A finite state machine framework for robust analysis and control of hybrid systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 107-115).Hybrid systems, describing interactions between analog and discrete dynamics, are pervasive in engineered systems and pose unique, challenging performance verification and control synthesis problems. Existing approaches either lead to computationally intensive and sometimes undecidable problems, or make use of highly specialized discrete abstractions with questionable robustness properties. The thesis addresses some of these challenges by developing a systematic, computationally tractable approach for design and certification of systems with discrete, finite-valued actuation and sensing. This approach is inspired by classical robust control, and is based on the use of finite state machines as nominal models of the hybrid systems. The development does not assume a particular algebraic or topological structure on the signal sets. The thesis adopts an input/output view of systems, proposes specific classes of inequality constraints to describe performance objectives, and presents corresponding 'small gain' type arguments for robust performance verification. A notion of approximation that is compatible with the goal of controller synthesis is defined. An approximation architecture that is capable of handling unstable systems is also proposed.(cont.) Constructive algorithms for generating finite state machine approximations of the hybrid systems of interest, and for efficiently computing a-posteriori bounds on the approximation error are presented. Analysis of finite state machine models, which reduces to searching for an appropriate storage function, is also shown to be related to the problem of checking for the existence of negative cost cycles in a network, thus allowing for a verification algorithm with polynomial worst-case complexity. Synthesis of robust control laws is shown to reduce to solving a discrete, infinite horizon min-max problem. The resulting controllers consist of a finite state machine state observer for the hybrid system and a memoryless full state feedback switching control law. The use of this framework is demonstrated through a simple benchmark example, the problem of stabilizing a double integrator using switched gain feedback and binary sensing. Finally, some extensions to incremental performance objectives and robustness measures are presented.by Danielle C. Tarraf.Ph.D

Advances in the Theory of Fixed-time Stability with Applications in Constrained Control and Optimization

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.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168071/1/kgarg_1.pd

A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Mathematical control of complex systems

Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited