Three-dimensional impact angle guidance law based on robust repetitive control

Research Article Three‑dimensional impact angle guidance law based on robust repetitive control Wenguang Zhang1 · Wenjun Yi1 · Jun Guan1 · Yue Qu1 Received: 27 June 2019 / Accepted: 31 October 2019 / Published online: 5 November 2019 © Springer Nature Switzerland AG 2019 Abstract This paper presents a robust repetitive control (RC) design applied to three-dimensional homing guidance of missiles with impact angle constraint. The proposed guidance law is substantially a composite control method, which is constructed through a combination of RC and sliding mode control. More specifically, the RC exerts advantages to drive the state tracking error converge to zero, then sliding mode control is triggered, making the system be robust in terms of noise and disturbance. The effectiveness of the proposed guidance law is validated through simulation. Keywords Guidance law · Impact angle · Repetitive control List of symbols R Relative distance between the missile and the target 𝜙 Pitch line-of-sight angle (PLOS) 𝜃 Yaw line-of-sight angle (YLOS) e⃗r Unit vector along the LOS e⃗𝜙 Unit vector along the PLOS e⃗𝜃 Unit vector along the YLOS a⃗ T = wr e⃗r + w𝜃 e⃗𝜃 + w𝜙 e⃗𝜙 Acceleration vector of the target a⃗ M = u𝜃 e⃗𝜃 + u𝜙 e⃗𝜙 Acceleration vector of the missile R̈ Relative acceleration along to LOS 𝜙̈ Angular acceleration along to LOS 𝜃̈ Angular acceleration of 𝜃 Ṙ Relative velocity between the missile and the target 𝜃̇ Angular velocity of 𝜃 𝜙̇ Angular velocity of 𝜙 1 Introduction Intercepting maneuvering targets with a small miss-distance is not the only task of the guidance law design in some applications, for example, antitank or antiship missiles, which are also required to approach the target from a predetermined impact angle in order to increase the warhead effectiveness [1, 2]. Hence, it is necessary to design guidance law with impact angle constraint. During the guidance process, the guidance system continuously measures the relative position information, and sends command to the flight control system. The kinematics equation of the missile-target pursuit dynamic behavior is found to be uncertain nonlinear multiple-input multiple-output (MIMO) system with cross-coupling [3]. In the past, Proportional Navigation Guidance law (PNG) was widely used in homing guidance area [4]. Along with the progress of computer science and mathematics, a lot of nonlinear control methods have been applied to this issue [5–8]. Among them, sliding mode control (SMC) was widely adopted by researchers for its unique properties, for example, it is robust to parameter variations and external disturbance * Wenguang Zhang, | 1National Key Laboratory Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China. SN Applied Sciences (2019) 1:1549 | https://doi.org/10.1007/s42452-019-1601-2 Vol.:(0123456789) Research Article SN Applied Sciences (2019) 1:1549 | https://doi.org/10.1007/s42452-019-1601-2 [9]. But SMC suffers from some drawbacks, which are: the upper bound of uncertainties must be known, and the existence of chattering phenomenon, which may cause the excitation of unmodeled dynamics [10]. Recently, a new guidance law design based on Iterative Learning Control (ILC) is proposed in [11], and the numerical experiments show that the proposed method is capable of reducing the time to reach the head-on condition to interception. However, impact angle constraint is not taken into consideration in this paper. Besides, there exists robustness problem of ILC [12]. In this paper, we propose a robust repetitive control strategy for guidance mission of homing missiles. The RC is combined with sliding mode control in order to acquire both of their advantages. Specifically, the RC is utilized to guarantee the reachability of the sliding mode, and then the sliding mode control is committed to enhance the robustness of the system. Simulations under different scenarios are performed, and the validation of the proposed method is verified. This paper is organized as follows: in Sect. 2, the dynamics of target-missile relative motion is illustrated, and the object of the guidance law with impact angle constraint is addressed. In Sect. 3, the robust repetitive control is designed in the framework of sliding mode control. Numerical experiments are performed to demonstrate the effectiveness of the proposed method in Sect. 4. At last, concluding remarks are summarized in Sect. 5. Fig. 1  Three-dimensional pursuit-evasion geometry Assumption 1 [14] Assume that the missile intercepting the target by impact happens when R = R0 ≠ 0, and there exist two positive constants Rmin and Rmax , which satisfy Rmin < R < Rmax. Let 𝜃d and 𝜙d be the desired final LOS angles in elevation and azimuth, respectively. By accepting the concept that zeroing the LOS angle rate will lead a perfect interception and taking the terminal angle constraint into consideration, the control object is to design a guidance law in such a way that 𝜃 → 𝜃d , 𝜙 → 𝜙d , 𝜃̇ → 0, 𝜙̇ → 0 can be fulfilled asymptotically [15]. 2 Problem formulation 3 Composite guidance law design In the actual interception, the target-missile relative motion takes place in a three dimensional environment. It can be denoted in the spherical LOS coordinate system as Fig. 1 shows. The 3D pursuit dynamic system can be expressed as follows [13]: Let e𝜃 and e𝜙 denote the tracking error of 𝜃 and 𝜙, respectively, which are defined as e𝜃 = 𝜃 − 𝜃d , e𝜙 = 𝜙 − 𝜙d . Then a sliding surface dynamics can be defined as follows: R̈ = R𝜙̇ 2 + R𝜃̇ 2 cos2 𝜙 + aTr − aMr (1) σ𝜃 (t) = c1 e𝜃 (t) + c2 ė 𝜃 (t) a a 2Ṙ 𝜃̇ 𝜃̈ = − + 2𝜙̇ 𝜃̇ tan 𝜙 + T 𝜃 − M𝜃 R Rcos𝜙 Rcos𝜙 (2) σ𝜙 (t) = c3 e𝜙 (t) + c4 ė 𝜙 (t) aT 𝜙 aM𝜙 2Ṙ 𝜙̇ 𝜙̈ = − − 𝜃̇ 2 sin 𝜙cos𝜙 + − R R R (3) In fact, only the accelerations normal to the missile’s velocity are available in the terminal guidance phase. Therefore, only Eqs. (2) and (3) are used in guidance law design. Vol:.(1234567890) 3.1 Derivation of sliding surface (4) (5) where ci , (i = 1, 2, 3, 4) are coefficients of a Hurwitz polynomial. Computing the time derivative of Eqs. (4) and (5) and considering Eqs. (2) and (3) gives � � � � ⎛ c 𝜃̇ + c − 2Ṙ 𝜃̇ + 2𝜙̇ 𝜃̇ tan 𝜙 + aT 𝜃 − aM𝜃 ⎞ 1 2 σ d Rcos𝜙 Rcos𝜙 � ⎟ 𝜃 � R =⎜ ⎜ c 𝜙̇ + c − 2Ṙ 𝜙̇ − 𝜃̇ 2 sin 𝜙cos𝜙 + aT 𝜙 − aM𝜙 ⎟ dt σ𝜙 4 ⎠ ⎝ 3 R R R (6) Research Article SN Applied Sciences (2019) 1:1549 | https://doi.org/10.1007/s42452-019-1601-2 3.2 Robust RC guidance law design 3.3 Stability analysis Let us consider the auxiliary control terms In this section, we will prove the system represented by Eq. (11) is stable by Lyapunov stable theory [16]. Because the dynamic models of 𝜎𝜃 and 𝜎𝜙 have the similar forms, here we only take 𝜎𝜃 as an example to show the process of proof. Firstly, the tracking error of 𝜎𝜃 is defined as e1 (t) = y1 − 𝜎𝜃 , and the initial condition of e𝜃 can be characterised by the following assumption: � aM𝜃 aM𝜙 � ̇ 1 ⎛ c1 𝜃−u R cos 𝜙 − 2Ṙ (...truncated)