Minimizing the Euclidean Condition Number

This paper considers the problem of determining the row and/or column scaling of a matrix A that minimizes the condition number of the scaled matrix. This problem has been studied by many authors. For the cases of the ∞-norm and the 1-norm, the scaling problem was completely solved in the 1960s. It is the Euclidean norm case that has widespread application in robust control analyses. For example, it is used for integral controllability tests based on steady-state information, for the selection of sensors and actuators based on dynamic information, and for studying the sensitivity of stability to uncertainty in control systems. Minimizing the scaled Euclidean condition number has been an open question—researchers proposed approaches to solving the problem numerically, but none of the proposed numerical approaches guaranteed convergence to the true minimum. This paper provides a convex optimization procedure to determine the scalings that minimize the Euclidean condition number. This optimization can be solved in polynomial-time with off-the-shelf software

Stability and Performance Analysis of Systems Under Constraints

All real world control systems must deal with actuator and state constraints. Standard conic sector bounded nonlinearity stability theory provides methods for analyzing the stability and performance of systems under constraints, but it is well-known that these conditions can be very conservative. A method is developed to reduce conservatism in the analysis of constraints by representing them as nonlinear real parametric uncertainty

Avery Final Report: Identification and Cross-Directional Control of Coating Processes

Coating refers to the covering of a solid with a uniform layer of liquid. Of special industrial interest is the cross-directional control of coating processes, where the cross-direction refers to the direction perpendicular to the substrate movement. The objective of the controller is to maintain a uniform coating under unmeasured process disturbances. Assumptions that are relevant to coating processes found in industry are used to develop a model for control design. We show how to identify the model from input-output data. This model is used to derive a model predictive controller to maintain flat profiles of coating across the substrate by varying the liquid flows along the cross direction. The model predictive controller computes the control action which minimizes the predicted deviation in cross-directional uniformity. The predictor combines the estimate obtained from the model with the measurement of the cross-directional uniformity to obtain a prediction for the next time step. A filter is used to obtain robustness to model error and insensitivity to measurement noise. The tuning of the noise filter and different methods for handling actuator constraints are studied in detail. The three different constraint-handling methods studied are: the weighting of actuator movements in the objective function, explicitly adding constraints to the control algorithm, i.e. constrained model predictive control, and scaling infeasible control actions calculated from an unconstrained control law to be feasible. Actuator constraints, measurement noise, model uncertainty, and the plant condition number are investigated to determine which of these limit the achievable closed loop performance. From knowledge of how these limitations affect the performance we find how the plant could be modified to improve the process uniformity. Also, because identification of model parameters is time-consuming and costly, we study how accurate the identification must be to achieve a given level of performance. The theory developed throughout the paper is rigorously verified though simulations and experiments on a pilot plant. The effect of interactions on the closed loop performance is shown to be negligible for this pilot plant. The measurement noise and the actuator constraints are shown to have the largest effect on closed loop performance

Modeling and Analysis of Drug-Eluting Stents With Biodegradable PLGA Coating: Consequences on Intravascular Drug Delivery

Increasing interests have been raised toward the potential applications of biodegradable poly(lactic-co-glycolic acid) (PLGA) coatings for drug-eluting stents in order to improve the drug delivery and reduce adverse outcomes in stented arteries in patients. This article presents a mathematical model to describe the integrated processes of drug release in a stent with PLGA coating and subsequent drug delivery, distribution, and drug pharmacokinetics in the arterial wall. The integrated model takes into account the PLGA degradation and erosion, anisotropic drug diffusion in the arterial wall, and reversible drug binding. The model simulations first compare the drug delivery from a biodegradable PLGA coating with that from a biodurable coating, including the drug release profiles in the coating, average arterial drug levels, and arterial drug distribution. Using the model for the PLGA stent coating, the simulations further investigate drug internalization, interstitial fluid flow in the arterial wall, and stent embedment for their impact on drug delivery. Simulation results show that these three factors, while imposing little change in the drug release profiles, can greatly change the average drug concentrations in the arterial wall. In particular, each of the factors leads to significant and yet distinguished alterations in the arterial drug distribution that can potentially influence the treatment outcomes. The detailed integrated model provides insights into the design and evaluation of biodegradable PLGA-coated drug-eluting stents for improved intravascular drug delivery.National Institutes of Health (U.S.) (NIBIB 5RO1EB005181

Robust Control for a Noncolocated Spring-Mass System

Robust control laws are presented for an undamped pair of coupled manes with a noncolocated sensor and actuator. This simple problem captures many of the features of more complex aircraft and space structure vibration control problems. The control problem is formulated in the structured singular value framework, which addresses the stability robustness to parameter variations directly. Controllers are designed by D-K iteration (commonly called μ-synthesis), and the resulting high-order controllers are reduced using Hankel model reduction. Design specifications such as settling time, actuator constraints insensitivity to measurement noise, and parameter uncertainty are achieved by the resulting controllers. Design Problems #1 and #2 were considered in [2]. Design Problem #4 in [11] will be considered in this paper

A mechanistic model for drug release in PLGA biodegradable stent coatings coupled with polymer degradation and erosion

Biodegradable poly(d,l-lactic-co-glycolic acid) (PLGA) coating for applications in drug-eluting stents has been receiving increasing interest as a result of its unique properties compared with biodurable polymers in delivering drug for reducing stents-related side effects. In this work, a mathematical model for describing the PLGA degradation and erosion and coupled drug release from PLGA stent coating is developed and validated. An analytical expression is derived for PLGA mass loss that predicts multiple experimental studies in the literature. An analytical model for the change of the number-average degree of polymerization [or molecular weight (MW)] is also derived. The drug transport model incorporates simultaneous drug diffusion through both the polymer solid and the liquid-filled pores in the coating, where an effective drug diffusivity model is derived taking into account factors including polymer MW change, stent coating porosity change, and drug partitioning between solid and aqueous phases. The model is used to describe in vitro sirolimus release from PLGA stent coating, and demonstrates the significance of simultaneous sirolimus release via diffusion through both polymer solid and pore space. The proposed model is compared to existing drug transport models, and the impact of model parameters, limitations and possible extensions of the model are also discussed.National Institute for Biomedical Imaging and Bioengineering (U.S.) (NIBIB, contract grant number: 5RO1EB005181

Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties

Computation of exact ellipsoidal bounds on the state trajectories of discrete-time linear systems that have time-varying or time-invariant linear fractional parameter uncertainties and ellipsoidal uncertainty in the initial state is known to be NP-hard. This paper proposes three algorithms to compute ellipsoidal bounds on such a state trajectory set and discusses the tradeoffs between computational complexity and conservatism of the algorithms. The approach employs linear matrix inequalities to determine an initial estimate of the ellipsoid that is refined by the subsequent application of the skewed structured singular value ν. Numerical examples are used to illustrate the application of the proposed algorithms and to compare the differences between them, where small conservatism for the tightest bounds is observed.Institute for Advanced Computing Applications and Technologie

Robust Control Structure Selection

Screening tools for control structure selection in the presence of model/plant mismatch are developed in the context of the Structured Singular Value (μ) theory. The developed screening tools are designed to aid engineers in the elimination of undesirable control structure candidates for which a robustly performing controller does not exist. Through application on a multicomponent distillation column, it is demonstrated that the developed screening tools can be effective in choosing an appropriate control structure while previously existing methods such as the Condition Number Criterion can lead to erroneous results