The decision by the bacterium Bacillus subtilis to produce spores has been an area of intense study in the molecular cell biology field. Though mechanisms describing how the decision is made are becoming better understood, the reasons why the decision is made are still nebulous. Spore formation is known to be a survival mechanism, but the circumstances under which it is preferred over other mechanisms have not been carefully examined. In an attempt to address this issue quantitatively, we introduce simple models and a control framework to compare two bacterial survival strategies: sporulation and a simple reduction in metabolic activity. Our findings provide evidence that the decision to sporulate may be the outcome of an underlying optimal control problem to contend with expected future environmental challenges.
In this paper, we take a minimax approach to the problem of computing a worst-case linear mean squared error (MSE) estimate of X given Y , where X and Y are jointly distributed random vectors with parametric uncertainty in their distribution. We consider two uncertainty models, PA and PB. Model PA represents X and Y as jointly Gaussian whose covariance matrix Λ belongs to the convex hull of a set of m known covariance matrices. Model PB characterizes X and Y as jointly distributed according to a Gaussian mixture model with m known zero-mean components, but unknown component weights. We show: (a) the linear minimax estimator computed under model PA is identical to that computed under model PB when the vertices of the uncertain covariance set in PA are the same as the component covariances in model PB, and (b) the problem of computing the linear minimax estimator under either model reduces to a semidefinite program (SDP). We also consider the dynamic situation where x(t) and y(t) evolve according to a discrete-time LTI state space model driven by white noise, the statistics of which is modeled by PA and PB as before. We derive a recursive linear minimax filter for x(t) given y(t).
Data Collaboration is a framework designed to make inferences from experimental observations in the context of an underlying model. In the prior studies, the methodology was applied to prediction on chemical kinetics models, consistency of a reaction system, and discrimination among competing reaction models. The present work advances Data Collaboration by developing sensitivity analysis of uncertainty in model prediction with respect to uncertainty in experimental observations and model parameters. Evaluation of sensitivity coefficients is performed alongside the solution of the general optimization ansatz of Data Collaboration. The obtained sensitivity coefficients allow one to determine which experiment/parameter uncertainty contributes the most to the uncertainty in model prediction, rank such effects, consider new or even hypothetical experiments to perform, and combine the uncertainty analysis with the cost of uncertainty reduction, thereby providing guidance in selecting an experimental/theoretical strategy for community action.
A systematic consistency analysis and optimization procedure is applied to models of representative ozone, OH, and HO2 observations in the mesosphere and upper stratosphere. The approach considers both measurement and rate parameter uncertainties. The results show some data point inconsistencies and the inability of the accepted photochemical mechanism to predict observations without unfavored large alterations of many rate constants from their consensus values. Optimization results do favor larger rate constants for OH + O, and photolytic ozone and OH production.
We present an approach to uncertainty propagation in dynamic systems, exploiting information provided by related experimental results along with their models. The approach relies on a solution mapping technique to approximate mathematical models by polynomial surrogate models.We use these surrogate models to formulate prediction bounds in terms of polynomial optimizations. Recent results on polynomial optimizations are then applied to solve the prediction problem. Two examples which illustrate the key aspects of the proposed algorithm are given. The proposed algorithm offers a framework for collaborative data processing among researchers.
This paper introduces a practical data-driven method to discriminate among large-scale kinetic reaction models. The approach centers around a computable measure of model/data mismatch. We introduce two provably convergent algorithms that were developed to accommodate large ranges of uncertainty in the model parameters. The algorithms are demonstrated on a simple toy example and a methane combustion model with more than 100 uncertain parameters. They are subsequently used to discriminate between two models for a contemporarily studied biological signaling network.
This thesis considers Lyapunov based control analysis and synthesis methods for continuous time nonlinear systems with polynomial vector fields. We take an optimization approach of finding polynomial Lyapunov functions through the use of SOS programming and the application of the Positivstellensatz theorem.
There are three main areas considered in this thesis: local stability analysis, local performance analysis, and global and local controller and observer synthesis.
For local stability analysis, we present SOS programs that enlarge a provable region of attraction for polynomial systems. We propose using pointwise maximum and minimum of polynomials to reduce the number of decision variables and to obtain larger inner bounds on the region of attraction. This idea is illustrated most notably with a Van der Pol equations example. We also extend this region of attraction inner bound enlargement problem to polynomial systems with uncertain dynamics by considering both parameter-dependent and independent Lyapunov functions. Besides using the pointwise maximum of such functions, we also propose gridding the uncertain parameter space to further reduce the size of the SOS program. The significance of the gridding method is made apparent with two examples. A related stability region analysis problem of finding a tight outer bound for attractive invariant sets is also studied. We also present some computation statistics on a region of attraction benchmark example with arbitrary data and increasing problem size.
We study two local performance analysis problems for polynomial systems. The first is on finding outer bounds for the reachable set due to disturbances with L2 and L-infinity bounds. A SOS based refinement of the outer bound is proposed and illustrated with a previously studied example. The second problem is on finding an upper bound for the L2 to L2 gain and its refinement. Interesting results are obtained when this method is applied to an adaptive control example.
For controller synthesis, we present SOS programs for finding global and local Control Lyapunov Functions. For observer synthesis, we formulate SOS programs that search for polynomial observers using Lyapunov based methods. Examples are provided to demonstrate these synthesis methods.
It is hoped that the optimization based methods in this thesis will complement existing nonlinear analysis and design methods.
We consider nonlinear systems with polynomial vector fields and pose two classes of system theoretic problems that may be solved by sum of squares programming. The first is disturbance analysis using three different norms to bound the reachableset. The second is the synthesis of a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of the state feedback synthesis for handling systems with input saturation. Both classes of problems are demonstrated using two-state nonlinear systems.
The numerical approach of data collaboration is extended to address the mutual consistency of experimental observations. The analysis rests on the concept of a dataset, which represents an organization of pertinent experimental observations, their uncertainties, and mechanistic knowledge of the subject of interest. The numerical foundation of data collaboration lies in constrained optimization, utilizing solution mapping tools and robust control algorithms. A rigorous measure of dataset consistency is developed, and Lagrange multipliers are used to identify factors that influence consistency. The new analysis is demonstrated on a real-world example, taken from the field of combustion. In performing the consistency test, the new procedure identifies two major outliers of the dataset, which were corrected upon re-examination of the raw experimental data. The results of the analysis suggest a sequential procedure with step-by-step identification of outliers and inspection of the causes. Altogether, the new numerical approach offers an important tool for assessing experimental observations and model building.
Robust filtering under different assumptions and formulations are considered. Robustfilter design for systems described by time-varying linear fractional transformation (LFT) uncertain models is reformulated as linear matrix inequalities (LMIs) via upper bound techniques. The contribution is the treatment of norm bounded (both structured and unstructured) LFT uncertainty using LMI (rather than Riccati) methods. Furthermore, in the norm bounded unstructured uncertainty case, our results are less conservative than those by methods based on Riccati equations. Robust filter design for systems with time-invariant parameter uncertainties in a polytope is also considered, using parameter dependent Lyapunov functions to solve the problem. In both cases, we use upper bounds rather than the actual performance objectives.
We also exploit that the robust filter design problem (with model uncertainty and noise) is convex in the filter as an operator. The implication is that robust filter design can be carried out directly, rather than minimizing an upper bound of the objective function. We show that finite dimensional approximations can be used to obtain suboptimal solutions with any degree of accuracy. A design algorithm is proposed, which is made up of successive dinite dimensional approximations. This algorithm requires a worst case analysis result. A conceptual branch & bound algorithm is outlined, and a practical algorithm is give.
Polynomial state feedback controller synthesis for systems subject to actuator saturation is also considered. We are interested in two kinds of problems. The first one is to design a controller to enlarge a domain of attraction (DOA), and the second is for disturbance rejection. Sum of squares (SOS) programming is the computational tool. These synthesis problems are not convex, and ad-hoc algorithms are proposed. For linear systems with saturation , algorithms here can be used to improve available results.
We present a intuitive and self-contained formulation of a stability preserving receding horizon control strategy for a system where limited preview information is available for the disturbances. The simplicity of the derivation is due to (and its benefits somewhat offset by) a set of stringent and highly structured assumptions. The formulation uses a suboptimal value function for terminal cost, and relies on optimization strategies that only require a trivial improvement property, allowing implementation as an "anytime" algorithm. The nature of this strategy's performance is clarified with linear examples.
The subject of this report is a methodology for the transformation of (experimental) data into predictive models. We use a concrete example, drawn from the field of combustion chemistry, and examine the data in terms of precisely defined modes of scientific collaboration. The numerical methodology that we employ is founded on a combination of reponse surface technique and robust control theory. The numerical results show that an essential element of scientific collaboration is collaborative processing of data, demonstrating that combining the entire colelction of data into a joint analysis extracts substantially more of theinformation content of the data.
Construction of a Control Lyapunov Function (CLF) for nonlinear system is generally a difficult problem, but once a CLF is found, stabilization of the system is straight-forward. In this paper, we present an algorithm that searches for CLFs for polynomial systems that are affine in control using sums of squares programming. We also present an algorithm for searching local CLFs for the same class of nonlinear system when global asymptotic stabilization is not possilbe.
We consider the estimation of unknown signals in structured models that are interconnections of known linear dynamic systmes and unkown static maps, and contain unmeasured exogenous disturbances. a main motivation for analyzing such an issue is a system indentification problem in which such an interconnection exists, and the static maps are to identified when the inputs and/or outputs of the maps themselves are not available, and instead we must investigate them by interacting with the larger system.
Our technique is to formulate criteria and then search for estimates of the unmeasured signals based on three main types of criteria, these being that they are consisten with the linear dynamic system, that stochastic assumptions for disturbance processes are met, and that input-output pairs of the static maps are consistent with there being a static relationship between them. After revealing the basic approach to estimating signals, some time is spent on each of these three main parts of the estimation problem, presenting alternatives and implementation details.
The "staticness" consideration is a main contribution, and we present various possibilities for enforcing it. These are what make our formulation different from some other common estimation methods such as the Kalman filter or a least squares formulation.
Computational considerations are important because the entities being estimated are signals, and so the number of decision variables is necessarily large. We focus on solution elements which are compatible with efficient convex programming methods, and what can be done when they are not. We show examples and evaluate performance and usability of the method.
We consider the esitmation of unknonwn signals in structured models that are interconnections of known linear dynamic sytems and unkown static maps, and contain unmeasured exogenous disturbances. A main motivation for considering this is a system identification probrlem in which such an interconnection exists, and the static maps are to be identified when the inputs and/or outputs of the maps themselves are not available. out approach is to search for estimates of the unmeasured signals based on three main types of criteria, these being that they are consistent with the linear dynamic system, that stochastic assumptions for disturbance processes are met, and that input-output pairs of the static maps are consistent with there being a static relationship between them. We consider various candidate criteria for enforcing the staticness consideration; they are essentially smoothness or regularizing criteria. These are what make our formulation different from other common estimation methods, for instance Kalman smoothing. We compare and contrast different mehtods using an example.
We propose a convex optimization method for optimal robust linear filter design. This is based on the observation that the design problem, which is infinite dimensional, is convex in the filter. It is shown that finite dimensional relaxations can be used to get arbitrary close to the optimal solution. The design procedure constitutes successive finite dimensional approximations, involing worst case analysis to get converging upper and lower bounds. our approach differs from standard robust filtering techniques. usually, these minimize a specific choice of upper-bound of the objective function. The choice is usually well-motivated, but partially made for computational simplicity. The computational demands put forth in this paper are much larger.
We consider nonlinear systems with polynomial vector fields and present two algorithms based on sum of squares programmin, that may answer system theoretic questions. The first algorithm provides a bound for the reachable set of a system driven by a unit-energy disturbance, while the second synthesizes a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of the second algorithm for handling systems with input saturation. Both algorithms are demonstrated using two-state nonlinear systems.
This report introduces a robust design technique for micro=electro-mechanical systems (MEMS) subject to inherent geometric and matrial uncertainties. Although design has played an important role in MEMS development, little has been done to account for the uncertainties associated with MEMS fabrication. Current fabrication techniques have poor dimensional tolerances and material properties at the microscale are not well characterized. The robust design problem we pose is to minimize the expected variance between the actual and target system performance.
After a few assumptions, the robust design problem can be written as a constrained minimization. We consider a subset of these problems and design an algorithm to minimize rational polynomial functions subject to polynomial inequality constraints. In this report we give a dtailed outline of the algorithm, verify its performance on two standard optimization test problems.
Finally, we conclude with two MEMS design problems that illustrate our robust design technique and optimization algorithm. These examples show that the robust designs nominally meet the target performance and are significantly less sensitive to geometric uncertainties than typical designs.
We present a simple, self-contained formulation of a performance enhancing, stability preserving, receding horizon control strategy for a system where preview information is available for the disturbances. The simplicity of the derivation is due to (and its benefits some-what offset by) a set of stringent and highly structured assumptions. The formulation has two notable features: it uses a suboptimal value function for terminal cost, and relies on optimization strategies that only require a trivial improvement property, allowing implementation as an "anytime" algorithm. An example of tracking control of an air-to-air missle illustrates the possible benefits of this control methodology.
We present an approach to uncertainty propagation in dynamic systems, exploiting information provided by related experimental results along with their models. Our computational procedure draws from ideas and tools that are now common in robust control theory. A case study on well-known database of methane combustion experiments and models demonstrates the viability of our proposed method.
We consider a nonlinear state transformation that allows us to work with non-quadratic polynomial Lyapunov functions. We use these polynomials to form Lyapunov functions to demonstrate simultaneous stability for a finite collection of linear systems. Under a weak definiteness condition, our main result, Theorem 3, shows that the minimum degree polynomial Lyapunov function that demonstrates simultaneous stability for a collection of linear systems can be written as a homogeneous polynomial.
In this paper, the robust H2 and H∞ filter design problems are considered, where the uncertainties, unstructured or structured, are norm bounded and represented by linear fractional transformation (LFT). The main result is that after upper-bounding the objectives, the problems of minimizing the upper bounds are converted to finite dimensional convex optimization problrms involving linear matrix inequalities (LMIs). These are extensions of the results for systems with polytopic uncertainty. it is also shown that for the unstructured, norm bounded uncertainty case, the results here are less conservative than former results, where Riccati equation approach are used. a numerical example is given to illustrate the results.
This paper explores the feasibility of constructing an autonomous sensor array on a standard silicon wafer. Such a sensor-wafer wouldinclude integrated electronics, power, and communications, and would be cpapable of being placed into a standard production process step, or short sequence of steps. During the processing of the sensor-wafer, various process parameterswould be measured and recorded. There areseveral uses for such a senor wafer, including equipment characterization and design, process calibration, and equipment qualification and diagnosis. In this paper, various sensor architectures, power supplies, communications methods, and isolation techiques are dicussed, and particular choices are made. Several proof-of-concept designs that measure film-thickness and temperature are dicussed, and test results are reviewed for each design.
This report considers a nonlinear state transformation that possesses certain properties that make it amenable to controls problems. With this state transformation, we can form matrix representations of high degree multivariable polynomials, which allows us to use techniques of linear algebra and quadratic polynomials to gain a greater understanding of these higher degree polynomials.
Representing polynomials as matrices gives us an LMI test to see if a polynomial is a sum of squares polynomial as well as providing a generalization of the S-procedure to include higher degree polynomial under the nonlinear transform also gives a mthod for fitting data to both sum of squares and general polynomials.
Lastly considered in this report are sum of squares polynomial Lyapunov functions to demonstrate simultaneous stability for a finite collection of linear systems. We show that the minimum degree sum of squares Lyapunov function that demonstrates simultaneous stability for a collection of linear systems can be written as a homogeneous polynomial, subject to a weak definiteness condition. The resulting approach to finding a sum of squares Lyapunov function allows us to improve the performance of a bench mark example problem, as well as consider the stability of an observer that receives intermittent information from the plant.
This dissertation is concerned with the development of novel wireless sensor technologies appropriate for semiconductor manufacturing applications. More speci cally, the feasibility of placing sensors directly onto the surface of a standard silicon wafer is explored. Such a wafer-mounted sensor system would be fully integrated. It would include driver electronics, a power supply, and a communication system, in addition to the sensing elements. As a result, measurements can be made in-situ to extract the process state. With such sensor systems, processes can be automatically optimized, equipment can be e ciently diagnosed, and traditional test wafers can be replaced by more e ective "smart" sensor wafers.
The status quo of metrology methods in use in the semiconductor industry is first discussed. From this discussion, a compelling case for wireless sensor systems is made. Next, the impediments associated with engineering this type of system are discussed, and possible solutions are proposed. The remainder of the dissertation describes the design, fabrication, and testing of two types of sensors for plasma etch processes.
First, a film thickness sensor for polysilicon etch processes is presented. This sensor measures the resistance of a thin polysilicon film, and uses this information to infer the film's thickness. Changes in the measured thickness, due for example to etching, can be directly sensed by this device. The sensor system incorporates a temperature sensor, both for measuring wafer surface temperature, and for compensation of the lm thickness sensor against thermal variations. Design, fabrication, and testing results of this sensor are presented.
Next, a thermal flux sensor for plasma etch processes is developed. In plasma etch processes, there are many sources for the heat delivered to the wafer. The two most significant sources are the ion flux heating and the surface chemical reaction heating. The sensor discussed in this work is capable of separately measuring both effects, for use with equipment design, diagnostics, for control. Design, fabrication, and testing results of this sensor are presented.
Finally, future directions for this research topic are covered. In particular, alternate sensors, improved isolation, and novel uses for the sensor data are discussed.
This paper explores the feasibility of integrating in-situ sensors onto the surface of a silicon wafer, with the objective of placing this wafer into a processing tool to obtain real time measurements. This technique has numerous benefits: increased measurement speed, reduced sensor introduction cost, and increased spatial and temporal information. Various sensors and sensor wafers have been developed and tested in a variety of processing tools. Repeatable, real time measurements in harsh environments such as high temperature and plasma have been obtained.
This report describes a practical off-line approach to system identification of very high order, lightly-damped, multivariable systems. The focus is on the practical and computational aspects of this problem, together with an explicit discussion of the choices that the modeler must make interactively.
This paper explores the feasibility of constructing an autonomous sensor array on a standard silicon wafer. Such a sesnor-wafer would include integrated electronics, power, and communications, and would be capable of being placed into a standard production run. During the processing of the sensor-wafer, various process parameters would be measured and recorded. There are several uses for such a sensor wafer, including equipment characterization and design, process calibration, and equipment problem diagnosis. in this paper, various sensor architectures, power supplies, communications methods, and isolation techiques are discussed, and particular choices are made. Three proof-of-concept designs utilizing a four-point probe film-thickness measurement scheme are discussed, and test results are reviewed for each design
This paper describes a practical off-line approach to system identification of very high-order, lightly-damped, multivariable systems. In particular, we address the practical and computational aspects of this problem. We also discuss the various choices that the modeler must make interactively. Our eventual objective is to enable "automated" modeling for such systems by minimizing the burden on the modeler.
We begin with sampled measurements of the system's frequency response to obtain an initial model. The frequency domain data could be supplied from finite-element simulations, or from tradition sine-sweep experiments. This model is further refined based on time-domain data as it becomes available. The method is designed to identify very high-order systems by solving several low-order identification problems using the iterative least squares algorithm of Santhanam-Koerner, coupled with the multi-band strategy of Bayard. The individual solutions are combined to yield a transfer-function matrix model of the overall system. A succinct state-space realization of this estimate may be obtained using standard model reduction methods.
This approach has made it possible to readily estimate systems with hundreds of modes, thousands of frequency data points, and hundreds of input/output channels using standard PC hardware.
In this paper we first motivate the use of autonomous micro-sensor arrays for use in semiconductor manufacturing. Following this, we discuss three crictical issues that must be addressed in order to realize our goal of building these micro-sensor arays. We then describe our on-going development efforts of fabricating spatially resolved etch-rate and temperature sensors.
This paper is concerned with the problem of identifying static nonlinear maps in a general, structured interconnected system. These static nonlinear maps are nonparametric in that they do no have a natural parameterization that is known or suggested from an analytical understanding of the underlying process. Our technique involves selecting the nonlinear maps so as to maximize the "smoothness" or "staticness" of these maps while respecting the available input-output data and the noise model. These techniques avoid bias problems that arise whn imposing artificial parameterizations on the nonlinearities. Computationally, these methods reduce to iterative least squares problems together with Kalman smoothing. Preliminary examples reveal the promise of these techniques.
In this paper we first motivate the use of autonomous micro-sensor arrays for use in semiconductor manufacturing. Following this, we discuss three critical issues that must be addressed in order to realize our goal of building these micro-sensor arrays. We then describe our on-going development efforst of fabricating spatially resolved etch-rate and temperature sensors.
This thesis documented a comprehensive study of loudspeaker modeling and control. A lumped-parameter model for a voice-coil loudspeaker in a vented enclosure was presented that derived from a consideration fo physical principles. in addition, a low-frequency (20 Hz to 100 Hz), feedback control method designed to improve the nonlinear performance of the loudspeaker and a suitable performance measure for use in deisng and evaluation were proposed. Data from experiments performed on a variety of actual loudspeakers confirmed the practicality of the thoery developed in this work.
The lumped-parameter loudspeaker model, although simple, captured much of the nonlinear behavior of the loudspeaker. In addition, the model formulation allowed a straightforward application of modern control system methods and lent itself well to modern parametric identification techniques.
The nonlinear performance of the loudspeaker system was evaluated using a suitable distortion measure that was proposed and compared with other distortion measures currently used in practice. Futhermore, the linearizing effect of feeback using a linear controller (both static and dynamic) was studied on a class of nonlinear systems. The results illustratedthat the distortion reduction was potentially significant and useful upper bound on the closed-loop distortion was found based on the sensitivity function of the system's linearizaiton.
A feedback scheme based on robust control theory was chosen for application to the loudspeaker system. Using the pressure output of the loudspeaker system for feedback, the technique offered significatn advantages over those previously attempted.
Illustrative examples were presented that proved the applicability of the theory developoed in this dissertation to a variety of loudspeaker systems. The examples included a vented loudspeaker model and actual loudspeakers enclosed in both vented and sealed configurations. In each example, preditable and measurable distortion reduction at the output of the closed-loop system was recorded.
A linear, finite-dimensional plant, with state-space parameter dependence, is controlled using a parameter-dependent controller. The parameters whose values are in a compact set, are known in real time. Their rates of variation are bounded and known in real time also. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measure in induced L2 norm. Our approach uses a bounding technique based on parameter-dependent Lyapunov function, and then solves the control synthesis problem by reformulating the existence conditions into an semi-infinite dimensional convex optimization. We propose finite dimensional approximations to get sufficient conditions for successful controller design.
A linear, finite-dimensional plant, with rational state-space parameter dependence, is controlled using a parameter-dependent controller. The parameters are known to take on values in a unit ball, and are known in real-time. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measured in induced l2 norms. The approach taken uses the optimally scaled small gain theorem, and solves the control syntesis problem by reformulating the existence conditions into a finite dimensional convex optimization.
The structured singular value function (μ) is defined with respect to a given uncertainty set. This function is continuous if the uncertainties are allowed to be complex. However, if some uncertainties are required to be real then it can be discontinuous. It is shown that μ is always upper semicontinuous and condidtions are derived under which it is also lower semicontinuous. With these results, the real-parameter robustness problem is re-examined. A related (though not equivalent) problem is formulated, which is always continuous, and relationship between the new problem and the original real μ problem is made explicit. A numerical example and results obtained via this related problem are presented.
A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ. The μ-based methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state-space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state-space methods.