Certain Class Of Nonlinear Systems Research Articles

Identification of bilinear systems

A new technique for the identification of general bilinear systems is presented. The model reference adaptive systems (MARS) technique together with the direct method of Liapunov are used to construct the identifier for the system parameters. The plant is assumed to be multivariable, continuous and all of the states are accessible for measurements. The proposed technique can be applied on a certain class of nonlinear systems in which the nonlinearities could be approximated by quadratic terms without a significant loss in the overall accuracy. The technique is used to identify the parameters of a binary distillation column.

Engineering applications of the Chow-Yorke algorithm

The Chow-Yorke algorithm is a scheme for developing homotopy methods that are globally convergent with probability one. Homotopy maps leading to globally convergent algorithms have been created for Brouwer fixed-point problems, certain classes of nonlinear systems of equations, the nonlinear complementarity problem, some nonlinear two-point boundary-value problems, and convex optimization problems. The Chow-Yorke algorithm has been successfully applied to a wide range of engineering problems, particularly those for which quasi-Newton and locally convergent iterative techniques are inadequate. Some of those engineering applications are surveyed here.

On Stabilization of Classes of Nonlinear Systems with State Observers

This paper investigates the stability of the overall closed-loop system which is composed of an originally unstable nonlinear system, a state observer and a linear gain state feedback controller. From the practical implementation point of view, only a reduced-order nonlinear state observer and a linear identity state observer are delt with. Therefore it is assumed that the observation equation is linear and the system is autonomous. Firstly, it is shown by utilizing a kind of separation properties that an unstable nonlinear system which can not be stabilized with a linear observer is able to be stabilized globally with a globally asymptotically stable reduced-order nonlinear observer. Secondly, it is pointed out that for certain classes of nonlinear systems, even a linear identity observer can be used to provide global asymptotic stability. Lyapunov methods are used for each proof, which are suggested to apply to obtaining more general stability criterion for a wider class of large-scale composite systems than presently available ones.

A Model for Reference Adaptive Autopilot Ships. Practical Results

This paper describes the application of Model Reference Adaptive Control (MRAS) to automatic steering of ships. The main advantages in this case are the simplified controller adjustment which yields safer operation and the decreased fuel cost. After discussion of the mathematical models of process and disturbances, criteria for optimal steering are defined. Algorithms are given for direct adaptation of the controller gains, applicable after set point changes, as well as for identification and adaptive state estimation, to be used when the input is constant. Solutions for applying MRAS to a certain class of non-linear systems are dealt with. Full-scale trials at sea and tests with a scale model in a towing tank are described. It is shown that the autopilot designed indeed has the desired properties. Fuel savings up to five percent in comparison to conventional PID control are demonstrated. These savings are mainly possible because of the adaptive state estimator.

Machines in a category

After a brief historical review of early attempts to apply category theory to place sequential machines and control systems in a unified framework, we present three contributions to the theory of machines in a category: the dynamical interpretation functor which relates the response of a machine to a sequence of inputs with the reachability map in the system category; an application of the Krull-Schmidt theorem to provide a parallel decomposition for machines with polynomial state-transition; and a generalized notion of Hankel matrix which enables one to present finiteness conditions for linear systems even when defined over noncommutative rings, and for certain classes of nonlinear systems characterized by adjoint functors.

Existence of regular synthesis for general control problems

The concept of regular synthesis has been introduced by Bolt’anski in his classical paper [I] (cf. also [2]) on the sufficiency of the Pontrjagin maximum principle for time-optimal control problems. It has been used as an assumption on a closed-loop control to generate open-loop optimal controls. Using the theory of subanalytic sets it has been proved in [3] that every normal linear system admits a regular time-optimal synthesis. Subsequently, in [S], Sussmann has been able to dispose of the normality condition and to extend the theorem to a certain class of nonlinear systems. All of the mentioned papers deal entirely with the time-optimal control problem for systems which are linear in the control, with polyhedral control domains. The present paper constitutes an extension towards optimal control problems with general performance criteria and general control domains. The abstract theorem proved in this paper is modelled after an important class of problems-linear-quadratic optimal control problems with linear control constraints. This problem will be dealt with in a forthcoming paper. An important requirement in Bolt’anski’s definition of regular synthesis which is followed in [3,4] as well as [8, 91 is that the optimal trajectories enter the switching surfaces (called cells) transversally. This requirement has to be dropped not only in the linear-quadratic optimal control problem but also in the linear time-optimal control problem with a control domain having a piecewise smooth curvilinear boundary, as the following example demonstrates: Consider the system

Canonical Realization of Bilinear Input/Output Maps

The idea of extending the comprehensive theory of state space realization for linear systems to certain classes of nonlinear systems has been the subject of much recent research in mathematical system theory. One such class are those systems with a bilinear input/output map, first considered by Kalman [Pattern recognition properties of multi-linear machines, IFAC Symposium on Technical and Biological Problems of Control, Yerevan, Armenian SSR, September, 1968] in 1968. Kalman’s paper raised many questions concerning the canonical realization of such an input/output map; in particular it demonstrated that the proposed realization procedure could lead to a nonreachable realization. More recently, Sontag and Rouchaleau [On discrete-time polynomial systems, J. Non-linear Analysis, Theory, Methods, and Appli-cations, 1 (1976), no. 1, pp. 55–64] have shown that the required concept to be applied in this case is “quasi-reachability,” i.e. that the state set of the realization is the closure (in the Zariski topology) of the reachable set. The present paper is a further contribution to this theory, providing a necessary and sufficient condition for quasi–reachability of a realization of a bilinear input/output map. Unexpectedly, this is given by the easily verifiable algebraic criterion of reachability of a corresponding linear system. This leads to a constructive procedure, analogous to the linear case, for reducing a realization of the bilinear map to a quasi–reachable one, thereby reducing the state dimension of the realization.

On weak non-linearity of models of physical systems

The introduction of the concept of weak non-linearity is motivated by the effort to determine a certain class of non-linear systems whose properties important in technical appliations coincide with those of asymptotically stable linear systems. A model of the phzsical system described by Eq. (1) is called weakly non-linear (quasilinear) if any two solutions $x_1(t), x_2(t)$ of Eq. (1) sarisfy Eq. (2). A model described by Eq. (1) is weakly non-linear if there exists a way of expressing the right-hand side $f(x,t)$ of this equation in the form (3) so that the inequality (8) is satisfied.

Gradient Based Model Reference Adaptive Autopilots for Ships

A new gradient based method for the design of stable model reference adaptive control (MRAC) systems is presented and applied in an adaptive autopilot for ships.The method is not restricted to linear plants and models, but the plant and the model may belong to a certain class of nonlinear systems. When the state variable filter concept is applied, only the plant output response has to be measured. The state variable filters can be modified in order to reduce biasing due to measurement noise.The MRAC system is proven to be asymptotically stable by the second method of Liapunov. Hybrid simulation results of the autopilot design are presented.

An identification procedure for discrete multivariable systems

A new procedure is presented for the identification of discrete linear multivariable systems. Using Lyapunov's direct method the asymptotic stability of the overall system is established when the inputs are sufficiently general. It is shown that the procedure may also be extended for the identification of a certain class of nonlinear systems.

A frequency domain stability criterion for non-linear feedback systems

A criterion for the investigation of the stability of a certain class of non-linear system with sinusoidal input is proposed. It is derived from the concepts of the describing function method and the circle criterion. If the non-linear system satisfies the restrictions of describing function analysis and if the characteristic of the non-linearity is bound in a sectory/eɛ[a, b], there exists a critical region in the G inverse polar plane which contains ail possible describing functions of non-linearities bounded in this sector. Sufficient condition for the stability of the system is assured if the locus of the linear element does not intersect the critical region.

Decoupling certain classes of nonlinear systems by nonlinear state feedback

Decoupling certain classes of nonlinear systems by nonlinear state feedback

Correction to "On the optimization of a certain class of nonlinear systems"

Correction to "On the optimization of a certain class of nonlinear systems"

On the optimization of a certain class of nonlinear systems

It is shown that a fairly general class of nonlinear control laws is optimal (with respect to a certain class of functionals) for a large class of nonlinear systems. The work generalizes previous studies on the optimization of asymmetrical space vehicles.

On the Almost Sure Stability of Linear Dynamic Systems With Stochastic Coefficients

In this paper a Lyapunov type of approach is used to obtain sufficient conditions guaranteeing the almost sure stability of linear dynamic systems with stochastic coefficients. The results are generalized to include a certain class of nonlinear system and also to guarantee the almost sure boundedness of the forced oscillations of linear dynamic systems with stochastic coefficients.

Generalized recurrence relations in the analysis of nonlinear systems

Involved in solving a certain class of nonlinear systems with nonlinear derivatives, among other things, are the general recurrence relations presented herewith. The theory is applied to systems involving circuits and feedback systems. Incidental to the paper, a relationship is developed between circuits and feedback systems.

Taylor-Cauchy Transforms for Analysis of a Class of Nonlinear Systems

This paper presents a new transform calculus for analyzing a certain class of nonlinear systems. The method, which can be justified rigorously by the partition theory, essentially transforms a nonlinear differential equation having certain conditions imposed on its linear, nonlinear, and driving terms into an algebraic equation. The latter is easily solved recursively owing to its symmetry and convolution properties. The transform pair is based on a combination of the Cauchy integral theorem and Taylor's series in complex form. To illustrate the method a number of examples are solved and a table of transforms is included. Because the results of this transform technique are the same as those given by the partition method under certain circumstances, the two are compared. It is then seen that the solution can be obtained uniquely and exactly. The Taylor-Cauchy transform method can be compared with the Laurent-Cauchy transform method, given in a companion paper, for the solution of linear systems described by differential-difference and sum equations.

Recurrence relations in the solution of a certain class of nonlinear systems

This paper (a modified version of the material in an unpublished paper of the same name by the author presented at the AIEE 1957 Summer General Meeting) introduces aspects of a generalized mathematical theory for the analysis of a certain class of nonlinear systems. (Since the original presentation of this paper other work has appeared,1?5 which enlarges on the comments made here and extends the present work.) It is shown that under a very broad class of conditions, consistent with physical realization, a class of systems exists for which it is possible to develop exact and unique mathematical solutions. These conditions are important since nonlinear differential equations do not in general possess such unique properties although linear differential equations do.

A stability criterion for nonlinear systems

IN ANOTHER PAPER1 Wolf reports on the recurrence relations in the solution of a certain class of nonlinear systems. In Wolf's dissertation2 a mathematical theory for the analysis of a class of nonlinear systems has been developed. This paper reports on a stability criterion based on the aforementioned dissertation and its application to nonlinear as well as linear feedback control systems.