined with the SMC and TBG functions is proposed.Inspired by the TBG function, a novel sliding manifold with predefined-time convergence is constructed as: , ( ) ( ) To achieve the guidance objective, the following adaptive predefined-time anti-saturation guidance law with prescribed performance is proposed.

Stability analysis
Theorem 1. Considering the guidance system in Equation ( 4), if the guidance law is selected as Equation ( 14) with the sliding mode defined in Equation ( 13) , the adaptive law in Equation ( 16), the anti-saturated compensator in Equation ( 15), and the TBG function in Equation ( 8), the following conclusions can be obtained.
1) The sliding mode S , the anti-windup compensator state χ , and the adaptive variables 3) The LOS tracking error 1 e always satisfies the prescribed performance given in Equation (3) during the whole guidance stage.
Proof: The proof is divided into two steps, when Lyapunov candidate function, and take its time derivative.We can obtain: T d d (17) We substitute the Young's inequality-based relation into Equation ( 17) and obtain: .
According to Lemma 1, 1 V will converge into the following region 1 V Ω within a predefined time 1 f T , which means that the sliding mode S and the variables χ and 2 d are bound.Therefore, Conclusion 1) has been proved.
It shall be noted that choosing some small parameters s k , , and  ensures the sufficiently small convergent region , we rewrite Equation (13) as: , we will design as the candidate Lyapunov candidate function.Its time derivative yields: According to Lemma 1, 2i V will ultimately converge into a small region around zero within a fixed time , according to Equation (13), we will take its time derivative of V will be non-increasing if 1i    , and 2i V will eventually converge to the origin within a fixed time  4), the LOS tracking error 1 e will always satisfy the prescribed performance, and conclusion 3) is valid.The proof has been completed.

Simulation results
To validate the effectiveness of the proposed guidance law, a scenario in which an interceptor engages a sinusoidal maneuvering target is simulated, whose amplitude is  As can be seen from Figure 1, all the tracking errors of LOS angles are maintained in a user-defined prescribed function, where both the steady and the transit control performance are greatly improved.Figure 1 (d) shows that almost no maneuvers of interceptors are needed after the overall predefined time because the goal of zeroing the LOS angles has already been realized by introducing two TBG functions in the design of guidance laws.Such property is vital to improving the guidance accuracy.The evolution of LOS angles is presented in Figure 1 (e), where the control variables converge to zeros within a given time.In Figure 1 (f) and (g), a similar convergence can be found for the sliding mode and the transformed errors.The adaptive variables of external disturbances are shown in Figure 1 (h), which enables a robust prescribed performance engagement against a maneuvering target with predefined time convergence.The simulation results have verified the effectiveness of the proposed method.

Conclusion
In this work, a novel anti-saturated SMC-based guidance law with a predefined time convergence is proposed.A new sliding mode is constructed based on TBG functions, whose convergent time can be preset in advance.To ensure the tracking error within a prescribed bound, a transformed second-order system is designed.Based on Lyapunov's theory, a strict theoretical proof is presented to guarantee that all tracking errors converge to a given prescribed bounds within a predefined time.The numerical simulations represent the effectiveness and superiority of the proposed guidance law.Future work will seek to apply the results for a cooperative interception against a maneuvering target.
on the above analysis, it concludes that ( ) t μ is bounded during the whole guidance process.Due to the relation in Equation (

LdFigure 1 .
Figure 1.Simulation results for an interceptor engaging a sinusoidal maneuvering target under different predefined times.The simulation results are shown in Figure 1, where Figures 1 (a) and (b) show an effective interception achieved under different predefined times.Figure 1 (b) implies that the terminal guidance time can be shortened by selecting the proper predefined time.As can be seen from Figure1, all the tracking errors of LOS angles are maintained in a user-defined prescribed function, where both the steady and the transit control performance are greatly improved.Figure1 (d)shows that almost no maneuvers of interceptors are needed after the overall predefined time because the goal of zeroing the LOS angles has already been realized by introducing two TBG functions in the design of guidance laws.Such property is vital to improving the guidance accuracy.The evolution of LOS angles is presented in Figure1 (e), where the control variables converge to zeros within a given time.In Figure1(f) and (g), a similar convergence can be found for the sliding mode and the transformed errors.The adaptive variables of external disturbances are shown in Figure1(h), which enables a robust prescribed performance engagement against a maneuvering target with predefined time convergence.The simulation results have verified the effectiveness of the proposed method.