Inevitability of Phase-locking in a Charge Pump Phase Lock Loop using Deductive Verification

Phase-locking in a charge pump (CP) phase lock loop (PLL) is said to be inevitable if all possible states of the CP PLL eventually converge to the equilibrium, where the input and output phases are in lock and the node voltages vanish. We verify this property for a CP PLL using deductive verification. We split this complex property into two sub-properties defined in two disjoint subsets of the state space. We deductively verify the first property using multiple Lyapunov certificates for hybrid systems, and use the Escape certificate for the verification of the second property. Construction of deductive certificates involves positivity check of polynomial inequalities (which is an NP-Hard problem), so we use the sound but incomplete Sum of Squares (SOS) relaxation algorithm to provide a numerical solution

Towards Physical Hybrid Systems

Some hybrid systems models are unsafe for mathematically correct but physically unrealistic reasons. For example, mathematical models can classify a system as being unsafe on a set that is too small to have physical importance. In particular, differences in measure zero sets in models of cyber-physical systems (CPS) have significant mathematical impact on the mathematical safety of these models even though differences on measure zero sets have no tangible physical effect in a real system. We develop the concept of "physical hybrid systems" (PHS) to help reunite mathematical models with physical reality. We modify a hybrid systems logic (differential temporal dynamic logic) by adding a first-class operator to elide distinctions on measure zero sets of time within CPS models. This approach facilitates modeling since it admits the verification of a wider class of models, including some physically realistic models that would otherwise be classified as mathematically unsafe. We also develop a proof calculus to help with the verification of PHS.Comment: CADE 201

Pointwise Asymptotic Stability

The talk presents some concepts and results from systems and control theory, focusing on convergence to and stability of a continuum of equilibria in a dynamical system. The well-studied and understood asymptotic stability of a compact set requires Lyapunov stability of the set: solutions that start close remain close to the set, and that every solution converge to the set, in terms of distance. Pointwise asymptotic stability of a set of equilibria requires Lyapunov stability of each equilibrium, and that every solution converge to one of the equilibria. This property is present, for example, in continuous-time steepest descent and convergent saddle-point dynamics, in optimization algorithms generating convergent Fejer monotone sequences, etc., and also in many consensus algorithms for multi-agent systems. The talk will present some background on asymptotic stability and then discuss necessary and sufficient conditions for pointwise asymptotic stability in terms of set-valued Lyapunov functions; robustness of this property to perturbations; and how the property can be achieved in a control system by optimal control.Non UBCUnreviewedAuthor affiliation: Loyola University ChicagoFacult

Convexity, convergence and feedback in optimal control

Thesis (Ph. D.)--University of Washington, 2000The results of this thesis are oriented towards the study of convex problems of optimal control in the extended piecewise linear-quadratic format. Such format greatly extends the classical linear-quadratic regulator problem and allows for the treatment of control constraints, including state-dependent ones. The Hamiltonian system associated with a control problem, the optimal feedback mapping, and the value function are objects of main interest. Several tools of nonsmooth and convex analysis are developed, including a new approximation scheme for convex functions, characterizations of a saddle function through the properties of it's conjugate, and a new distance formula for monotone operators. The optimal feedback mapping for control problems is given, in terms of subdifferentials of the corresponding Hamiltonian and of the value function. The Hamiltonian system is employed to investigate the regularity properties of the value function for the problem in question. Conditions for differentiability of the value function and single-valuedness of the feedback in an extended linear-quadratic control problem are stated, in terms of the matrices and constraint sets defining the problem. Application of convex analysis to differential games yields explicit formulas for equilibrium controls and a generalized Hamiltonian equation describing an equilibrium trajectory

Direct design of robustly asymptotically stabilizing hybrid feedback

A direct construction of a stabilizing hybrid feedback that is robust to general measurement error is given for a general nonlinear control system that is asymptotically controllable to a compact set

Smooth patchy control Lyapunov functions

International audienceA smooth patchy control Lyapunov function for a nonlinear system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy certain further increase or decrease conditions. We prove that such a control Lyapunov function exists for any asymptotically controllable nonlinear system. We also show a construction, based on such a control Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise