In chemical processes, online measurements of all the process variables and parameters required for process control, monitoring and optimization are seldom available. The use of soft sensors or observers is, therefore, highly significant as they can estimate unmeasured state variables from available process measurements. However, for reliable estimation by a soft sensor, the process measurements have to be placed at locations that allow reconstruction of process variables by the soft sensors. This dissertation presents a new technique for computing an optimal measurement structure for state and parameter estimation of stable nonlinear systems. The methodology can compute locations for individual sensors as well as networks of sensors where a trade-off between process information, sensor cost, and information redundancy is taken into account. The novel features of the approach are (1) that the nonlinear behavior that a process can exhibit over its operating region can be taken into account, (2) that the technique is applicable for systems described by lumped or by distributed parameter models, (3) that the technique reduces to already established methods, if the system is linear and only some of the objectives are examined, (4) that the results obtained from the procedure can be easily interpreted, and (5) that the resulting optimization problem can be decomposed, resulting in a significant reduction of the computational effort required for its solution. The other issue addressed in this dissertation is designing soft sensors for a given measurement structure. In case of high-dimensional systems, the application of conventional soft sensor or observer designs may not always be practical due to the high computational requirements or the resulting observers being too sensitive to measurement noise. To address these issues, this dissertation presents reduced-order observer design techniques for state estimation of high-dimensional chemical processes. The motivation behind these approaches is that subspaces, which are close to being unobservable, cannot be correctly reconstructed in a realistic setting due to measurement noise and inaccuracies in the model. The presented approaches make use of this observation and reconstruct the parts of the system where accurate state estimation is possible.

The leading cause of death in patients who survive subarachnoid hemorrhage (SAH) is stroke as a result of cerebral arterial vasospasm1. Such vasospasms involve a vasoactive response, but they remain enigmatic and no clinical treatment has proven effective in prevention or reduction2. Arteries remodel in response to diverse mechanical loads and chemical factors. Following SAH, the surrounding vasculature is exposed to a radically altered chemo-mechanical environment. It is our hypothesis that chemical stimuli associated with the formation of an extravascular blood clot dominates the maladaptive growth and remodeling response early on, thus leading to important structural changes. However, it is not clear which of the many chemical factors are key players in the production of vasospasm. Before an accurate picture of the etiology of vasospasm can be produced, it is imperative to gain a better understanding of the non-pathogenic cerebral vasculature. In particular, the rabbit basilar artery is a well established model for vasospasm. However, surprisingly little is known about the mechanical properties of the rabbit basilar artery. Using an in vitro custom organ culture and mechanical testing device, acute and cultured basilar arteries from male White New Zealand specific pathogen free rabbits underwent cyclic pressurization tests at in vivo conditions and controlled levels of myogenic tone. Sections of basilar arteries were imaged for collagen fiber orientation at 0, 40 and 80 mmHg at in vivo stretch conditions using nonlinear optical microscopy. The nonlinear stress-strain curves provide baseline characteristics for acute and short-term culture basilar arteries. The active and passive testing creates a framework for interpreting the basal tone of arteries in our culture system. Nonlinear optical microscopy second harmonic generation provides unique microstructural information and allows imaging of live, intact arteries while maintaining in vivo geometries and conditions. Collagen fibers were found to be widely distributed about the axial direction in the adventitial layer and narrowly distributed about the circumferential direction in the adventitial layer. The quantified collagen fiber angles within the artery wall further support the development of accurate mathematical models.

Tool chatter in longitudinal turning is addressed with a new perspective using a complex machining model describing the coupled tool-workpiece dynamics subject to nonlinear regenerative cutting forces, instantaneous depth-of-cut (DOC) and workpiece whirling due to material imbalance. The workpiece is modeled as a system of three rotors: unmachined, being machined and machined, connected by a flexible shaft. The model enables workpiece motions relative to the tool and tool motions relative to the machining surface to be three-dimensionally established as functions of spindle speed, instantaneous DOC, rate of material removal and whirling. Excluding workpiece vibrations from the cutting model is found improper. A rich set of nonlinear behaviors of both the tool and the workpiece including period-doubling bifurcation and chaos signifying the extent of machining instability at various DOCs is observed. Presented numerical results agree favorably with physical experiments reported in the literature. It is found that whirling is non-negligible if the fundamental characteristics of machining dynamics are to be fully understood. The 3D model is explored along with its 1D counterpart, which considers only tool motions and disregards workpiece vibrations. Numerical simulations reveal diverse behaviors for the 3D coupled and 1D uncoupled equations of motion for the tool. Most notably, observations made with regard to the inconsistency in describing stability limits raise the concern for using 1D models to obtain stability charts. The nonlinear 3D model is linearized to investigate the implications of applying linear models to the understanding of machining dynamics. Taylor series expansion about the operating point where optimal machining conditions are desired is applied to linearize the model equations of motion. Modifications are also made to the nonlinear tool stiffness term to minimize linearization errors. Numerical experiments demonstrate inadmissible results for the linear model and good agreement with available physical data in describing machining stability and chatter for the nonlinear model. Effects of tool geometry, feed rate, and spindle speed on cutting dynamics are also explored. It is observed that critical DOC increases with increasing spindle speed and small DOCs can induce cutting instability -- two of the results that agree qualitatively well with published experimental data.

This work is concerned with the theoretical study and the implementation of algorithms for solving two particular types of nonlinear optimization problems, namely unconstrained and simple-bound constrained optimization problems. For unconstrained optimization, we develop a new algorithm which uses a filter technique and a trust-region method in order to enforce global convergence and to improve the efficiency of traditional approaches. We also analyze the effect of approximate first and second derivatives on the performance of the filter-trust-region algorithm. We next extend our algorithm to simple-bound constrained optimization problems by combining these ideas with a gradient-projection method. Numerical results follow the proposed methods and indicate that they are competitive with more classical trust-region algorithms.

The exchange rate target zone has been paid much attention in the early 1990 initially by Krugman (1991).It expressed when exchange rate surpasses the band of exchange rate that implicitly or explicitly determined by the central bank, the central Bank will intervene the foreign exchange by buying or selling foreign exchange to ensure the exchange rate staying inside the band, otherwise, the exchange rate will be allowed to fluctuate inside the band freely.According to Krugman (1991), when economic system faces random disturbances, the exchange rate target zone regime is helpful to narrow down the exchange rate volatility contrast to that in the floating exchange rate regime. That is, the exchange rate target zone has more essential stability,which is called ``honeymoon effect". In recent decade, Krugman's exchange rate target zone model has been tested empirically.In this thesis, the smooth transition autoregression with target zone (STARTZ) proposed originally by Lundbergh and Ter"{a}svirta (2006) and logistic smooth transition regression with two thresholds (LSTR2) are used to make comparisons for in-sample fitness and out-of-sample forcastability.Furthermore, we also test two important assumptions of the exchange rate target zone model: the credibility assumption and marginal interventions. The data are constructed with 755 daily spot exchange rates, denominated in Eurpean Currency Unit (ECU), from January 14, 1987 to December 29, 1989, in German, France, and Italy.We split the sample into in-sample (570 observations), and out-of-sample (185 observations), and make use of STARTZ-GARCH and LSTR2-STGARCH to fit the in-sample regimes, and apply Rapach and Wohard (2006)'s Bootstapping to generate the out-of-sample forecasts. Finally,we make use of Diebold and Mariano (1995)'s predictive accuracy tests to compare the out-of-sample forecastability between STARTZ and LSTR2 models.According to the empirical results, we can find that LSTR2 model has not bad performance in fitting the in-sample and forecasting the out-of-sample data compared to STARTZ model.

NonLinear Principal Component Analysis (NLPCA) using three-hidden-layer feed-forward neural networks can produce solutions that over-fit the data and are non-unique. These problems have been dealt with by subjective methods during the network training. This study shows that these problems are intrinsic due to the three-hidden-layer architecture. A simplified two-hidden-layer feed-forward neural network that has no encoding layer and no bottleneck and output biases is proposed. This new, compact NLPCA model alleviates these problems without employing the subjective methods and is called quasi-objective. The compact NLPCA is applied to the zonal winds observed at seven pressure levels between 10 and 70 hPa in the equatorial stratosphere to represent the Quasi-Biennial Oscillation (QBO) and investigate its variability and structure. The two nonlinear principal components of the dataset offer a clear picture of the QBO. In particular, their structure shows that the QBO phase consists of a predominant 28.4-month cycle that is modulated by an 11-year cycle and a longer-period cycle. The significant difference in variability of the winds between cold and warm seasons and the tendency for a seasonal synchronization of the QBO phases are well captured. The one-dimensional NLPCA approximation of the dataset provides a better representation of the QBO than the classical principal component analysis and a better description of the asymmetry of the QBO between westerly and easterly shear zones and between their transitions. The compact NLPCA is then applied to the Arctic Oscillation (AO) index and aforementioned zonal winds to investigate the relationship of the AO with the QBO. The NLPCA of the AO index and zonal-winds dataset shows clearly that, of covariation of the two oscillations, the phase defined by the two nonlinear principal components progresses with a predominant 28.4-month periodicity, plus the 11-year and longer-period modulations. Large positive values of the AO index occur when westerlies prevail near the middle and upper levels of the equatorial stratosphere. Large negative values of the AO index arise when easterlies occupy over half the layer of the equatorial stratosphere.

In this thesis we study the problem of stabilizing smooth embedded submanifolds in the state space of smooth, nonlinear, autonomous, deterministic control-affine systems. Our motivation stems from a realization that important applications, such as path following and synchronization, are best understood in the set stabilization framework. Instead of directly attacking the above set stabilization problem, we seek feedback equivalence of the given control system to a normal form that facilitates control design. The process of putting a control system into the normal form of this thesis is called transverse feedback linearization. When feasible, transverse feedback linearization allows for a decomposition of the nonlinear system into a “transverse” and a “tangential” subsystem relative to the goal submanifold. The dynamics of the transverse subsystem determine whether or not the system’s state approaches the submanifold. To ease controller design, we ask that the transverse subsystem be linear time-invariant and controllable. The dynamics of the tangential subsystem determine the motion on the submanifold. The main problem considered in this work, the local transverse feedback linearization problem (LTFLP), asks: when is such a decomposition possible near a point of the goal submanifold? This problem can equivalently be viewed as that of finding a system output with a well-defined relative degree, whose zero dynamics manifold coincides with the goal submanifold. As such, LTFLP can be thought of as the inverse problem to input-output feedback linearization. We present checkable, necessary and sufficient conditions for the existence of a local coordinate and feedback transformation that puts the given system into the desired normal form. A key ingredient used in the analysis is the new notion of transverse controllability indices of a control system with respect to a set. When the goal submanifold is diffeomorphic to Euclidean space, we present sufficient conditions for feedback equivalence in a tubular neighbourhood of it. These results are used to develop a technique for solving the path following problem. When applied to this problem, transverse feedback linearization decomposes controller design into two separate stages: transversal control design and tangential control design. The transversal control inputs are used to stabilize the path, and effectively generate virtual constraints forcing the system’s output to move along the path. The tangential inputs are used to control the motion along the path. A useful feature of this twostage approach is that the motion on the set can be controlled independently of the set stabilizing control law. The effectiveness of the proposed approach is demonstrated experimentally on a magnetically levitated positioning system. Furthermore, the first satisfactory solution to a problem of longstanding interest, path following for the planar/vertical take-off and landing aircraft model to the unit circle, is presented. This solution, developed in collaboration with Luca Consolini and Mario Tosques at the University of Parma, is made possible by taking a set stabilization point of view.

Long bridges tend to develop large deformations under the action of intense dynamical loads such as wind or earthquakes. Unless these deformations are controlled in some fashion, the structure might suffer damage or even collapse. One possible solution to this problem is to apply external forces to the bridge through suspension cables. This work presents an active robust control scheme to suppress the vibrations caused by the vertical ground motion due to an earthquake of a cable-stayed bridge. It is proven both mathematically and through computer simulation that the active nonlinear controller is capable of reducing the amplitude of the vibrations to an arbitrarily small size. This may save the bridge structure during a strong earthquake. It is shown that the control scheme performs satisfactorily even if parts of the system fail during an earthquake. An alternative method to derive the control law using finite elements is also presented. / Graduation date: 1993

The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

In this thesis we study the problem of stabilizing smooth embedded submanifolds in the state space of smooth, nonlinear, autonomous, deterministic control-affine systems. Our motivation stems from a realization that important applications, such as path following and synchronization, are best understood in the set stabilization framework. Instead of directly attacking the above set stabilization problem, we seek feedback equivalence of the given control system to a normal form that facilitates control design. The process of putting a control system into the normal form of this thesis is called transverse feedback linearization. When feasible, transverse feedback linearization allows for a decomposition of the nonlinear system into a “transverse” and a “tangential” subsystem relative to the goal submanifold. The dynamics of the transverse subsystem determine whether or not the system’s state approaches the submanifold. To ease controller design, we ask that the transverse subsystem be linear time-invariant and controllable. The dynamics of the tangential subsystem determine the motion on the submanifold. The main problem considered in this work, the local transverse feedback linearization problem (LTFLP), asks: when is such a decomposition possible near a point of the goal submanifold? This problem can equivalently be viewed as that of finding a system output with a well-defined relative degree, whose zero dynamics manifold coincides with the goal submanifold. As such, LTFLP can be thought of as the inverse problem to input-output feedback linearization. We present checkable, necessary and sufficient conditions for the existence of a local coordinate and feedback transformation that puts the given system into the desired normal form. A key ingredient used in the analysis is the new notion of transverse controllability indices of a control system with respect to a set. When the goal submanifold is diffeomorphic to Euclidean space, we present sufficient conditions for feedback equivalence in a tubular neighbourhood of it. These results are used to develop a technique for solving the path following problem. When applied to this problem, transverse feedback linearization decomposes controller design into two separate stages: transversal control design and tangential control design. The transversal control inputs are used to stabilize the path, and effectively generate virtual constraints forcing the system’s output to move along the path. The tangential inputs are used to control the motion along the path. A useful feature of this twostage approach is that the motion on the set can be controlled independently of the set stabilizing control law. The effectiveness of the proposed approach is demonstrated experimentally on a magnetically levitated positioning system. Furthermore, the first satisfactory solution to a problem of longstanding interest, path following for the planar/vertical take-off and landing aircraft model to the unit circle, is presented. This solution, developed in collaboration with Luca Consolini and Mario Tosques at the University of Parma, is made possible by taking a set stabilization point of view.