Boolean functions of an odd number of variables with maximum algebraic immunity

In this paper, we study Boolean functions of an odd number of variables with maximum algebraic immunity. We identify three classes of such functions, and give some necessary conditions of such functions, which help to examine whether a Boolean function of an odd number of variables has the maximum algebraic immunity. Further, some necessary conditions for such functions to have also higher nonlinearity are proposed, and a class of these functions are also obtained. Finally, we present a sufficient and necessary condition for Boolean functions of an odd number of variables to achieve maximum algebraic immunity and to be also 1-resilient.     

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.     

Simplification of z -transfer functions by continued fractions †

A computer-oriented procedure for the simplification of a z -transfer function is presented. The method consists of (1) transformation of the z-transfer function into the w domain by the bilinear transformation, w=(z? l)/)z+ 1), (2) continued fraction expansion of the w -transfer function into the Cauer second form, (3) keeping the first several quotients and discarding others, (4) converting the truncated continued fraction into z-transfer function of low order. An example is used to illustrate the rapid rate of convergence.     

Euler-Lagrange conditions and estimation of states for a discrete-time linear system with time-delay†

In this paper, the Euler-Lagrange conditions for the extremization of a performance functional for a discrete time linear system with delay fire derived and this is then used for the estimation of states for the same system. The estimation problem is represented as a Two Point Boundary Value (TPBV) problem which is then solved numerically by using the steepest descent technique.     

Design of dominant-type control systems with large parameter variations†

This paper presents a design procedure for dominant-type systems with large plant parameter variations. The principal contribution is that a fourth-order approximation is used in the dominant region instead of a third-order, which up to now had been the most advanced method. The s-domain specifications of the system are assumed to be in the form of an acceptable dominant closed-loop pole region with bounds on the location of the ‘far-off’ closed-loop poles. The design philosophy is to place compensation zeros within the acceptable dominant closed-loop pole region such that the dominant closed-loop poles remain within their prescribed region despite the large variation in the plant parameters. The design procedure is for plants with simultaneous independent variation in the gain factor and a pair of poles. The design is such as to minimize the sensitivity of the system to internal noise.     

On approximating the step response of second order all-pole transfer functions by the step-response of second-order transfer functions with two poles and a negative real zero†

The geometry and application of K-functions is studied which provide a generalization of the Liapunov direct method in stability theory to allow the control of motion. A sufficiency theorem is proved that provides a means for quantitative design and investigation of dynamical systems.     

Step response of a class of second-order non-minimum phase transfer functions†

This paper investigates the problem of controlling a discrete-time linear system with jump parameters. A review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems. Dynamic programming has been applied by many researchers and it was observed that no closed-form analytical solution could be constructed because of the ‘dual’ aspects of the controller. The main contribution of the present work is an algorithm, suitable for computer implementation, for the optimal dual control. The construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite-dimensional optimization procedure to obtain the optimal control as a function of the statistics. This is achieved by first developing a suitable recursive realization of a ‘filter’ which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations. Several examples are presented to illustrate the use of the algorithm for constructing optimal and suboptimal controllers.     

A transformation of non-linear dynamical systems with a single singular singularly perturbed differential equation†

Singularly perturbed state differential equations of the form [xdot] = f(x, z, t, ?), x(t₀, ?) = x₀(?); μ(?)? = g(x, z, t, ?), z(t₀ ?) = z₀(?) with lim μ(?) = 0; ?, μ > 0 are considered, where the nominal equation 0 = g(x, z, t, 0)^{? → ∞} does not have to be solvable for z. A fairly general transformation of the above system into a form [xdot]^* = f ^*(x^*, z, t; z(1),...,z^(d?1), ? ); μ^*(?)z^(d)= g^*(x^*. z(⁰),...z^(d?1), t; ?), with dim x^* = dim x ?(d ? 1), d ? 1 is proposed. The transformed system stands a better chance of being analysed by existing methods (especially by those proposed by Hoppensteadt (1971) and Hoppensteadt and Mi ranker (1976)) than the original singular singularly perturbed form. Informative examples are presented.     

Stabilization of a relay system containing three-position element and dead time†

An exact method of analysis of non-linear systems containing a three-position relay element and a dead-time element is given. Such a system shows two limit cycles, an unstable limit cycle and a stable limit cycle. Expressions are derived to evaluate the period and amplitude of the oscillations.

Using the phase-plane concept it is shown that the given system can be stabilized by adding a zero to the second-order transfer function of the rational part.     

Estimation of states and parameters for a discrete-time system using quasilinearization†

An attempt is made here to demonstrate the usefulness of quasilinearization technique for the estimation of states and parameters of a discrete-time system. The estimation problem is viewed as a least-squared-errors problem which is then transformed into a Two Point Boundary Value (TPBV) problem by the use of the Discrete Maximum Principle. The TPBV problem is then solved by the quasilinearization technique.     

Non-linear transformations of stochastic processes associated with Fourier transforms of band-limited positive functions†

The passage of Gaussian, processes (both stationary and non-stationary) through a device characterized by a convolution transform is studied. The output of such a device is a non-stationary stochastic process of harmonizable type. This process along with its finite Fourier transform are examined in detail. Some sampling theorems are also stated. A stochastic series representation in terms of prolate spheroidal wavefunctions is derived.     

A method of determining the crosscorrelation functions for a class of non-linear systems†

A use is made of the method of finding output autocorrelation functions of continuous time-invariant nonlinear systems, when they are represented by their functional expansion. This enables the input-output crosscorrelation functions to be calculated by the use of multi-dimensional Laplace transforms.

A simple example for a pseudo random binary sequence input is presented.     

Convergence of certain training algorithms for adaptive threshold elements with imperfect weight components†

A proof of the convergence of an adaptive algorithm based on the concept of characteristic vector is presented. It is shown that the adaptive algorithm will converge even when imperfections existing in the components of the ATE and the imperfections are within certain finite limits.     

Performance indices for closed-loop linear systems optimally controlled by a single input variable†

In this paper an explicit expression for a general performance index associated with single-input optimal closed-loop systems is derived. Several uses and implications of this expression are discussed.     

On the transient solution of the unforced duffing equation with large damping†

The refined elliptic-function method previously developed (Soudack and Barkham 1970) is applied to the unforced Duffing equation with large damping. The theoretical development for this case, and an example, are presented. The Kryloff-Bogoliuboff solution is shown for comparison with the present result.

The advantage of this approach, well illustrated by the example, is in its ability to predict the solution phase. The approach is therefore superior to quasi-linear methods, such as the Kryloff-Bogoliuboff method, which rapidly fails to give a good approximation due to phase errors.     

A method of obtaining the dual-input describing functions for a periodic input†

A combined analytic and graphic procedure is described for obtaining the dual-input describing functions for a class of non-linear elements subjected to a periodic input. The procedure is illustrated by considering the non-linearity to be a simple contactor involving hysteresis and the describing functions obtained for the element are presented.     

Making a polynomial Hurwitz-invariant by choice of feedback gains†

An integrated design procedure is developed for a modified Smith predictor and associated controller for linear time-delay systems having transfer functions of the form k ₁, A exp (—sT)/B, where A and B are monic polynomials in s of degree n — l and n, respectively. A is Hurwitz and B has a single right-half-plane root at s = λ. For l=1,2,3, an augmented PI controller guarantees asymptotic stability for λT less than an l-dependent limit. The procedure for l = 3 is extended to l = 4 with the introduction of derivative action into the controller. Design arguments are on root locus topology, and on Nyquist analysis applied to an auxiliary system.     

Synthesis of linear digital controllers using dead-beat observers for systems with inaccessible states†

A procedure is given to synthesize linear digital controllers for uniformly observable linear time-varying discrete data systems where some of the state variables are not available for feedback. Dead-beat observer theory is used and the controller generates the exact optimal control law in at most q?1 sampling periods, where q is the observability index of the system.     

Closed-form approximation of the periodic response of a system with a saturating non-linearity†

A technique for approximating harmonic, periodic solutions of a class of non-linear equations involving u saturation non-linearity is applied to a fourth-order equation which describes an electrical circuit containing a non-linear amplifying element. The computational effort required to obtain the closed-form approximation which this method yields is compared with that required for a numerical solution by the conventional methods of quasilinearization and patching. The various methods use comparable amounts of computer time, but the other classical methods do not offer the advantage of the closed-form approximation which our technique provides.     

Minimal-time control of linear-distributed systems with a composite-norm constraint†

This paper deals with the time-optimal control of a class of linear-distributed systems with a composite-norm constraint. The method of i-theory of moments is used to obtain the solution. This gives rise to two auxiliary problems: (i) an infinite set of Fredholm integral equations, and (ii)a minimization problem in the dual space.