In this paper, we introduce the notion of periodic safety, which requires that the system trajectories periodically visit a subset of a forward-invariant safe set, and utilize it in a multi-rate framework where a high-level planner generates a reference trajectory that is tracked by a low-level controller under input constraints. We introduce the notion of fixed-time barrier functions which is leveraged by the proposed low-level controller in a quadratic programming framework. Then, we design a model predictive control policy for high-level planning with a bound on the rate of change for the reference trajectory to guarantee that periodic safety is achieved. We demonstrate the effectiveness of the proposed strategy on a simulation example, where the proposed fixed-time stabilizing low-level controller shows successful satisfaction of control objectives, whereas an exponentially stabilizing low-level controller fails.
This paper presents conditions for ensuring forward invariance of safe sets under sampled-data system dynamics with piecewise-constant controllers and fixed time-steps. First, we introduce two different metrics to compare the conservativeness of sufficient conditions on forward invariance under piecewise-constant controllers. Then, we propose three approaches for guaranteeing forward invariance, two motivated by continuous-time barrier functions, and one motivated by discrete-time barrier functions. All proposed conditions are control affine, and thus can be incorporated into quadratic programs for control synthesis. We show that the proposed conditions are less conservative than those in earlier studies, and show via simulation how this enables the use of barrier functions that are impossible to implement with the desired time-step using existing methods.
The centralized power generation infrastructure that defines the U.S. electric grid is slowly moving to the distributed architecture due to the explosion in use of renewable generation and distributed energy resources (DERs), such as residential solar, wind turbines and battery storage. Furthermore, variable pricing policies and profusion of flexible loads entail frequent and severe changes in power outputs required from the individual generation units. In view of these challenges, a fixedtime convergent, fully distributed economic dispatch algorithm for scheduling optimal power generation among a set of DERs is proposed. The proposed algorithm addresses both load balance and generation capacity constraints. The algorithm subsumes and betters the existing economic dispatch algorithms on three fronts: (a) It is shown that the distributed algorithm converges to optimal dispatch solution in a fixed-time, regardless of the initial states of every generation unit. Additionally, the time of convergence can be prespecified by the user. (b) Regardless of the physical topology of the power network, the proposed framework allows for timevarying communication topology. (c) Finally, it is shown that the proposed algorithm is robust to uncertain information resulting from noisy communication with neighbors. Several case studies are presented that corroborate computational and robustness aspects of the proposed distributed economic dispatch algorithm.
IIn this work, we study finite-time stability of hybrid systems with unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of a class of hybrid systems to be finite-time stable. More specifically, we show that even if the value of the Lyapunov function increases during continuous flow, i.e., if the unstable modes in the system are active for some time, finite-time stability can be guaranteed if the finite-time convergent mode is active for a sufficient amount of cumulative time. This is the first work on finite-time stability of hybrid systems using multiple Lyapunov functions. Prior work uses a common Lyapunov function approach, and requires the Lyapunov function to be decreasing during the continuous flows and non-increasing at the discrete jumps, thereby, restricting the hybrid system to only have stable modes, or to only evolve along the stable modes. In contrast, we allow Lyapunov functions to increase both during the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes.
Continuous-time optimization is currently an active field of research in optimization theory; prior work in this area has yielded useful insights and elegant methods for proving stability and convergence properties of the continuous-time optimization algorithms. This paper proposes novel gradient-flow schemes that yield convergence to the optimal point of a convex optimization problem within a fixed time from any given initial condition for unconstrained optimization, constrained optimization, and min-max problems. It is shown that the solution of the modified gradient flow dynamics exists and is unique under certain regularity conditions on the objective function, while fixed-time convergence to the optimal point is shown via Lyapunov-based analysis. The application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient-dominance, a relaxation of strong convexity. Then, a modified Newton’s method is presented that exhibits fixed-time convergence under some mild conditions on the objective function. Building upon this method, a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed. More specifically, constrained optimization problems formulated as min-max problems are considered, and a novel method for computing the optimal solution in fixed-time is proposed using the Lagrangian dual. Finally, the general min-max problem is considered, and a modified scheme to obtain the optimal solution of saddle-point dynamics in fixed time is developed. Numerical illustrations that compare the performance of the proposed method against Newton’s method, rescaled-gradient method, and Nesterov’s accelerated method are included to corroborate the efficacy and applicability of the modified gradient flows in constrained and unconstrained optimization problems.
This paper develops a novel Continuous-time Accelerated Proximal Point Algorithm (CAPPA) for l1-minimization problems with provable fixed-time convergence guarantees. The problem of l1-minimization appears in several contexts, such as sparse recovery (SR) in Compressed Sensing (CS) theory, and sparse linear and logistic regressions in machine learning to name a few. Most existing algorithms for solving l1-minimization problems are discrete-time, inefficient and require exhaustive computer-guided iterations. CAPPA alleviates this problem on two fronts:(a) it encompasses a continuous-time algorithm that can be implemented using analog circuits;(b) it betters LCA and finite-time LCA (recently developed continuous-time dynamical systems for solving SR problems) by exhibiting provable fixed-time convergence to optimal solution. Consequently, CAPPA is better suited for fast and efficient handling of SR problems. Simulation studies are presented that corroborate computational advantages of CAPPA.
In this paper, the problem of generating safe trajectories for multi-agent systems in the presence of wind and dynamic obstacles is considered. A robust controller is designed to counteract a class of state disturbances that can be thought of as wind disturbance for aerial vehicles. The considered disturbance is unmatched, bounded with known bounds, with no assumptions on the regularity properties or the distribution of the disturbance. It is also assumed that only partial states are observed, and finite-time state-estimator-based finite-time state-feedback control is used to generate the system trajectories. It is shown that, even with limited and erroneous sensing, agents are capable of avoiding collisions with moving obstacles and with each other. The designed protocol is distributed, scalable with the number of agents, and of provable safety and convergence guarantees.
In this paper, we present a control framework for a general class of control-affine nonlinear systems under spatiotemporal and input constraints. First, we present a new result on fixed-time stability, i.e., convergence within a fixed time independently of the initial conditions, in terms of a Lyapunov function. We show robustness of the proposed conditions in terms of fixed-time stability guarantees in the presence of a class of additive disturbances. Then, we consider the problem of designing control inputs for a general class of nonlinear, control-affine systems to achieve forward invariance of a safe set, as well as convergence to a goal set within a prescribed (i.e., user-defined) time. We show that the aforementioned problem based on spatiotemporal specifications can be translated into a temporal logic formula. Then, we present a quadratic program (QP) based formulation to compute the control input efficiently. We show that the proposed QP is feasible, and discuss the cases when the solution of the QP solves the considered problem of control design. In contrast to prior work, we do not make any additional assumptions on existence of a Lyapunov or a Barrier function for the feasibility of the QP. We present two case studies to corroborate our proposed methods. In the first example, the adaptive cruise control problem is considered, where a following vehicle needs to obtain a desired goal speed while maintaining a safe distance from the lead vehicle. For the second example, we consider the problem of robot motion planning for a two-agent system, where the objective of the robots is to visit a given sequence of sets in a prescribed time sequence while remaining in a given safe set and maintaining safe distance from each other.
In this paper, the fixed-time stability of a novel proximal dynamical system is investigated for solving mixed variational inequality problems. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that the solution of the proposed proximal dynamical system exists in the classical sense, is uniquely determined and converges to the unique solution of the associated mixed variational inequality problem in a fixed time. As a special case, the proposed proximal dynamical system reduces to a novel fixed-time stable projected dynamical system. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumptions of strong monotonicity are relaxed to that of strong pseudomonotonicity. Connections to convex optimization problems are discussed, and commonly studied dynamical systems in the continuous-time optimization literature are shown as special cases of the proposed proximal dynamical system considered in this paper. Finally, several numerical examples are presented that corroborate the fixed-time convergent behavior of the proposed proximal dynamical system.
This paper presents a method to solve distributed optimization problem within a fixed time over a time-varying communication topology. Each agent in the network can access its private objective function, while exchange of local information is permitted between the neighbors. This study investigates first nonlinear protocol for achieving distributed optimization for timevarying communication topology within a fixed time independent of the initial conditions. For the case when the global objective function is strictly convex, a second-order Hessian based approach is developed for achieving fixed-time convergence. In the special case of strongly convex global objective function, it is shown that the requirement to transmit Hessians can be relaxed and an equivalent first-order method is developed for achieving fixed-time convergence to global optimum. Results are further extended to the case where the underlying team objective function, possibly non-convex, satisfies only the Polyak-Łojasiewiz (PL) inequality, which is a relaxation of strong convexity.