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RESEARCH PAPERS

J. Basic Eng. 1969;91(2):137-138. doi:10.1115/1.3571041.
Abstract
Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):139-147. doi:10.1115/1.3571042.

This paper describes a technique which extends the application of the Kalman linear state variable feedback controller to practical nonlinear systems. This technique involves the linearization of the system model with residual nonlinearities retained. Extra linear states are then used to represent the residual nonlinearities in formulating the optimal controller. The use of the technique is illustrated by generating a controller for a simplified model of a hydroelectric plant. The advantages are shown by a comparison between the response of the nonlinear system with a controller generated using all of the nonlinearities as extra states and one using only one of the nonlinearities as an extra state. This technique could be used to design a nonlinear analog controller with fixed coefficients or an extremely simple control routine for an online digital computer.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):149-154. doi:10.1115/1.3571047.

A simplified approach to the solution of linear optimal control problems with quadratic performance indexes is described in this paper. The phase-variable canonical form is used to develop a new type of optimal system equivalence. This concept leads to a substantial simplification of the matrix Riccati equation. The simplified matrix Riccati equation is of the same form for any problem of a given order, say, n, and contains only n nonzero forcing functions. That is, it always corresponds to a set of constant-coefficient scalar differential equations; in various nth-order problems the n nonzero forcing functions and the terminal conditions simply assume different forms. In a very strong sense, this simplified matrix Riccati equation is the simplest possible Riccati equation arising from optimization problems. The method is developed for general time-varying systems with finite terminal time. It is developed also for the important special case of time-invariant systems with infinite terminal time.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):155-160. doi:10.1115/1.3571051.

In problems of optimal control, the final time T may be fixed or it may be unrestricted. For the unrestricted final time case, an additional necessary condition that the Hamiltonian be zero is added to the conditions for optimality used for the fixed time case. In this paper, it will be shown that this necessary condition may lead to a local maximum of the performance criterion with respect to final times as well as a local minimum. This paper first develops a computational algorithm using only the H = 0 condition, and then develops a sufficient condition for a local minimum with respect to final time and a computational algorithm employing this condition. Numerical examples are given to illustrate all results.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):161-167. doi:10.1115/1.3571052.

The optimal open loop control of systems described by a set of linear partial differential equations is investigated. The performance index is of quadratic type and the mean square error is considered as a special case. Energy type inequality constraints are imposed on the control inputs. The problem is formulated as a minimization problem in Hilbert space. The necessary and sufficient conditions for a minimum are obtained and it is proved that these conditions yield the global minimum. It is shown how the solution to the constrained problem can be obtained from the solution of the unconstrained problem. The optimal control functions satisfy Fredholm integral equations with symmetric kernels. The paper presents an example where the solution is obtained by eigenfunction expansion.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):168-172. doi:10.1115/1.3571053.

A proposed apparatus for controlling temperatures from −190 deg to +650 deg with an estimated accuracy of 0.001 deg C is described. The apparatus utilizes helium as the heat transfer medium. The selection of the gain constants of the controller depends upon the system parameters. The hypothetical case of a solid cylindrical block with an integral plus proportional controller is considered, the differential equations are set up, and a graphical scheme is presented for the selection of the controller constants.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):173-178. doi:10.1115/1.3571054.

Filtering equations are derived for processes described by linear partial differential equations with known homogeneous boundary conditions. Both discrete-time and continuous-time measurements are treated. As in the case of linear systems with time delays, the filtering and variance equations become partial differential equations for processes with continuous measurements. A numerical solution to the nonlinear variance equation is obtained for a particular diffusion process.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):179-184. doi:10.1115/1.3571055.

This paper describes an experimental method of establishing an ordinary differential equation to represent, or describe, a nonlinear physical system. Each nonlinear function that appears in the differential equation is approximated by a piecewise continuous collection of simple power series expansions. A given elementary expansion is used to represent a function over only a small fraction of the total range of the corresponding argument. These expansions are characterized by a few coefficients which are determined, during the identification process, by a steep descent adjustment procedure. It is assumed that the system may be excited by a specified input and that neither the input nor the output is significantly corrupted by noise. Systems of any order, and with any number of unknown constant coefficients and continuous, single-valued nonlinear functions may be identified with this procedure. Prior knowledge required to implement the method includes the form and order of a differential equation to describe the unknown system, and the location and arguments of functions in this equation assumed or known to be nonlinear. Examples are given to illustrate the efficacy of the method.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):185-189. doi:10.1115/1.3571056.

A problem in the optimal control of a nuclear rocket requires the minimization of a functional subject to an integral equation constraint and an integrodifferential inequality constraint. A theorem giving first-order necessary conditions is derived for this problem in the form of a multiplier rule. The existence of multipliers and the arbitrariness of certain variations is shown. The fundamental lemma of the calculus of variations is applied. A simple example demonstrates the applicability of the theorem.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):190-194. doi:10.1115/1.3571057.

The optimal control problem for a broad class of distributed parameter systems defined by vector parabolic partial differential equations is considered. The problem is solved by discretizing the spatial domain and then treating the (large) resultant set of ordinary differential equations as a set of independent subsystems. The subsystems are determined by decomposition of the total system into lower-dimensional problems and the necessary conditions for optimality of the overall system are then satisfied by an iterative procedure. With this treatment, the optimal control problem can be solved for larger systems (or finer spatial discretizations) than would otherwise be feasible. An example is given for a system described by a nonlinear parabolic partial differential equation in one space dimension.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):195-199. doi:10.1115/1.3571058.

Suppose imprecise observations are made on imprecisely defined nonlinear processes, and one wishes to estimate the state of the process at certain fixed instants of time lying within the interval of observation. Furthermore, assume that it is required to update these estimates as additional observations become available. This is precisely the problem of sequential interpolation. The equations of the sequential interpolating filter, when a least-squares estimation criterion is used, are obtained in this paper. The interpolation problem is first shown to be equivalent to a two-point boundary-value problem. The two-point boundary-value problem is converted to an initial-value problem using invariant imbedding. The initial-value problem leads directly to a sequential filter.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):201-205. doi:10.1115/1.3571061.

The merit of a multiplicative control mode is demonstrated by means of an optimal-control synthesis procedure that is applied to a bilinear regulator problem. Not only is the bilinear regulator superior to the linear regulator for step inputs, but it exhibits far better performance for sinusoidal inputs. Though bilinear control systems are very common in nature, for the most part, they have been overlooked in engineering despite their advantages.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):207-210. doi:10.1115/1.3571064.

Necessary and sufficient conditions are obtained for the stability of the following second order linear system:

ẋ = θ(t)x,   θ(t) = θt+i=1lTi
and
θ(t)  = A1,   0<t<T1
  = A2,   T1<t<T1+T2
     ⋮
  = Al,   i=1l−1Ti<t<i=1lTi
in terms of the eigenvalues and elements of the matrices A i , i = 1, 2[[ellipsis]]l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):211-216. doi:10.1115/1.3571067.
Abstract
Topics: Fluids , Testing
Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):217-226. doi:10.1115/1.3571070.

The method of characteristics has been used in a variety of graphical, analytical, and numerical ways as a powerful tool in the solution of hyperbolic partial differential equations. The availability of digital computers permits the basic method to be applied to a greatly extended class of problems represented by semihyperbolic equations. This general extension is illustrated by problems of unsteady fluid flow in rigid tubes with the effects of frequency or history-dependent wall shear and heat transfer.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):228-237. doi:10.1115/1.3571074.

The problem of assigning a physically meaningful measure of the “quality” of controllability and observability to dynamical systems which are completely controllable and/or completely observable is considered. One particular analytical measure of quality, for a class of linear dynamical systems, is proposed and an effective computational procedure for maximizing the proposed quality, with respect to certain adjustable structural parameters of the dynamical system, is described. Three illustrative examples are worked in detail.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):239-244. doi:10.1115/1.3571077.

A method is presented to identify partial differential equations and associated boundary conditions of a distributed parameter system. The method is applicable to linear, nonlinear, and time-varying systems. The technique requires that the form of the differential equation and boundary conditions be known up to a set of constant parameters. Finite differences are used to approximate derivatives. Identification is carried out by using normal operating data. The accuracy of the identification depends upon the approximation errors and measurement noise. Methods for decreasing approximation errors are presented.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):246-249. doi:10.1115/1.3571081.

Definitions of system sensitivity for linear single variable systems have been extended, in the past, to linear multivariable systems in the form of a sensitivity matrix. The role of the sensitivity matrix in multivariable feedback control systems is studied further in this paper. The sensitivity matrix serves the dual function of governing the effects of plant parameter variation on the system transfer matrix and governing the effects of disturbances on the system output. The design implications of this are considered and it is shown that certain controllability/observability conditions are necessary if the system design is to be effective. By appropriate design of the loop gain matrix, L(s), a desired insensitivity to system error sources may be achieved. Unless the system has certain controllability/observability properties insensitivity cannot be achieved. It is shown that L(s) must have the property of functional reproducibility which is a relatively strong controllability/observability requirement.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):251-256. doi:10.1115/1.3571084.

The optimization of bounded feedback gains with respect to an arbitrary integral performance criterion is performed using nonlinear programming. The gradient projection method has been applied to several examples of fifth order, linear, and nonlinear, with good results.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):257-262. doi:10.1115/1.3571087.

The results of a comprehensive study of the comparative performance characteristics of geometrically similar vented and unvented bistable amplifiers, together with their actual dimensions, are presented. The Reynolds number in the tests ranged from 9,750 to 60,000, Mach number from 0.07 to 0.42, and the power jet velocity from 75 to 460 ft/sec. Each amplifier as conceived and designed was capable of giving a maximum of geometric flexibility which enabled a systematic evaluation of the shape and location of the splitter plate and Coanda-walls. It was found, within the range of Reynolds and Mach numbers tested, that certain gain characteristics and the range of operation of a given unvented amplifier overlap, within a narrow range of P0 /Ps and Q0 /Qs values, with the corresponding gain characteristics and the range of operation of the vented amplifier. It was also found that a convex-walled amplifier with proper geometry exhibits considerably better performance characteristics than are normally associated with such devices.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):264-272. doi:10.1115/1.3571091.

A momentum integral analysis is presented for the incompressible, steady, axisymmetric flow in a short vortex chamber of the type commonly used in vortex valves. The analysis is developed with the aid of flow visualization photographs and considers the interaction which occurs between the main vortex core flow and the viscous chamber end wall boundary layers. The radial pressure distributions predicted by the analysis compare favorably with measured end wall static pressure distributions.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):277-283. doi:10.1115/1.3571095.

A distributed parameter thermal and stress model is developed for a nuclear rocket. The resultant equations for the optimal control problem are a pair of coupled, bilinear, partial differential equations. The thermal stress constraint forms an inequality which is a function of both the state and the control. The initial conditions are steady state, and the terminal condition is that the coolant flow obtain a fixed, higher level. The distributed parameter system is discretized in the space dimension to give an arbitrary order set of ordinary differential, state equations. It is shown how a result based on the Weierstrass necessary condition and derived by Berkovitz from the calculus of variations using a slack variable technique may be applied. This condition is shown to require the optimal control to be “boundary control” with no switching. The optimal control program must make the inequality constraint an equality at some location throughout the transient. Based on the result that boundary control is the optimal control, an algorithm is developed to compute the optimal control program. The algorithm was programmed on a digital computer and numerical results are given for the optimal flow program and the resultant stress distributions for various cases.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):284-288. doi:10.1115/1.3571096.

An intracorporeal left ventricular bypass, incorporating an electrically driven roller pump, has operated successfully in unrestrained calves for periods up to 11 days. A pneumatically powered prototype, designed for eventual human application, extends potential effective pumping periods to two months in tests simulating implanted conditions. Mechanical design and materials maintain clotting and cell damage below traumatic levels and no subsequent ill effects are indicated.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):290-294. doi:10.1115/1.3571099.

A scheme is outlined for a useful way to think about the complex biological organism, man. It is based on physiological findings that the regulating and control functions in the system make use of active processes, exhibiting oscillatory properties [1]. The resulting homeostatic regulation, which was the key concept proposed by Bernard, Sechenov, and Cannon for the living system [2], emerges from mediation of these oscillators. Because of its dynamic character, the scheme is renamed homeokinesis [3]. The concept may be extended to man’s behavioral complex. In outline, it touches on all the time or frequency domains in life—that is, of the many episodes in man.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):295-304. doi:10.1115/1.3571100.

Throughout the life of the vertebrates, the core of the central nervous system, the reticular formation, has retained the power to commit the whole animal to one mode of behavior rather than another. Its anatomy, or wiring diagram, is fairly well known, but to date no theory of its circuit action has been proposed that could possibly account for its known performance. Its basic structure is that of a string of similar modules, wide but shallow in computation everywhere, and connected not merely from module to adjacent module, but by long jumpers between distant modules. Analysis of its circuit actions heretofore proposed in terms of finite automata or coupled nonlinear oscillators has failed. We propose a set of nonlinear, probabilistic, hybrid computer concepts as guidelines for specifying the operational schemata of the foregoing modules. Using the smallest numbers and greatest simplifications possible, we arrive at a reticular formation model consisting of 12 anastomatically coupled modules stacked in columnar array. A simulation test of its behavior shows that despite its 800-line complexity, it still behaves as an integral unit, rolling over from stable mode to stable mode as directed by its succession of input 60-tuples.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):305-312. doi:10.1115/1.3571101.

Communication within biological systems involves either or both of two kinds of channels, each with a different kind of signal. The two types of signals are: (a) nerve impulse, or (b) hormones. These channels appear in control loops that provide external control over many plant processes that characterize the living states. One of these chemical control arrangements, the adrenal glucocorticoid system, has been studied by means of computer simulation in order to assess its role in the regulation of energy fluxes in animals. The dominant time-constant of the controller system ranges from ten minutes to one hour in higher animals and man, and the system follows a circadian rhythm of one cycle per 24 hours at its input. The controller system is nonlinear; its bandpass characteristics are a function of its initial operating level, and cannot be concisely stated. However, the simulation indicates that energy transformations occurring within animals in the frequency domain of one cycle every 24 hours down to one cycle every hour could be entrained or driven by the adrenal glucocorticoid system.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):313-320. doi:10.1115/1.3571102.

The thyroid-pituitary system of warm-blooded animals does appear to behave as an effective regulatory system in which the controlled variable is the level of free (unbound) thyroid hormone in the plasma. A negative feedback loop illustrating the concept of the free hormone level in plasma acting on the pituitary gland to regulate the secretion of thyroid-stimulating hormone (TSH) is an essential feature of the biologist’s line-and-arrow diagram of this system and some function to express this action appears in any mathematical description of the system. However, the relationship between the rate of secretion of thyroid hormone and the concentration of free hormone in the plasma to which the pituitary is exposed is neither simple nor direct. The animal is a highly complex and variable plant, affecting the distribution and metabolism of thyroid hormone. Any adequate and realistic model of the system must consider the features of the plant in detail. A mathematical formulation of the process of hormone binding to carrier proteins in plasma is presented. Simulation of even a limited aspect of a complex biological system as it exists in real life may provide valuable insight into the overall behavior of the system.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1969;91(2):321-324. doi:10.1115/1.3571103.

This paper first reviews the basic elements of the reproduction system in female mammals and indicates the adaptive significance of the cyclicity of structure and function always observed in this system. A detailed description of the reproductive cycle in the laboratory rat is then presented, and a theoretical model of the control of the rat estrous cycle is described. The purpose of the model is to “explain” the mechanism of cyclic function of reproduction in the rat. Some examples of the heuristic value of the model are then outlined, including the beginning of computer simulation.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Basic Eng. 1969;91(2):325-327. doi:10.1115/1.3571104.
Abstract
Commentary by Dr. Valentin Fuster

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