Study of non-linear differential systems through transformation techniques†

A study of a certain type of non-linear differential equation of 1st and 2nd order has been made. It is shown that these equations are reduceable to solvable form through the transformation techniques, provided the coefficients occurring in the differential equations satisfy some connective conditions.     

Asymptotic observers for non-linear systems†

Observer design for non-linear systems is discussed. Some recent approaches are based on state and output change of coordinates to transform a non-linear system into a particular observer form, from which an asymptotic observer can be designed ensuring the asymptotic stability of the error dynamics in the new coordinates. In this paper, the stability properties of the error dynamics are studied in the original coordinates. With some examples, it is shown how the asymptotic stability in the new coordinates does not imply, in general, the asymptotic stability in the original ones. Some general results are stated and proved to guarantee the asymptotic stability of the error dynamics in the original coordinates.     

Diagonalization and inverses for non-linear systems†

The problem of decoupling and inverting dynamic systems is considered. Attention is focused on the problems that arise as available linear techniques ore applied to nonlinear systems. The decoupling of non-linear multi-variate differential equations is illustrated in a series of three examples.     

Synthesis of control policies for manufacturing systems using eigenvalue-assignment techniques †

In this paper, eigenvalue–assignment techniques are used to synthesize control policies for a manufacturing system. This system is modelled as a diserete–time multivariate dynamical system with a disturbance input duo to sales. The state variables of the system are the rotes of flow of parts at the various work stations, the backlogs of parts, and the inventory level of the finished product. The control inputs ore the man–hours–per–week required for the various work processes involved in the manufacturing system.     

Transformation groups for weakly non-linear periodic systems†

The method of infinitesimal transformations is applied to weakly non-linear periodic systems described by n first-order coupled differential equations. The transformation groups, which leave the equations invariant, are determined by a perturbation method, Those transformations roduce the systems to non-linear autonomous form. The equivalence of the results of the method, within the first-order approximation, to other averaging and perturbation techniques is illustrated by an example of a second-order system.     

Infinitesimal transformations for non-linear time varying systems†

In this paper, Lie's method of infinitesimal transformations is used to generate some general classes of second-order non-linear non-autonomous differential equations which can be reduced to first-order systems. The relevance of some recently published, transformation methods to this method is established. Although the discussion is restricted to second-order systems, the method can be used for higher-order systems. Some special cases, where the differential equations can be completely integrated, are discussed.     

Adaptive non-linear Bayesian estimation for bounded-noise systems†

An adaptive non-linear filtering algorithm is developed for handling linear, time-invariant, discrete, dynamical systems with bounded input and measurement noise disturbances. The new result yields the conditional-mean state estimate by combining the Bayesian decision rule, a reachable set concept and a technique of approximating probability density functions by their moments. The adaptive feature is needed because only the input noise bounds, rather than the complete input density function, are assumed to be known. Numerical comparisons on a second-order system are presented to demonstrate that the non-linear filtering algorithm adapts itself to yield considerably more accurate results than those obtained by applying the best linear filter.     

Stabilization of non-linear systems†

The problem of stabilizing a discrete-time non-linear system is considered. For a rather large class of common stabilizable non-linear systems, a procedure leading to the stabilization of a given non-linear system Σ belonging to that class is derived. In this procedure, a pair of compensators is constructed, consisting of a precompensator and an output feedback compensator, which, when connected in closed loop around the system Σ, yield a closed-loop system that is internally stable for bounded input sequences. The procedure allows the construction of infinitely many different pairs of such compensators, thus facilitating the choice of a convenient one.     

The stochastic observability for noisy non-linear stochastic systems†

The purpose of this paper is to present sufficient conditions for the stochastic observability of noisy non-linear systems. First, the stochastic observability is defined and then an estimation algorithm of the system state is constructed for the given noisy non-linear stochastic system. Secondly, by using a stochastic Lyapunov-like approach, sufficient conditions of the stochastic observability are shown for non-linear systems with state dependent noise and also with state independent noise. Finally, for the purpose of supporting the theoretical aspects developed here, results of digital simulation studies are demonstrated.     

On the state estimation for non-linear dynamic systems †

The object of this paper is to present an approximate technique for state estimation of non-linear dynamical systems under noisy observations. The conditional cumulant is introduced, by which the conditional probability density can be characterized. A set of dynamical equations satisfied by conditional cumulants is derived, and an approximate method is proposed for computing the cumulants. The relation of the cumulant method to the stochastic linearization technique is also discussed. Finally the state estimation problem for linear stochaatic system with state-dependent disturbance is solved to illustrate the use of the Gaussian approximation.     

Stability criteria for a class of non-linear systems†

A graphical method for investigating the absolute stability of a particular class of systems with many non-linearities is given. The technique is an extension to previous results for the class of systems.     

State-feedback control of non-linear systems†

A design method for state-feedback controllers for single-input non-linear systems is proposed. The method makes use of the transformations of the non-linear system into ‘controllable-like’ canonical forms. The resulting non-linear state feedback is designed in such a way that the eigenvalues of the linearized closed-loop model are invariant with respect to any constant operating point. The method constitutes an alternative approach to the design methodology recently proposed by Baumann and Rugh. Also a review of different transformation methods for non-linear systems is presented. An example and simulation results of different control strategies are provided to illustrate the design technique.     

Modified maximum principle for non-linear open-loop PFM control systems†

The problem of finding the optimum control function for pulse frequency modulated (PFM) control systems is considered in this paper. The modified maximum principle (MMP) is extended to cover non-linear open-loop systems. A method for obtaining the optimum control function is presented and illustrated by an example.     

Method of variation of parameters for overdamped non-linear systems†

A simple analytical method is developed to obtain the time response of second-order non–linear overdamped systems with a small non–linearity based on the Krylov— Bogoliubov method of variation of parameters. The solutions obtained for different initial conditions for a second–order system whose linear equation has real roots show good agreement with those obtained by numerical integration.     

Sequential estimation of states for non-linear lumped parameter systems†

This paper considers the sequential estimation of plants described by non-linear differential equations. The statistical nature of the disturbance being unknown, the problem is treated as an optimal control problem with least-squares criterion. It is seen that the criterion function satisfies the Bellman equation at terminal time, which is considered as a running variable. The sequential estimator equations are directly obtained by using Pearson's approximation solution.     

A general stability criterion for non-linear time-varying feedback systems†

In this paper a stability theorem is proved for systems with a non-linear time-varying feedback element. It involves a trade-off between restrictions on the non-linear characteristic, on its rate of variation, and on the transfer function of the forward path element. It clarifies the relationship between many existing frequency domain stability criteria, such as the Popov criterion, the Nyquist criterion, and the circle criteria.     

Stability criteria for control systems with one non-linear element†

Liapunov functions are used to develop frequency domain criteria for a class of systems in which the given non–linear characteristic is bounded in both sector and slope. The criteria include the special case when order of numerator and denominator is equal in the linear transfer functions of the class of systems.     

On optimization of non-linear stochastic systems†

Results of application of the perturbation approach along with stochastic approximations to the problem of optimizing non-linear stochastic systems subject to quadratic index are discussed. The perturbation approach is first introduced with the help of linear regulator systems. This is combined subsequently with statistical linearization in order to derive sub-optimal solutions for the problem of optimizing a non-linear system. Finally, a new stochastic approximation technique, that permits more accuracy than the linearization, is suggested for the non-linear problem. Numerical examples are presented to compare the results of the various approximation methods and also to illustrate the possible advantages of non-linearization.