Stability of singularly perturbed hybrid systems with restricted systems evolving on boundary layer manifolds

We present a singular perturbation theory applicable to systems with hybrid boundary layer systems and hybrid reduced systems {with} jumps from the boundary layer manifold. First, we prove practical attractivity of an adequate attractor set for small enough tuning parameters and sufficiently long time between almost all jumps. Second, under mild conditions on the jump mapping, we prove semi-global practical asymptotic stability of a restricted attractor set. Finally, for certain classes of dynamics, we prove semi-global practical asymptotic stability of the restricted attractor set for small enough tuning parameters and sufficiently long period between almost all jumps of the slow states only

Singularly Perturbed Stochastic Hybrid Systems: Stability and Recurrence via Composite Nonsmooth Foster Functions

We introduce new sufficient conditions for verifying stability and recurrence properties in singularly perturbed stochastic hybrid dynamical systems. Specifically, we focus on hybrid systems with deterministic continuous-time dynamics that exhibit multiple time scales and are modeled by constrained differential inclusions, as well as discrete-time dynamics modeled by constrained difference inclusions with random inputs. By assuming regularity and causality of the dynamics and their solutions, respectively, we propose a suitable class of composite nonsmooth Lagrange-Foster and Lyapunov-Foster functions that can certify stability and recurrence using simpler functions related to the slow and fast dynamics of the system. We establish the stability properties with respect to compact sets, while the recurrence properties are studied only for open sets

Robust Hybrid Systems for Control, Learning, and Optimization in Networked Dynamical Systems

The deployment of advanced real-time control and optimization strategies in socially-integratedengineering systems could significantly improve our quality of life whilecreating jobs and economic opportunity. However, in cyber-physical systems such assmart grids, transportation networks, healthcare, and robotic systems, there still existseveral challenges that prevent the implementation of intelligent control strategies.These challenges include the existence of limited communication networks, dynamicand stochastic environments, multiple decision makers interacting with the system,and complex hybrid dynamics emerging from the feedback interconnection of physicalprocesses and computational devices.In this dissertation, we study the problem of designing robust control and optimizationalgorithms for cyber-physical systems using the framework of hybrid dynamicalsystems. We propose different theoretical frameworks for the design and analysis offeedback mechanisms that optimize the performance of dynamical systems without requiringan explicit characterization of their mathematical model, i.e., in a model-freeway. The closed-loop system that emerges of the interconnection of the plant with thefeedback mechanism describes, in general, a set-valued hybrid dynamical system. Thesetypes of systems combine continuous-time and discrete-time dynamics, and they usuallylack the uniqueness of solutions property. The framework of set-valued hybriddynamical systems allows us to study many complex dynamical systems that emerge indifferent engineering applications, such as networked multi-agent systems with switching graphs, non-smooth mechanical systems, dynamic pricing mechanisms in transportationsystems, autonomous robots with logic-based controllers, etc. We proposea step-by-step approach to the design of different types of discrete-time, continuous-time,hybrid, and stochastic controllers for different types of applications, extendingand generalizing different results in the literature in the area of extremum seeking control,sampled-data extremization, robust synchronization, and stochastic learning innetworked systems. Our theoretical results are illustrated via different simulations andnumerical examples

Online Optimization of Switched LTI Systems Using Continuous-Time and Hybrid Accelerated Gradient Flows

This paper studies the design of feedback controllers that steer the output of a switched linear time-invariant system to the solution of a possibly time-varying optimization problem. The design of the feedback controllers is based on an online gradient descent method, and an online hybrid controller that can be seen as a regularized Nesterov's accelerated gradient method. Both of the proposed approaches accommodate output measurements of the plant, and are implemented in closed-loop with the switched dynamical system. By design, the controllers continuously steer the system output to an optimal trajectory implicitly defined by the time-varying optimization problem without requiring knowledge of exogenous inputs and disturbances. For cost functions that are smooth and satisfy the Polyak-Lojasiewicz inequality, we demonstrate that the online gradient descent controller ensures uniform global exponential stability when the time-scales of the plant and the controller are sufficiently separated and the switching signal of the plant is slow on the average. Under a strong convexity assumption, we also show that the online hybrid Nesterov's method guarantees tracking of optimal trajectories, and outperforms online controllers based on gradient descent. Interestingly, the proposed hybrid accelerated controller resolves the potential lack of robustness suffered by standard continuous-time accelerated gradient methods when coupled with a dynamical system. When the function is not strongly convex, we establish global practical asymptotic stability results for the accelerated method, and we unveil the existence of a trade-off between acceleration and exact convergence in online optimization problems with controllers using dynamic momentum. Our theoretical results are illustrated via different numerical examples

Spatial Formation Control

In this thesis, we study robust spatial formation control from several aspects. First, we study robust adaptive attitude synchronization for a network of rigid body agents using various attitude error functions defined on SO(3). Our results are particularly useful for networks with large initial attitude difference. We devise an adaptive geometric approach to cope with situations where the inertia matrices are not available for measurement. We use the Frobenius norm as a measure for the difference between the actual values of inertia matrices and their estimated values, to construct the individual adaptive laws of the agents. Compared to the previous methods for synchronization on SO(3) such as those which are based on quaternions, our proposed approach does not contain any attitude representation ambiguity. As the final part of our studies from the attitude synchronization aspect, we analyze robustness to external disturbances and unmodeled dynamics, and propose a method to attenuate such effects. Simulation results illustrate the effectiveness of the proposed approach. In the next part of the thesis, we study the distributed localization of the extremum point of unknown quadratic functions representing various physical or artificial signal potential fields. It is assumed that the value of such functions can be measured at each instant. Using high pass filtering of the measured signals, a linear parametric model is obtained for system identification. For design purposes, we add a consensus term to modify the identification subsystem. Next, we analyze the exponential convergence of the proposed estimation scheme using algebraic graph theory. In addition, we derive a distributed identifiability condition and use it for the construction of distributed extremum seeking control laws. In particular, we show that for a network of connected agents, if each agent contains a portion of the dithering signals, it is still possible to drive the system states to the extremum point provided that the distributed identifiability condition is satisfied. In the final part of this research, several robust control problems for general linear time invariant multi-agent systems are studied. We consider the robust consensus problem in the presence of unknown Lipschitz nonlinearities and polytopic uncertainties in the model of each agent. Next, this problem is solved in the presence of external disturbances. A set of control laws is proposed for the network to attain the consensus task and under the zero initial condition, achieves the desired H-infinity performance. We show that by implementing the modified versions of these control laws, it is possible to perform two-time scales formation control

Learning-Based Controller Design with Application to a Chiller Process

In this thesis, we present and study a few approaches for constructing controllers for uncertain systems, using a combination of classical control theory and modern machine learning methods. The thesis can be divided into two subtopics. The first, which is the focus of the first two papers, is dual control. The second, which is the focus of the third and last paper, is multiple-input multiple-output (MIMO) control of a chiller process. In dual control, the goal is to construct controllers for uncertain systems that in expectation minimize some cost over a certain time horizon. To achieve this, the controller must take into account the dual goals of accumulating more information about the process, by applying some probing input, and using the available information for controlling the system. This is referred to as the exploration-exploitation trade-off. Although optimal dual controllers in theory can be computed by solving a functional equation, this is usually intractable in practice, with only some simple special cases as exceptions. Therefore, it is interesting to examine methods for approximating optimal dual control. In the first paper, we take the approach of approximating the value function, which is the solution of the functional equation that can be used to deduce the optimal control, by using artificial neural networks. In the second paper, neural networks are used to represent and estimate hyperstates, which contain information about the conditional probability distributions of the system uncertainties. The optimal dual controller is a function of the hyperstate, and hence it should be useful to have a representation of this quantity when constructing an approximately optimal dual controller. The hyperstate transition model is used in combination with a reinforcement learning algorithm for constructing a dual controller from stochastic simulations of a system model that includes models of the system uncertainties. In the third paper, we suggest a simple reinforcement learning method that can be used to construct a decoupling matrix that allows MIMO control of a chiller process. Compared to the commonly used single-input single-output (SISO) structures, these controllers can decrease the variations in some system signals. This makes it possible to run the system at operating points closer to some constraints, which in turn can enable more energy-efficient operation