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.     

Robustness of absolute stability†

The stability of a class of single-input, single-output singularly perturbed systems formed by a linear time-invariant feedforward block with a sector bounded time varying feedback is considered. It is shown that if the reduced order ‘ slow ’ subsystem is absolutely stable and the parasitics are asymptotically stable and sufficiently fast then the full system is absolutely stable. Bounds on the singular perturbation parameter for uniform asymptotic stability and absolute stability are obtained.     

A criterion for the bounded-input bounded-output stability of a discrete-data system with a slope-restricted non-linearity†

This paper presents and proves a criterion for the bounded-input bounded-output (BIBO) stability of a class of discrete-data systems with a slope-restricted non-linearity. The proof is based on the principle of contraction mapping. A numerical example is given to illustrate the application of the criterion, and to discuss its implication in relation to some existing results.     

On the stability of a class of non-linear continuous systems†

A procedure to find the existence and domain of asymptotic stability for a class of systems described by a set of non-linear partial differential equations is presented in the context of semigroup theory.

No a priori bound is imposed on the non-linearity which is assumed to be a quad ratic (bilinear) or multilinear form.

The linearized system can be non-self-adjoint and need not be asymptotically stable equation by equation.

The solution of the stability problem is carried out. in two steps :

(a) the development of an L₂-like norm that bounds above the sup norm, a requirement to obtain a useful estimate of the non-linear term, and

(a)the computation of an estimate of the L₂-likc norm of the semigroup generated by the linearized system in terms of its usual L₂ norm.     

Global stability and performance of a simplified adaptive algorithm†

Most self-tuning and adaptive control algorithms usually use reference models, controllers, or identifiers of about the same order as the controlled plant. Since the dimension of the plants in the real world may be very large or unknown, implementation of adaptive control procedures may be difficult, or sometimes impossible. In this paper we prove global stability for a simple adaptive algorithm that can use low-order model reference and controllers, since no observers or identifiers are used in the adaptation process. The algorithm is basically fitted for systems that are denominated as ‘almost positive real’. It is shown that, at the price of bounded rather than vanishing output tracking errors, the simple algorithm can be applied in systems that can be stabilized via constant output feedback. These procedures are believed to reduce considerably the effort required for implementation of adaptive control in practical applications, especially in multivariable large-scale systems.     

On the stability of control systems with randomly time-varying characteristics†

The purpose of this paper is to study the stability conditions of randomly time-varying control systems.

First we define the stability, in the stoehastic sense, of the control system with randomly time-varying characteristics and secondly we obtain the conditions for global stability in the mean for a control system which is governed in each time interval by two different differential equations with probability p and 1-q respectively.

Finally we examine the stability condition for the concrete control systems by experiments of the Monte Carlo method.     

Domain of stability for a class of non-linear partial differential equations†

The domain of stability in the weak topology of a parabolic partial differential equation with negative feedbacks in two independent variables is determined explicitly in terms of the Weierstrass p-function. Locations of singularities are found for the equianharmonic case.     

On the stability of interconnected systems†

This paper investigates the stability of systems which can be regarded as composed of interconnected subsystems. A sufficient condition for inputs-output L ^p stability, in terms of the L ^p gains of the subsystems and their interconnections is derived. For the case of L ² stability, it is compared with other criteria for asymptotic stability, obtained by Lyapunov techniques, and shown to give better results for a certain class of systems.     

The computation of stability regions for systems with many singular points†

A method is presented for the generation of Liapunov functions for the analysis of non-linear systems with many singular points. Criteria are given for non-oscillatory behaviour, Stability regions for all stable equilibria can be found, using a single Liapunov function of the proposed type.     

Connective stability of large-scale stochastic systems†

In this work, we will introduce the notion of connective stability in the mean for large-scale stochastic systems under structural perturbations. Sufficient conditions for this kind of stability are obtained by the decomposition-aggregation method in the context of vector Liapuiiov functions.     

Effects of controller dynamics on the stability of a class of optimal control systems†

In this paper, the effects of controller dynamics on the stability characteristics of a class of multivariate optimal control systems are investigated. The results of this investigation indicate that the asymptotic stability of systems of this class may well be lost because of the practical impossibility of implementing control laws derived on the basis of the classical optimal control theory of Kalman (1960),     

Non-asymptotic stability of complex valued differential systems†

Sufficient conditions for the existence of orbital motion and non-asymptotic, non-orbital motion in the solutions of constant coefficient complex variable differential equations are obtained through Liapunov's direct method. Matrix analogues of those conditions assure the corresponding type of motion of interconnected systems whose complex coefficient matrices are simultaneously diagonalizable.     

Comparison of numerical methods in stability analysis†

The results of a numerical investigation of a number of techniques for stability analysis of non-linear systems are presented. The methods compared are the Ingwerson and Zubov procedures for construction of Liapounov functions, a number of methods for obtaining optimum Liapounov functions, tracking functions, the alternating extreme radius path, and the method of Luus and Lapidus.     

L2-bounded stability for a stochastic system†

Both types of linear and non-linear systems with random coefficients are treated in this paper. Four theorems are proved by using the stochastic Liapunov functional, which give the sufficient condition in matrix form for exponential asymptotic stability in the largo with probability one and L₂-bounded stability with probability one.     

A generalized stability criterion for a class of second-order non-linear control systems†

A condition for the existence of a limit cycle for a class of second-order non-linear control systems with a type zero linear plant is derived. The limit cycle stability is assured when F(a, b, α, β)/E curve has a positive slope at the point of intersection of F(a, b, α, β) and K loci. A typical second-order non-linear system is analysed for the purpose of illustrating the method of finding the amplitude and frequency of sustained oscillation. The method of studying the effects of variation of K, α and β on the amplitude and frequency of oscillation of the system is indicated. With regard to this, the proposed method has certain advantages over the usual method of analysis.     

On the asymptotic stability of a particular class of distributed parameter feedback control systems†

In this paper the stability problem for a particular class of non-linear feedback control systems governed by a parabolic partial differential equation is considered. A sufficient condition for global asymptotic stability of the null equilibrium state is derived by the method of comparison functions.     

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.     

Stability of a Hysteretic System with Random Signal Inputs†

The stability of a feedback system with a hysteretie relay controller, when forced by a Gaussian signal, is analysed by applying the quasi-linearization approach.A method for computing the sinusoidal and random signal gains of a non-linear element, based on the evaluation of the input—output cross power of the clement, is first explained. Knowledge of these two gains for an ideal relay is then used to find the corresponding gains for a hysteretic relay. These gains are finally used for analysing the effects of both high-frequency and low-frequency random signals on the auto-oscillations of a second-order system with the hysteretic relay. Results of simulator studies are also presented.     

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.