Advances in the Theory of Fixed-time Stability with Applications in Constrained Control and Optimization
Creators: name: Garg, Kunal
Publisher: name: My University
Year: 2021
Subjects: subject: Constrained control
subject: Fixed-time stability
subject: Optimization
subject: Multi-agent control
subject: Optimization based control synthesis
subject: Aerospace Engineering
subject: FOS: Mechanical engineering
subject: Engineering

description: Driving the state of dynamical systems to a desired point or set is a problem of crucial practical importance. Various constraints are present in real-world applications due to structural and operational requirements. Spatial constraints, i.e., constraints requiring the system trajectories to evolve in some textit{safe} set, while visiting some goal set(s), are typical in safety-critical applications. Furthermore, temporal constraints, i.e., constraints pertaining to the time of convergence, appear in time-critical applications, for instance, when a task must complete within a fixed time due to an internal or an external deadline. Moreover, imperfect knowledge of the operational environment and/or system dynamics, and the presence of external disturbances render offline control policies impractical and make it essential to develop methods for online control synthesis. Thus, from the implementation point-of-view, it is desired to design fast optimization algorithms so that an optimal control input, e.g., min-norm control input, can be computed online. As compared to exponential stability, the notion of fixed-time stability is stronger, with the time of convergence being finite and is bounded for all initial conditions. This dissertation studies the theory of fixed-time stability with applications in multi-agent control design under spatiotemporal and input constraints, and in the field of continuous-time optimization. First, multi-agent control design problems under spatiotemporal constraints are studied. A vector-field-based controller is presented for distributed control of multi-agent systems for a class of agents modeled under double-integrator dynamics. A finite-time controller that utilizes the state estimates obtained from a finite-time state observer is designed to guarantee that each agent reaches its goal location within a finite time while maintaining safety with respect to other agents as well as dynamic obstacles. Next, new conditions for fixed-time stability are developed to use fixed-time stability along with input constraints. It is shown that these new conditions capture the relationship between the time of convergence, the domain of attraction, and the input constraints for fixed-time stability. Additionally, the new conditions establish the robustness of fixed-time stable systems with respect to a class of vanishing and non-vanishing additive disturbances. Utilizing these new fixed-time stability results, a control design method using convex optimization is presented for a general class of systems having nonlinear, control-affine dynamics. Control barrier and control Lyapunov function conditions are used as linear constraints in the optimization problem for set-invariance and goal-reachability requirements. Various practical issues, such as input constraints, additive disturbance, and state-estimation error, are considered. Next, new results on finite-time stability for a class of hybrid and switched systems are proposed using a multiple-Lyapunov-functions framework. The presented framework allows the system to have unstable modes. Finally, novel continuous-time optimization methods are studied with guarantees for fixed-time convergence to an optimal point. Fixed-time stable gradient flows are developed for unconstrained convex optimization problems under conditions such as strict convexity and gradient dominance of the objective function, which is a relaxation of strong convexity. Furthermore, min-max problems are considered and modifications of saddle-point dynamics are proposed with fixed-time stability guarantees under various conditions on the objective function.