Accepting grammars and systems

We investigate several kinds of regulated rewriting (programmed, matrix, with regular control, ordered, and variants thereof) and of parallel rewriting mechanisms (Lindenmayer systems, uniformly limited Lindenmayer systems, limited Lindenmayer systems and scattered context grammars) as accepting devices, in contrast with the usual generating mode. In some cases, accepting mode turns out to be just as powerful as generating mode, e.g. within the grammars of the Chomsky hierarchy, within random context, regular control, L systems, uniformly limited L systems, scattered context. Most of these equivalences can be proved using a metatheorem on so-called context condition grammars. In case of matrix grammars and programmed grammars without appearance checking, a straightforward construction leads to the desired equivalence result. Interestingly, accepting devices are (strictly) more powerful than their generating counterparts in case of ordered grammars, programmed and matrix grammars with appearance checking (even programmed grammarsm with unconditional transfer), and 1lET0L systems. More precisely, if we admit erasing productions, we arrive at new characterizations of the recursivley enumerable languages, and if we do not admit them, we get new characterizations of the context-sensitive languages. Moreover, we supplement the published literature showing: - The emptiness and membership problems are recursivley solvable for generating ordered grammars, even if we admit erasing productions. - Uniformly limited propagating systems can be simulated by programmed grammars without erasing and without appearance checking, hence the emptiness and membership problems are recursively solvable for such systems. - We briefly discuss the degree of nondeterminism and the degree of synchronization for devices with limited parallelism

3 sampled-data control of nonlinear systems

This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research

Stochastic nonlinear control: A unified framework for stability, dissipativity, and optimality

In this work, we develop connections between stochastic stability theory and stochastic optimal control. In particular, first we develop Lyapunov and converse Lyapunov theorems for stochastic semistable nonlinear dynamical systems. Semistability is the property whereby the solutions of a stochastic dynamical system almost surely converge to (not necessarily isolated) Lyapunov stable in probability equilibrium points determined by the system initial conditions. Then we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems. Specifically, we provide a simplified and tutorial framework for stochastic optimal control and focus on connections between stochastic Lyapunov theory and stochastic Hamilton-Jacobi-Bellman theory. In particular, we show that asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both stochastic stability and optimality. Moreover, extensions to stochastic finite-time and partial-state stability and optimal stabilization are also addressed. Finally, we extended the notion of dissipativity theory for deterministic dynamical systems to controlled Markov diffusion processes and show the utility of the general concept of dissipation for stochastic systems.Ph.D

Membership for limited ET0L languages is not decidable

In this paper, we show how to encode arbitrary enumerable set of numbers given by register machines within limited EPT0L systems and programmed grammars with unconditional transfer.This result has various consequences, e.g.the existence of nonrecursive sets generable by 1lET0L systems or by programmed grammars with unconditional transfer. Moreover, ordered grammars are strictly less powerful than 1lET0L systems

Guaranteed safe switching for switching adaptive control

Adaptive control algorithms may not behave well in practice due to discrepancies between the theory and actual practice. The proposed results in this manuscript constitute an effort in providing algorithms which assure more reliable operation in practice. Our emphasis is on algorithms that will be safe in the sense of not permitting destabilizing controllers to be switched in the closed-loop and to prevent wild signal fluctuations to occur. Coping with the connection or possible connection of destabilizing controllers is indeed a daunting task. One of the most intuitive forms of adaptive control, gain scheduling, is an approach to control of non-linear systems which utilizes a family of linear controllers, each of which provides satisfactory control for a different operating point of the system. We provide a mechanism for guaranteeing closed-loop stability over rapid switching between controllers. Our proposed design provides a simplification using only finite number of pre-determined values for the controller gain, where the observer gain is computed via a table look-up method. In comparison to the original gain scheduling design which our procedure builds on, our design achieves similar performance but with much less computational burden. Many multi-controller adaptive switching algorithms do not explicitly rule out the possibility of switching a destabilizing controller into the closed-loop. Even if the new controller is ensured to be stabilizing, performance verification with the new controller is not straightforward. The importance of this arises in iterative identification and control algorithms and multiple model adaptive control (MMAC). We utilize a limited amount of experimental and possibly noisy data obtained from a closed-loop consisting of an existing known stabilizing controller connected to an unknown plant-to infer if the introduction of a prospective controller will stabilize the unknown plant. We propose analysis results in a nonlinear setting and provide data-based tests for verifying the closed-loop stability with the introduction of a new nonlinear controller to replace a linear controller. We also propose verification tools for the closed-loop performance with the introduction of a new stabilizing controller using a limited amount of data obtained from the existing stable closed-loop. The simulation results in different practical scenarios demonstrate efficacy and versatility of our results, and illustrate practicality of our novel data-based tests in addressing an instability problem in adaptive control algorithms

Adaptive model predictive control

The problem of model predictive control (MPC) under parametric uncertainties for a class of nonlinear systems is addressed. An adaptive identi er is used to estimate the pa- rameters and the state variables simultaneously. The algorithm proposed guarantees the convergence of parameters and the state variables to their true value. The task is posed as an adaptive model predictive control problem in which the controller is required to steer the system to the system setpoint that optimizes a user-speci ed objective function. The technique of adaptive model predictive control is developed for two broad classes of systems. The rst class of system considered is a class of uncertain nonlinear systems with input to state stability property. Using a generalization of the set-based adaptive estimation technique, the estimates of the parameters and state are updated to guarantee convergence to a neighborhood of their true value. The second involves a method of determining appropriate excitation conditions for nonlin- ear systems. Since the identi cation of the true cost surface is paramount to the success of the integration scheme, novel parameter estimation techniques with better convergence properties are developed. The estimation routine allows exact reconstruction of the systems unknown parameters in nite-time. The applicability of the identi er to improve upon the performance of existing adaptive controllers is demonstrated. Then, an adaptive nonlinear model predictive controller strategy is integrated to this estimation algorithm in which ro- bustness features are incorporated to account for the e ect of the model uncertainty. To study the practical applicability of the developed method, the estimation of state vari- ables and unknown parameters in a stirred tank process has been performed. The results of the experimental application demonstrate the ability of the proposed techniques to estimate the state variables and parameters of an uncertain practical system.Departamento de Ingeniería de Sistemas y AutomáticaMáster en Investigación en Ingeniería de Procesos y Sistemas Industriale

Optimality of maximal-effort vaccination

It is widely acknowledged that vaccinating at maximal effort in the face of an ongoing epidemic is the best strategy to minimise infections and deaths from the disease. Despite this, no one has proved that this is guaranteed to be true if the disease follows multi-group SIR (Susceptible-Infected-Recovered) dynamics. This paper provides a novel proof of this principle for the existing SIR framework, showing that the total number of deaths or infections from an epidemic is decreasing in vaccination effort. Furthermore, it presents a novel model for vaccination which assumes that vaccines assigned to a subgroup are distributed randomly to the unvaccinated population of that subgroup. It suggests, using COVID-19 data, that this more accurately captures vaccination dynamics than the model commonly found in the literature. However, as the novel model provides a strictly larger set of possible vaccination policies, the results presented in this paper hold for both models

Control Theory: A Mathematical Perspective on Cyber-Physical Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently the control field is facing new challenges motivated by application domains that involve networks of systems. Examples are interacting robots, networks of autonomous cars or the smart grid. In order to address the new challenges posed by these application disciplines, the special focus of this workshop has been on the currently very active field of Cyber-Physical Systems, which forms the underlying basis for many network control applications. A series of lectures in this workshop was devoted to give an overview on current theoretical developments in Cyber-Physical Systems, emphasizing in particular the mathematical aspects of the field. Special focus was on the dynamics and control of networks of systems, distributed optimization and formation control, fundamentals of nonlinear interconnected systems, as well as open problems in control

Distributed control in the smart grid

This thesis addresses some of the challenges that arise when the new smart grid paradigm is applied to power systems. In particular, novel control strategies are designed to deal in a decentralized matter with the increasing complexity of the network. Two main areas are investigated: participation to frequency control of variable-speed wind turbines and management of large populations of competing agents (e.g. micro-storage devices and "smart appliances") that exchange energy with the system. The first part of this work presents two different techniques that allow wind turbines to provide frequency response: following the trip of a large power plant, the turbines population increases its aggregate generated power, reducing the resulting drop in frequency. A first method models the wind turbines as stochastic hybrid systems: the generators switch randomly between two operative modes characterized by different efficiency and generated power at equilibrium. Transitions are driven by frequency-dependent switching functions: single generators behave randomly while large populations perform deterministically, changing the total power in response to frequency variations. The second proposed control strategy allows a prescribed increase in generation, distributing the control effort among the individual turbines in order to maximize the duration of frequency support or minimize the resulting kinetic energy losses. The second part of the thesis deals with large populations of agents which determine their operation strategy in response to a broadcast price signal. Micro-storage devices performing energy arbitrage are initially considered: each agent charges/discharges during the day in order to maximize its profit. By approximating the number of devices as infinite, modelling the population as a continuum and describing the problem through a differential game with infinite players (mean field game), it is possible to avoid synchronicity phenomena and determine an equilibrium for the market. Finally, the similar case of flexible demand is analyzed, with price-responsive appliances that schedule their power consumption in order to minimize their energy cost. Necessary and sufficient conditions for the existence of a Nash equilibrium are provided, extending the results by introducing time-varying constraints on the power rate and considering partial flexibility of the devices.Open Acces