Switched systems as models of dynamic treatment scheduling

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Switched systems as models of dynamic treatment scheduling

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Title: Switched systems as models of dynamic treatment scheduling
Author: Wang, Ning
Abstract: One interesting problem in treating a disease with multiple drug therapies is the timing of the drug treatment. In this thesis, we study the switching time of the drug therapies by using the theories of switching system developed by Dayawansa and Martin [2]. The model we are interested in is developed in the following manner. We assume some 2 by 1 vector, x, to represent the state of the patient in the treatment, then the length of the vector is being measured and that is scaled so that the length 0 represent death and 1 represent remission. For each treatment we assume the model of the form xn+1 = Axn where A is 2 by 2 matrix with |A| 6= 0. Then for two treatments, we have the combined model xn+1 = (nA + (1 − n)B)xn where |A||B| 6= 0 and n 2 {0, 1}. We consider n to be a control variable which represents the treatment to be applied. Then for each sequence of n, the switched linear system can be written as the product of random matrices A and B. In this thesis we give conditions that a discrete time switched linear system must satisfy if it is stable. We do this by calculating the mean and covariance of the set of matrices obtained by using all possible switches. The theory of switched linear systems has received considerable attention in the systems theory literature in the last two decades. However, for discrete time switched systems the literature is much older going back to at least the early 1960��s with the publication of the paper of Furstenberg and Kesten in the area of products of random matrices, or if you like the random products of matrices. The way that we have approached this problem is to consider the switched linear system as evolving on a partially ordered network that is, in fact, a tree. This allows us to make use of the developments of 50 years of study on random products that exists in the statistics literature. A nice byproduct of this research is that we use K��onig��s theorem of finatary trees. This may be the first use of this theorem in systems and control. In this thesis, based on the proof of several theories, we apply a 2 by 2 matrices example, to the timing of drug treatment problem and we can find a particular switching treatments regime to drive the system away from the origin, which means the patients could be cured.
URI: http://hdl.handle.net/2346/17399
Date: 2009-05

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