Showing papers in "Mathematics of Control, Signals, and Systems in 1989"

Approximation by superpositions of a sigmoidal function

Abstract: In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real variables with support in the unit hypercube; only mild conditions are imposed on the univariate function. Our results settle an open question about representability in the class of single hidden layer neural networks. In particular, we show that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with only a single internal, hidden layer and any continuous sigmoidal nonlinearity. The paper discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks.

A bisection method for computing the H ∞ norm of a transfer matrix and related problems

Abstract: We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H∞ Armijo line-search problem with no more computation than is required to compute a single H∞ norm.

Sufficient lyapunov-like conditions for stabilization

Abstract: In this paper we study the stabilizability problem for nonlinear control systems. We provide sufficient Lyapunov-like conditions for the possibility of stabilizing a control system at an equilibrium point of its state space. The stabilizing feedback laws are assumed to be smooth except possibly at the equilibrium point of the system.

On the control of discrete-event dynamical systems

Abstract: We study a class of problems related to the supervisory control of a discrete-event system (DES), as formulated by Ramadge and Wonham, and we focus on the computational effort required for their solution. While the problem of supervisory control of a perfectly observed DES may be easily solved by dynamic programming, the problem becomes intractable (in the sense of complexity theory) when imperfectly observed systems are considered.

Feedback stabilization of a linear control system in hilbert space with an a priori bounded control

Abstract: This paper derives a feedback controlf(t), ‖f(t)‖E≦r,r>0, which forces the infinite-dimensional control systemdu/dt=Au+Bf, u(0)=u o ≠H to have the asymptotic behavioru(t)→0 ast→∞ inH. HereA is the infinitesimal generator of aC o semigroup of contractionse At on a real Hilbert spaceH andB is a bounded linear operator mapping a Hilbert space of controlsE intoH. An application to the boundary feedback control of a vibrating beam is provided in detail and an application to the stabilization of the NASA Spacecraft Control Laboratory is sketched.

On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation

Abstract: This paper characterizes the class of closed and (M, N)-recognizable languages in terms of certain structural aspects of relevant automata. This characterization leads to algorithms that effectively compute the supremal (M, N)-recognizable sublanguage of a given language. One of these algorithms is used, in an alternating manner with an algorithm which yields the supremal (∑u, N)-invariant resulting algorithm is proved. An example illustrates the use of these algorithms.

Impulse control of piecewise-deterministic processes

Abstract: In a recent paper we presented a numerical technique for solving the optimal stopping problem for a piecewise-deterministic process (P.D.P.) by discretization of the state space. In this paper we apply these results to the impulse control problem. In the first part of the paper we study the impulse control of P.D.P.s. under general conditions. We show that iteration of the single-jump-or-intervention operator generates a sequence of functions converging to the value function of the problem. In the second part of the paper we present a numerical technique for computing optimal impulse controls for P.D.P.s. This technique reduces the problem to a sequence of one-dimensional minimizations. We conclude by presenting some numerical examples.

A state space approach to the problem of adaptive pole assignment

Abstract: An adaptive pole-placement algorithm for the class of single-input/ single-output systems of ordern is proposed. The asymptotic properties of the algorithm do not depend on persistently exciting signals. Excitation is used only initially to avoid pole-zero cancellation of the parameter estimates. The main result is that the asymptotic behavior of the system equals the behavior we would have obtained on the basis of knowledge of the true system. This does not imply full identification of the desired control law, so we propose the descriptive termweak self-tuning.

Oblique projections: Formulas, algorithms, and error bounds

Abstract: When an orthogonal projection is to be computed using data acquired by distributed sensors, it is often necessary to process each sensor's data locally and then transmit the results to a central facility for final processing. The most efficient way to do this is to compute oblique projections locally. This choice makes the final processing a matter of summing the oblique projections. In this paper we derive new formulas, and iterative algorithms and associated error bounds, for oblique projections in arbitrary Hilbert spaces.

Block noninteracting control with stability via static state feedback

Abstract: Necessary and sufficient conditions for the existence of a static state feedback that achieves noninteraction and internal stability are obtained. This is accomplished by first characterizing the set of all controllability subspaces (distributions) that can arise as solutions to the noninteracting control problem. This characterization is then used to identify a fixed internal dynamics that is common to every noninteractive closed loop. The stability properties of this dynamics is shown to be the key factor in the problem of achieving noninteracting control with internal stability.

Adaptive Algorithms with Filtered Regressor and Filtered Error

Abstract: This paper presents a unified framework for the analysis of several discrete time adaptive parameter estimation algorithms, including RML with nonvanishing stepsize, several ARMAX identifiers, the Landau-style output error algorithms, and certain others for which no stability proof has yet appeared. A general algorithmic form is defined, incorporating a linear time-varying regressor filter and a linear time-varying error filter. Local convergence of the parameters in nonideal (or noisy) environments is shown via averaging theory under suitable assumptions of persistence of excitation, small stepsize, and passivity. The excitation conditions can often be transferred to conditions on external signals, and a small stepsize is appropriate in a wide range of applications. The required passivity is demonstrated for several special cases of the general algorithm.

On the quantitative characterization of approximate decentralized fixed modes using transmission zeros

Abstract: The spectrum of a linear time-invariant multivariable system, using decentralized linear time-invariant controllers, can only be assigned to a symmetric set of complex numbers that include the decentralized fixed modes (DFM). Hence only systems with stable DFM can be stabilized. Although the concept of DFM characterizes when a decentralized controller can stabilize a system, it gives no indication of howhard it is to effect such a stabilization. A system is considered hard to stabilize if large controller gains are required. Modes that are hard to shift are termedapproximate decentralized fixed modes. In this paper two new assignability measures which quantify the difficulty of shifting a mode are derived. The first is coordinate invariant and is based on the distance between a mode and a set of transmission zeros. The second is coordinate dependent and is based on the minimum singular value of a set of transmission zero matrices.

A systolic algorithm for riccati and lyapunov equations

Abstract: Riccati and Lyapunov equations can be solved using the recursive matrix sign method applied to symmetric matrices constructed from the corresponding Hamiltonian matrices. In this paper we derive an efficient systolic implementation of that algorithm where theLDLT andUDUT decompositions of those symmetric matrices are propagated. As a result the solution of a class of Riccati and Lyapunov equations can be obtained inO(n) time steps on a bidimensional (triangular) grid ofO(n2) processors, leading to an optimal speedup.

Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations

Abstract: In this paper we consider the problem of optimal regulation of large space structures in the presence of flexible appendages. For simplicity of presentation, we consider a spacecraft consisting of a rigid bus and a flexible beam. The complete dynamics of the system is given by a coupled set of ordinary and partial differential equations. We show that the solution of this hybrid system is defined in a product space of appropriate finite- and infinite-dimensional spaces. We develop necessary conditions for determining the control torque and forces for optimal regulation of attitude maneuvers of the satellite along with simultaneous suppression of elastic vibrations of the flexible beam.

Model reduction by phase matching

Abstract: This paper discusses procedures for approximating high-order rational power spectrum matrices and minimum phase stable transfer function matrices by lower-order objects of the same type. The basis of the approximation is to secure closeness of a high-order and low-order minimum phase stable transfer function matrix in phase, and to infer from this, closeness in magnitude. A suitable definition of multivariable phase is needed. Particular cases of the approximation procedure which are already known are cast in a general framework, which is also shown to include relative error approximation. A number of error bounds are given. Extensions to approximation of nonminimum phase transfer function matrices are also provided.

Chattering linear systems: A model of rapidly oscillating coefficients

Abstract: We introduce linear control systems, termed chattering systems, which model instantaneous oscillations in the control parameters. Such systems serve as a limit case of systems with rapidly oscillating control parameters, which can be analyzed as perturbations from the chattering model. Several optimization and regulation problems for chattering systems are examined, along with the robustness property: the possibility of employing the solutions of the chattering case in the rapidly oscillating approximations. The theory is demonstrated on an example of an armature-controlled dc motor.

Interpolation type problems in the class of positive trigonometric polynomials of fixed order

Abstract: This paper is devoted to a family of interpolation type problems for positive trigonometric polynomials of a given ordern. Via the Riesz-Fejer factorization theorem, they can be viewed as natural generalizations of the partial autocorrelation problem for discrete time signals of lengthn+1. The relevant variables for a specific problem are well-defined linear combinations of the coefficients of the underlying trigonometric polynomial. An efficient method is obtained to characterize the feasibility region of the problem, defined as the set of points having these variables as coordinates. It allows us to determine the boundary of that region by computing the extreme eigen values and the corresponding eigenvectors of certain well-defined Hermitian Toeplitz matrices of ordern+1. The method is an extension of one proposed by Steinhardt to solve the coefficient problem for positive cosine polynomials (which belongs to the family). Other interesting applications are the Nevanlinna-Pick interpolation problem for polynomial functions, and the simple interpolation problem for positive trigonometric polynomials. The close connection between the generalized Steinhardt method and classical techniques based on the polarity theorem for convex cones and on the Hahn-Banach extension theorem are established.

Exact solutions to some minimum time problems with inequality state constraints

Abstract: A heuristic method is developed for generating exact solutions to certain minimum time problems, with inequality state and control constraints. The control equation is linear and autonomous, with scalar-valued control. The state constraints are also linear inequalities. Assuming knowledge of a finite sequence, in which state and/or control constraints become active along an optimal path, the maximum principle is reduced to a set of equations and inequalities in a finite number of unknowns. A solution to the equations and inequalities determines both the solution path and a proof of its optimality. Certain types of constraint sequences lead to overdetermined equation systems, and this fact is interpreted in terms of the qualitative behavior of solutions to these problems. Two path-planning problems are solved, as illustrations of the solution technique.

Singularities and normal forms of observed dynamics

Abstract: We define and study a notion of a singular point for observed dynamical systems with no controls. Lists of normal forms for such singularities are given in both general and generic cases. One class of normal forms corresponds to singularities which appear in catastrophe theory.