Sampled-data Control System Design Using Reverse Element

In this paper, the authors deal with one method for the control of sampled-data control systems. In contrast with ordinary sampled-data control systems, the polarity of the control signal is reversed at several instants in every sampling period. By deciding the instants properly, the finite settling time response can be obtained. Especially, for a step input, it is shown that the system error can be reduced to zero in one sampling period, irrespective of the order of the controlled element. Furthermore, the compensator is simpler than that for ordinary sampled-data control systems, because it consists only of a sampler, a hold element, and a reverse element which reverses the polarity of the control signal. Even for a controlled element with a symmetric saturation characteristic, the settling time has a finite value. Moreover, it is shown that the reverse element can be used along with an ordinary compensator containing delay elements, and that the settling time can be made shorter. The response for a random input is also analyzed for a typical sampled-data control system with a reverse element

Approximate Solution of Mathieu's Differential Equation

This paper presents a method for the approximate solution of the differential equations of the Mathieu-Hill type. This method is based on the analytical method of the Periodically Interrupted Electric Circuits. The second-order differential equations with periodic coefficients, considered in this paper, are represented by the general form : d²y/dz²+f(z)y=0, where f(z) is a single-valued periodic function of fundamental period Zᵀ, when f(z)=a+16q cos 2z, it is known as Mathieu's differential equation. Based on the procedure in this paper, the periodic function f(z) is subdivided into m functions, f₁(z), ··· , fᵣ(z), ··· , fₘ(z), each of which has a different interval zᵣ, (r=l, 2, ···, m) for one period zᴛ of f(z). Namely the function fᵣ(z) represents the linear approximation of f(z) in each interval, that is, fᵣ(z)=2cz+d, 0≦z≦zᵣ, (r=1, 2, ···, m) where the values of c and d are constant. From this practical linear approximation, the present method is adequate for the determination of the approximate solution of the differential equations of the Mathieu-Hill type and this method has certain advantages, especially for the stability of the solution and also the transient solution. The stability chart for Mathieu's differential equation is obtained and plotted for the ranges of -3≦a≦34 and 0≦q≦2. This result is very well coincident with Ince's numerical one computed for the range of q=0 to 5.0. The obtained solutions and their numerical results may be extensively accurate. And the procedure considered in this paper is useful for the mathematical analysis of a large class of physical problems

Analysis of the Harmonic Producer Circuits

This study introduces a mathematical method for the analysis of the harmonic producer circuits with a ferromagnetic saturable core coil, taking the effect of magnetic hysteresis into consideration. This method is based on the theorems of the Periodically Interrupted Electric Circuits of the Third Genus, and on the application of the Digital Computer (KDC-1). First we describe in general how to apply this analytical method to the basic circuit of harmonic producers to clarify its behaviour as accurately as possible, and next to the harmonic producer circuit containing an additional capacitor. One numerical example is presented to show the performance of the harmonic producer circuit in this paper

An Approximate Analysis of the Transient Stability of One- or Two-Machine Systems

In this paper, first, we correct Hano's approximate analysis of power system stability in which he has neglected to consider the initial angular displacement and velocity of machines when he solved his approximate differential equation of angular motion of one or two machines in power systems. Next we propose better procedures, based on approximating the trigonometric function in the original nonlinear differential equation of angular motion of the machines by more appropriate triangles than Hano's, or by trapezoids. Then developing these approximate procedure, we derive a sort of stability criterion of one- or two- machine systems, the simple formulae for the critical switching time and so on, when the circuit breakers are reclosed or not reclosed after the fault has been cleared. At last, comparing the calculated results of some transient stability problems by the approximate procedures with those by the conventional step-by-step method, we ascertain that the approximate analysis of system stability, especially the trapzoid-approximation is a good approximate analysis of system stability

Low-frequency Oscillation of a Bounded Plasma in an External Magnetic Field

Low-frequency plasma waves with non-axisymmetric modes are generally analyzed under the existence of an ion beam. A boundary condition, that a cylindrical plasma is coaxially immersed in a cylindrical current sheet with a vacuum clearance, is taken into account. Oscillating electromagnetic field and electron and ion motions associated with the waves are also examined. Natural oscillations with a free boundary and cylindrical plasma waves surrounded by a conducting cylinder are found to exist. Ion cyclotron waves of axisymmetric modes found by Stix can be derived as a special case of these modes

An Analysis of Non-Linear Sampled-Data Feedback Control Systems

Higher order sampled-data feedback systems which contain a saturating element or a backlash element are investigated in this paper. This study introduces a new approach to the analysis of non-linear sampled-data control systems. At first the authors describe a new analytical method for such systems using the theorem of Periodically Interrupted Electric Circuits and how to apply the Digital Computer (KDC-1) to this theorem. The method presented here can be applied to any higher order systems with any non-linear elements by making use of the digtal computer simulation of the above theorem and the non-linear element. Some illustrative examples are given to clarify the method involved. One example of the third order sampled-data feedback system with a saturating element is investigated in case of initial conditions being given without external forces, a unit step function and sine wave inputs being given. The examples show that in the case of a step input as well as initial conditions existing, slight variations of initial values result in difierent modes of periodic oscillations, while in the case of a sinusoidal input, a slight modification of non-linear characteristics results in forced oscillations in one case and in sub-harmynic oscillations in another. Two illustrative examples of the second order system with a backlash element are considered in the case where the linear system is followed by the backlash or follows the backlash. Some results obtained by numerical computations are presented to show the performance of the system dynamics on the basis of the new analytical method presented here

Analysis of Sampled-Data Feedback Control Systems with Finite Pulse Width

A method for analysis of sampled-data feedback control systems with finite pulse width is presented in this work. The analysis is based on introducing a new technique with some approximations by the theories of the Periodically Interrupted Electric Circuits. The results make it possible to obtain the transient and steady-state response of such systems containing a sampler with any pulse width. Furthermore, by utilizing this method, the stability criterion is readily developed and its application to systems under consideration make clear the effects of the sampler with finite pulse width on the performance of sampled-data feedback control systems, which are of considerable importance in view of its engineering applications. Some illustrative examples are given to clarify the method involved and its numerical results are presented to show the performance of the system dynamics. Moreover the results in this investigation are examined by the simulation of the control system in question by means of the analog computer

The Steady-State and Transient Characteristics of the Chopper-Modulated Circuit Having Four Circuit Modes

The problem considered in this paper is one, in general, of making clear the characteristic of chopper-modulated circuits from the theoretical point of view. The chopper-modulated circuit under consideration is assumed to have four circuit modes. In a previous investigation, we have already discussed a chopper-modulated circuit having common two circuit modes, and also have developed the valuable analytical method for a periodically interrupted electric circuit having two circuit modes excited by a complex sinusoidal input. Here we shall generally extend the foregoing analytical method for a periodically interrupted electric circuit having m circuit modes driven by a complex sine wave input and basing on this method, we shall discuss a chopper-modulated circuit having four circuit modes. Only a type of practical interest is treated, that is, the transformer coupled chopper-modulated circuit which is frequently used with conventional chopper amplifiers, and other types which are not considered here. The method considered in this paper is an available technique explained to make clear the steady-state and transient performance of these types