Matthew S. Holzel

A comparison of least squares algorithms for estimating Markov parameters

The purpose of this work is to compare model structures and identification algorithms for estimating Markov parameters in the presence of uncorrelated and correlated input, process, and output noise. We consider several least-squares variants with ARX and μ-Markov model structures, which are compared with white noise identification signals.

Adaptive Control of the NASA Generic Transport Model Using Retrospective Cost Optimization

We provide a detailed description of retrospective-cost based adaptive control, which is a discrete-time adaptive control law for stabilization, command following, and disturbance rejection that is effective for systems that are unstable, MIMO, and/or nonminimum phase. The adaptive control algorithm includes guidelines concerning the modeling information needed for implementation. This information includes the sign of the high-frequency gain as well as the nonminimum-phase zeros. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant’s spectral radius, the required information can be approximated by a sufficient number of Markov parameters. No additional information about the poles or zeros need be known, and no matching conditions are required. We apply this adaptive control technique to NASA’s Generic Transport Model to illustrate disturbance rejection under unknown, reduced controller authority.

Real-Time Frequency Response Estimation from Flight Data

A nonparametric method for estimating frequency responses in realtime using a method based on recursive least squares in the time domain was developed and studied. The proposed method uses sinusoidal functions at selected frequencies known to be contained in the input in conjunction with recursive least squares to rapidly estimate Fourier series coefficients for input and output time series, and thereby estimate frequency responses. Practical problems that arise when applying the discrete Fourier transform in recursive form were identified using simple examples. A general expression for accurate realtime calculation of the covariance matrix for recursive least squares parameter estimation was developed and used to calculate valid uncertainty bounds for realtime frequency response estimates. Simulation data generated with optimized multisine inputs were used to investigate the accuracy of the proposed method for realtime frequency response estimation, and to validate the calculated error bounds. The approach was also applied in realtime to flight data from a subscale jet transp ort aircraft. Comparisons of realtime frequency response estimates and error bounds with results from conventional postflight batch analysis showed that results from the realti me method were in statistical agreement with postflight batch estimates, with valid error bounds that properly indicated the quality of the realtime frequency response estimates.

Sensor-Only Noncausal Blind Identification of Pseudo Transfer Functions

Abstract Motivated by passive health monitoring applications, we consider blind identification where only sensor measurements are available. The goal is to identify a pseudo transfer function (PTF) between two sensors in the presence of an unknown initial state and unknown exogenous input. For this problem, we choose one sensor to be the pseudo input to the system and we delay the second sensor, treating it as the pseudo output.We show that the order of the pseudo-transfer function is no larger than one higher than the order of the system. We demonstrate this method on a two-degree-of-freedom mass-spring-damper system and validate the identified PTFs by comparing them with analytical results.

On the accuracy of least squares algorithms for estimating zeros

We investigate the accuracy of least-squares-based algorithms for estimating system zeros in the presence of known or unknown order and known or unknown relative degree. Specifically, we use least-squares to estimate the parameters of ARX and μ-Markov models from which zero estimates are calculated directly using the numerator polynomial as well as indirectly using the truncated Laurent expansion or the eigensystem realization algorithm (ERA). To employ the truncated Laurent expansion or ERA, we consider the Markov parameters estimated from the μ-Markov model. Lastly, we investigate the spurious zeros of the μ-Markov model and truncated Laurent expansion to determine to what extent these zeros behave in a predictable manner.

Adaptive control using retrospective cost optimization with RLS-based estimation for concurrent Markov-parameter updating

We present a discrete-time adaptive control law that is effective for systems that are MIMO and either minimum phase or nonminimum phase. The adaptive control algorithm provides guidelines concerning the modeling information needed for implementation. This information includes a sufficient number of Markov parameters to capture the sign of the high-frequency gain as well as the nonminimum-phase zeros. No additional information about the poles or zeros need be known. In this paper, recursive least-squares estimation is used for concurrent Markov parameter estimation. We present numerical examples to illustrate the algorithms effectiveness in handling nonminimum-phase zeros as plant changes occur.

Consistent identification of Hammerstein systems using an ersatz nonlinearity

We develop a method for identifying SISO Ham merstein systems with an unknown static nonlinearity, linear dynamics, white input noise and colored output noise. We use least squares with a μ-Markov model to estimate the Markov parameters of the linear time-invariant dynamical system. Since the input to the linear system is not available, we use a substitute (ersatz) nonlinearity to transform the input for use in the regressor matrix. We prove that the Markov parameters of the system can be estimated consistently up to a constant scalar as the amount of data increases. This method is demonstrated with several numerical examples.

Sensor-to-sensor identification of Hammerstein systems

Traditional system identification uses measurements of the inputs, but when these measurements are not available, alternative methods, such as blind identification, output-only identification, or operational modal analysis, must be used. Yet another method is sensor-to-sensor identification (S2SID), which estimates pseudo transfer functions whose inputs are outputs of the original system. A special case of S2SID is transmissibility identification. Since S2SID depends on cancellation of the input, this approach does not extend to nonlinear systems. However, in the present paper we show that, for the case of a two-output Hammerstein system, the least-squares estimate of the PTF is consistent, that is, asymptotically correct, despite the presence of the nonlinearities.

SVD-based computation of zeros of polynomial matrices

We present an algorithm for determining the zeros of polynomial matrices of arbitrary order, normal rank, and dimension. Specifically, we use the singular value decomposition to reduce the problem to an eigenvalue problem.

Direct Optimal Controller Identification for Uncertain Systems using Frequency Response Function Data

Abstract Here we present a new approach to optimal controller identification which unifies system identification and optimal control theory. Starting with empirical, open-loop frequency response function (FRF) data from a system, it is shown that the optimal controller can be identified directly without performing the intermediary steps of system identification and controller design. The primary benefit is that we are able to work directly with the measured data and the uncertainties inherent in it. Further, we go on to show a method of incorporating the empirical FRF uncertainty into the cost for robustness against plant uncertainty. This method leads to a more precise identification of H2 and LQG controllers since it avoids the residual errors associated with performing the traditional intermediary step of system identification, while concurrently accounting for measured system uncertainty.