Donald M. Wiberg

The extended Kalman filter as a pulmonary blood flow estimator

Pulmonary blood flow is represented as a time-varying parameter in a simple model of the bodily uptake of gases that are soluble in blood. The pulmonary blood flow is estimated on-line using Ljungs modification of the extended Kalman filter. The experimental technique for estimation of pulmonary blood flow is based on Zwarts soluble gas method, modified to permit time-varying ventilation and pulmonary perfusion, and also to permit binary dynamic forcing of the inhaled soluble gas. Computer simulations were performed to determine the convergence time and to choose values of algorithmic parameters. Experiments done in humans gave agreement to within the accuracy of that measured invasively by a thermal dilution technique. The method shows promise for use in many biomedical applications, including patient monitoring, physiological experimentation, and clinical heart function tests.

An online parameter estimator for quick convergence and time-varying linear systems

A recursive algorithm called 3-OM is presented to estimate parameters and noise variances for discrete-time linear stochastic systems. The unprojected version of 3-OM is globally convergent with probability 1 to minima of the asymptotic negative log-likelihood function. 3-OM approximates the quick convergence attained by the optimal nonlinear filter used as a parameter estimator. The state-space form of 3-OM permits application to time-varying linear systems and to online tuning of a Kalman filter.

An ordinary differential equation technique for continuous-time parameter estimation

An ordinary differential equation technique is developed via averaging theory and weak convergence theory to analyze the asymptotic behavior of continuous-time recursive stochastic parameter estimators. This technique is an extension of L. Ljungs (1977) work in discrete time. Using this technique, the following results are obtained for various continuous-time parameter estimators. The recursive prediction error method, with probability one, converges to a minimum of the likelihood function. The same is true of the gradient method. The extended Kalman filter fails, with probability one, to converge to the true values of the parameters in a system whose state noise covariance is unknown. An example of the extended least squares algorithm is analyzed in detail. Analytic bounds are obtained for the asymptotic rate of convergence of all three estimators applied to this example. >

A convergent approximation of the continuous-time optimal parameter estimator

Continuous-time linear stochastic systems that are bilinear in the state and parameters are considered. A specific approximation to the optimal nonlinear filter used as a recursive parameter estimator is derived by retaining third-order moments and using a Gaussian approximation for higher order moments. With probability one, the specific approximation is proved to converge to a minimum of the likelihood function. The proof uses the ordinary differential equation technique and requires that the trajectories of the slow system be bounded on finite time intervals and that the fixed parameter fast system by asymptotically stable. The fixed parameter fast system is proved to be asymptotically stable if the parameter update gain is small enough. Essentially, the specific approximation is asympotically equivalent to the recursive prediction error method, thus inheriting its asymptotic rate of convergence. A numerical simulation for a simple example indicates that the specific approximation has better transient response than other commonly used convergent parameter estimators. >

Bulk wind estimation and prediction for adaptive optics control systems

We present a wind-predictive controller for astronomical adaptive optics (AO) systems that is able to predict the motion of a single windblown layer in the presence of other, more slowly varying phase aberrations. This controller relies on fast, gradient-based optical flow estimation to identify the velocity of the translating layer and a recursive mean estimator to account for turbulence that varies on a time scale much slower than the operating speed of the AO loop. We derive the Cramer-Rao lower bound for the wind estimation problem and show that the proposed estimator is very close to achieving theoretical minimum-variance performance. We also present simulations using on-sky data that show significant Strehl increases from using this controller in realistic atmospheric conditions.

Geometric view of adaptive optics control.

The objective of an astronomical adaptive optics control system is to minimize the residual wave-front error remaining on the science-object wave fronts after being compensated for atmospheric turbulence and telescope aberrations. Minimizing the mean square wave-front residual maximizes the Strehl ratio and the encircled energy in pointlike images and maximizes the contrast and resolution of extended images. We prove the separation principle of optimal control for application to adaptive optics so as to minimize the mean square wave-front residual. This shows that the residual wave-front error attributable to the control system can be decomposed into three independent terms that can be treated separately in design. The first term depends on the geometry of the wave-front sensor(s), the second term depends on the geometry of the deformable mirror(s), and the third term is a stochastic term that depends on the signal-to-noise ratio. The geometric view comes from understanding that the underlying quantity of interest, the wave-front phase surface, is really an infinite-dimensional vector within a Hilbert space and that this vector space is projected into subspaces we can control and measure by the deformable mirrors and wave-front sensors, respectively. When the control and estimation algorithms are optimal, the residual wave front is in a subspace that is the union of subspaces orthogonal to both of these projections. The method is general in that it applies both to conventional (on-axis, ground-layer conjugate) adaptive optics architectures and to more complicated multi-guide-star- and multiconjugate-layer architectures envisaged for future giant telescopes. We illustrate the approach by using a simple example that has been worked out previously [J. Opt. Soc. Am. A 73, 1171 (1983)] for a single-conjugate, static atmosphere case and follow up with a discussion of how it is extendable to general adaptive optics architectures.

A spatial non-dynamic LQG controller: Part I, application to adaptive optics

The theory of a spatial non-dynamic LQG controller developed in Part II is applied to astronomical adaptive optics (AO). The AO control system is described and reduced to an exact finite dimensional model by projecting into observable and controllable subspaces. A dynamic optimal minimum variance controller is derived, using the estimated covariance of the residuals rather than assuming a Komolgorov structure function. This controller is compared with a non-dynamic optimal controller and with the usual null-seeking controller

Gaussian-optimal on-line parameter estimation

The gaussian-optimal on-line parameter estimator is presented in both continuous and discrete time. This estimation algorithm is based on the optimal non-linear filter equations under the approximation that both state and parameters have gaussian distributions. Only the first and second conditional moments need be updated, because the fourth order moments are computed from the second order and no other order moments are needed. The convergence of this on-line parameter estimator is analyzed by the method of Ljung.

Canonical equations for boundary feedback control of stochastic distributed parameter systems

The plant considered is described by a set of time-invariant linear partial differential equations with spatially generalized Wiener process disturbance of the state and point measurement. Boundary control is used to minimize the expected value of a functional that is quadratic in the state and control. The well-known separation into linear optimal state estimator and linear optimal deterministic controller is heuristically shown using an extension of the Caratheodory lemma to obtain sufficient conditions. Each resulting Hamilton-Jacobi equation is shown equivalent to a pair of linear canonical partial differential equations. Use of the canonical equations leads to exact analytical solutions under certain conditions by means of closed loop modes. This permits a paper and pencil analysis of the engineering properties of the control of a large class of practical distributed systems with boundary controls and point measurements. An example is given of the boundary control of a one space dimension diffusion equation with input noise at the other boundary and noisy measurement taken at an internal point. Because an analytical solution is obtained, the effect of the location of this internal measurement point can be analyzed.

The MIMO Wiberg estimator

The Wiberg estimator is an online parameter and state estimation algorithm that is a third-order moment approximation to the optimal estimator and has a global convergence point at the true values of the parameters. Previously, it has been derived only in the single-input, single output (SISO) case in observability form. In this work, it is derived for the multiple-input, multiple-output (MIMO) case in a state-space form representing a large class of systems that are bilinear in state and parameters. This form of the Wiberg estimator is then shown to be asymptotically equivalent to the state-space form of the recursive prediction error method.<<ETX>>