Robust finite-time guidance against maneuverable targets with unpredictable evasive strategies

https://doi.org/10.1016/j.ast.2018.04.004Get rights and content

Abstract

This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the interceptor and the target, but also by external un-modeled factors. By robustly stabilizing a line-of-sight rate, this paper proposes an integrated continuous finite-time disturbance observer/bounded continuous finite-time stabilizer strategy. The design of this integrated strategy has two points: 1) effect of a target maneuver is modeled as disturbance and then is estimated by the second-order homogeneous observer; 2) the finite-time stabilizer is actively coupled with the observer. Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are discussed. Numerical comparison studies demonstrate the guidance performance.

Introduction

Interception of a maneuverable target is one of essential questions in the study of homing guidance. With the advancement of artificial intelligence, propulsion, composite material, etc, evasive capability of the target has been increasing at a rapid pace, and poses a challenging problem. This new guidance problem is referred to as Unpredictable Maneuverable Target Interception (UMTI) herein; that is, the interceptor cannot predict the evasive strategy of the target. The existing evasive strategies consist of conventional maneuver models [1], [2] and optimal evasive strategies [3], [4], [5]. The conventional maneuver models, such as a step maneuver, depend on prescribed maneuvers. The optimal evasive strategies are completely determined by the relative motion information of the interceptor and the target. These two types cannot be applied to the UMTI since both of them do not explicitly take account of the external un-modeled factors, particularly involving the deterministic yet unknown. In addition, considering that maneuvering capability of the target is comparable to that of the interceptor, the interceptor should be capable of exploiting its limit maneuverable capability, that is, control saturation.

This paper is concerned with designing a robust guidance law to achieve the UMTI. In the literature, a number of guidance laws have been investigated to deal with the problems of the maneuverable target interception. These guidance laws can be roughly classified into two categories: relative-motion-prediction-based guidance law (RMP-GL) and manifold-stabilization-based guidance law (MS-GL). The RMP-GL usually obtains an optimal solution subject to a predetermined interception engagement (such as an ideal collision triangle scenario) using prediction of the relative motion between the interceptor and the target. The design tools of the RMP-GLs mainly include optimal control theory and differential game theory. For example, in [6], assuming an explicit model of the target maneuver and a first-order missile dynamics, the interception problem is formulated as a linear quadratic control problem; in [4], a nonlinear 3D-vector guidance law is designed, which is an optimal strategy pair in the sense of the saddle-point inequality. The RMP-GLs, however, cannot be applied to the UMTI since the prediction of the intercept motion (including a time-to-go and an intercept point) is difficult to carry out due to significant uncertainties of the target motion.

With respect to the MS-GLs, they render the interceptor-target relative motion around a prescribed manifold by compensating for (or suppressing) adverse effect of various disturbances and uncertainties related to the target maneuver. The design tools mainly consist of adaptive control, sliding mode control, high-gain control, etc. In [7], the authors parameterize upper bounds of the target acceleration and then develop the resulting parameter adaptive laws, thus handling the target maneuver. As such, the control saturation and the speed of the parameter adaptation are two fundamental concerns in adaptive control design [8], [9], [10]. To address these two concerns, in [11], a set of relative-state-dependent basis functions is selected to represent the target acceleration, and then a weight vector adaptive law is derived using optimal modification technique. Although the adaptive guidance laws work well in the above-mentioned scenarios, they have three drawbacks herein: 1) the guidance performance heavily depends on the convergence of adaptive parameters, while these parameters are easily prone to diverge when the jerk of the maneuverable target is considerable; 2) the robustness to the target maneuver relies on the parameterization of the target maneuver, while it is very hard to accurately parameterize the target maneuver under study; 3) the order of the closed-loop guidance system will inevitably increase as the dimension of the adaptive parameters grows.

The sliding mode control is used to suppress the effect of the target maneuver in [12], [13]. In [14], the high-order sliding mode control is adopted to estimate and to compensate for the effect of the target maneuver. In this regard, the sliding mode guidance laws are upper-bound-dependent. For the UMTI, the upper bounds of the target acceleration are required to be chosen relatively big for the sake of completely canceling the adverse effects of the target maneuver. Consequently, the resulting control is conservative and may induce severe chattering due to various kinds of modeling imperfections [15]. As another way, a smooth sliding mode guidance law is proposed in [16], in which an adaptive law is used to estimate the upper bound of the target maneuver. However, the adaptive parameter may diverge under the situation of control saturation.

In [17], the high-gain control is used to ensure input-to-state stability for a closed-loop LOS rate dynamics, and thereby the residual error of the LOS rate can be made sufficiently small in the presence of the target maneuver. In [18], a finite-time guidance is proposed to nullify the LOS rate, and the convergence boundary layer is theoretically analyzed. The guidance performance of the high-gain control, however, may deteriorate due to measurement noise when the system bandwidth is excessively enlarged. In fact, it is difficult to make a reasonable trade-off between the disturbance suppression and the noise attenuation without the priori information of the evasive strategy of the target.

Following the technical route of the MS-GLs, a robust finite-time guidance (RFTG) law is presented in this paper. An integrated continuous finite-time disturbance observer (CFTDO)/bounded continuous finite-time stabilizer (BCFTS) strategy is proposed. The CFTDO is designed to estimate the disturbances that are the effects of the target maneuvers by making the observation-error dynamics behave as a second-order homogeneous system. The BCFTS uses a typical first-order homogeneous system to specify the finite-time stability of the nominal guidance system in consideration. Both the CFTDO and the BCFTS are based on non-smooth yet continuous feedback control. Such non-smooth feedback control has a favorable characteristic: its equivalent control gain increases as the feedback error decreases. In contrast to existing smooth feedback controls, the non-smooth feedback control may have better convergence and robustness [19], [20] and is suitable to be applied to the UMTI. Given these virtues, a few guidance laws based on the non-smooth feedback control have been designed. For example, the finite-time convergence of the LOS rate is realized in [12], but the effect of the target maneuver is suppressed using the conventional sliding mode control; in [21], using the high-gain control, the authors present a guidance law with finite-time input-to-state stability (FTISS). Note that the robustness of these two guidance laws is routinely guaranteed by the sliding mode control or the high-gain control. As discussed previously, these two control methods have their own drawbacks when adopted in the UMTI.

To address the preceding drawbacks, the integrated strategy is put forward with two key ideas: 1) the effect of the target maneuver is modeled as the disturbance and is estimated using the CFTDO; 2) the design of the BCFTS is coupled with that of the CFTDO. An unique characteristic of this integrated strategy is that it can explicitly deal with the interplay between the CFTDO and the BCFTS. This characteristic is different from the existing finite-time disturbance observer-based control methodologies (e.g., [22], [23], [24], [25], [26]), which design the observer and the finite-time controller in a decoupled fashion: they are based on a zero-observation-error assumption. However, the observation errors cannot be perfectly canceled in practice, especially concerning the UMTI. Nevertheless, the proposed RFTG law can avoid such a restricted assumption because of the integrated design adopted herein.

In addition, the proposed integrated strategy has several appreciable advantages in the current scenario. The BCFTS, which uses the control saturation to quickly stabilize the LOS rate, can take advantage of the limit of the maneuver capability of the interceptor. Regarding the CFTDO, it can rapidly estimate the disturbance of interest without suffering from the parameter convergence issues. Through cooperation between the BCFTS and the CFTDO, the resultant RFTG law is independent of both the explicit maneuver models and the upper bounds of the target maneuvers. Further, the RFTG law can guarantee locally FTISS in the presence of bounded derivative of the disturbance, implying the LOS rate can be rendered within a bounded error as quickly as possible. Numerical comparison results demonstrate the proposed RFTG law can guarantee the high-precision miss-distance.

This paper is organized as follows. We begin by formulating a new short-range interception problem. In particular, a notion of the unpredictable evasive strategies is introduced. The next section presents the robust finite-time guidance law. Then, the convergence analysis of the LOS rate is conducted in non-saturation and saturation cases. Finally, numerical comparison studies are performed to assess the guidance performance.

Section snippets

Problem formulation

The design and analysis of the RFTG law is based on the following assumptions:

Assumption 1

The motion of the interceptor and the target is in a planar plane.

Assumption 2

Both the interceptor and the target perform maneuvers orthogonal to their velocity vectors.

Assumption 3

The speeds of the interceptor and the target, VM and VT, are constant, and VM>VT.

The equations of the relative motion between the interceptor and the target are given as follows [27]:r˙=VMcos(λγM)+VTcos(λγT),λ˙=VMsin(λγM)rVTsin(λγT)r,ω˙=2r˙ωrcos(λγM)a

Robust finite-time guidance design

In this section, the RFTG law is presented: first, the CFTDO is designed to estimate the effect of the target maneuvers; second, the integrated CFTDO/BCFTS strategy is proposed to robustly stabilize the LOS rate. Note that the design of both the CFTDO and the BCFTS is based on homogeneity technology, which is able to construct homogeneous dynamic systems with favorable finite-time stability. The fundamental notions used herein are briefly introduced in Appendix A.1, and Appendix A.2.

Convergence analysis of LOS rate

This section presents the main theoretical results of the proposed RFTG law. Using homogeneity technique, global finite-time stability is established for the CFTDO. Based on Lyapunov theory, the convergence analysis of the LOS rate is conducted in the linear region of the saturation function in (10) and in the case of the control saturation, respectively.

Simulations

In this section, numerical comparison results for the RFTG law and the other three guidance laws are presented. Based on a target maneuver model of the UMTI, case studies are also carried out where guidance performance is systematically assessed for the proposed RFTG law.

Conclusion

In this paper, the integrated continuous finite-time disturbance observer (CFTDO)/bounded continuous finite-time stabilizer (BCFTS) strategy is proposed to approach the problem of intercepting a maneuvering target with an unpredictable evasive strategy. The local FTISS of the proposed RFTG law is established in the case of the bounded derivative of the effect of the target maneuver. Numerical comparisons demonstrate the favorable guidance performance of the RFTG law in terms of the homing

Conflict of interest statement

There is no conflict of interest.

References (33)

  • P. Zarchan

    Representation of realistic evasive maneuvers by the use of shaping filters

    J. Guid. Control Dyn.

    (1979)
  • R. Atir et al.

    Target maneuver adaptive guidance law for a bounded acceleration missile

    J. Guid. Control Dyn.

    (2010)
  • V. Turetsky et al.

    Target evasion from a missile performing multiple switches in guidance law

    J. Guid. Control Dyn.

    (2016)
  • S. Gutman et al.

    3d-nonlinear vector guidance and exo-atmospheric interception

    IEEE Trans. Aerosp. Electron. Syst.

    (2015)
  • R.H. Chen et al.

    Homing missile guidance and estimation under agile target acceleration

    J. Guid. Control Dyn.

    (2007)
  • I. Rusnak et al.

    Guidance law against spiraling target

    J. Guid. Control Dyn.

    (2016)
  • Cited by (11)

    • A switching-based state-scaling design for prescribed-time stabilization of nonholonomic systems with actuator dead-zones

      2021, Aerospace Science and Technology
      Citation Excerpt :

      In addition, finite-time stable systems generally perform good properties of robustness and disturbance rejection [13]. These facts excite the growing interest of studying finite-time control over recent years [14–23]. However, the settling time functions acquired in the aforementioned results are dependent on initial system conditions, which prohibits their practical engineering applications to a certain extent, since the desirable characteristics in a preset time cannot be derived without knowledge of initial conditions.

    • A finite-time 3D guidance law based on fixed-time convergence disturbance observer

      2020, Chinese Journal of Aeronautics
      Citation Excerpt :

      Numerous guidance solutions have been proposed, such as finite-time robust guidance law,1,14 angle constrained guidance law15,16 and adaptive guidance solutions,17,18 Disturbance Observer Based Control (DOBC) is another broadly applied approach to solve the guidance problems. For example, guidance laws based on high-order Robust Exact Disturbance Observer (REDO),3,19 non-homogeneous high-order REDO,20 Extended State Observer (ESO)4,21–23 and so on. However, the convergence rate of the abovementioned observers mainly depends on the magnitude of the gains, and the transient process of the observers may adversely affect system performance.

    • Highly maneuvering target interception via robust generalized dynamic inversion homing guidance and control

      2020, Aerospace Science and Technology
      Citation Excerpt :

      In this interception model, the guidance law integrated autopilot dynamics through second-order dynamics, unknown target's maneuvers using a sliding mode observer, and measured information of the guidance system. In [31], the authors presented a finite-time solution to the problem of short-range interception of evasive targets. Robust stabilization of the line-of-sight rate was obtained by coupling a disturbance observer with a finite-time stabilizer.

    • Guidance strategies for interceptor against active defense spacecraft in two-on-two engagement

      2020, Aerospace Science and Technology
      Citation Excerpt :

      Couples of classical guidance laws were proposed for this engagement such as the pure pursuit (PP), proportional navigation (PN), augmented proportional navigation (APN) approaches. In addition, some guidance laws based on finite-time control strategy, multi-objective optimization, fractional calculus algorithm, and sliding mode control approach were investigated in some certain cases [2–6]. Besides the one-on-one interception scenario, multi-agent pursuit-evasion is also an interesting topic being studied in recent years.

    • Three-dimensional terminal angle constrained robust guidance law with autopilot lag consideration

      2019, Aerospace Science and Technology
      Citation Excerpt :

      Using bounds of the target acceleration and jerk and consisting of two adaptive terms, an adaptive finite time convergent guidance law for maneuvering target is presented in [22]. To cope with unpredictable target evasive strategies, Zhang et al. [23] proposed a robust finite-time guidance law which modeled the target maneuver as a disturbance and introduced a second order homogeneous observer to estimate it. In [24], a prescribed performance function based control method is presented to drive the sliding mode variable to zero, lead to smooth guidance commands and fulfill the terminal LOS angle constraint.

    View all citing articles on Scopus
    View full text