Showing papers in "IEEE Transactions on Automatic Control in 1975"

A discrete state-space model for linear image processing

Abstract: The linear time-discrete state-space model is generalized from single-dimensional time to two-dimensional space. The generalization includes extending certain basic known concepts from one to two dimensions. These concepts include the general response formula, state-transition matrix, Cayley-Hamilton theorem, observability, and controllability.

Routh approximations for reducing order of linear, time-invariant systems

Abstract: A new method of approximating the transfer function of a high-order linear system by one of lower order is proposed. Called the "Routh approximation method" because it is based on an expansion that uses the Routh table of the original transfer function, the method has a number of useful properties: if the original transfer function is stable, then all approximants are stable; the sequence of approximants converge monotonically to the original in terms of "impulse response" energy; the approximants are partial Pade approximants in the sense that the first k coefficients of the power series expansions of the k th-order approximant and of the original are equal; the poles and zeros of the approximants move toward the poles and zeros of the original as the order of the approximation is increased. A numerical example is given for the calculation of the Routh approximants of a fourth-order transfer function and for illustration of some of the properties.

Pole assignment by gain output feedback

Abstract: This short paper deals with the problem of pole assignment with incomplete state observation. It is shown that if the system is controllable and observable, and if n \leq r + m - 1 , an almost arbitrary set of distinct closed-loop poles is assignable by gain output feedback, where n, r , and m are the numbers of state variables, inputs and outputs, respectively. This result improves considerably the ones obtained so far about this problem. Different from the conventional approach using the characteristic equation, an approach based on the properties of the eigenspaces of the closed-loop dynamics is used in this short paper, which gives a new light on the various problems in the linear system theory. It is also shown, as a direct consequence of this result, that the minimum order of the dynamic compensator required for almost arbitrary pole assignment of overall closed-loop system is not greater than n - m - r + 1 .

Approximate non-Gaussian filtering with linear state and observation relations

Abstract: Two approaches to the non-Gaussian filtering problem are presented. The proposed filters retain the computationally attractive recursive structure of the Kalman filter and they approximate well the exact minimum variance filter in cases where either 1) the state noise is Gaussian or its variance small in comparison to the observation noise variance, or 2) the observation noise is Gaussian and the system is one step observable. In both cases, the state estimate is formed as a linear prediction corrected by a nonlinear function of past and present observations. Some simulation results are presented.

Observing the states of systems with unmeasurable disturbances

Abstract: In order to estimate the state of a linear time-invariant multivariable system by using a Luenberger observer, it is generally assumed that all system inputs are measurable. This correspondence presents a simple and useful procedure for the construction of minimal-order state estimators for systems with unmeasurable disturbances.

Output feedback stabilization and related problems-solution via decision methods

Abstract: Given an unstable finite-dimensional linear system, the output feedback problem is, first, to decide whether it is possible by memoryless linear feedback of the output to stabilize the system, and, second, to determine a stabilizing feedback law if such exists. This paper shows how this and a number of other linear system theory problems can be simply reformulated so as to allow application of known algorithms for solution of the existence question, with the construction problem being solved by some extension of these known algorithms. The first part of the output feedback problem is solvable with a finite number of rational operations, and the second with a finite number of polynomial factorizations. Other areas of application of the algorithm are described: Stability and positivity tests, low-order observer and controller design, and problems related to output feedback. Alternative computational procedures more or less divorced from the known algorithms are also proposed.

On pole assignment in linear multivariable systems using output feedback

Abstract: The general problem of pole assignment in a linear, time-invariant multivariable system via output feedback is considered. It is shown that given a controllable-observable system ( C,A,B ) in which A \in R^{n \times n} , rank B = m , rank C = r , then for almost all ( B,C ) pairs, \min (n,m + r-1) poles can be assigned arbitrarily close to \min (n, m + r-1 ) specified symmetric values by using output feedback.

Approach to performance and sensitivity multiobjective optimization: The goal attainment method

Abstract: This short paper is concerned with computational methods for solving optimization problems with a vector-valued index function (vector optimization). It uses vector optimization as a tool for analyzing static control problems with performance and parameter sensitivity indices. The first part of this short paper presents a new computational method, the goal attainment method, which overcomes some of the limitations and disadvantages of methods currently available. The second part presents an integrated, multiobjective treatment of performance and sensitivity optimization based on a vector index approach. A numerical example in electric power system control is included, with analysis and results demonstrating the use of the goal attainment method and application of the approach to performance and sensitivity optimization.

Square-root algorithms for least-squares estimation

Abstract: We present several new algorithms, and more generally a new approach, to recursive estimation algorithms for linear dynamical systems. Earlier results in this area have been obtained by several others, especially Potter, Golub, Dyer and McReynolds, Kaminski, Schmidt, Bryson, and Bierman on what are known as square-root algorithms. Our results are more comprehensive. They also show bow constancy of parameters can be exploited to reduce the number of computations and to obtain new forms of the Chandrasekhar-type equations for computing the filter gain. Our approach is essentially based on certain simple geometric interpretations of the overall estimation problem. One of our goals is to attract attention to non-Riccati-based studies of estimation problems.

A Riccati equation for block-diagonalization of ill-conditioned systems

Abstract: A simple transformation, originally introduced for singularly perturbed systems, is now applicable to a larger class of time-invariant systems.

Convergence of discretization procedures in dynamic programming

Abstract: The computational solution of discrete-time stochastic optimal control problems by dynamic programming requires, in most cases, discretization of the state and control spaces whenever these spaces are infinite. In this short paper we consider a discretization procedure often employed in practice. Under certain compactness and Lipschitz continuity assumptions we show that the solution of the discretized algorithm converges to the solution of the continuous algorithm, as the discretization grids become finer and finer. Furthermore, any control law obtained from the discretized algorithm results in a value of the cost functional which converges to the optimal value of the problem.

Optimal continuous finite preview problem

Abstract: This short paper deals with the preview problem, in which a controller can use future information as well as present and past information in deciding the control. The problem of following command signals is considered for the continuous time system. It is assumed that the controller can make use of the finite preview information with respect to command signal from the present time t up to t la time units in the future. t la is called the preview length and usually t_{la} , the duration of the problem, which distinguishes the present problem from the optimal tracking problem. It is shown how to utilize the local future information obtained by finite preview to minimize an optimality criterion evaluated over the problem duration. An example is given to show advantages of making use of future information and the existence of an effective preview length.

Time optimal behavior of human saccadic eye movement

Abstract: Optimal control theory takes into account constraints such as energy and time economies which are relevant to the understanding of biological design. The versional eye tracking system responsible for the extremely rapid and precise movements called saccades, which occur, for example, during reading, seemed a likely biological system in which to test for time optimality. A homeomorphic detailed physiological model was constructed based on quantitative muscle, neuronal and oculomotor characteristcs. It is a sixth-order nonlinear representation which considers reciprocal innervation and the asymmetrical force-velocity relationship of the agonist-antagonist muscle pair that moves the eye. Simulations were done by digital computer, and responses of the model to first-, second-, and third-order time optimal control signals were observed; the major portions of the response trajectories were essentially the same. The model response was then compared with measured human saccadic eye movements, and it was found that this experimental data agreed most completely with the model driven by first-order time optimal control signals. Additionally, electromyographic studies in man and neurophysiological experiments in animals agree in showing that the nervous controller signals during saccades are also of the first-order type. Thus, we can conclude that the saccadic eye movements studied are accurately depicted by a model that yields time optimal responses.

Design of piecewise constant gains for optimal control via Walsh functions

Abstract: This paper presents a technique for determinating time-varying feedback gains of linear systems with quadratic performance criteria. The gains are approximated by the piecewise constants which axe naturally determined by Walsh functions. After introducing Walsh functions in the beginning we develop an operational matrix for solving state equations. Then using the operational matrix we solve the piecewise constant gains problem.

Feedback between stationary stochastic processes

Abstract: A simple formulation is given for the notion of feedback between two stationary stochastic processes in terms of the canonical representation of the joint process. The definition presented here has an equivalent formulation in terms of filtering theory, and provides statistical criteria for the detection of feedback. A simulation example is presented and an application to the United Kingdom unemployment-gross domestic product relation is described.

An adaptive state estimation solution to the maneuvering target problem

Abstract: A new approach to tracking a maneuvering target is presented. Modeling the randomly varying target as a semi-Markov process and then incorporating the statistics into the design of an adaptive state estimator produced an estimation scheme that greatly reduced the large bias errors that usually appear when a target makes an unexpected, large-scale maneuver.

Structure determination and parameter identification for multivariable stochastic linear systems

Abstract: This paper will consider the problem of using output data to identify a constant, multivariable, stochastic linear system which has unknown dimension, system matrices, and noise covariances. In order to obtain consistent parameter estimates, we use the innovations representation for the output process, in which the system matrices are chosen in a certain (invariant) canonical form. A systematic procedure is described for estimating the system structure and parameters of the innovations representation. Simulation results are presented to illustrate the identification method. A large-sample error analysis of the identification method is also given.

A branching algorithm for discriminating and tracking multiple objects

Abstract: A recursive branching algorithm for multiple-object discrimination and tracking consists of a bank of parallel filters of the Kalman form, each of which estimates a trajectory associated with a certain selected measurement sequence. The measurement sequences processed by the algorithm are restricted to a tractable number by combining similar trajectory estimates, by excluding unlikely measurement/state associations, and by deleting unlikely trajectory estimates. The measurement sequence selection is accomplished by threshold tests based on the innovations sequence and state estimates of each filter. Numerical experiments performed using the algorithm illustrate how the accuracy of the a priori state estimates and trajectory model influences the selectivity of the algorithm.

Describing functions revisited

Abstract: The well-known graphical describing function procedure can be simply modified to provide a completely reliable method for predicting whether or not certain kinds of nonlinear feedback system can oscillate. The modified method is easy to use and quantifies, in a natural way, the usual intuitive ideas about describing function reliability. In addition to the usual graphs, the user has to draw a band which measures the amount of uncertainty introduced by the approximations inherent in the method; in return for this extra work, the method gives error bounds for oscillation predictions, as well as ranges of frequency and amplitude over which oscillation is impossible. The main restriction is that the nonlinear element must be single valued and have bounded slope.

The stabilizing solution of the discrete algebraic riccati equation

Abstract: This short paper extends the existence theory of the stabilizing solution of the discrete algebraic Riccati equation. The main result provides necessary and sufficient conditions for the existence of such a solution, free of any a priori conditions on the problem data. For the special case of the optimal regulator problem, standard results are recovered as special cases.

Control of systems with jump Markov disturbances

Abstract: -Control of stochastic differential equations of the form dot{x}=f^{r(t)}(t,x,u) in which r(t) is a fiie-state Markov p n m s is discussed Dynamic programming optimalityconditions are shown to be necessary and sufficient for oplimality. A stochastic minimom principle whose adjoints satisfy deterministic integral equations is defiied and shorn to be necessary and snffiaent for optimality.

Observers for systems characterized by semigroups

Abstract: The theory of observers is generalized from finite dimensional linear systems to abstract linear systems characterized by semigroups on Banach spaces. Sufficient conditions are given for both identity and reduced-order observers to exist for the abstract system. It is shown that the spectrum of a closed-loop control system using an observer is the union of the spectrum of the observer and the spectrum of the closed-loop system with state feedback. The observer theory for the abstract system is used to show that observability is a sufficient condition for the existence of an observer for a system modeled by a linear functional differential equation.

Identification of autoregressive moving-average parameters of time series

Abstract: A procedure for sequentially estimating the parameters and orders of mixed autoregressive moving-average signal models from time-series data is presented. Identification is performed by first identifying a purely autoregressive signal model. The parameters and orders of the mixed autoregressive moving-average process are then given from the solution of simple algebraic equations involving the purely autoregressive model parameters.

A mixed method for multivariable system reduction

Abstract: A mixed method which combines dominant-eigenvalue concept and matrix-continued-fraction approach is proposed to obtain a stable reduced model from a stable high degree multivariable system.

Counterexamples to general convergence of a commonly used recursive identification method

Abstract: A recursive algorithm for parametric identification of discrete-time systems known as Panuska's method, the approximate maximum likelihood method or the extended matrix method, is analyzed. Making use of recently developed theory for asymptotic analysis of recursive stochastic algorithms, dynamic systems, and autoregressive moving average (ARMA) processes are constructed for which this algorithm does not converge. The manner in which the counterexamples are constructed yields insight into the algorithm and provides ideas how to improve the convergence properties.

Superiority of transfer function over state-variable methods in linear time-invariant feedback system design

Abstract: The objectives and achievements of state-variable methods in linear time-invariant feedback system synthesis are examined. It is argued that the philosophy and objectives associated with eigenvalue realization by state feedback, with or without observers, are highly naive and incomplete in the practical context of control systems. Furthermore, even the objectives undertaken have not really been attained by the state-variable techniques which have been developed. The extremely important factors of sensor noise and loop bandwidths are obscured by the state-variable formulation and have been ignored in the state-variable literature. The basic fundamental problem of sensitivity in the face of significant plant parameter uncertainty has hardly received any attention. Instead, the literature has concentrated primarily on differential sensitivity functions and even those results are so highly obscured in the state-variable formation as to lead to incorrect conclusions. In contrast, the important practical considerations and constraints have been clearly revealed and considered in the transfer function formulation. Differential sensitivity results are simple and transparent. For single input-output systems, there exists an exact design technique for achieving quantitative sensitivity specifications in the face of significant parameter uncertainty, which is optimum in an important practical sense. This problem is much more difficult and has not been completely solved for multivariable systems, but it has at least been realistically attacked by some transfer function methods. Finally, the concepts of controllability and observability so much emphasized in the state-variable literature are examined. It is argued that their importance in this problem class has been greatly exaggerated. On the one hand, transfer function methods can be used to check for their existence. On the other hand, nothing is lost when they are ignored, if the synthesis problem is treated as one with parameter uncertainty by transfer function methods.

An iterative method for the identification of nonlinear systems using a Uryson model

Abstract: A new model for identifying nonlinear systems is presented. The multipath structure with each path consisting of a polynomial followed by linear dynamics is a direct extension of the single path Hammerstein model. An iterative algorithm for obtaining the dynamics from finite length input and noisy output data records is presented and shown to converge for a class of inputs including colored Gaussian processes. Computer simulations demonstrate the feasibility of the model and algorithm.

A lower bound on the estimation error for Markov processes

Abstract: A lower bound on the minimal mean-square error in estimating nonlinear Markov processes is presented. The bound holds for causal and uncausal filtering. The derivation is based on the Van Trees' version of the Cramer-Rao inequality.

Design of minimal order stable observers for linear functions of the state via realization theory

Abstract: --The design of a minimal order stable observer and a minimal order observer with arbitrary poles that estimates a vector linear function of the state of a multivariable system is discussed. It is shown that both problems can be solved in a straightforward manner using partial realization theory, and several new results are given. These include a strong bound for the dimension of the minimal order stable observer and a simple necessary condition to design the minimal order observer with arbitrary poles that estimates a vector linear function of the state of a multiple-output system. Necessary and sufficient conditions are given for designing a minimal order observer with arbitrary poles for the case of estimating a vector linear function of the state of a single-output system and the case of estimating a scalar linear function of the state of a multiple-output system. A procedure to carry out the design in each of these cases is described. No restrictions whatsoever (except stability) are placed on the possible values of the observer poles. A significant observation of this paper is that the dynamics of the observer are constrained (in all cases) only by the gain matrix in the feedback law to be estimated and the output structure of the given system.

New results on the controllability and observability of general composite systems

Abstract: Using Rosenbrock's frequency domain approach [1], necessary and sufficient conditions are obtained for a general series-parallel or feedback composite system to be controllable or/and observable. The conditions are given in terms of the state equations of the composite system and are very simple to compute. Using these results, it is shown that a general series-parallel feedback cascade system is controllable or/and observable for "almost all" interconnection matrices, provided some mild conditions on the subsystems are imposed.