New Adaptive Mesh Refinement Strategy for Entry Guidance via Sequential Convex Programming

In this paper, the highly nonlinear planetary-entry optimal control problem is formulated as a sequence of convex problems to facilitate rapid solution. The nonconvex control constraint is avoided by introducing a new state variable to the original three-dimensional equations of motion. The nonconvex objective function and path constraints are convexified by first-order Taylor-series expansions, and the nonconvex terms in the dynamics are approximated by successive linearizations. A successive solution procedure is developed to find an approximated solution to the original problem, and its convergence is discussed. In each iteration, a convex optimization problem is solved by the state-of-the-art interior-point method with deterministic convergence properties. Finally, the proposed method is verified and compared to a general-purpose optimal control solver by numerical solutions of minimum terminal-velocity and minimum heat-load entry problems. The sequential method converges to accurate solutions with faster speed than the general-purpose solver using MATLAB on a desktop computer with a 64-bit operating system and an Intel Xeon E3-1225 V2 3.2 GHz processor, which demonstrates its potential real-time application for computational guidance.

This paper proposes a near‐optimal air‐to‐ground missile guidance law with impact angle and impact velocity constraints based on sequential convex programming. A realistic aerodynamic model is introduced into the problem formulation, such that traditional optimization theory cannot obtain an analytical solution to the optimization problem under state constraints. The original problem is considered as an optimization problem, and the angle of attack is replaced with the angle of attack rate as a new control variable to reconstruct the problem and simplify the solving process. Next, the independent variable is changed in the differential equations to linearize and discretize the problem such that the reconstructed problem can be solved using sequential convex programming. The results obtained by numerical simulations confirmed that the proposed algorithm is valid and faster than the general‐purpose nonlinear optimal control problem solver. Finally, it was verified that different impact angles and impact velocities were achieved.

Entry trajectory optimization for hypersonic vehicles has been formulated as constrained optimal control problems, which are difficult to solve because of the existing high nonlinearity and nonconvexity. In recent years, convex optimization has shown promise for real-time onboard applications with the state-of-the-art interior-point methods. In this paper, line-search and trust-region techniques are introduced to fundamentally improve the performance of the sequential convex programming method for entry trajectory optimization. Two improved algorithms are developed: line-search sequential convex programming and trust-region sequential convex programming. In addition, a new trajectory generation method is proposed by taking advantage of the predictor–corrector method to find an initial 3-D trajectory for the developed successive algorithms. As such, the convergence of the solution process is improved. To demonstrate the effectiveness and performance of the newly proposed algorithms, numerical simulations are presented for maximum-terminal-velocity and minimum-heat-load entry problems. Results show that these two problems are not well solved using the basic sequential convex programming algorithm. However, they can be efficiently solved by the two improved algorithms in a few seconds in MATLAB, showing a significant improvement over general-purpose solvers, such as GPOPS.

This paper proposed an optimal time-varying proportional navigation guidance law based on sequential convex programming. The guidance law can achieve the desired impact angle and impact time with look angle and lateral acceleration constraints. By treating the multiconstraints’ guidance problem as an optimization problem and changing the independent variable to linearize the problem and constraints, the original nonlinear and nonconvex problem is transformed into a series of convex optimization problem so that it can be quickly solved by sequential convex programming. Numerical simulations compared to nonlinear programming and traditional analytical guidance law demonstrate the effectiveness and efficiency of the proposed algorithm. Finally, the proposed guidance law is verified to satisfy different impact time periods and impact angle constraints.

This paper presents a distributed, guidance and control algorithm for reconfiguring swarms composed of hundreds to thousands of agents with limited communication and computation capabilities. This algorithm solves both the optimal assignment and collision-free trajectory generation for robotic swarms, in an integrated manner, when given the desired shape of the swarm (without pre-assigned terminal positions). The optimal assignment problem is solved using a distributed auction assignment that can vary the number of target positions in the assignment, and the collision-free trajectories are generated using sequential convex programming. Finally, model predictive control is used to solve the assignment and trajectory generation in real time using a receding horizon. The model predictive control formulation uses current state measurements to resolve for the optimal assignment and trajectory. The implementation of the distributed auction algorithm and sequential convex programming using model predictive control produces the swarm assignment and trajectory optimization (SATO) algorithm that transfers a swarm of robots or vehicles to a desired shape in a distributed fashion. Once the desired shape is uploaded to the swarm, the algorithm determines where each robot goes and how it should get there in a fuel-efficient, collision-free manner. Results of flight experiments using multiple quadcopters show the effectiveness of the proposed SATO algorithm.

We analyze the convergence properties of two Newton-type algorithms for the solution of unconstrained nonlinear optimization problems with convex substructure: Generalized Gauss-Newton (GGN) and Sequential Convex Programming (SCP). While both algorithms are identical to the classical Gauss-Newton method for the special case of nonlinear least squares, they differ when applied to more general convex outer functions. We show under mild assumptions that GGN and SCP have locally linear convergence with the same contraction rate. The convergence or divergence rate can be characterized as the smallest scalar that satisfies two linear matrix inequalities. We further show that bad convergence or divergence at a given local minimum can be a desirable property in the context of estimation problems with symmetric likelihood functions, because it avoids that the algorithm is attracted by statistically undesirable local minima. Both algorithms and their convergence properties are illustrated with a numerical example.

ABSTRACTTrajectory planning of formation rendezvous of multiple unmanned aerial vehicles (UAVs) is formulated as a mixed-integer optimal control problem, and an efficient hierarchical planning approach based on the Dubins path and sequential convex programming is proposed. The proposed method includes the assignment of rendezvous points (high level) and generation of cooperative trajectories (low level). At the high level, the assignment of rendezvous points to UAVs is optimized to minimize the total length of Dubins-path-based approximate trajectories. The assignment results determine the geometric relations between the UAVs’ goals, which are used as equality constraints for generating trajectories. At the low level, trajectory generation is treated as a non-convex optimal control problem, which is transformed to a non-convex parameter optimization and then solved via sequentially performing convex optimization. Numerical experiments demonstrate that the proposed method can generate feasible trajectories and can outperform a typical nonlinear programming method in terms of efficiency.

This work finds a lower bound on the average dwell-time (ADT) of switching signals such that a continuous-time, graph-based, switched system is globally asymptotically stable, input-to-state stable, or integral input-to-state stable. We first formulate the lower bound on the ADT as a nonconvex optimization problem with bilinear matrix inequality constraints. Because this formulation is independent of the choice of Lyapunov functions, its solution gives a less conservative lower bound than previous Lyapunov-function-based approaches. We then design a numerical iterative algorithm to solve the optimization based on sequential convex programming with a convex-concave decomposition of the constraints. We analyze the convergence properties of the proposed algorithm, establishing the monotonic evolution of the estimates of the average dwell-time lower bound. Finally, we demonstrate the benefits of the proposed approach in two examples and compare it against other baseline methods.

This study focuses on the key challenge of achieving high-precision guidance for spacecraft: real-time trajectory optimization. The highly efficient sequential convex programming (SCP) is a potential method to address this issue, but its convergence is difficult to guarantee. To enhance the convergence performance of SCP, this paper proposes a high-dimensional merit-function-based SCP method. The main contribution of this paper is threefold. Firstly, the traditional one-dimensional penalty function used to ensure the convergence of SCP is extended to a high-dimensional merit function, which avoids using the penalty parameter that is difficult to determine in advance. Secondly, to match the high-dimensional merit function, a feasibility-then-optimality convergence strategy is proposed, which is essentially an application of the iterative direction descent property in higher dimensions. Finally, based on the high-dimensional merit function and the feasibility-then-optimality strategy, an iterative algorithm is developed, and its convergence analysis is provided under certain assumptions. In numerical simulations, the effectiveness and robustness of the proposed algorithm are validated using the reentry trajectory optimization problem of a lunar capsule.

The present paper is dedicated to the investigation of optimal control problems whose behavior is described by quasilinear differential equations of first order on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem of the existence and uniqueness of a generalized solution in the space \(C_{\mu }(\overline{G})\) is proved for quasilinear differential equations; necessary conditions of optimality are obtained in terms of the principle of maximum; the Bitsadze–Samarski boundary value problem is proved for a linear differential equation of first order; the existence of a solution in the space \(C_{\mu }^{p}(\overline{G})\) is proved and an a priori estimate is derived. A theorem on the necessary and sufficient condition of optimality is proved for a linear optimal control problem.

Difference inclusions provide a discrete-time analogue of differential inclusions, which in turn play an important role in the theories of optimal control, implicit differential equations, and invariance and viability, to name a few. In this paper we: (i) introduce a framework suitable for the study of difference inclusions for which the state evolves on a manifold; (ii) use this framework to develop necessary conditions for optimality for a broad class of discrete-time problems of dynamic optimization in which the state evolves on a manifold M. The necessary conditions for optimality we derive include the case for which the state $$q_i$$ is subject to constraints $$q_i \in S_i \subseteq M$$ , for $$S_i$$ a closed set. The resulting necessary conditions for optimality appear as discrete-time versions of the Euler–Lagrange inclusion studied by Ioffe (in Trans Am Math Soc 349(7):2871–2900, 1997), Ioffe and Rockafellar (in Calc Var Partial Differ Equ 4(1):59–87, 1996), Mordukhovich (in SIAM J Control Optim 33(3):882–915, 1995), Mordukhovich (in Variational analysis and generalized differentiation II: applications. Springer, Berlin, 2006), and Vinter and Zheng (in SIAM J Control Optim 35(1):56–77, 1997) generalized in a natural way to the case in which the state is evolving on a manifold.

Most direct methods solve finite time horizon optimal control problems with nonlinear programming solver. In this paper, we propose a numerical method for solving nonlinear optimal control problem with state and control inequality constraints. This method used quasilinearization technique and Haar wavelet operational matrix to convert the nonlinear optimal control problem into a quadratic programming problem. The linear inequality constraints for trajectories variables are converted to quadratic programming constraint by using Haar wavelet collocation method. The proposed method has been applied to solve Optimal Control of Multi-Item Inventory Model. The accuracy of the states, controls and cost can be improved by increasing the Haar wavelet resolution.

In this work, we discuss normal forms of necessary conditions of optimality (NCO) for optimal control problems subject to pathwise state constraints and for problems in the calculus of variations with inequality constraints. It is known that standard forms of the NCO may fail to provide information that is useful to identify optimal solutions, namely when the multiplier associated with the objective function takes the value zero. The normal forms of the NCO guarantee that the conditions remain always informative, which is of importance in critical applications where decisions based on optimization are taken, such as autonomous systems. Based on a previous nondegenerate maximum principle for optimal control problems, we extend the strengthness of these conditions to normality while applying them to the particular case of calculus of variations problems. We compare our results with existent normal forms of NCO for dynamic optimization problems and conclude that, when applied to calculus of variations problems, we may say that, under similar conditions, we can apply such result to a wider class of problems, having less regularity on the data.

This paper presents a sequential convex programming (SCP) algorithm designed to achieve dynamically feasible solutions for highly nonlinear optimal control problems over sparse time grids. This framework solves a sequence of convex subproblems until convergence to a solution of the original nonconvex problem. The proposed algorithm has the following features to enhance speed and accuracy: (1) a deviation variable model to improve the scaling of each convex subproblem; (2) an exact discretization technique that is inverse-free and permits arbitrarily-precise satisfaction of the dynamics over large time horizons; (3) a virtual buffer approach to eliminate artificial infeasibility that allows each subproblem to be represented as a quadratic program (QP); and (4) a variable time grid that leaves the timesteps between nodes (and thus the time-horizon) as free variables for the optimizer. Discretization is performed via a ``stitching condition'' required for the deviation variable approach. This SCP algorithm demonstrates high-fidelity satisfaction of the dynamics for particularly nonlinear systems, such as hypersonic entry vehicles, while harnessing solution speeds from quadratic programming amenable to real-time implementations. To remove jitter from the resulting control solution, a control rate limit constraint can easily be applied across each timestep due to the continuous first-order-hold parameterization of the control. To demonstrate this performance, a comprehensive study is performed for a 3-degree-of-freedom (DoF) trajectory optimization on multiple hypersonic reentry scenarios formulated as optimal control problems. When compared against state-of-the-art methods, the optimal solutions obtained by this SCP algorithm have reduced residual errors between the solution trajectory and a one-shot integration of the full nonlinear dynamics for hypersonic reentry.

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