0018-9251 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2019.2938126, IEEE Transactions on Aerospace and Electronic Systems Finite-Horizon Robust Suboptimal Control based Impact Angle Guidance Shashi Ranjan Kumar and Arnab Maity Abstract—This paper presents impact angle guidance based on finite-horizon robust optimal control. This approach, a fusion of newly proposed finite-time state dependent-Riccati equation and integral sliding-mode-control, solves finite-horizon tracking problem for input-affine nonlinear systems with specified ter- minal conditions. It also ensures robustness against external disturbances and uncertainties from beginning. The technique for ensuring specified final-time involves deriving closed-form control expression in terms of approximate solution of differential Riccati equation, which further reduces to solutions of algebraic Riccati and Lyapunov equations. This yields computationally efficient algorithm. To alleviate chattering, the super-twisting algorithm is incorporated with proposed technique. Guidance strategy is derived after converting impact angle problem to tracking one using proposed algorithm. Efficacy of the proposed guidance is vindicated for various engagement geometries, and also compared with an existing guidance strategy. Index Terms—Impact angle guidance, Robust optimal control, Finite-time SDRE, Integral sliding mode control I. I NTRODUCTION Guidance strategy plays a key role in ensuring not only zero miss distance but also to satisfy terminal constraints. Terminal constraint, such as impact angle, is important to increase the warhead effectiveness, the penetration capability, and reducing collateral damages. For practical situations, it is beneficial to achieve the desired impact angle within a finite time. In literature, there exist sliding mode control (SMC) based guidance laws for aligning interceptors to a desired direction within a finite-time, and ensuring robustness against external disturbances and uncertainties [1]–[4]. In [1] and [2], the guidance laws were proposed based on the terminal SMC (TSMC) and finite-time convergence stability, respectively. In [3], [4], the guidance laws were derived using the nonsingular TSMC, avoiding possible singularities during implementation, for maneuvering and stationary targets, respectively. Although the guidance laws in [1]–[4] ensure finite time convergence, but the time of convergence was not fixed. As the engagements are of finite-time, the control over time of convergence could be of paramount importance. This necessitates an efficient finite-time convergent guidance strategy to complete the mis- sion in an optimal sense. In [5], the guidance law based on the integral SMC and LOS shaping was proposed, which guaran- tees convergence of error to zero at interception. In [6], SMC based guidance with look angle constraints, using bearing only information, was proposed. A finite time convergent guidance using the backstepping was proposed in [7]. Shashi Ranjan Kumar (srk@aero.iitb.ac.in), and Arnab Maity (arnab@aero.iitb.ac.in) are Assistant Professors in Intelligent System & Control Lab at the Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai, India. It is well known that the optimal control approaches are per- haps well suited for reducing operation cost, structural loads, minimizing control effort, and achieving desired terminal con- ditions simultaneously [8]–[11]. This fact provided a thrust to design such guidance [12]–[17], using linearized engagement geometry. However, they require time-to-go estimates during implementation, which is a formidable challenge. Moreover, an optimal control theory based trajectory op- timization formulation often leads to a two-point boundary value problem, which in turn lead to the numerical iterative methods. These methods may not be feasible for real-time implementation. To address this issue, the approximate dy- namic programming, followed by adaptive critic approach, is a real-time design technique, where the optimal control problem is solved using two neural networks [18], [19]. A potential drawback of this approach is that the domain in which state trajectory is likely to lie has to be known for training of networks. This is rather difficult to predict prior to the control design. Another promising approach is the linear quadratic regulator [20] for linear systems or state dependent Riccati equation (SDRE) [21], [22] for input-affine nonlinear systems, to solve infinite-horizon optimal control problems. Typically, the guidance design demands a finite-horizon problem formu- lation [23], and hence, these methods may be not suitable for optimal guidance design with stringent final conditions. Recently, a finite-horizon SDRE (Finite-SDRE) approach is proposed in [24]–[26]. However, this approach in [24]–[26] suffers from singularity problem of a matrix inverse, when the solution of differential Riccati equation (DRE) converges to the solution of algebraic Riccati equation (ARE). To avoid this problem, a solution approach, which needs to be heuristically selected, has been suggested, which may not be effective for real-life practical problems, especially in missile guidance problem. If the above aspects can be addressed, the option of using finite-horizon optimal control to develop a powerful optimal guidance can also be explored. This motivates us to develop finite-time SDRE (FT-SDRE) approach for solving the problem of finite-horizon optimal tracking control for input-affine nonlinear systems with spec- ified terminal conditions, while addressing above mentioned constraints. The proposed approach is based on a modification in SDRE method [21], [22] using the solution methodology of finite-horizon optimal regulator problem for linear system [27], [28]. A brief discussion of this idea was presented in [29]. Since the solution of time-varying Riccati equation needs to be computed at each time step due to the nature of finite- time nonlinear control problem, the FT-SDRE is important for such problem. Its philosophy involves deriving closed-form solution, in terms of an approximate solution of the matrix DRE for a specified final-time. The solution can be further