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The moments of the first-passage time of a narrow-band stochastic process produced by a linear oscillator excited by wide-band noise are presented for the case of a “safe” region defined by a circular barrier in the phase plane. As long as damping is small the moments produced will be approximately the same as those for the two-sided barrier problem. An iterative set of partial differential equations governing the moments, the first being the Pontriagin-Vitt equation for the mean first-passage time, are solved by using a Galerkin approximation. No restrictions are required on the barrier size.

In this paper the dynamic interactions between mechanical and reactive modes of machines in a multi-machine power system are investigated. The dynamic interactions are expressed in terms of machine angles, transient voltages, and coefficients which are functions of system parameters and operation conditions. The effect of the dynamic interaction can be examined by the electric torque loci on the A6-Aw phase plane. It is found that the interaction between the mechanical modes of machines has significant effect on stability characterized by the natural mechanical oscillation frequency and damping of individual machines.

Summary A method for increasing the diagnostic capability of the clinical electrocardiogram has been further developed. The coordinated display of voltage against the time derivative of voltage (dv/dt), i.e., PPC, was found to be remarkably sensitive to subtle aberrations in QRS contours not easily visualized in the standard electrocardiographic portrayal of voltage against time. A standard twelve-lead electrocardiogram and phase-plane loops were displayed and photographed on an oscillographic recorder, the latter by placing voltage (V) on the vertical axis and the first time derivative of voltage on the horizontal axis. Data storage on magnetic tape and computer analysis of the data were also carried out. Data from 100 “normal” (as determined from clinical ECG, history, and physical examination) subjects demonstrated evident, repeatable notching patterns in the PPC. The repeatability of these patterns was demonstrated from beat-to-beat in each lead of every individual, and was found to be consistent even when the readings were taken over long periods of time. The PPC's of different individuals showed consistent common notching patterns which would indicate that anatomical and physiological bases exist to explain this phenomenon.

— The imagery of phase object is possible through the phase-contrast technique of Zernike. This technique effects a unitary transformation of the phase variation into amplitude variation, and thus makes the transparent object visible. Such objects usually are assumed to have a small phase variation, and hence the phase-contrast images will have a small contrast. It is known that an enhancement in the contrast of images of low contrast objects is feasible through the use of an amplitude contrast technique. This implies that a simultaneous use of the phase and amplitude contrast technique should increase the sensitivity of the imaging techniques, thus enabling the lowering of the limits of minimum delectability as well as the resolution of phase variation in a given object. This question has een studied in this paper. The effect of a partially coherent illumination of the object has been included in this study.

A study has been made of atrial transmembrane potentials of rabbit and guinea pig hearts. In addition to the voltage-time record the first time derivative of the action potential was displayed as a function of voltage, yielding a phase-plane trajectory. A number of parameters of the action potential were determined from the trajectory, assuming cable behaviour. The validity of this assumption is discussed, and the effects of temperature reduction determined.

The problem of minimal time control of xenon spatial oscillations in nuclear power reactors is investigated and the effect of constraints on the control variable (rod position), spatial offset magnitude and rate of change, which is closely related to the rate of change of the local power density, is studied. The concept of spatial offset phase plane is extended to include direct reference to the measured spatial offset. The optimal constrained and unconstrained trajectories and switching curves are analyzed in the phase plane. Work done on the Carnegie-Mellon University digital reactor simulator confirmed the success of the derived theoretical results. Operational strategies that embody both the strength of the theory and the simplicity needed for practical application are recommended.

In an earlier publication (Jupp, 1972), a solution of the Ideal Resonance Problem is exhibited explicitly in terms of the ‘mean’ elements; to second order in the small parameter in the case of libration, and to first order in the case of deep circulation. Both representations possess a singularity when the ‘mean’ modulus of the Jacobi elliptic functions is unity; this corresponds to the separatrix of the phase plane of the dynamical system.

The dynamics of a neutrally buoyant float descending from the surface to its level of neutral buoyancy in a stable linearly stratified fluid are considered theoretically and experimentally. When the descent is large compared to the float size, the dynamics can be modeled by balancing the acceleration, buoyancy and drag forces, using a quadratic velocity drag form. A general analytical solution can be found in the phase plane, and several typical solutions are computed. Results of visual and quantitative experiments in a salt-stratified tank were in agreement with the theoretical results.

An analysis of Goodwin's fundamental nonlinear oscillator in the light of the requirement that concentrations must be intrinsically non-negative reveals that most of the phase plane is “forbidden”, and that there is one distinguished closed curve that has some of the properties of a limit cycle.

The motion of a particle constrained to move on a circular path within a body that is spinning and coning is examined. The equation governing this motion is expressed as a nonlinear, nonautonomous ordinary differential equation, and the behavior of solutions of this equation is investigated using phase plane techniques. Several classes of phase plane portraits are identified, and boundaries that separate system parameters giving rise to portraits of each class are established. Practical applications for the information thus generated include the analysis and design of certain passive balancing devices used to attenuate wobble in spinning spacecraft and the study of simplified models of nutation dampers and thermal control devices consisting of circular tubes partially filled with fluid.

A technique is described for time resolving, to nanosecond precision, the velocity distribution of a periodically varying electron beam and for generating a direct plot of the entire phase plane distribution function. Calculations are presented to indicate the maximum time rate‐of‐change of the distribution function which may be accurately observed with this technique. The high‐frequency response obtainable with this technique makes it a powerful diagnostic tool for use in studying the interaction of an electron beam with plasmas and, in particular, the trapping phenomenon in beam‐plasma interactions.

A simple algorithm for the time-optimal control of chemical processes during setpoint changes, in processes which can be described by a second-order lag plus dead time model, is described. Knowledge of the unsteady state model parameters is not required because the algorithm uses a dimensionless phase plane on which the switching curves are independent of system parameters for a given forcing function. The algorithm gives the parameters of a second-order lag plus dead time model as a by product of the setpoint change. It is easily tuned and is relatively insensitive to changes in the process dynamics. The algorithm does not require a large computer or long computing times and has been implemented on both analog and digital computers in controlling computer simulated systems.

It is well known that the equations governing the motion of a freely-rotating rigid body possess an exact analytical solution, involving Jacobi's elliptic functions. Andoyer (1923) and Deprit (1967) have shown that the problem may be very usefully reduced to a one-degree-of-freedom Hamiltonian system. When two of the body's principal moments of inertia are very nearly equal, the Hamiltonian system has the same form as the Ideal Resonance Problem. In earlier publications (Jupp, 1969, 1972, 1973), the author has constructed formal power-series solutions of the latter problem. In this article, the general solution of the Ideal Resonance Problem is employed to formulate a second-order formal series solution of the problem of a freely-rotating rigid body which has two of its principal moments of inertia differing by a small quantity. This solution is firstly expressed in terms of the mean elements, and then in terms of the initial conditions. The latter solution is global in nature being applicable over the whole phase plane. It is demonstrated that the exact solution and the second-order formal series solution, written in terms of the initial conditions, differ by terms of at most third order in the small parameter, over the whole domain of possible motions. This serves as an important check on the general results published in the earlier articles.

By combining the instantaneous phase shift of a twosinusoid-input describing function with a phase response of the tuned circuit it has been possible to develop the locking equation for single-stage LC injection-locked oscillators ILO. The locking range as well as steady-state phase shift of an ILO are calculated for subharmonic, fundamental, and harmonic locking, the dc transfer characteristic of a nonlinear device being approximated, at the dc operating point, by a third-order polynomial. The phase plane representation is included. Experimental results are given to support some of the theoretical relations derived.

The Lotka-Volterra nonlinear differential equations for two competing species P and Q contain six independent parameters. Their general analytic solutions, valid for arbitrary values of the parameters, are at present unknown. However, when two or more of these parameters are interrelated, it is possible to obtain the exact solutions in the P , Q phase plane, and six cases of solvability are given in this paper. The dependence of the solutions on the parameters and the initial conditions can thus be readily investigated.

We introduce the concepts of the degree and the order of synchronism on the basis of a mathematical model of the emergence of synchronization in the form of an asymptotically stable integral torus in the phase plane. We investigate the existence conditions for synchronisms in a dynamic system described by differential equations with rapidly rotating phases. As an application we examine synchronisms in a system of quasi-Hamiltonian objects. In recent years the phenomena of synchronization and resonance in dynamic systems have been subjected to intensive study, in particular, in connection with the question of the synchronization of satellites [1, 2] and of mechanical vibrators [3]. On the mathematical side the appearance of synchronization is closely connected with the theory of differential equations with rapidly rotating phases. Here in the first place we must mention the works specified in [4–9].

Quasi-continuous temperature control is needed when electric space heating is applied to temperature-dependent processes. In such applications, multiposition systems cannot supply the accurate control often required, and quasi-continuous systems have economic and operating advantages over many continuous systems. The reference input of a quasi-continuous system can be a trimmed, optimized, open-loop command derived from the incoming and outgoing quality levels of the process. Feedforward correction for process lag, and overriding commands, can easily be incorporated into such systems. Quasi-continuous characteristics contain a number of discontinuities. Time can be saved by representing these characteristics by a continuous profile. A multitude of intermediate values of manipulated variable can be obtained by new and inexpensive methods. When the heating load must remain intact, phase-discontinuity methods can be used which approximate level-discontinuity control. For fast response processes, phase-controlled SCR's (silicon-controlled rectifiers) can be used. The T-T (temperature-time) diagrams have limitations as representations of quasi-continuously controlled processes. Time domain information can be obtained from the phase plane where quasi-continuous performance resembles a spiral. Nonlinearities can be advantageously introduced into quasi-continuous profiles. Performance is improved by derivative feedback. It is possible to predict quasi-continuous system behavior by means of a 3-dimensional actuating-signal display.

A mathematical model is derived for the voltage step-down dc-to-dc converter in which a hysteretic bistable trigger circuit is used to regulate the output voltage. Normalized second-order differential equations are derived for the output-voltage error, or output-voltage ripple, measured with respect to a constant reference. The method of successor functions is applied to the piecewise analytic phase plane trajectory for the errors and the conditions leading to a limit cycle are initially formulated in two transcendental equations. With the objective of obtaining an analytic solution for the period of the limit cycles these two equations are then replaced by approximate algebraic equations which are solvable in general terms. The approximations are carefully based on properties that are typical of all converters of this type, and lead to quite simple but accurate expressions for the period of the limit cycle in terms of arbitrary system parameters. Expressions for the amplitude of the limit cycle are also given, and its stability is tested. A numerical example based on an actual representive system is given. Certain unusual characteristics of the limit cycle as a function of certain system parameters are pointed out.

The interaction of three electrostatic waves in a collisionless plasma is treated to fourth order neglecting the wave—particle interaction (damping). Using the principles of energy conservation, and invariance under time reversal, conditions on the coupling constants are derived, enabling the solution to be expressed as a function of only two coupling constants. Phase plane diagrams of the solutions are sketched showing that the only singular points are stable equili bria. It is suggested how the theory may be applied to the explanation of the ‘floating spike’ resonance observable when one-half the hybrid frequency is the difference between the plasma and gyro frequencies (Hagg & Muldrew 1968).

A negative resistance connected to a ``suitable'' resonant circuit (a current‐controlled negative resistance connected to a series‐tuned circuit, or a voltage‐controlled negative conductance connected to a parallel‐tuned circuit) constitutes an almost‐harmonic oscillator. In this paper the ``unsuitable'' connections (a current‐controlled negative resistance connected to a parallel‐tuned circuit, or a voltage‐controlled negative conductance connected to a series‐tuned circuit) are analyzed qualitatively in terms of variables leading to a single‐valued phase plane. Physical considerations, expressed in a jump condition, serve to connect different regions of analyticity of the differential equations of the circuit in this plane. The ``unsuitable'' connections cannot sustain almost‐harmonic oscillations, and their theoretically possible relaxation oscillations are unstable; they yield bistable circuits.

The optimal control problem, in the presence of uncertainty in the plant, is formulated as a game between the uncertainty and the control variables. This approach gives an optimal control strategy which is effective even under the "worst" conditions of uncertainty. The optimal control of a second-order plant with uncertainty in frequency brings out several interesting features. In particular, the existence of a barrier demarcating the controllable and uncontrollable regions in the phase plane is revealed, which is absent in the corresponding one-sided optimization.

A survey of some elementary macroeconomic models is presented which indicates the potential that exists for the application of control theory in economics. An initial linear model is developed systematically to embrace an increasing degree of economic sophistication leading to a non-linear formulation, the need for which is soon made apparent. Examples are given to demonstrate the use of characteristic root analysis, of the root locus technique to examine parameter sensitivity and of phase plane and stability concepts. At the same time, important economic interpretations which can be drawn from the results are indicated.

The two-dimensional autonomous system in the general form is considered. The solution is in terms of the phase trajectories in the phase plane. In this paper, a force field over the phase plane is formulated such that the path of a particle in this force field coincides with the path of the trajectories, provided the initial condition is properly chosen. Hence, the trajectories can be obtained by solving the path of the particle subject to the formulated force field. It is further developed that, for any given system, it is theoretically possible to formulate a conservative force field to give the same property. Consequently, it is found out that the trajectories can be solved either analytically by a simple substitution or graphically by means of the family of trajectories which are orthogonal to the trajectories of the original system. Various examples including the generalized Van der Pol equation are given gs illustrations.

A phase plane technique, which takes advantage of convenient relationships in other planes (for example, displacement versus time) to effect graphical solutions for a non-linear time varying second order differential equation, is developed. Several special cases of this general second order equation are considered. The special case of most practical importance occurs when the differential equation is permitted to become piece-wise linear. To demonstrate the simplicity of the technique, for the piecewise linear case, an example involving saturation in an inertially damped position servomechanism is given.

A decision algorithm which simulates the rapid adaptive behavior of human controllers following sudden changes in plant dynamics is developed. The control of a VTOL aircraft in hover following failure of the stability augmentation system is used as a specific example. The decision algorithm is based on the assumption that the human controller recognizes certain pattern features in the error/error-rate phase plane. Experimental data, obtained from pilots facing four possible alternatives following the time of failure, are presented. The proposed decision algorithm is developed, and digital simulation results are discussed. A theoretical justification for the algorithm, based on statistical decision theory, is presented in the Appendix.

Poincare formulated a general problem of resonance in the case of a dynamical system which is reducible to one degree of freedom. He introduced the concept of a global solution; in essence, this means that the domain of the solution(s) covers the entire phase plane, comprising regions of libration and circulation.

This paper deals with the term constant time loci (C.T.L.) in the phase-plane and shows that in piecewise linear systems, the isoclines are also the C.T.L. A relation between the slope of the phase trajectory at any point on the phase-plane and the time taken by the system to reach that point is established, assuming any one of the isoclines as the t = 0 line. As a result, in the case of piecewise linear systems, time solutions, i.e., x vs. t, can be directly determined from the phase-plane plot. An example demonstrates the efficacy of this new approach.

The paper introduces two intrinsic criteria for runaway in fixed bed tubular reactors based on the occurrence of characteristic points in the temperature profile along the reactor. Their transposition in the partial pressure-temperature phase plane allows the derivation of some very simple formulae for the prediction of the critical values for the operating variables. The results are compared with those obtained from Barkelew's empirical criterion. The paper also treats the related problem of a hot spot ehich has to be limited for reasons after the runaway. The simple formulae mentioned above can also be used in this case to determine the operating conditions.

Nonlinear servo analysis is applied to a free running regulator. Steady state performance is determined using a modified sinusoidal describing function approach. Gain-phase plots provide a graphical analog. Transient analysis is accomplished using phase plane techniques. Complete analysis - steady state and transient -- is performed for a basic regulator and for an active output filter version. Results are applicable to specific requirements.

In this paper an analysis of a phase look loop (PLL) having an active filter instead of the passive one in the loop has been made using the phase plane technique. System equations are derived and expressions for the capture range and the acquisition time have been obtained. Verification of the results of the analysis has been made by analogue simulation. The results obtained support the analysis.

This paper presents a generalization of the parameter plane method in that it considers the case when the characteristic equation coefficients are nonlinear functions of the system adjustable parameters. The generalized method is then applied to the system analysis in which the coefficients are linear functions of two parameters and their product. As a simple and rapid procedure for factoring polynomials in the parameter plane, the method is used in the design of linear continuous multivariable control systems. A non-linear system with two nonlinearities is also considered whereby the stability and existence of limit cycles are investigated.

Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.

A comparative study of the performance of sampled data system with a variety of hold circuits is presented. The transient response and stability of the systems considered have been investigated using the incremental phase plane technique. The comparison is extended to include two common types of non-linear elements.

In those cases where dynamic tests are to be interpreted in terms of the molecular structure of rubber, there is a definite need for a method of analysis which is wholly free from mathematical assumptions. Such a method of analysis is possible by the use of phase planes of velocity plotted against displacement. This is so because the slope of the trajectory at any point of the motion of any single degree of freedom system can be written solely in terms of physical quantities, namely, driving force, viscoelastic reaction and momentum. The analysis involves no hidden mathematical assumption. Methods of deriving stress strain relationships from phase plane trajectories are described. This phase plane method is discussed in some detail. Examples are given showing the inadequacy of the usual linear analysis in the case of rubberlike materials. Analysis shows that dynamic test machines should either apply a known motion and record a force-deflection curve or apply a force of known wave form and record the resulting motion as a phase plane trajectory. If force and motion are not kept independent of each other in this manner there is an inbuilt mathematical relationship which, more often than not, influences and falsifies the results. An important conclusion is that only when the nature of the non-linearities, or what is the same thing, the character of the functions which correctly describe the actual behavior of a vibrating system have been established by sound analysis of test data, should the phenomena be ascribed to the physical properties of the material. Because of the important consequence of the acceptance of some common mathematical steps in any analysis, certain of these are critically examined and their relevance discussed.

When the signal component of the input to a linear unity feedback sampled-data system is additively contaminated by a relatively wide-band noise process, it has been shown previously that the mean-square values of the true and apparent error quantities are minimized by essentially the same value of loop gain. This paper presents an investigation into the stability of a proportional-rate extremum regulator, which utilizes this fact to establish the optimum gain setting from finite estimates of the measurable mean-square apparent error. Assuming a parabolic operating characteristic, the stability and dynamic behavior of the adaptive loop is studied using path tangent curves in the incremental phase plane. The ability of the regulator to establish the optimum gain setting for a given system is verified experimentally for a number of different operating conditions. In addition, the experimental results show that the transient disturbances generated by the corrections to the gain parameter may adversely affect the regulator stability if they exist over a sufficient fraction of the measurement period.

This paper presents a simple technique for improving the pull-in capability of phase-lock loops. This technique, called derived rate rejection or DRR, differs from those which use an external AFC loop in simplicity of implementation and design rationale, although the end result is the same. If, as is usually the case, a coherent detector accompanies the phase-lock loop, the implementation of the DRR technique requires only the addition of a switch. The switching logic results from a superficial consideration of the nonlinear equation for the phase-lock loop and its solution in the phase plane. The switch does not affect the normal behavior of the loop after lock has been attained. Results of computer studies show the improvement realizable for the following configurations: 1) Proportional-plus-integral control. 2) Proportional-plus-imperfect integral control. For an initial frequency error of five times the linearized phase lock-loop natural frequency, the improvement in pull-in time is a factor of two. For an initial frequency error of ten times the phase-lock loop natural frequency, the improvement in pull-in time is a factor of ten.

It is known that the switching curve which defines the synthesis in the problem of determining time optimal trajectories to the origin for second-order differential systems consists of two basic lobes together with their ‘reflections’ through certain conjugate time intervals. The switching curve may thus possess corners. It is here shown that these corners lie on the x-axis in the phase plane for a particular class of problems.

The paper deals with a control system which has a plant consisting of two integrators with a saturable control input. With a command signal (which may be a step plus a ramp) the optimum switching curve and the optimum controller for two different performance criteria (‘Minimum integral square of error’ and ‘Minimum integral modulus of error’) are found explicitly by purely algebraic and analytic methods which start from Pontryagin's Maximum Principle and involves Taylor's series expansion of the Hamiltonian system of equations between two arbitrary successive switch points on the forward path of an arbitrary optimum trajectory. The equations of the switching curves in the phase plane obtained by this method are verified with the known results already determined by Brennan and Roberts (1962), Eggleston (1963), Fuller (1960 a) and Wonham (1963) by other methods.

A review of high intensity phenomena observed in the CPS is presented. At injection an attempt is made to describe the combined effect of space charge and resonances, which result in a loss of 70% of the injected beam intensity and a reduction by a factor of 3 of the central core brilliance. Results of measurements on the effect of a localised bad vacuum are given. The situation at transition is briefly described. Seven different instabilities observed in the CPS in longitudinal or transverse phase plane are analysed. The compensation techniques are described in all cases. Whenever possible an attempt has been made to give some qualitative explanation of the mechanism.

A method of optimizing neutron flux distribution to obtain maximum power generation from a given core is presented. Constraining conditions governing maximum heat flux and control rod absorption are taken into consideration. It is shown that the problem is reduced to that of optimal control. In the case of one group model the phase plane technique is effective for an intuitive analysis of the problem. It is verified that the solution so obtained satisfies Pontry-agin's principle. The solution is one of “bang-pang type”. The result is extended to the case of two group model, assuming that the solution will again be of bang-pang type. A comparison is made between the one group and the two group models.

An increasing familiarity of power system engineers with frequency domain techniques has motivated the search for stability criteria for power systems that could be applied in the frequency domain. Following the advent of phase plane methods for second -order systems and the second method of Lyapunov, the Popov method gained popularity. This investigation attempts to apply an extension of Popov's method to determine the transient stability of a single machine infinite bus system under a large transient disturbance. Circle criterion provides simple rules for examining both the linear and non-linear portions of the power system in the frequency domain and for arriving at conditions to be imposed on these portions in order to ensure absolute stability. An upper and lower bound is established for the non-linear gain by treating the system's mathematical model as a set of non-linear algebraic equations and limiting values of gains are estimated by solving these equations. A numerical example is solved to illus...

An analysis based on weighted linear approximations is developed for certain non-linear ordinary differential equations, and is applied to the description of singular points and trajectories in the phase plane. This technique is compared with the analysis of Poincare based on Taylor series linearization where the latter is applicable. Numerical results are obtained for a quadratic friction problem, and compared with the first-order Krylov-Bogoliubov-Mitropolsky result. The new technique is also successfully applied to problems of Lienard, Keil and Malgarini, and to a special case of a non-isolated singular point, where Taylor series linearizations are inapplicable or yield qualitatively incorrect results.

The phase motion in cyclic accelerators in the presence of a proper field of a bunch is investigated. Self-consistent stationary distributions of the density and the corresponding distributions of the potential are determined. A general expression for the charge limit is obtained, which does not depend on the distribution of particles in the phase plane. It is shown that even in the absence of an external rf field it is possible to have a steady self-phased state due to the action of the proper field.

A procedure for the analysis of the trigger sensitivity and of the switching speed of flip-flops of second order is developed. In particular the phase plane topology is investigated and the transition time evaluated. The method is applied first to a system with the simplest possible differential equation, then extended to a point contact emitter input transistor flip-flop.

This paper deals with the concentration and temperature profiles in a non-ideal stirred continuous reactor with two feeds. The conditions for analogy between the C and T profiles are discussed theoretically. It is emphasized that not only an analogy in the fundamental balance equations but also an analogy in boundary and initial conditions should be achieved to obtain an analogy in C-T profiles. The separation of feed streams appears to reduce considerably the number of cases for which the analogy of C-T profiles holds. Further, some of the results of local simultaneous measurements of C and T in a non-ideal stirred continuous reactor with two feeds are presented. The method of measurement and the reactor vessel are described and the experimental programme is outlined. Photographs of the T-α phase plane on an oscilloscope screen are shown and the results obtained from these measurements are summarized.

The damped sinusoidal oscillation of glycolytic intermediates of S. carlsbergensis (ATCC 4228) shows a close synchrony of FDP and RPN kinetics measured fluorometrically. A crossover point of the oscillatory metabolite pattern is found between G-6-P (F-6-P) and FDP identifying PFK as the control site. The phase relations of G-6-P and FDP are of especial interest; an open figure in the phase plane plot is obtained during maximal oscillations; the figure is more nearly closed as the oscillations subside. A reaction mechanism involving activation of PFK by one of its reaction products is proposed as the basic oscillator mechanism.

The direct method of Liapunov is applied to the problem of power system stability. The method is first compared with the phase plane technique and equal-area criterion, with all three approaches giving identical results for the equivalent 1-machine system. A Liapunov function for the 3-machine system is developed, and conservative prediction of stability is possible if the machine velocities and angles are known at the time of final system disturbance. The method is easily extended to multimachine systems. Further work is needed in the area of easier determination of better Liapunov functions and the inclusion of such items as resistance, regulator action, governor action, and damping torques.

A simple ?minimum-fuel,? digital attitude control system requiring no rate gyros is described; the system is applicable to practically any type of orbiting space vehicle. An inertial guidance computer furnishes a quantized attitude angle to the control system. With this attitude information and a measure of time, the control computer accurately computes the vehicle's average rate. A novel switching zone arrangement in the phase plane results in decisions controlling on-off reaction jets that adjust the vehicle's attitude. Jet selection and duration of the jets' operation are determined as a function of the vehicle's angular position and average rate in relation to the switching zones in the phase plane. The system operates so that computed rates are very accurate when accurate rates are required and less accurate when accuracy is not important. The end result is a system that converges to a low-rate limit cycle while consuming near-minimum fuel. The parameters controlling the switching logic are prestored memory constants which are electronically loadable and changeable. This flexibility makes the system compatible with a wide variety of payloads and missions.