A Preconditioned Inexact Active-Set Method for Large-Scale Nonlinear Optimal Control Problems

We provide a global convergence proof of the recently proposed sequential homotopy method with an inexact Krylov--semismooth-Newton method employed as a local solver. The resulting method constitutes an active-set method in function space. After discretization, it allows for efficient application of Krylov-subspace methods. For a certain class of optimal control problems with PDE constraints, in which the control enters the Lagrangian only linearly, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.Comment: 26 page

Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.Comment: 25 pages, 4 figure

On Reduced Input-Output Dynamic Mode Decomposition

The identification of reduced-order models from high-dimensional data is a challenging task, and even more so if the identified system should not only be suitable for a certain data set, but generally approximate the input-output behavior of the data source. In this work, we consider the input-output dynamic mode decomposition method for system identification. We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified systems

상승단계 발사체의 최적 궤적 생성 및 강건 제어 기법

학위논문 (박사)-- 서울대학교 대학원 : 공과대학 기계항공공학부, 2018. 2. 김현진.This research focused on trajectory generation and control of a flexible launch vehicle during ascent flight. An important issue of a launch vehicle design is generating optimal trajectory during its atmospheric ascent flight while satisfying constraints such as aerodynamic load. These constraints become more significant due to wind disturbance, especially in the maximum dynamic pressure region. On the other hand, modern launch vehicles are becoming long and slender for the reduction in structure mass to increase payload. As a result, they possess highly flexible bending modes in addition to aerodynamically unstable rigid body characteristics. This dissertation proposes a rapid and reliable optimization approach for trajectory generation via sequential virtual motion camouflage (VMC) and non-conservative robust control for an unstable and flexible launch vehicle. First, an optimal trajectory is generated in a rapid and reliable manner through the introduction of the virtual motion camouflage. VMC uses an observed biological phenomenon called motion camouflage to construct a subspace in which the solution trajectory is generated. By the virtue of this subspace search, the overall dimension of the optimization problem is reduced, which decreases the computational time significantly compared to a traditional direct input programming. Second, an interactive optimization algorithm is proposed to find a feasible solution easier. For this, the constraint correction step is added after VMC optimization. Since VMC is a subspace problem, a feasible solution may not exist when subspace is not properly constructed. In order to address this concern, a quadratic programming (QP) problem is formulated to find a direction along which the parameters defining the subspace can be improved. Via a computationally fast QP, specific parameters (such as prey and reference point) used in VMC can be refined quickly and sequentially. As a result, the proposed interactive optimization algorithm is less sensitive to the initial guess of the optimization parameters. Third, a non-conservative 2-DOF H infty controller for an unstable and flexible launch vehicle is proposed. The objectives of the control system are to provide sufficient margins for the launch vehicle dynamics and to enhance the speed of the closed-loop response. For this, a robust control approach is used. The key of the control design is to overcome conservativeness of the robust control. The baseline controllers using the optimal control such as LQG and LQI are designed prior to a robust controller. These optimal controllers are used to find a desirable shape of the sensitivity transfer function in order to reduce conservativeness of the robust control. After implementation and analysis of the baseline controllers, an improved sensitivity weighting function is defined as a non-conventional form with different slopes in the low frequency and around crossover frequency, which results in performance enhancement without loss of robustness. A two-degree-of-freedom H infty controller is designed which uses feedback and feedforward control together to improve tracking performance with the proposed sensitivity weighting function as a target closed-loop shape. The resulting H infty controller stabilizes the unstable rigid body dynamics with sufficient margins in the low frequency, and also uses gain stabilization in addition to phase stabilization to handle the lightly damped bending modes in the high-frequency region.1 Introduction 1 1.1 Background and motivations 1 1.2 Literature survey 3 1.2.1 Optimal trajectory generation for a launch vehicle 3 1.2.2 Controller design for a flexible launch vehicle 5 1.3 Research objectives and contributions 6 1.4 Thesis organization 7 2 Launch Vehicle Dynamics 9 2.1 Frame and coordinate 9 2.2 Rigid body motion 9 2.3 Aerodynamic forces and moments 12 2.4 Gravity force 14 2.5 Thrust forces and moments 14 2.6 Flexible bending modes 15 3 Optimal Trajectory Generation 16 3.1 VMC based trajectory optimization 16 3.1.1 Nonlinear constrained trajectory optimization problem 17 3.1.2 VMC formulation 17 3.2 VMC based trajectory optimization applied to the launch vehicle 21 3.2.1 Relationship between launch vehicle dynamics and VMC 21 3.2.2 Selection of reference point and virtual prey motion 23 3.2.3 Trajectory optimization via VMC 25 3.2.4 Sequential VMC: constraint correction 27 3.2.5 Comparison study 29 3.3 Numerical simulations 31 3.3.1 Case 1: No wind disturbance 36 3.3.2 Case 2: Z-axis wind disturbance 39 3.3.3 Case 3: Y -axis wind disturbance 43 3.3.4 Case 4: Z and Y -axes wind disturbance 48 3.3.5 Performance comparison 51 4 Robust Control 57 4.1 Launch vehicle model description 57 4.1.1 Rigid body model 58 4.1.2 Flexible modes and Actuator 59 4.1.3 System properties and design specications 63 4.2 Baseline controllers design 65 4.2.1 Set-point LQG 65 4.2.2 Integral LQG 69 4.3 Robust controller design 74 4.3.1 H infinity control theory 74 4.3.2 Two-degree-of freedom H infinity controller 76 4.3.3 Selection of weighting functions: Wp and Wu 77 4.3.4 Synthesis results 82 4.3.5 Comparison study 88 4.4 Numerical simulation 94 5 Conclusions 98 Abstract (in Korean) 106Docto