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.     

Vulnerability of dynamic systems†

The purpose of this paper is to formulate and partially resolve the problem of vulnerability of dynamic systems in the context of the theory of directed graphs. A system is considered vulnerable if its structural properties such as input (output) reachability or structural controllability (observability) is destroyed by a perturbation characterized as removal of a line, or set of lines, from the corresponding graph. Graph-theoretic procedures are developed to identify the minimal sets of lines which are essential for preserving the structural properties of the system.     

Discrete-time convolution feedback systems †

This paper considers multi-input multi-output discrete-time feedback systems characterized by y = G?e and e = u — y. Theorem I shows that if the closed-loop impulse response H is stable in the sense that H ? ? ¹ _n×n (p), then [Gtilde] (z) =Ñ(z) [ [Dtilde](z)] ^? where Ñ(z), ;[Dtilde](z) are in ? ¹ _n×n (p) Theorem III gives necessary and sufficient conditions for H ? ? ¹ _n×n (p) Finally Theorem III gives necessary and sufficient conditions for stability when [Gtilde] (z) has a finite number of multiple poles in ∣z∣ > p : the case where the leading term of the Laurent expansion at each of these poles is singular is treated in detail.     

Optimal control of neutral systems†

A method is presented whereby an optimal control may be obtained for a linear time-varying neutral system. The performance measure is quadratic with fixed finite upper limit. A form is derived for the coat functional, and the optimal control is obtained as the solution to a set of differential equations. A numerical technique for the solution of the equations is given and existence and uniqueness are proven.     

Approximation of discrete linear systems†

Given a multi–input, multi–output linear time–invariant discrete–time system or its weighting matrices, equations are derived to characterize a system of lower dimension which optimally approximates the given system with respect to the sum of squared output errors of the impulse response. These equations are shown to imply that in principle a modified B. L. Ho algorithm can be used to find the approximation. The connection with projection methods of approximation is demonstrated, Some comments on computational difficulties are included.     

Feedback systems and multiple time-scales†

We study feedback design for systems with multiple time-scale structure uncovering an intrinsic time-scale structure for a state feedback system with full state output. From this we construct a multiple time-scale asymptotic observer and consider a system with state feedback via a two time-scale asymptotic observer. Finally we obtain more general results for a restricted class of systems.     

Constrained controllability of discrete-time systems†

The constrained controllability of the discrete-time system x_k+1=A(k)x_k+B(k)u,_k is considered where the control uk is termed admissible if it satisfies specified magnitude constraints. Constrained controllability is concerned with the existence of an admissible control which steers the state x to a given target set from a specified initial state

Conditions for checking constrained controllability to a given target set from a specified initial state are presented. These conditions involve solving finite-dimensional optimization problems and can be checked via numerical computation. In addition, conditions for checking global constrained controllability to a given target set are presented. A system is globally constrained controllable if for every initial state, there exists an admissible control that steers the system to the target.

If a given discrete-time system is constrained controllable, it may be desirable to obtain an admissible control that steers the system to the target from a specified initial state. Such a control is called a steering control. Results for computing steering controls are also presented

This paper is concluded with a numerical example. In this example, it is shown that the constrained controllability of a continuous-time system which has been discretized is dependent on the discretization time. The set of states which can be steered to the target changes as the discretization time changes. Furthermore, the example shows that a discrete-time steering control cannot always be obtained by discretizing a continuous-time steering control; the steering control for the discrete-time system must be obtained directly from the discrete-time model.     

Stability of pulse-width-modulated feedback systems†

Sufficient conditions for the asymptotic stability in the sense of Lagrange or ultimate bonncloclness of pulse-width-modulated control systems are developed in this paper using the Second Method of Liapunov. This method is not limited to higher-order systems and is applicable to plants having transfer functions with real or complex poles. The conditions have been obtained by constraining the first difference of the quadratic form of a Liapunov function to be negative definite by expanding it in a Taylor series.     

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.     

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.     

Identification of stochastic distributed-parameter systems†

The parameter estimation problem is treated for a general class of distributed-parameter systems disturbed by random inputs. The policy of transforming the problem into an equivalent lumped-parameter form amenable to existing statistical parameter estimation methods is followed. A computed example illustrates the effectiveness of the theory.     

Sensitivity-state models for linear systems†

In this paper time-domain models for sensitivity analyses of linear systems have been developed. The procedures for developing those models are quite similar to the formulation of state models which are based on graph-theoretic concepts. Sensitivity analyses with respect to variations in component values, the initial conditions and the driving functions appear as three different facets of the same analytical procedure. These models admit a very convenient interpretation in terms of networks which properly can be exploited for digital computer studies. Additionally, attention is focused on developing sensitivity models for three specific types of interconnection, viz. parallel, cascade and feedback. On the basis of these models, the effect of system structure on sensitivity is studied.     

Disturbance rejection in linear systems †

The problem of making the transfer function between disturbances and controlled outputs identically zero by using state feedback, feedforward control and dynamic compensation in u linear timo invariant system is treated and a constructive solve ability condition is presented.     

Posicast and negicast control systems†

A possibility of obtaining dead beat response of a class of high-order control systems by adopting the technique of a single step control of the input step command is described in this paper. For achieving the desired response by such input processing techniques, the possibility demands that configurations of high-order systems be of suitable forms and the method of control necessitates positive casting of the system input like the posicast control technique in ease of oven-order systems and negative casting of the input in case of odd–order systems. The system configurations and the forms of the processed input signals in those cases have separately been investigated. The speed of the response avoiding saturated operation of such compensated systems has been studied. The effect of an improper switching operation in casting the system input has also been investigated. The results of the above studios made on the simulated systems have been given.     

Observability of minimal,distal systems†

In a recent series of papers, Drager and Martin have shown global observability for a variety of particular classes of systems (including both discrete and continuous time systems). The underlying philosophy behind their methods is apparently that if a flow is ergodic and the system observation has a sufficiently special value, the system is completely observable. In this paper we prove observability for observed systems whose underlying flow is minimal and distal. While our results hold also in discrete time, we illustrate this result for invariant ergodic systems on nilmanifolds, i.e. those invariant flows with an infinitesimal generator that is ‘irrational modulo the commutator’. One well-known example of such a flow is the irrational flow on a torus studied by Drager and Martin.     

Modelling and simulation of stochastic systems†

In the field of automatic control, systems with stochastic inputs are frequently encountered. The purpose of this paper is twofold: (I) to treat the problem of mathematically modelling a general class of stochastic systems, and (2) to simulate the optimal control for a special case of this class of stochastic systems, specifically a linear time-invariant system with stochastic inputs and stochastically perturbed observations. The general class of systems considered in the modelling includes systems having non-linear and time-varying state dependence and additive white noise which is added in a non-linear and time-varying manner.     

Minimal controllability for systems with delays†

In this paper we consider the problem of minimal controllability for a general class of linear spectrally controllable systems. We show that the smallest number of controls that are necessary in order that spectral controllability may hold depends only upon the matrices of the uncontrolled system and that it is equal to the smallest number of outputs necessary for the dual problem of ‘spectral observability’; moreover, any spectrally controllable system can be made minimally controllable by acting upon the control matrix only, and any spectrally observable system can be made minimally observable by acting upon the output matrix only. In the sequel this result is explained in the cases of the delayed systems and of some kind of generalized systems.     

The root-loci of four pole systems†

The paper shows how the general shape of the root–locus of a four polo system is related to the relative positions of its open loop poles. The various forms of the locus are displayed on two maps. One map shows the loci of systems having two or four real open loop poles while the second map deals with the case of two pairs of complex poles. For each general shape of the locus analytical expressions are given for real axis break points, imaginary axis crossings, and maximum gain consistent with closed loop stability.     

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.     

On linear modelling of non-linear systems†

In many cases of interest, systems described by non-linear differential equations are modelled by linear equations for purposes of synthesizing feedback controllers. A theorem relating the stability properties of the linear model and the non-linear system is presented and discussed.