Set Membership Overapproximation of Hybrid Systems with Applications to System Analysis and Estimator and Controller Design

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The introduction of assistive/autonomous features in cyber-physical systems, e.g., self-driving vehicles, have paved the way to a relatively new field of system analysis for safety-critical applications, along with the topic of controlling systems with performance and safety guarantees. The different…

The introduction of assistive/autonomous features in cyber-physical systems, e.g., self-driving vehicles, have paved the way to a relatively new field of system analysis for safety-critical applications, along with the topic of controlling systems with performance and safety guarantees. The different works in this thesis explore and design methodologies that focus on the analysis of nonlinear dynamical systems via set-membership approximations, as well as the development of controllers and estimators that can give worst-case performance guarantees, especially when the sensor data containing information on system outputs is prone to data drops and delays. For analyzing the distinguishability of nonlinear systems, building upon the idea of set membership over-approximation of the nonlinear systems, a novel optimization-based method for multi-model affine abstraction (i.e., simultaneous set-membership over-approximation of multiple models) is designed. This work solves for the existence of set-membership over-approximations of a pair of different nonlinear models such that the different systems can be distinguished/discriminated within a guaranteed detection time under worst-case uncertainties and approximation errors. Specifically, by combining mesh-based affine abstraction methods with T-distinguishability analysis in the literature yields a bilevel bilinear optimization problem, whereby leveraging robust optimization techniques and a suitable change of variables result in a sufficient linear program that can obtain a tractable solution with T-distinguishability guarantees. Moreover, the thesis studied the designs of controllers and estimators with performance guarantees, and specifically, path-dependent feedback controllers and bounded-error estimators for time-varying affine systems are proposed that are subject to delayed observations or missing data. To model the delayed/missing data, two approaches are explored; a fixed-length language and an automaton-based model. Furthermore, controllers/estimators that satisfy the equalized recovery property (a weaker form of invariance with time-varying finite bounds) are synthesized whose feedback gains can be adapted based on the observed path, i.e., the history of observed data patterns up to the latest available time step. Finally, a robust kinodynamic motion planning algorithm is also developed with collision avoidance and probabilistic completeness guarantees. In particular, methods based on fixed and flexible invariant tubes are designed such that the planned motion/trajectories can reject bounded disturbances using noisy observations.
Date Created
2023
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The Optimization of CEF Thermodynamic Modeling

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Metal oxides are crucial materials that can be applied to sustainable processes for heat storage or oxygen pumping. In order to be able to apply metal oxides to industrial processes, an effective model of the metal oxide’s reduction thermodynamics is…

Metal oxides are crucial materials that can be applied to sustainable processes for heat storage or oxygen pumping. In order to be able to apply metal oxides to industrial processes, an effective model of the metal oxide’s reduction thermodynamics is required. To do this, Wilson et al., (2023) developed a compound energy formulism (CEF) algorithm to form these models. The algorithm in its current form can effectively form model thermodynamics; however, the data set required for this model is extensive and large, leading to high costs of modeling a metal oxide. Furthermore, the algorithm faces further difficulties with uneven data densities within the set, leading to poorer fits for low density data. To assist in alleviating the cost associated with data collection, data-omitting strategies were performed to find unimportant points, or points that formed models that had good fits to the original model when removed. After conducting these tests, many points and trends were found to be crucial to keep within the data set, but due to uneven data density, no definitive conclusions could be made on how to reduce the algorithm’s data set. The tests gave evidence that points in high data density regions could be removed from the data set due to only the fact that there existed nearby points to provide essential information to closely interpolate/extrapolate the missing data. Although this project currently did not meet the goal of reducing the data set, preliminary findings of what points could be non-crucial to the data set were identified. Future testing with the proposed weighting methods will be conducted to determine what data can be safely removed from the set to form models that properly reflect the metal oxide’s properties.

Date Created
2023-05
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Implementation and Comparison of H∞ Observers for Time-Delay Systems

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In this thesis, different H∞ observers for time-delay systems are implemented and

their performances are compared. Equations that can be used to calculate observer gains are mentioned. Different methods that can be used to implement observers for time-delay systems are illustrated.…

In this thesis, different H∞ observers for time-delay systems are implemented and

their performances are compared. Equations that can be used to calculate observer gains are mentioned. Different methods that can be used to implement observers for time-delay systems are illustrated. Various stable and unstable systems are used and H∞ bounds are calculated using these observer designing methods. Delays are assumed to be known constants for all systems. H∞ gains are calculated numerically using disturbance signals and performances of observers are compared.

The primary goal of this thesis is to implement the observer for Time Delay Systems designed using SOS and compare its performance with existing H∞ optimal observers. These observers are more general than other observers for time-delay systems as they make corrections to the delayed state as well along with the present state. The observer dynamics can be represented by an ODE coupled with a PDE. Results shown in this thesis show that this type of observers performs better than other H∞ observers. Sub-optimal observer-based state feedback system is also generated and simulated using the SOS observer. The simulation results show that the closed loop system converges very quickly, and the observer can be used to design full state-feedback closed loop system.
Date Created
2018
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A convex approach for stability analysis of partial differential equations

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Description
A computational framework based on convex optimization is presented for stability analysis of systems described by Partial Differential Equations (PDEs). Specifically, two forms of linear PDEs with spatially distributed polynomial coefficients are considered.

The first class includes linear coupled PDEs…

A computational framework based on convex optimization is presented for stability analysis of systems described by Partial Differential Equations (PDEs). Specifically, two forms of linear PDEs with spatially distributed polynomial coefficients are considered.

The first class includes linear coupled PDEs with one spatial variable. Parabolic, elliptic or hyperbolic PDEs with Dirichlet, Neumann, Robin or mixed boundary conditions can be reformulated in order to be used by the framework. As an example, the reformulation is presented for systems governed by Schr¨odinger equation, parabolic type, relativistic heat conduction PDE and acoustic wave equation, hyperbolic types. The second form of PDEs of interest are scalar-valued with two spatial variables. An extra spatial variable allows consideration of problems such as local stability of fluid flows in channels and dynamics of population over two dimensional domains.

The approach does not involve discretization and is based on using Sum-of-Squares (SOS) polynomials and positive semi-definite matrices to parameterize operators which are positive on function spaces. Applying the parameterization to construct Lyapunov functionals with negative derivatives allows to express stability conditions as a set of LinearMatrix Inequalities (LMIs). The MATLAB package SOSTOOLS was used to construct the LMIs. The resultant LMIs then can be solved using existent Semi-Definite Programming (SDP) solvers such as SeDuMi or MOSEK. Moreover, the proposed approach allows to calculate bounds on the rate of decay of the solution norm.

The methodology is tested using several numerical examples and compared with the results obtained from simulation using standard methods of numerical discretization and analytic solutions.
Date Created
2016
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Parallel optimization of polynomials for large-scale problems in stability and control

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In this thesis, we focus on some of the NP-hard problems in control theory. Thanks to the converse Lyapunov theory, these problems can often be modeled as optimization over polynomials. To avoid the problem of intractability, we establish a trade…

In this thesis, we focus on some of the NP-hard problems in control theory. Thanks to the converse Lyapunov theory, these problems can often be modeled as optimization over polynomials. To avoid the problem of intractability, we establish a trade off between accuracy and complexity. In particular, we develop a sequence of tractable optimization problems - in the form of Linear Programs (LPs) and/or Semi-Definite Programs (SDPs) - whose solutions converge to the exact solution of the NP-hard problem. However, the computational and memory complexity of these LPs and SDPs grow exponentially with the progress of the sequence - meaning that improving the accuracy of the solutions requires solving SDPs with tens of thousands of decision variables and constraints. Setting up and solving such problems is a significant challenge. The existing optimization algorithms and software are only designed to use desktop computers or small cluster computers - machines which do not have sufficient memory for solving such large SDPs. Moreover, the speed-up of these algorithms does not scale beyond dozens of processors. This in fact is the reason we seek parallel algorithms for setting-up and solving large SDPs on large cluster- and/or super-computers.

We propose parallel algorithms for stability analysis of two classes of systems: 1) Linear systems with a large number of uncertain parameters; 2) Nonlinear systems defined by polynomial vector fields. First, we develop a distributed parallel algorithm which applies Polya's and/or Handelman's theorems to some variants of parameter-dependent Lyapunov inequalities with parameters defined over the standard simplex. The result is a sequence of SDPs which possess a block-diagonal structure. We then develop a parallel SDP solver which exploits this structure in order to map the computation, memory and communication to a distributed parallel environment. Numerical tests on a supercomputer demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors, and analyze systems with 100+ dimensional state-space. Furthermore, we extend our algorithms to analyze robust stability over more complicated geometries such as hypercubes and arbitrary convex polytopes. Our algorithms can be readily extended to address a wide variety of problems in control such as Hinfinity synthesis for systems with parametric uncertainty and computing control Lyapunov functions.
Date Created
2016
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System identification using discontinuous data sets and PID loop-shaping control of a vertical take-off and landing drone

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Vertical taking off and landing (VTOL) drones started to emerge at the beginning of this century, and finds applications in the vast areas of mapping, rescuing, logistics, etc. Usually a VTOL drone control system design starts from a first principles…

Vertical taking off and landing (VTOL) drones started to emerge at the beginning of this century, and finds applications in the vast areas of mapping, rescuing, logistics, etc. Usually a VTOL drone control system design starts from a first principles model. Most of the VTOL drones are in the shape of a quad-rotor which is convenient for dynamic analysis.

In this project, a VTOL drone with shape similar to a Convair XFY-1 is studied and the primary focus is developing and examining an alternative method to identify a system model from the input and output data, with which it is possible to estimate system parameters and compute model uncertainties on discontinuous data sets. We verify the models by designing controllers that stabilize the yaw, pitch, and roll angles for the VTOL drone in the hovering state.

This project comprises of three stages: an open-loop identification to identify the yaw and pitch dynamics, an intermediate closed-loop identification to identify the roll action dynamic and a closed-loop identification to refine the identification of yaw and pitch action. In open and closed loop identifications, the reference signals sent to the servos were recorded as inputs to the system and the angles and angular velocities in yaw and pitch directions read by inertial measurement unit were recorded as outputs of the system. In the intermediate closed loop identification, the difference between the reference signals sent to the motors on the contra-rotators was recorded as input and the roll angular velocity is recorded as output. Next, regressors were formed by using a coprime factor structure and then parameters of the system were estimated using the least square method. Multiplicative and divisive uncertainties were calculated from the data set and were used to guide PID loop-shaping controller design.
Date Created
2015
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A Control Systems Engineering Approach for Adaptive Behavioral Interventions: Illustration With a Fibromyalgia Intervention

Description

The term adaptive intervention has been used in behavioral medicine to describe operationalized and individually tailored strategies for prevention and treatment of chronic, relapsing disorders. Control systems engineering offers an attractive means for designing and implementing adaptive behavioral interventions that…

The term adaptive intervention has been used in behavioral medicine to describe operationalized and individually tailored strategies for prevention and treatment of chronic, relapsing disorders. Control systems engineering offers an attractive means for designing and implementing adaptive behavioral interventions that feature intensive measurement and frequent decision-making over time. This is illustrated in this paper for the case of a low-dose naltrexone treatment intervention for fibromyalgia. System identification methods from engineering are used to estimate dynamical models from daily diary reports completed by participants. These dynamical models then form part of a model predictive control algorithm which systematically decides on treatment dosages based on measurements obtained under real-life conditions involving noise, disturbances, and uncertainty. The effectiveness and implications of this approach for behavioral interventions (in general) and pain treatment (in particular) are demonstrated using informative simulations.

Date Created
2014-09-01
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Optimized Treatment of Fibromyalgia Using System Identification and Hybrid Model Predictive Control

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The term adaptive intervention is used in behavioral health to describe individually tailored strategies for preventing and treating chronic, relapsing disorders. This paper describes a system identification approach for developing dynamical models from clinical data, and subsequently, a hybrid model…

The term adaptive intervention is used in behavioral health to describe individually tailored strategies for preventing and treating chronic, relapsing disorders. This paper describes a system identification approach for developing dynamical models from clinical data, and subsequently, a hybrid model predictive control scheme for assigning dosages of naltrexone as treatment for fibromyalgia, a chronic pain condition. A simulation study that includes conditions of significant plant-model mismatch demonstrates the benefits of hybrid predictive control as a decision framework for optimized adaptive interventions. This work provides insights on the design of novel personalized interventions for chronic pain and related conditions in behavioral health.

Date Created
2014-12-01
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Carbonate-ceramic dual-phase membranes for high temperature carbon dioxide separation

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Emission of CO2 into the atmosphere has become an increasingly concerning issue as we progress into the 21st century Flue gas from coal-burning power plants accounts for 40% of all carbon dioxide emissions. The key to successful separation…

Emission of CO2 into the atmosphere has become an increasingly concerning issue as we progress into the 21st century Flue gas from coal-burning power plants accounts for 40% of all carbon dioxide emissions. The key to successful separation and sequestration is to separate CO2 directly from flue gas (10-15% CO2, 70% N2), which can range from a few hundred to as high as 1000°C. Conventional microporous membranes (carbons/silicas/zeolites) are capable of separating CO2 from N2 at low temperatures, but cannot achieve separation above 200°C. To overcome the limitations of microporous membranes, a novel ceramic-carbonate dual-phase membrane for high temperature CO2 separation was proposed. The membrane was synthesized from porous La0.6Sr0.4Co0.8Fe0.2O3-d (LSCF) supports and infiltrated with molten carbonate (Li2CO3/Na2CO3/K2CO3). The CO2 permeation mechanism involves a reaction between CO2 (gas phase) and O= (solid phase) to form CO3=, which is then transported through the molten carbonate (liquid phase) to achieve separation. The effects of membrane thickness, temperature and CO2 partial pressure were studied. Decreasing thickness from 3.0 to 0.375 mm led to higher fluxes at 900°C, ranging from 0.186 to 0.322 mL.min-1.cm-2 respectively. CO2 flux increased with temperature from 700 to 900°C. Activation energy for permeation was similar to that for oxygen ion conduction in LSCF. For partial pressures above 0.05 atm, the membrane exhibited a nearly constant flux. From these observations, it was determined that oxygen ion conductivity limits CO2 permeation and that the equilibrium oxygen vacancy concentration in LSCF is dependent on the partial pressure of CO2 in the gas phase. Finally, the dual-phase membrane was used as a membrane reactor. Separation at high temperatures can produce warm, highly concentrated streams of CO2 that could be used as a chemical feedstock for the synthesis of syngas (H2 + CO). Towards this, three different membrane reactor configurations were examined: 1) blank system, 2) LSCF catalyst and 3) 10% Ni/y-alumina catalyst. Performance increased in the order of blank system < LSCF catalyst < Ni/y-alumina catalyst. Favorable conditions for syngas production were high temperature (850°C), low sweep gas flow rate (10 mL.min-1) and high methane concentration (50%) using the Ni/y-alumina catalyst.
Date Created
2011
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