Applications of trajectory-based analysis in optimization and control

Abstract

The synergy between optimization and control is a long standing tradition. In fact, this synergy is becoming more and more apparent because of the multi-disciplinary character of the most pressing, current engineering problems along with constant developments of these two fields. Historically, optimization methods have helped the control community to achieve their design goals formalized in some sort of objective function. On the other hand, control theory has provided a setting to interpret complicated aspects of optimization algorithms. In this thesis, we address three problem instances that lie on the boundary of optimization and control. We employ tools from one field to address a problem in the other field. Fundamentally, our proposed methods share a similar character: their analysis techniques are trajectory-based. In simple words, our proposed methods exploit the trajectories generated by the dynamics that represent each problem instance. The first problem focuses on a 2nd-order, damped differential equation (ODE). This ODE along with its numerous variations have been used to develop or analyze various optimization algorithms, known as fast methods. As an alternative to the existing methods, we first amend the underling ODE with two types of state-dependent inputs, and then extend the resulting controlled dynamics to two hybrid control systems. Employing a trajectory-based analysis, both control laws are constructed to guarantee exponential convergence in a suboptimality measure. To show that the trajectories generated by each hybrid control system are well-posed, we demonstrate Zeno-freeness of solution trajectories in both cases. Furthermore, we propose a mechanism to determine a time-discretization step-size such that the resulting discrete-time hybrid control systems are exponentially stable. Event-based implementation of control laws have received a lot of attention during the past decade. The reason for this interest is the hope to reduce the conservatism involved in the traditional periodic implementation. In the second problem of this thesis, we introduce an event-based sampling policy for a constraint-tightening, robust model predictive control (RMPC) method. The triggering mechanism is a sequence of hyper-rectangles constructed around the optimal state trajectories. In particular, the triggering mechanism's nature makes the proposed approach a suitable choice for plants without a centralized sensory node. A key feature of the proposed method is its complete decoupling from the RMPC method's parameters, facilitating a meaningful comparison between the periodic and aperiodic implementation policies. Furthermore, we provide two types of convex formulations to design the triggering mechanism. The last problem we focus on in this thesis is also related to the event-based implementation of a control law. However, the main aim here is to propose an entity that can be utilized by a real-time engineer to schedule tasks in a networked structure. A common entity provided in the literature related to event-triggering approaches is the minimal inter-execution time (to show the avoidance of a Zeno behavior in the closed-loop system). Nonetheless, such a quantity is extremely conservative when used for scheduling purposes. In this problem, we consider an $\mathcal{L}_2$-based triggering mechanism introduced in the literature and propose a framework to construct a timed safety automaton that can capture the triggering instants generated by this mechanism. In our analysis, we borrow some tools from stability analysis of delayed systems along with reachability analysis to construct the desired timed safety automaton.