Michael K. Sain

Invertibility of linear time-invariant dynamical systems

The question of inverting linear time-invariant dynamical systems has been of interest to control engineers for many years. An example of the nonstate-variable approach and application of this concept is the very well known work of Bode and Shannon in 1950. With the integration of state-variable descriptions and techniques into systems problems, the question has recently reappeared in a somewhat more complicated guise. Basically, the problem of inverting such dynamical systems is to determine the existence of the inverse, its properties, and its construction in terms of the matrices which characterize its state description. The first general study of existence appears to be due indirectly to Brockett and Mesarovic in 1965, and the first general construction seems to have been proposed by Youla and Dorato in 1966. Neither of these works was intended to develop a substantial insight into the properties of the inverse. The present work introduces the concept of the inherent integration associated with a dynamical system, i.e., the number of integrations which no inverse dynamical system can remove unless ideal differentiators are introduced. The existence of the inverse is discussed in terms of a determination of the inherent integration, and the construction which realizes this minimum number of integrations is given. The existence tests introduced are at worst one-half as complex as that of Brockett and Mesarovic and the construction offers a substantial improvement in conceptual simplicity over that of Youla and Dorato. The results are made possible by recognizing an essential equivalence with an associated problem in real sequential circuits and appear to be applicable to related problems in sensitivity, estimation, and game theory.

Controlling buildings: a new frontier in feedback

The protection of civil structures, including their material contents and human occupants, is without doubt a worldwide priority of the most serious importance. Such protection may range from reliable operation and comfort, on the one hand, to survivability on the other. Examples of such structures leap to ones mind, and include buildings, offshore rigs, towers, roads, bridges, and pipelines. In like manner, events that cause the need for such protective measures are earthquakes, winds, waves, traffic, lightning, and-today, regrettably-deliberate acts. Indications are that control methods will be able to make a genuine contribution to this problem area, which is of great economic and social importance. In this article, we review the rapid developments which have been occurring in the area of controlled civil structures, including full-scale implementations, actuator types and characteristics, and trends toward the incorporation of more modern algorithms and technologies.

Codes, automata, and continuous systems: Explicit interconnections

Close relationships are established between convolutional codes and zero-state automata and between cyclic codes and zero-input automata. Furthermore, techniques of automata theory and continuous system theory are used to elaborate on the coding problem; and approaches from coding and automata are used to establish and interpret typical structural conditions in continuous systems. The investigation incorporates basic coding concepts into the currently emerging common basis for automata and continuous systems, and it gives explicit examples of the resulting benefits accruing to each of these areas from the others.

Reliability-based measures of structural control robustness

Abstract Because of the uncertainty inherent in engineering structures, consistent probabilistic stability/performance measures are essential to accurately assessing and comparing the robustness of structural control systems. An approach is presented herein for calculating such probabilistic measures for a controlled structure. First and second order reliability methods (FORM/SORM) are shown to be appropriate for the required calculations. The concepts are illustrated through several examples of seismically excited structures with active protective systems.

Controlling buildings: A new frontier in feedback

The protection of civil structures, including their material contents and human occupants, is without doubt a worldwide priority of the most serious current importance. Such protection may range from reliable operation and comfort, on the one hand, to survivability on the other. Examples of such structures leap to ones mind, and include buildings, offshore rigs, towers, roads, bridges, and pipelines. In like manner, events that cause the need for such protective measures are earthquakes, winds, waves, traffic, lightning, and-today, regrettably-deliberate acts. Indications are that control methods will be able to make a genuine contribution to this problem area, which is of great economic and social importance. In this article, we review the rapid recent developments which have been occurring in the area of controlled civil structures, including full-scale implementations, actuator types and characteristics, and trends toward the incorporation of more modern algorithms and technologies.

Feedback-feedforward control of structures under seismic excitation☆

Abstract While the base acceleration resulting from a seismic activity on a civil engineering structure is not known a priori, it can be measured in real time. In this paper, it is shown that this extra information can be used for achieving a better control of the structure with little additional effort. The approach taken is to augment the equations of motion for the structural system with an appropriate model of the earthquake excitation based on filtering a Gaussian white noise process. The augmented equations of motion are used to determine a control which utilizes both feedback and feedforward compensation. The feedback loop incorporates measurements of the response of the structure into the control law. The information from both the structure and the earthquake excitation model is utilized in the feedforward control law with an observer designed to estimate the states of the eartquake model based upon the base acceleration measurements. A quadratic performance index is used as a measure of optimality of the control algorithms. Results are given which indicate that the proposed method offers advantages in performance over the control method employing only state feedback and that it is also able to improve upon results of the recently develped instntaneous control algorithms. It is also shown that the nostationarity in the earthquake excitation can often be neglected in practical design of linear systems. Finally, a discussion is given of how the method might be coupled with equivalent linearization techniques and extended for use with nonlinear structures.

Control of linear systems according to the minimal variance criterion--A new approach to the disturbance problem

The problem of minimizing the ensemble average of a performance index in the presence of control noise has received substantial attention in the literature. This work considers a generalization of viewpoint, in which the index variance is minimized while its expectation is constrained. Necessary and sufficient relations are derived for linear, time-invariant systems and disturbances having rational spectra. The open-loop, optimal-feedback solution is specified by its characteristic equation and boundary conditions for Gaussian noises and plants with distinct eigenvalues. The canonic structure of a noise-free plant incorporating covariance data from the disturbance process is shown to have fundamental significance in the optimal solution. Several examples are presented.

Performance-measure densities for a class of LQG control systems

When linear-quadratic-Gaussian (LQG) control problems are solved, the performance measure takes on a probability density function Of a type which has received very little attention in the literature because of the nonzero-mean multidimensional character of the state process. This paper combines techniques of the fast Fourier transform, recent results of Baggeroer and Schwartz, and the classic methods of Karhunen-Loeve to present what seem to be the first published curves of such densities. As a vehicle for discussion, a one-parameter family of LQG control systems of the minimal variance type serves both to provide examples of the new density curves and to display certain controlled behaviors of the moments of the densities.

Nonlinear blackbox modeling of MR-dampers for civil structural control

Protecting civil engineering structures from severe impacts like strong earthquakes has demanded intensive research in the past two decades. One of the most promising devices proposed for structural protection is the magnetorheological (MR) fluid dampers. To fully explore their potentials in the real-time feedback control implementations, accurate and robust modeling of the devices is a prerequisite. This paper first proposes a general nonlinear blackbox structure to model the MR damping behavior on the displacement-velocity phase plane. Two constructive parameter estimation algorithms are subsequently developed which are based on the recent mathematical advances in wavelets and ridgelets analysis. Compared with the traditional physical modeling, this research aims at improving model numerical stability and model structure generality. The achievement of these objectives is evaluated in the modeling of an experimental MR-damper in a base-isolation structural control system.

On the zeros and poles of a transfer function

Abstract The poles and zeros of a linear transfer function can be studied by means of the pole module and the transmission zero module. These algebraic constructions yield finite dimensional vector spaces whose dimensions are the number of poles and the number of multivariable zeros of the transfer function. In addition, these spaces carry the structure of a module over a ring of polynomials, which gives them a dynamical or state space structure. The analogous theory at infinity gives finite dimensional spaces which are modules over the valuation ring of proper rational functions. Following ideas of Wedderburn and Forney, we introduce new finite dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or surjective. A new exact sequence relates the global spaces of zeros, the global spaces of poles, and the new generic zero spaces. This sequence gives a structural result which can be summarized as follows: “The number of zeros of any transfer function is equal to the number of poles (when everything is counted appropriately).” The same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general (possibly improper) transfer functions to controlled invariant and controllability subspaces (including such spaces at infinity).