# Browsing by Subject "Nonlinear systems"

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Item The analysis of nonlinear systems driven by almost periodic inputs(2003) Van Zyl, Gideon Johannes; Sandberg, Irwin W.Show more We start in Chapter 1 by motivating the research, defining almost periodic and asymptotically almost periodic functions, reviewing known properties of these functions and introducing notation to be used in the other chapters. In Chapter 2 we show how input-output stability theory bears on the problem of obtaining a frequency-domain stability criterion that can be used to design periodically driven varactor (nonlinear capacitor) circuits with guaranteed stability. We present an example of how a varactor frequency doubler with guaranteed stability may be designed using this criterion. In Chapter 3 we give an analytical basis for evaluating the spectral coefficients for a large family of systems. This involves a convergent iterative process and certain bounds on the errors incurred in truncating the process. In Chapter 4 we consider the equations of a large class of nonlinear circuits driven by asymptotically almost periodic inputs, and give an analytical basis for the use of harmonic balance to find steady-state solutions. More specifically, we show that in a certain setting there is a unique solution to the problem of obtaining a harmonic balance approximation, and that the approximations approach the actual solution as additional spectral components are included. The results in Chapters 2–4 all involve a key circle-criterion hypothesis. In Chapter 5 we give an example that shows that this hypothesis cannot be relaxed significantly. In Chapter 6 we give a generalization of the techniques in the first chapters that allows us to extend the theory to include non-diagonal nonlinearities. We give an example that shows the generalization is useful even in the case that the nonlinearity is diagonal. In Chapter 7 we derive an integral equation that describes a nonlinear resistor in parallel with a nonlinear capacitor driven by a Thevenin equivalent source. We show how to obtain a contraction mapping operator on the set of square integrable functions and show the optimality of certain constants used in the construction of the contraction mapping operator. We end in Chapter 8 by looking at an alternative method for determining the stability of steady state regimes in nonlinear circuits. We point out some problems in the theory and suggest a solution for one problem, but conclude that this method has some serious shortcomings.Show more Item The application of visualization methods to educational data sets with inspiration from statistical and fluid mechanics(2014-05) Bendinelli, Anthony James; Marder, Michael P., 1960-Show more This dissertation focuses on the development of visualization methods that enable us to examine longitudinal data in a unique way. We take inspiration from statistical and fluid mechanics to represent our data as a "flow" through time. Our visualizations represent vector fields (or flow plots), streamlines, and trajectories, and they are constructed in a similar manner to how one might analyze the aggregate motion of particles in a fluid. However, the subject of our research extends beyond ordinary fluid mechanics. We will use our visualizations to examine statewide standardized test scores in Texas from 2003 to 2011. The nature of the data makes it a perfect match for our methodology, since students' test scores tend to change over time in a semi-deterministic but nonlinear manner. Furthermore, our methods represent a departure from the standard ways of analyzing educational data. By visualizing the changes in students' test scores over a nine-year period, we discovered that our flow plots were changing with the eventual graduating class of 2012. The change in our visualizations was caused by an educational policy known as the Student Success Initiative, or SSI. The policy forced students to pass their standardized tests in 5th and 8th grade, or risk being held back a grade. To help with this process, students who initially failed were given extra instruction and additional opportunities to take the test. SSI was implemented in such a way that it would affect the class of 2012 and beyond, although we did not know of the program's existence until our plots had been developed. SSI had a successful impact on the educational career of Texas students; a far greater percentage of students were able to pass the 5th and 8th grade standardized tests after SSI was implemented. The striking feature of SSI, however, is that it also significantly improved test scores in 6th, 7th, 9th, and 10th grade. Despite its success at improving test scores over many years and grades, the program was eventually defunded. This was partially due to an inability to construct a lengthy longitudinal analysis of the program's influence. Our methodology would have conclusively shown the effectiveness of the SSI policy. Despite the defunding of the SSI, I am confident our methodology can be extended to illustrate changes in other data systems. These systems may or may not be related to education; our code and techniques are designed to be as universal as possible. We will explore several extensions to other data sets at the end of this dissertation.Show more Item Control system identification using additive test signals(1963) Watt, Joseph Tee, 1933-; Not availableShow more System characteristic identification using normal operating signals and additive test signals is discussed. The advantages of additive test signals are enumerated. A system which senses and adapts to changes in a nondominant pole was simulated on an analog computer. A two-parameter adaptive control using a single high-frequency sinusoidal test signal to detect changes in gain and pole locations is proposed. When a random probe is used, it is possible to obtain a set of coefficients theoretically capable of characterizing nonlinear systems. The design problems encountered in applying this method to simple linear passive networks are considered. Crosscorrelation identification of linear systems using periodic discrete interval binary noise is discussed.Show more Item Structural analysis, design and optimization of nonlinear control systems using the linear algebraic equivalence of nonlinear controllers(2003) Gwak, Kwan-woong; Masada, Glenn Y.Show more Designing a nonlinear control scheme addressing major design issues in a unified way is a difficult task. However, structural information of a system/controller allows one to diagnose and identify performance characteristics at the design stage, to select better solution methods, and identify and eliminate potential problems, thereby improving the design of the controllers. In this research, structural analysis, design and optimization of nonlinear control systems are presented using the linear algebraic equivalence of nonlinear controllers (LAENC) to address the design issues of nonlinear control systems in a systematic way. It is shown that feedback linearization and sliding mode control possess the very useful structural feature, the linear algebraic equivalence, on which many well-developed linear algebraic solution techniques can be applied directly. Using this LAENC, input-constrained nonlinear optimal controllers are proposed. Investigation of the nonlinear control structure by applying the singular value decomposition to the LAENC shows that control inputs/outputs are composed of a linear combination of finite number of mode shapes and that the linear combination coefficients for the control input determine the final distribution of the control inputs. It is also shown that the contribution of each output mode towards the target vector is determined by colinearity, and colinearity can be used as a criterion for mode truncation optimization. The proposed algorithms are applied to the temperature control of an enclosed and radiation-dominant thermal system. The pure nonlinear controllers generate input-constraint-violating and large-oscillating solutions due to the ill-conditionedness, hence regularization methods that reduce the effects of ill-conditionedness are embedded into the nonlinear controller designs based on the LAENC. Also, the proposed input-constrained nonlinear optimal controller is applied and successful results are obtained. Using structural analysis, control input modes that cause input-constraint-violation and that contribute little to the total performance but use large control energy are truncated. As a result, input-constraint-satisfying solutions with a reduction in control effort are obtained.Show more