James B. Farison

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.

Identification and control of linear discrete systems

Real-time identification and control of linear discrete systems with Gauss-Markov random parameters and conditioned quadratic cost function are considered. Algorithms for explicit calculation of the identification and control are given for systems with perfect state measurement and are illustrated by example applications. The control strategy generated by the cost function is an identitication-adaptive controller which is continually revised as new data are received. The identification equations develop an explicit model of the system which is available for other purposes if desired, and can be used apart from the control context.

Spatially invariant image sequences

The authors define linearly additive spatially invariant image sequences and present an explicit mathematical model for describing them. In such a sequence, all objects are positionally invariant in each image of the sequence but have varying gray-scale contributions to the successive images of the sequence. The various components (features or processes) of the scene or object contribute additively to each image of the sequence, but each component has a characteristic variation (signature) from image to image due to the variation of the function, parameter or spectral band over the sequence. Objects with different spectral characteristics will have different image sequence signatures which can be used to distinguish them. Also presented are the general formulation, derivation, and explicit expression for the linear filter, called the simultaneous diagonalization (SD) filter, that calculates a single new image from the sequence such that a desired process is emphasized and any number of undesired processes is suppressed in the filtered image.

The matrix properties of minimum-time discrete linear regulator control

The properties of the multiple-input time-optimal regulator state-transition matrix for linear discrete systems are developed by using the matrix concepts of nilpotency, eigenvector analysis, and Jordan canonical form.

Improved stability robustness bounds using state transformation for linear discrete systems

Abstract Sufficient-condition bounds on the linear time-varying perturbations for robust stability of an asymptotically stable linear time-invariant discrete-time system have been reported recently. The development involves both Schwartz and triangular inequalities, and tends to give conservative results. This paper uses state transformation to obtain improvement in these bounds. Both unstructured and structured perturbation bounds are shown to improve using transformations. The proposed analysis is applied to a fourth-order macroeconomic system model.

On the Volterra-series functional identification of non-linear discrete-time systems

The problem of determining the kernels in the discrete Volterra-series representation of a time-invariant non-linear discrete-time system is considered. The identification problem is transformed to an optimal control problem, which is then solved using dynamic-programming formulation. Examples illustrating the use of the techniques are presented.

Noninvasive Measurement of Shear Rate in Autologous and Prosthetic Bypass Grafts

There is a major difference in thrombogenicity between lower extremity prosthetic and autologous vein bypass grafts, and arterial blood flow shear rate is known to influence thrombus formation. Despite this association, there has been little direct clinical observation of shear rates in bypass grafts. The authors developed a new noninvasive method to quantitate human arterial shear rate and used it in a pilot study to characterize differences in lower extremity bypasses. Shear rates were measured in 10 prosthetic and 14 autologous vein femoropopliteal bypass grafts. With CVI-M-mode color flow ultrasonography in resting supine patients, a velocity profile was recorded from a midgraft longitudinal section in the ultrasound beam direction. Shear rates were calculated by using a mathematical-graphic computer program at the anteromedial (near) and posterolateral (far) graft walls by averaging values immediately before and after peak systolic velocity (PSV). Comparison between prosthetic and autologous graft groups respectively revealed that differences in age (67 ± [S] vs 71 ± 10 yr), male gender (60% vs 43%), prevalence of hypertension (50% vs 71%), diabetes (40% vs 64%), smoking (50% vs 50%), hypercholesterolemia (30% vs 29%), coronary artery disease (60% vs 50%), and critical ischemia (60% vs 86%) did not reach statistical significance (p>0.19). Median PSVs were significantly less in prosthetic than in autologous vein bypasses (37 ± 13 vs 57 ±22 cm/s, p = 0.018). Prosthetic and autologous graft diameters were not statistically significantly different (6.3 ± 1.1 vs 5.6 ± 1.3 mm, p = 0. 18). Shear rates were significantly less in prosthetic than in autologous vein bypasses both at the near wall (382± 146 vs 698 +271 s-1, p=0.003) and at the far wall (551 ±235 vs 827 ±339 s-1, p=0.037). This mathematical model can be used to calculate shear rate from observed ultrasound flow patterns. Prosthetic bypass grafts had lower shear rates than autologous vein grafts.

Controller design for discrete systems subjected to disturbances

A transfer-function approach is presented for the problem of driving to zero in minimum time the output of a system G(z) subjected to a known form, unmeasurable disturbance sequence. The solution is of the same form as the minimum variance regulator for systems subjected to stochastic disturbances. When the disturbance sequence or spectral representation has the form A(z)/B(z), where A(z) and B(z) are polynomials in z −1, the transfer function ‘ seen ’ by the disturbance must have the form Na(z)B(z)/A(z), where Na(z) is determined by A(z), B(z) and n, the leading power of z −1 in G(z). This property is exploited by a new method for computing controller coefficients in which the number of unknowns is reduced by n compared to the commonly used method of undetermined coefficients. It can yield general solutions which are difficult, if not impossible, to determine by any other known method.

Improved robust stability bounds for discrete-time linear regulators with computational delays

Abstract Recently obtained stability robustness bounds on linear time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system are applied to linear regulators with computational delays. The bounds were developed using Lyapunov theory and singular value decomposition for unstructured and structured time-varying perturbations. The new structured perturbation bound may provide improved results for the elemental perturbations over a bound based on the unstructured perturbation bound of Ishihara (1988, Automatica, 24, 696–700), as indicated both by an example and by sufficient conditions.

Analysis of a class of non-linear discrete-time systems by the Volterra series†

Abstract The discrete form of the Volterra series is used to evaluate the response of a class of non-linear discrete-time systems described by an ordinary nonlinear difference equation with zero initial conditions. Recurrence relations for the Volterra kernels are developed in the time domain and in the transform domain as well.