Predicting oscillations in relay feedback systems, using fixed points of Poincar\'e maps, and Hopf bifurcations

The relay autotuning method identifies plant parameters, from oscillations of the plant under relay feedback. To predict the presence and nature of such oscillations, we apply the following two approaches: (a) analysis of the switching dynamics, while using an ideal relay, and (b) bifurcation analysis, while using a smooth approximation of the relay. For stable plants with positive DC gains, our analyses predict that: (i) a periodic orbit is guaranteed, for a class of non-minimum phase plants of relative degree one, whose step response starts with an inverse response, and (ii) for a wider class of plants, whose root locus diagrams cross the imaginary axis at complex conjugate values, limit cycles are merely suggested.Comment: submitted to the IEEE transactions on Automatic Contro

Packet based Inference and Control

Communication constraints in Networked Control systems are frequently limits on data packet rates. To efficiently use the available packet rate budgets, we have to resort to event-triggered packet transmission. We have to sample signal waveforms and transmit packets not at deterministic times but at random times adapted to the signals measured. This thesis poses and solves some basic design problems we face in reaching for the extra efficiency. We start with an estimation problem involving a single sensor. A sensor makes observations of a diffusion process, the state signal, and has to transmit samples of this process to a supervisor which maintains an estimate of the state. The objective of the sensor is to transmit samples strategically to the supervisor to minimize the distortion of the supervisor's estimate while respecting sampling rate constraints. We solve this problem over both finite and infinite horizons when the state is a scalar linear system. We describe the relative performances of the optimal sampling scheme, the best deterministic scheme and of the suboptimal but simple to implement level-triggered sampling scheme. Apart from the utility of finding the optimal sampling strategies and their performances, we also learnt of some interesting properties of the level-triggered sampling scheme. The control problem is harder to solve for the same setting with a single sensor. In the estimation problem for the linear state signal, the estimation error is also a linear diffusion and is reset to zero at sampling times. In the control problem, there is no equivalent to the error signal. We pay attention to an infinite horizon average cost problem where, the sampling strategy is chosen to be level-triggered. We design piece-wise constant controls by translating the problem to one for discrete-time Markov chain formed by the sampled state. Using results on the average cost control of Markov chains, we are able to derive optimality equations and iteratively compute solutions. The last chapter tackles a binary sequential hypothesis testing problem with two sensors. The special feature of the problem is the ability of each sensor to hear the transmissions of the other towards the supervisor. Each sensor is afforded on transmission of a sample of its likelihood ratio process. We restrict attention to level-triggered sampling. Although we are unable to demonstrate overall optimality of the asynchronous scheme we pursue, we are able to describe its advantages over other level-triggered schemes and of course the deterministic one. The chief merits of this thesis are the formulation and solution of some basic problems in multi-agent estimation and control. In the problems we have attacked, we have been able to deal with the differences in information patterns at sensors and supervisors. The main demerits are the ignoring of packet losses and of variable delays in packet transmissions. The situation of packet losses can however be handled at the expense of additional computations. To summarize, this thesis provides valuable generalizations of the works of Astrom and Bernhardsson and of Kushner on timing of Control actions and of Sampling observations respectively

Bounds on set exit times of affine systems, using Linear Matrix Inequalities

Efficient computation of trajectories of switched affine systems becomes possible, if for any such hybrid system, we can manage to efficiently compute the sequence of switching times. Once the switching times have been computed, we can easily compute the trajectories between two successive switches as the solution of an affine ODE. Each switching time can be seen as a positive real root of an analytic function, thereby allowing for efficient computation by using root finding algorithms. These algorithms require a finite interval, within which to search for the switching time. In this paper, we study the problem of computing upper bounds on such switching times, and we restrict our attention to stable time-invariant affine systems. We provide semi-definite programming models to compute upper bounds on the time taken by the trajectories of an affine ODE to exit a set described as the intersection of a few generalized ellipsoids. Through numerical experiments, we show that the resulting bounds are tighter than bounds reported before, while requiring only a modest increase in computation time.publishedVersio

A NOTE ON THE MEAN-SQUARE QUANTIZATION ERROR

A note on the mean-square quantization error for a scalar random variable with a probability density function which is bounded, piece-wise continuous, and has a finite second moment

Adaptive sampling for linear state estimation

When a sensor has continuous measurements but sends occasional messages over a data network to a supervisor which estimates the state, the available packet rate fixes the achievable quality of state estimation. When such rate limits turn stringent, the sensor’s messaging policy should be designed anew. What are the good causal messaging policies ? What should message packets contain ? What is the lowest possible distortion in a causal estimate at the supervisor ? Is Delta sampling better than periodic sampling ? We answer these questions for a Markov state process under an idealized model of the network and the assumption of perfect state measurements at the sensor. If the state is a scalar, or a vector of low dimension, then we can ignore sample quantization. If in addition, we can ignore jitter in the transmission delays over the network, then our search for efficient messaging policies simplifies. Firstly, each message packet should contain the value of the state at that time. Thus a bound on the number of data packets becomes a bound on the number of state samples. Secondly, the remaining choice in messaging is entirely about the times when samples are taken. For a scalar, linear diffusion process, we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion. We stick to finite-horizons and impose a hard upper bound N on the number of allowed samples. We cast the design as a problem of choosing an optimal sequence of stopping times. We reduce this to a nested sequence of problems, each asking for a single optimal stopping time. Under an unproven but natural assumption about the least-square estimate at the supervisor, each of these single stopping problems are of standard form. The optimal stopping times are random times when the estimation error exceeds designed envelopes. For the case where the state is a Brownian motion, we give analytically: the shape of the optimal sampling envelopes, the shape of the envelopes under optimal Delta sampling, and their performances. Surprisingly, we find that Delta sampling performs badly. Hence, when the rate constraint is a hard limit on the number of samples over a finite horizon, we should should not use Delta sampling

Intrusion Detection with Support Vector Machines and Generative Models

This paper addresses the task of detecting intrusions in the form of malicious programs on a host computer system by inspecting the trace of system calls made by these programs. We use "attack-tree" type generative models for such intrusions to select features that are used by a Support Vector Machine Classifier. Our approach combines the ability of an HMM generative model to handle variable-length strings, i.e. the traces, and the non-asymptotic nature of Support Vector Machines that permits them to work well with small training sets

Sampling of Diffusion Processes for Real-time Estimation

This paper addresses the causal sampling of observations of a diffusion process that results in a good quality continuous estimator based upon these samples. The optimal sampling scheme with a fixed number of samples is found by solving an optimal (multiple) stopping problem. This is solved explicitly in a special case. A class of threshold approximations is also described