History-dependent modified sliding mode interception strategies with maximal capture zone

Abstract

In order to construct the guidance strategy in a realistic nonlinear noise-corrupted interception endgame against a maneuverable target, a linearized zero-sum differential game is considered. Assuming perfect information in this game, sufficient conditions are established, which guarantee that a continuous interception strategy with memory (history-dependent) has the maximal capture zone. Two examples of such a strategy are analyzed: a modified super-twisting second-order sliding mode control and a modified integral sliding mode control. Simulation results of the original nonlinear interception endgame demonstrate that these strategies considerably reduce the chattering created by the classical game optimal bang-bang strategy without deteriorating the homing performance.

Introduction

The problem of intercepting a maneuverable target admits different mathematical formulations. In [1], [2], [3], the terminal phase of such interception was formulated as a linear perfect information pursuit-evasion differential game with bounded controls (without penalties on the control usage), where the interceptor is the pursuer and the target is the evader. The cost function in this game is the miss distance.

The planar version of such a game can be reduced to a scalar game, where the state variable is the zero-effort miss distance. The solution of this game [1] includes a singular region of the game space, where the optimal strategies are arbitrary (but bounded) and the game value is constant. Outside of this region, bang-bang type control strategies, i.e. bounded controls with the sign of the zero-effort miss distance, are optimal. It was shown in [1] that if some conditions are satisfied, the constant game value in the singular region is zero, i.e. the closure of this region becomes the capture zone of the game optimal pursuer strategy. By definition, the capture zone of a given strategy is the set of all initial positions from which this strategy guarantees capture robustly with respect to any admissible evader control satisfying the control constraint.

The implementation of an arbitrary (non-unique) optimal pursuer strategy in the singular region allows several options for the designer. The most evident option is to use the bang-bang strategy in the entire game space. This strategy has the maximal capture zone, i.e. it contains the capture zones of all other admissible pursuer strategies. Nevertheless, being a first-order sliding mode control, it leads to control chattering, which can create implementation difficulties [4], [5]. In particular, it leads to unacceptable wear of the actuators and an unnecessary over expenditure of the control. For this reason a number of studies were devoted to eliminate or at least to reduce considerably this phenomenon [6], [7], [8], [9], [10], [11]. Several alternative (linear, saturated linear and weakly nonlinear) strategies were proposed in order to avoid the chattering [12], [13], [14], [15], [16], [17].

These strategies guarantee capture at a prescribed time t_f with reduced chattering. In [12], necessary and sufficient conditions were formulated in order to assure that a saturated continuous feedback strategy has the maximal capture zone.

An alternative approach to avoid chattering is utilizing an interceptor strategy with memory, i.e. a strategy depending on a time history of state variables. Higher order sliding mode control algorithms [18], [9], [19], [20], [21], [22], [23] represent such a history-dependent strategy. Such strategies can bring the system onto the time axis before t_f and keep it there till t_f without chattering. Interception applications of a second-order sliding mode control can be found for example in [24], [25]. In these papers, the interception problem with nonlinear dynamics is considered. The disturbance is assumed to be a differentiable function of time with known Lipschitz constant of the derivative. The duration of the interception is not prescribed. For this problem, a second-order sliding mode guidance law is designed, providing control smoothness and finite time capture. Since control constraints are not specified in the model, this guidance strategy guarantees robust capture from the entire interception space. Another example of a history-dependent strategy is given by an integral sliding mode control [26], [27], [28], [29], [30]. An important feature of this control is tracking the trajectory of the undisturbed system, generated by a nominal control, from the very beginning of the control process. This ensures the insensitivity of the system motion to matched uncertainties.

The structure of the present paper, where the disturbance (the target control) can be also discontinuous, is the following. In the next section, the interception endgame scenario is stated. In a linear framework, which is justified for endgame analysis, the interception duration can be precalculated and prescribed in advance. The first and simplest solution for this problem, which was derived in [1], is briefly recalled in Section 3. It has the maximal capture zone, but yields a chattering control. In the sequel, a general interception strategy with memory (history-dependent) is considered. Sufficient conditions, guaranteeing the maximality of the capture zone for a history-dependent strategy, are established in Section 4. Based on this result, two history-dependent interception strategies are designed: a modification of the super-twisting second-order sliding mode control (Section 5) and a modification of the integral sliding mode control (Section 6). In order to evaluate the performance of these strategies in comparison with the bang-bang strategy [1] and a saturated linear strategy used in the singular region [31], Monte Carlo simulations are carried out in a realistic nonlinear noise-corrupted interception scenario (Section 7). These simulations also confirm the validity of the using a linear model for the study. Conclusions are presented in Section 8.