Anton H. C. Smith

Indirect Measurements via a Polynomial Chaos Observer

This paper proposes an innovative approach to the design of algorithms for indirect measurements based on a polynomial chaos observer (PCO). A PCO allows the introduction and management of uncertainty in the process. The structure of this algorithm is based on the standard closed-loop structure of an observer originally introduced by Luenberger. This structure is here extended to include uncertainty in the measurement and in the model parameters in a formal way. Possible applications of this structure are then also discussed

Dynamic Performance of a SCARA Robot Manipulator With Uncertainty Using Polynomial Chaos Theory

This short paper outlines how polynomial chaos theory (PCT) can be utilized for manipulator dynamic analysis and controller design in a 4-DOF selective compliance assembly robot-arm-type manipulator with variation in both the link masses and payload. It includes a simple linear control algorithm into the formulation to show the capability of the PCT framework.

Uncertainty and Worst-Case Analysis in Electrical Measurements Using Polynomial Chaos Theory

In this paper, the authors propose an analytical method for estimating the possible worst-case measurement due to the propagation of uncertainty. This analytical method uses polynomial chaos theory (PCT) to formally include the effects of uncertainty as it propagates through an indirect measurement. The main assumption is that an analytical model of the measurement process is available. To demonstrate the use of PCT to assess a worst-case measurement, the authors present two examples. The first one involves the use of PCT to estimate the possible worst case of a measurement due to the propagation of parametric uncertainty of a low-pass filter. This case study concerns the analysis of nonlinear effects on the propagation of uncertainty of a signal-conditioning stage used in power measurement. In this paper, the PCT method is applied to determine the probability density function (pdf) of magnitude and phase of the frequency response of the filter and their impact on the power measurement. Of particular interest is the use of PCT to determine the worst-case, expected-case, and best-case effects of the filter, avoiding the reconstruction of the complete pdf of the filter output. The results illustrate the potential of this method to determine the significant boundary of measurement uncertainty, even when the uncertainty propagates through a nonlinear nonpolynomial function. In the second example, the authors use PCT to perform a worst-case analysis for an indirect measurement of a loop impedance. For both examples, the PCT method is compared with the numerical Monte Carlo analysis and the analytical method described in the guide on uncertainty on measurements (GUM).

Robust Controller Using Polynomial Chaos Theory

This paper describes the application of polynomial chaos theory, PCT, to create a robust controller. It describes a two-degrees-of-freedom controller that uses the measured states as well as the estimated uncertainty of the measured states as feedback. Using the measured states as feedback rejects disturbances, while using the feedback of the uncertainty states compensates for the inadequacies of the feed-forward gain in the presence of parametric changes. The PCT controller utilizes a PCT observer to estimate the uncertainty on the measured states. Because the uncertain states are estimated in closed loop, run-time changes in parameters directly affect the uncertainty estimation. Therefore, estimating uncertainty can compensate for parameter changes in the model. This paper provides experimental results of this novel controller design methodology using a buck converter as an example

Indirect Measurements Via Polynomial Chaos Observer

This paper proposes an innovative approach to designing algorithms for indirect measurements based on a polynomial chaos observer (PCO). A PCO allows the introduction and management of uncertainty in a process. The structure of this algorithm is based on the standard closed-loop structure of the observer that is originally introduced by Luenberger. This structure is extended here to formally include uncertainty in the measurement and in the model parameters. Possible applications of this structure are discussed

Modeling of uncertainty and applications in monitoring and control of power electronics

This paper describes the application of polynomial chaos theory to the modeling, simulation, and control of power electronics systems. The result of this work is a circuit simulation method that is able to quantitatively account for the uncertainty of component parameters, and thereby reveal the effects of those uncertainties, during the design process. This paper introduces the mathematical background and then three, different applications: automatic system modeling under uncertainty, uncertainty-based control, uncertainty-based monitoring and diagnostics.

Uncertainty and Worst Case Analysis for a Low-Pass Filter Using Polynomial Chaos Theory

In this paper, the authors propose an analytical method to estimate the worst case in terms of uncertainty propagation in a measurement operation. The theory is exemplified with a specific application: power measurement with a signal conditioning stage including a low-pass filter. In AC power measurement the low pass filtering affects magnitude and phase of each harmonic component. The low pass filter is usually made of discrete components, which inherently contain parameter uncertainty. The effect of this uncertainty must be quantified to determine its effect on the quality of the power measurement. In this paper, the authors describe the use of Polynomial Chaos Theory (PCT) to quantify the effect of parameter uncertainty of a second order low pass filter such as the Sallen-Key filter. Furthermore, the authors demonstrate the use of PCT to obtain the probability density function (PDF) of the Sallen-Key filter given a certain parametric uncertainty. In particular it is shown how to use PCT to obtain the worst-case, expected and best-case output of the Sallen-Key filter without actually reconstructing the whole PDF. This result demonstrates the potential to determine significant boundary on the final measurement uncertainty.

Bounding the Dynamic Behavior of an Uncertain System via Polynomial Chaos-based Simulation

Parametric uncertainty can represent parametric tolerance, parameter noise or parameter disturbances. The effects of these uncertainties on the time evolution of a system can be extremely significant, mostly when studying closed-loop operation of control systems. The presence of uncertainty makes the modeling process challenging, since it is impossible to express the behavior of the system with a deterministic approach. If the uncertainties can be defined in terms of probability density function, probabilistic approaches can be adopted. In many cases, the most useful aspect is the evaluation of the worst-case scenario, thus limiting the problem to the evaluation of the boundary of the set of solutions. This is particularly true for the analysis of robust stability and performance of a closed-loop system. The goal of this paper is to demonstrate how the polynomial chaos theory (PCT) can simplify the determination of the worst-case scenario, quickly providing the boundaries in time domain. The proposed approach is documented with examples and with the description of the Maple worksheet developed by the authors for the automatic processing in the PCT framework.

Gas turbine emulator for testing of high-speed generators

This work demonstrates, in simulation, a method by which the speed-torque behavior of a gas turbine engine can be emulated by a synchronous motor. The method applies modelbased vector control to the drive of the synchronous motor in order to permit testing of a new high speed generator prior to the availability of the actual gas turbine engine. The credibility of this approach is established by simulating the step load response of the gas turbine engine emulator. Results show that the emulator is able to track the steady-state and transient speed behavior of a gas turbine engine following step increases and decreases of electrical load with a tracking error below 1 percent.

Confidence interval estimation using polynomial chaos theory

This paper proposes an analytical method to estimate the confidence interval of a measurement using polynomial chaos theory (PCT). In previous work, the authors proposed an approach based on polynomial chaos theory (PCT) to evaluate the worst-case in an indirect measurement process, which in some cases can be considered as 100% confidence interval estimate. In many practical cases though, it is important to be able to determine the confidence intervals. The confidence interval computed as the integral the probability density function (PDF), the cumulative distribution function (CDF), requires the availability of the PDF. An analytical approach to calculate of the PDF associated to a PCT polynomial has been previously proposed in literature. This analytical approach based on a regularized estimation of the joint PDF, yields a well-defined and well shaped PDF. The method proposed in this paper is based on the analytical reconstruction of PDF from the polynomial structure of the PCT expansion. Once the PDF is obtained, the integral can be calculated and an estimation of the confidence interval can be derived. This methodology is demonstrated using an indirect loop impedance measurement as an example.