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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Observability of constant states of linear systems†

This paper tackles, in essence, two problems. In the first a method of analysis by freezing in intervals of slowly time-varying systems is developed. Under this method the parameters of a time-varying system are frozen in consecutive intervals to their value at the beginning of the interval. Thus in each interval the system becomes time-invariant, or a constant coefficient one, and analytic methods can be applied. Such a mode of solution enables the use of steps which are considerably larger than those common in classical numerical methods. Error estimates are given for the various cases.

In the second part the freezing in intervals method is applied to systems with a time-averaged non-linearity. A non-linear system with time-averaging can be regarded as one composed of a time-varying system with a variable parameter. A nonlinear operation, such as squaring, is performed on the signal of the system and followed by low-pass filtering. This filtered signal controls, in turn, the variable parameter. Detailed examples with error analyses are given. Digital simulation by the above and a slower, commonly used, method are included.     

Stability of a class of interconnected systems†

Sufficient conditions are given for exponential stability of the equilibrium of systems of interconnected components taken from a class of nonlinear, time-varying, multi-terminal, differential, algebraic and mixed components. The analysis is based on the mathematical models of the unconstrained components and the linear constraints imposed by the interconnections between the components. A Liapunov function is constructed for the interconnected system from Liapunov functions for the individual components. A parameter vector appearing in the Liapunov function is selected in a prescribed optimal manner. The Liapunov function so constructed can be used to establish a lower bound on the rate of decay of the system. An example of a ninth-order, non-linear system is included to demonstrate the application to design.     

Optimization of linear systems of constrained configuration†

For the sake of simplicity it is often desirable to restrict the number of feedbacks in a controller. In this case the optimal feedbacks depend on the disturbance to which the system is subjected. Using a quadratic error integral as a measure of the response, three criteria of optimization are considered :

1. The response to a given initial disturbance.

2. The worst response to an initial disturbance of given magnitude.

3. The worst comparison with the unconstrained optimal system.

It is shown that for each of these criteria the gradient with respect to the feedbacks can be calculated by a uniform method. The solution may then be found either directly or by a descent procedure. The method is illustrated by an example.     

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.     

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.     

Sub-optimal bang-bang control of stochastic systems†

Much attention has been paid to the statistical linearization approach to the optimizing problem for stochastic systems. In this paper, a technique is presented for designing a sub-optimal bang-bang control law with a class of non-linear switching functions. The structural form of the non-linear switching function is prespecifled, while leaving free parameters to be chosen in a optimal fashion. Using statistical linearization, the free parameters are chosen so as to be independent of the external noise levels as well as to minimize the average cost. Detailed discussions are given of two typical examples.     

Sensitivity design of optimal linear systems†

A measure of sensitivity is introduced into the performance index of the optimal linear system. The sensitivity vector is adjoined to the system state vector and the overall system is optimized. A procedure for selecting the weighting matrix elements is given. Finally a design procedure for obtaining approximately optimal feedback controllers with bounds on sensitivity is given. The methods are illustrated with a first-order example.     

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.