A closed analytic form for the time maximum disturbance isochrones of second-order linear systems

It is shown that each time maximum disturbance isochrone (boundary of reachable set) for second-order linear systems with the most stressful bounded disturbance has a closed analytic form. Further it is shown that these isochrones can be constructed by purely geometric methods.     

State-local optimality of a bang–bang trajectory: a Hamiltonian approach

We give a sufficient condition for a bang–bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints. We underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time. Moreover, it is given through a finite-dimensional minimization problem, hence is suited for numerical verification. A geometric interpretation through the projection of the Hamiltonian flow on the state space is also given.     

Inequality path constraints in optimal control: a finite iteration -convergent scheme based on pointwise discretization

This paper presents a new result in the analysis and implementation of path constraints in optimal control problems (OCPs). The scheme uses the well-known concept of discretizing path constraints on a finite number of points, yielding a set of interior-time point constraints replacing the original path constraints. The approach replaces the original OCP by a sequence of OCPs which is shown to converge in a finite number of steps to the solution of the original path constrained problem with -accuracy. Numerical results, verifying the theoretical analysis, are presented. The method is shown to be effective and promising for future applications, particularly in control vector parameterization implementations.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

航天器姿态受限的协同势函数族设计方法

提出一种考虑航天器姿态约束的协同势函数设计方法, 在姿态全局收敛的同时, 保证姿态在机动过程中始终满足姿态约束. 首先, 建立航天器姿态指向约束模型, 并针对每一个指向约束设计软约束区域; 然后, 基于“角度扰动”方法设计协同势函数族; 接着, 通过设计协同势函数族内函数切换规律, 在软约束区域内构建满足姿态约束的势函数, 并给出区域内势函数临界点分布的调整方法; 最后, 将所得的势函数用于航天器的避障控制, 以比例−微分控制为例, 通过数值仿真, 验证该方法的有效性.     

Constrained optimal control of Navier–Stokes flow by semismooth Newton methods

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier–Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and prove q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.     

SOLVING ABSTRACT COOPERATIVE PATH‐FINDING IN DENSELY POPULATED ENVIRONMENTS

The problem of cooperative path‐finding is addressed in this work. A set of agents moving in a certain environment is given. Each agent needs to reach a given goal location. The task is to find spatial temporal paths for agents such that they eventually reach their goals by following these paths without colliding with each other. An abstraction where the environment is modeled as an undirected graph is adopted—vertices represent locations and edges represent passable regions. Agents are modeled as elements placed in the vertices while at most one agent can be located in a vertex at a time. At least one vertex remains unoccupied to allow agents to move. An agent can move into unoccupied neighboring vertex or into a vertex being currently vacated if a certain additional condition is satisfied. Two novel scalable algorithms for solving cooperative path‐finding in bi‐connected graphs are presented. Both algorithms target environments that are densely populated by agents. A theoretical and experimental evaluation shows that the suggested algorithms represent a viable alternative to search based techniques as well as to techniques employing permutation groups on the studied class of the problem.     

Controller synthesis with guaranteed closed-loop phase constraints

In this paper, we present an analysis and synthesis approach for guaranteeing that the phase of a single-input, single-output closed-loop transfer function is contained in the interval [−α,α] for a given α>0 at all frequencies. Specifically, we first derive a sufficient condition involving a frequency domain inequality for guaranteeing a given phase constraint. Next, we use the Kalman–Yakubovich–Popov theorem to derive an equivalent time domain condition. In the case where , we show that frequency and time domain sufficient conditions specialize to the positivity theorem. Furthermore, using linear matrix inequalities, we develop a controller synthesis approach for guaranteeing a phase constraint on the closed-loop transfer function. Finally, we extend this synthesis approach to address mixed gain and phase constraints on the closed-loop transfer function.     

Robust control of uncertain differential linear repetitive processes

For two-dimensional (2-D) systems, information propagates in two independent directions. 2-D systems are known to have both system-theoretical and applications interest, and the so-called linear repetitive processes (LRPs) are a distinct class of 2-D discrete linear systems. This paper is concerned with the problem of L₂–L_∞ (energy to peak) control for uncertain differential LRPs, where the parameter uncertainties are assumed to be norm-bounded. For an unstable LRP, our attention is focused on the design of an L₂–L_∞ static state feedback controller and an L₂–L_∞ dynamic output feedback controller, both of which guarantee the corresponding closed-loop LRPs to be stable along the pass and have a prescribed L₂–L_∞ performance. Sufficient conditions for the existence of such L₂–L_∞ controllers are proposed in terms of linear matrix inequalities (LMIs). The desired L₂–L_∞ dynamic output feedback controller can be found by solving a convex optimization problem. A numerical example is provided to demonstrate the effectiveness of the proposed controller design procedures.     

A simple derivation of ARE solutions to the standard control problem based on LMI solution

In this note, a simple derivation of the Riccati equation based solutions to the standard control problem, namely the well-known Glover–Doyle solution and DGKF solution is given based on LMI solution. It is hoped that this will be helpful in deepening the understanding of Riccati equation solutions.     

Control design in the time and frequency domain using nonsmooth techniques

Significant progress in control design has been achieved by the use of nonsmooth and semi-infinite mathematical programming techniques. In contrast with LMI or BMI approaches, these new methods avoid the use of Lyapunov variables, which gives them two major strategic advances over matrix inequality methods. Due to the much smaller number of decision variables, they do not suffer from size restrictions, and they are much easier to adapt to structural constraints on the controller. In this paper, we further develop this line and address both frequency- and time-domain design specifications by means of a nonsmooth algorithm general enough to handle both cases. Numerical experiments are presented for reliable or fault-tolerant control, and for time response shaping.