On the general theory of optimal processes†

A previous paper treating the problem of optimal control of systems whoso control signals belong to a pre-specified class of functions is discussed. The problem is re-examined for a broader class of systems and the results are formulated in a maximum principle of the Pontryagin type. The maximum principle states that a necessary condition for the optimal control is that the first variation of the time integral of the Hamiltonian is zero if the control function can vary in two opposite directions, is non-positive if the control function can vary in only one direction. The free end-point and the fixed end-point problems as well as the variable end-point problems are considered. The admissible controls are assumed to be piecewise continuous and bounded. The requirement of operative convexity as imposed in the related paper is lifted.     

On the cardinal number of complete sets of Boolean operations†

It is shown in the literature (using the Post—Yablonsky theorem) that a complete set of Boolean operations cannot have a cardinal number greater than four. It is the object of this paper to improve this bound and prove that a complete set can have a cardinal number of at most three or, in other words, there does not exist a complete (non–redundant) set of more than three Boolean operations. The proof given here is constructive, using the Post—Yablonsky theorem, truth tables and combinatorial set theory.     

Application of bifrequency phase equivalent reduction to system synthesis†

The paper discusses a method to synthesize high–order linear systems with the gain cross–over frequency (ω_eg), the phase cross-over frequency (ω_cp)and the phase margin (φm) as the given specifications, using the bifrequency phase equivalent reduction method.     

Application of pseudolinear partitioned filter to passive vehicle tracking†

The partitioned estimation method is applied to the bearing-only target tracking problem. A pseudolinear partitioned tracking filter is developed initially in the form of recursive processing. It is then shown that such a tracking filter consists mainly of two parts, a ‘ manoeuvre predictor ’, driven by the deterministic own-sensor manoeuvre input, and a ‘ psuedolinear partitioning fixed-point smoother ’, which gives the initial position and speed estimates. Furthermore, by taking into consideration the parallel processing mechanism, a pseudolinear partitioned tracking filter with data compression is proposed to average the bearing data contaminated by the measurement noise. The parametric relationship between r.m.s. estimation error, data compressing (or renovating) interval, measurement noise level, sensor manoeuvre structure and initial range estimate is presented through Monte Carlo simulations.     

Application of generalized Zubov's method to power system stability†

The generalized form of Zubov's partial differential equation (Szego 1962) is written using the dynamical equations of a power system. Then a Lyapunov function which satisfies the partial differential equation is obtained by using a transformation of state variables. The V function obtained is used to describe the region of stability. The application of the method to the power system stability problem is illustrated by considering a synchronous generator connected to an infinite bus, Three examples, using three different models for the synchronous machine, are given.     

Application of an operational calculus to certain time-varying systems†

This paper develops an operational calculus for the operator B_μ = t?μDt^1+μD with — 1 < μ<∞ along the approach of Mikusinski. The calculus is applied to the solution of certain time-varying networks and a boundary-value problem.     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

High-order eigenproblem sensitivity methods: theory and application to the design of linear dynamical systems†

In a previous paper (Crossley and Porter 1969) simple and explicit derivations were given for the first-order eigenvalue and eigenvector sensitivity coefficients for the Fundamental eigenproblem associated with the behaviour of linear dynamical systems governed by equations of the form [xdot] = Ax. In the present paper, similar techniques are used to derive the corresponding second-order eigenvalue and eigenvector sensitivity coefficients. The second-order theory is then applied to two examples in order to compare the results obtainable from this theory with the corresponding results obtainable from the first-order theory and from exact calculations.     

Application of an approximate time delay to a Posicast control system†

The performance of a method of producing relatively long time delays, using truncated Fourier series expansion, applied to a Posicast control system was investigated. The controlled system had a long natural period and was lightly damped. Both the Posicast control and the system were simulated on an analogue computer.

The response of the system to step inputs with Posicast showed dramatic improvement over the uncompensated system. The overall system approached the theoretical response of an ideal Posicast system. A series of runs was made with successively cruder approximations of the delay function. Response was not seriously degraded until the frequency of the truncated terms became comparable to the upper 3 dB frequency of the controlled system.

Other waveforms were tested, including sine, square, sawtooth and quasi-random signals. In all cases, the performance of the system was well behaved. The approximate method of delay, therefore, offers a means of implementing Posicast control with systems requiring relatively long delays of continuously variable inputs.     

Application of state space statistical linearization to optimal stochastic control of non-linear systems†

The formulas for different methods of state space version of statistical linearization are given and applied to sub-optimal stochastic control of non-linear systems, as recently suggested by Wonham and Cashman (1968). Comparison with their optimal results shows that there is no immediate reason to prefer one linearization method to another. Especially the conjecture of Kazakov does not hold in general, i.e. the mean coefficients of two different linearization methods must provide best results.     

Sampled-data theory applied to the modelling and control analysis of compression ignition engines—Part I†

Proposals for the use of electronic speed governing of diesel engines used for driving alternators has led to a re-examination of the dynamic behaviour of the engine for control purposes. If maximum advantage to the plant performance is to be achieved from using more complex and expensive governing devices, then better models of individual components in the power system for simulation and control studies may be necessary. This paper describes an investigation performed to model the engine in terms of a discrete system. The frequency domain properties of the model (for various numbers of cylinders) are examined and compared with the model usually assumed, as well as with the frequency response of an actual engine. A companion paper will extend this work using 2-transform techniques.     

Application of a modified quasilinearization technique to totally singular optimal control problems†

This paper presents an algorithm for the computation of totally singular optimal control problems. The algorithm is an extension of a modified quasilinearization technique which was originally developed for the solution of non-linear two-point boundary-value problems. A numerical example is included for illustration.     

Application of computational invariant theory to Kobayashi hyperbolicity and to Green–Griffiths algebraic degeneracy

A major unsolved problem (according to Demailly (1997)) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the source. The underlying group action is not reductive, but we provide a complete algorithm to generate all invariants, in arbitrary dimension n$n$ and for jets of arbitrary order k$k$.     

Asymptotic stability and boundedness of non-linear difference equations†

The problem of asymptotic stability and boundedness of non-linear difference equations is considered, In this paper we shall show some results which guarantee asymptotic stability for the trivial solution. Wo wish to present results which provide explicit regions of asymptotic stability. The results replace these words, 1 sufficiently small', by explicit estimates which are simple.

The concept of asymptotic stability to difference equations can be quite useful in the design of system response to arbitrary initial conditions.     

Distributional approach to the analysis of linear time-varying systems†

This paper is devoted to the analysis of linear time-varying systems using distribution theory. The technique is developed to obtain the output given the input as well as to obtain the impulse response from the knowledge of the output and input. All quantities are considered time varying. The basic approach used converts the integral equation into a distributional differential equation.     

A computational technique for the control of systems described by partial difference equations†

A penalty function method is applied to the determination of optimal controls for systems described by partial difference equations. The use of this penalty function method known as the ?-technique allows us to incorporate the dynamical constraints of the distributed parameter system into a new objective function. The conditions for the solution which minimizes the new objective function and the interrelationships between this solution and the solution which minimizes the original objective function are given. These interrelationships also suggest computational schemes for the synthesis of optimal controls for linear and non-linear distributed systems.     

Approximating reachable sets for a class of linear control systems†

We present a technique for overestimating the reachable set from the origin for a class of n-dimensional linear control systems. The proposed ‘box’ method is based upon decomposing the system into one and two-dimensional subsystems for which bounds on the new variables can readily be found. Using these bounds enables the construction of a n-dimensional parallelepiped containing the reachable set of the original system. Examples of this procedure are given as well as a comparison to an overapproximation afforded by a Lyapunov approach.     

A New Approach to the Analysis of Non-recurrent Ladder Networks†

A new approach to the analysis of the non-recurrent ladder network is presented. Using the converse of the conventional specification for the series and shunt branches, a rapid method of analysis is evolved. The rules governing the combination of the series admittances and shunt impedances are shown to be of an extremely simple nature and easier to apply than those pertaining to the conventional specification. In addition, unlike in the latter case, the same rules apply to both voltage ratio and transfer impedance.