Showing papers in "IEEE Transactions on Automatic Control in 1971"
TL;DR: In this paper, the identity observer, a reduced-order observer, linear functional observers, stability properties, and dual observers are discussed, along with the special topics of identity observer and reduced order observer.
Abstract: Observers which approximately reconstruct missing state-variable information necessary for control are presented in an introductory manner. The special topics of the identity observer, a reduced-order observer, linear functional observers, stability properties, and dual observers are discussed.
2,544Â citations
TL;DR: In this paper, the optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated, and the integrand of the performance criterion is allowed to be fully quadratically in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem.
Abstract: The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the terminal state is constrained to be zero are treated. The integrand of the performance criterion is allowed to be fully quadratic in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem. Frequency-domain and time-domain conditions for the existence of solutions are derived. The algebraic Riccati equation is then examined, and a complete classification of all its solutions is presented. It is finally shown how the optimal control problems introduced in the beginning of the paper may be solved analytically via the algebraic Riccati equation.
1,436Â citations
TL;DR: In this paper, the problem of estimating the state of a linear dynamic system using noise-corrupted observations, when input disturbances and observation errors are unknown except for the fact that they belong to given bounded sets, is considered.
Abstract: This paper is concerned with the problem of estimating the state of a linear dynamic system using noise-corrupted observations, when input disturbances and observation errors are unknown except for the fact that they belong to given bounded sets. The cases of both energy constraints and individual instantaneous constraints for the uncertain quantities are considereal. In the former case, the set of possible system states compatible with the observations received is shown to be an ellipsoid, and equations for its center and weighting matrix are given, while in the latter case, equations describing a bounding ellipsoid to the set of possible states are derived. All three problems of filtering, prediction, and smoothing are examined by relating them to standard tracking problems of optimal control theory. The resulting estimators are similar in structure and comparable in simplicity to the corresponding stochastic linear minimum-variance estimators, and it is shown that they provide distinct advantages over existing schemes for recursive estimation with a set-membership description of uncertainty.
698Â citations
TL;DR: In this paper, the role of the linear-quadratic stochastic control problem in engineering design is reviewed in tutorial fashion, motivated by considering the control of a non-linear uncertain plant about a desired input-output response.
Abstract: The role of the linear-quadratic stochastic control problem in engineering design is reviewed in tutorial fashion. The design approach is motivated by considering the control of a non-linear uncertain plant about a desired input-output response. It is demonstrated how a design philosophy based on 1) deterministic perturbation control, 2) stochastic state estimation, and 3) linearized stochastic control leads to an overall closed-loop control system. The emphasis of the paper is on the philosophy of the design process, the modeling issue, and the formulation of the problem; the results are given for the sake of completeness, but no proofs are included. The systematic off-line nature of the design process is stressed throughout.
611Â citations
TL;DR: In this paper, the authors modify existing linear regulator and servomechanism theories to take into account the presence of persistent fluctuating disturbances and systematically exploit any useful effects that may be present in the action of external disturbances.
Abstract: Modern linear regulator and servomechanism theories (of the deterministic type) either ignore system disturbances altogether or assume they can be represented by initial conditions on the plant state vector. Controllers designed by such theories may fail to meet performance specifications when the system is subjected to persistently acting disturbances. In this paper, we show how one can modify existing regulator and servomechanism theories to take into account the presence of persistent fluctuating disturbances. By this means one can design a deterministic controller which consistently maintains set-point regulation, or servotracking, in the face of a bread class of realistic external disturbances. In addition, we show how one can systematically exploit any useful effects that may be present in the action of external disturbances.
565Â citations
TL;DR: In this article, the square root approach is proposed to solve the problem of discrete filtering in the absence of a state estimate and an error covariance matrix from stage to stage, which is equivalent algebraically to the conventional Kalman approach.
Abstract: The conventional Kalman approach to discrete filtering involves propagation of a state estimate and an error covariance matrix from stage to stage. Alternate recursive relationships have been developed to propagate a state estimate and a square root error covariance instead. Although equivalent algebraically to the conventional approach, the square root filters exhibit improved numerical characteristics, particularly in ill-conditioned problems. In this paper, current techniques in square root filtering are surveyed and related by applying a duality association. Four efficient square root implementations are suggested, and compared with three common conventional implementations in terms of computational complexity and precision. The square root computational burden should not exceed the conventional by more than 50 percent in most practical problems. An examination of numerical conditioning predicts that the square root approach can yield twice the effective precision of the conventional filter in ill-conditioned problems. This prediction is verified in two examples. The excellent numerical characteristics and reasonable computation requirements of the square root approach make it a viable alternative to the conventional filter in many applications, particularly when computer word length is limited, or the estimation problem is badly conditioned.
467Â citations
TL;DR: The Kalman estimation technique is examined from the point of view of the asymptotic behavior of the errors, and both "true" and "apparent" divergence are demonstrated by a simple scalar system.
Abstract: The Kalman estimation technique is examined from the point of view of the asymptotic behavior of the errors in the estimates. It is shown that, under certain conditions, the mean-square errors may become unbounded with time, and that this divergence may or may not be correctable by increasing the intensity of process noise assumed in the filtering model General results are derived for multidimensional systems, and both "true" and "apparent" divergence are demonstrated by a simple scalar system. Divergence due to numerical inaccuracies is considered, and an example problem in orbital navigation is used to demonstrate divergence and its elimination.
383Â citations
TL;DR: In this article, the adaptive estimators are applied to the problem of state estimation with non-Gaussian initial state, to estimation under measurement uncertainty (joint detection-estimation) as well as to system identification.
Abstract: Optimal structure and parameter adaptive estimators have been obtained for continuous as well as discrete data Gaussian process models with linear dynamics. Specifically, the essentially nonlinear adaptive estimators are shown to be decomposable (partition theorem) into two parts, a linear nonadaptive part consisting of a bank of Kalman-Bucy filters and a nonlinear part that incorporates the adaptive nature of the estimator. The conditional-error-covariance matrix of the estimator is also obtained in a form suitable for on-line performance evaluation. The adaptive estimators are applied to the problem of state estimation with non-Gaussian initial state, to estimation under measurement uncertainty (joint detection-estimation) as well as to system identification. Examples are given of the application of the adaptive estimators to structure and parameter adaptation indicating their applicability to engineering problems.
329Â citations
TL;DR: In this paper, the current status of decoupling theory for linear constant multivariable systems is described in vector space terms and appropriate background concepts including invariant and controllability subspaces are discussed.
Abstract: The current status of decoupling theory for linear constant multivariable systems is described. The subject is treated in vector space terms and appropriate background concepts including invariant and controllability subspaces are discussed. Suggestions are given for translating vector space operations into matrix operations suitable for computation. The controllability subspace is used to formulate the restricted (static compensation) decoupling problem. Although the most general version of this problem is unsolved, there are known solutions for three special cases. A complete solution to the extended (dynamic compensation) decoupling problem is known. If a linear constant multivariable system can be decoupled at all, by any means whatever, then it can always be decoupled using linear dynamic compensation. The internal structure of a decoupled system is described in simple matrix terms. Using this representation, it is possible to characterize the system pole distributions which may be achieved while preserving a decoupled structure. A procedure is outlined for synthesizing a dynamic compensator of low order which will decouple a system. The procedure actually provides minimal order decoupling compensators for systems in which the number of open-loop inputs equal the number of outputs to be controlled.
319Â citations
TL;DR: In this paper, a noniterative method developed by Hsia for the particular case where the transfer function in the Hammerstein model has no zeros is extended to the general case when the transfer functions may have zeros.
Abstract: A noniterative method developed by Hsia for the particular case where the transfer function in the Hammerstein model has no zeros is extended to the general case where the transfer function may have zeros. Numerical examples show that the computation time by this method is considerably less than by the iteratire procedure proposed by Narendra and Gallman while the accuracy of the estimates is comparable.
294Â citations
TL;DR: In this paper the optimal discrete-time linear-quadratic regulator problem is carefully presented and the basic results are reviewed.
Abstract: In this paper the optimal discrete-time linear-quadratic regulator problem is carefully presented and the basic results are reviewed. Dynamic programming is used to determine the optimization equations. Special attention is given to problems unique to the discrete-time case; this includes, for example, the possibility of a singular system matrix and a singular control-effort weighting matrix. Some problems associated with sampled-data systems are also summarized, e.g., sensitivity to sampling time, and loss of controllability due to sampling. Computational methods for the solution of the optimization equations are outlined and a simple example is included to illustrate the various computational approaches.
TL;DR: In this article, a correlation technique which identifies a system in its canonical form is presented, which is capable of being implemented on-line and can be used in conjunction with the Kalman filter.
Abstract: Kalman gave a set of recursive equations for estimating the state of a linear dynamic system. However, the Kalman filter requires a knowledge of all the system and noise parameters. Here it is assumed that all these parameters are unknown and therefore must be identified before use in the Kalman filter. A correlation technique which identifies a system in its canonical form is presented. The estimates are shown to be asymptotically normal, unbiased, and consistent. The scheme is capable of being implemented on-line and can be used in conjunction with the Kalman filter. A technique for more efficient estimation by using higher order correlations is also given. A recursive technique is given to determine the order of the system when the dimension of the system is unknown. The results are first derived for stationary processes and are then extended to nonstationary processes which are stationary in the q th increment. An application of the results to a practical problem is presented.
TL;DR: In this correspondence an algorithm is presented for computing the steady-state optimal feedback law of the discrete-time invariant linear regulator that converges quadratically in a neighborhood of the steady state.
Abstract: In this correspondence an algorithm is presented for computing the steady-state optimal feedback law of the discrete-time invariant linear regulator that converges quadratically in a neighborhood of the steady state.
TL;DR: Realization theory for both time-invariant and time-variable linear systems is developed and its applicability to linear quadratic control and filtering is discussed in this paper, where the emphasis is on obtaining physically meaningful realizations and several procedures which accomplish this are detailed.
Abstract: Realization theory for both time-invariant and time-variable linear systems is developed and its applicability to linear quadratic control and filtering is discussed. For time-invariant systems a review of canonical structure theory is given and various properties such as minimality and equivalence are characterized in terms of the Hankel matrix. Realization theory for such systems is then developed based on the Hankel matrix and a new computational algorithm is presented. For time-variable systems the emphasis is on obtaining physically meaningful realizations and several procedures which accomplish this are detailed. For "constant rank" systems, a generalization of the Hankel matrix approach is also presented.
TL;DR: In this paper, the problem of keeping the state of a linear dynamic system in a specified region is investigated, and necessary and sufficient conditions for a solution to the problem and an algorithm that constructs the control are derived for open-loop and closed-loop control laws.
Abstract: A linear dynamic system with input and observation uncertainties is studied. The uncertainties are constrained to be contained in specified sets. No probabilistic structure is assumed. The problem of keeping the state of the system in a specified region is investigated. Necessary and sufficient conditions for a solution to the problem and an algorithm that constructs the control are derived for open-loop and closed-loop control laws. The algorithm is also approximated by a bounding ellipsoid algorithm. Two special control laws, linear and "linear-plus-dead-band," are studied, and the regions that contain the state and control are characterized. Ellipsoids that bound the state and control are also derived, and a simple example of linear and linear-plus-dead-band control laws is presented.
TL;DR: In this paper, the basic principles of least squares estimation are introduced and applied to the solution of some filtering, prediction, and smoothing problems involving stochastic linear dynamic systems.
Abstract: In this tutorial paper the basic principles of least squares estimation are introduced and applied to the solution of some filtering, prediction, and smoothing problems involving stochastic linear dynamic systems. In particular, the paper includes derivations of the discrete-time and continuous-time Kalman filters and their prediction and smoothing counterparts, with remarks on the modifications that are necessary if the noise processes are colored and correlated. The examination of these state estimation problems is preceded by a derivation of both the unconstrained and the linear least squares estimator of one random vector in terms of another, and an examination of the properties of each, with particular attention to the case of jointly Gaussian vectors. The paper concludes with a discussion of the duality between least squares estimation problems and least squares optimal control problems.
TL;DR: In this paper, the authors compared the performance of several non-linear filters for the real-time estimation of the trajectory of a reentry vehicle from its radar observations, including iterative-sequential filters, single-stage iteration filters, and second-order filters.
Abstract: This paper compares the performance of several non-linear filters for the real-time estimation of the trajectory of a reentry vehicle from its radar observations. In particular, it examines the effect of using two different coordinate systems on the relative accuracy of an extended Kalman filter. Other filters considered are iterative-sequential filters, single-stage iteration filters, and second-order filters. It is shown that a range-direction-cosine extended Kalman filter that uses the measurement coordinate system has less bias and less rms error than a Cartesian extended Kalman filter that uses the Cartesian coordinate system. This is due to the fact that the observations are linear in the range-direction-cosine coordinate system, but nonlinear in the Cartesian coordinate system. It is further shown that the performance of the Cartesian iterative-sequential filter that successively relinearizes the observations around their latest estimates and that of a range-direction-cosine extended Kalman filter are equivalent to first order. The use of a single-stage iteration to reduce the dynamic nonlinearity improves the accuracy of all the filters, but the improvement is very small, indicating that the dynamic nonlinearity is less significant than the measurement nonlinearity in reentry vehicle tracking under the assumed data rates and measurement accuracies. The comparison amongst the nonlinear filters is carried out using ten sets of observations on two typical trajectories. The performance of the filters is judged by their capability to eliminate the initial bias in the position and velocity estimates.
TL;DR: In this article, a quantitative model is developed for the response characteristics of the human operator, subject to his inherent psychophysical limitations, which can be used to predict task performance and human control characteristics.
Abstract: Modern control and estimation theory is used to establish a framework for the analysis of manned-vehicle systems. By assuming that the human behaves "optimally" in some sense, subject to his inherent psychophysical limitations, a quantitative model is developed for the response characteristics of the human operator. The resultant model can be used to predict task performance and human control characteristics. The model is described in detail and is used to predict quantities measured experimentally in both simple and reasonably complex manual control tasks. Remarkable agreement between measured and predicted quantities is obtained, demonstrating the value and potential of the optimization approach to manned-vehicle systems analysis.
TL;DR: Three areas are briefly reviewed: 1) control systems with human controller, 2) control system with man-machine controller, and 3) autonomous robot systems.
Abstract: A supplement to Fu [1] is presented. Three areas are briefly reviewed: 1) control systems with human controller, 2) control systems with man-machine controller, and 3) autonomous robot systems. Problems for further research are discussed.
TL;DR: In this article, the problem of designing compensators, the dimensions of which are fixed a priori, for linear systems is considered, and two types of compensators are considered: static (gain only) compensators which operate directly upon the output signals to generate the controls, and dynamic compensators of fixed dimension.
Abstract: This paper considers the problem of designing compensators, the dimensions of which are fixed a priori, for linear systems. Two types of compensators are considered: first, static (gain only) compensators which operate directly upon the output signals to generate the controls, and second, dynamic compensators of fixed dimension. The equations that define the parameters of such compensators are developed.
TL;DR: In this article, a review of the solution of the linear regulator problem for linear systems with state and control-dependent disturbances is presented, both the finite and infinite terminal time cases are treated.
Abstract: A review of the solution of the linear regulator problem for linear systems with state- and control-dependent disturbances is presented. Both the finite and infinite terminal time cases are treated. The solution to the complete state feedback case is given in detail and that for the output feedback case is noted. The general conclusion is that control-dependent noise calls for conservative control (small gains) while state-dependent noise calls for vigorous control (large gains). Of course it is the degree of this behavior that is important and this is given explicitly by the algorithms in this paper.
TL;DR: The use of the innovations allows us to obtain formulas and simple derivations that are remarkably similar to those used for the linear case thereby distinguishing clearly the essential points at which the nonlinear problem differs from the linear one.
Abstract: In Parts I and II of this paper, we presented the innovations approach to linear least-squares estimation in additive white noise. In the present paper, we show how to extend this technique to the nonlinear estimation (filtering and smoothing) of non-Gaussian signals in additive white Gaussian noise. The use of the innovations allows us to obtain formulas and simple derivations that are remarkably similar to those used for the linear case thereby distinguishing clearly the essential points at which the nonlinear problem differs from the linear one.
TL;DR: In this paper, the theory of optimal control for time delay systems and the quadratic performance criterion is presented from two points of view: 1) the geometric approach, which yields a maximum principle, and 2) the dynamic programming-Caratheodory approach.
Abstract: The theory of optimal control for time delay systems and the quadratic performance criterion is presented from two points of view: 1) the geometric approach, which yields a maximum principle, and 2) the dynamic programming-Caratheodory approach, which yields a feedback controller synthesis. The relationship of the two approaches is discussed, as well as extensions of the theory to non-linear problems and nonlinear performance criterion.
TL;DR: It is shown that it is often better to process statistically independent measurements in more than one batch and then to use sequential processing than to process them together via simultaneous processing.
Abstract: How practical is a Kalman filter? One answer to this question is provided by the computational requirements for the filter. Computational requirements-computational time per cycle (iteration) and required storage-determine minimum sampling rates and computer memory size. These requirements are provided in this paper as functions of the dimensions of the important system matrices for a discrete Kalman filter. Two types of measurement processing are discussed: simultaneous and sequential. It is shown that it is often better to process statistically independent measurements in more than one batch and then to use sequential processing than to process them together via simultaneous processing.
TL;DR: Theoretical results on the optimal control of linear time lag systems with respect to a quadratic criterion were provided in this article. But the results were not applied to the design of a feedback controller for a chemical process having transport lag.
Abstract: Theoretical results on the optimal control of linear time lag systems with respect to a quadratic criterion are shown to provide a practical feedback controller design procedure. A computational procedure is given for obtaining a stable linear feedback controller. A comprehensive illustration is given of the application of the quadratic criterion results to the design of a feedback controller for a chemical process having transport lag.
TL;DR: In this paper, it is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method.
Abstract: This paper deals with recent advances in developing direct methods for studying the transient stability problem of single-machine and multimachine power systems. The paper starts out with the construction of the mathematical model that is usually employed in the analyis of power system transient stability. Computer simulation methods are then briefly discussed, and it is indicated why accurate direct methods for transient stability investigations would be most welcome. It is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method. The main purpose of the paper is to give an exposition of the interesting results that have been obtained by applying Lyapunov's second method to the transient stability problem of single-machine and multimachine power systems. In the final portion of the paper some areas for further research are discussed.
TL;DR: In this article, optimal control with a quadratic performance index has been suggested as a solution to the problem of regulating an industrial plant in the vicinity of a steady state, but it is shown that such control is usually not feasible, and if feasible can have serious defects.
Abstract: -Optimal control with a quadratic performance index has been suggested as a solution to the problem of regulating an industrial plant in the vicinity of a steady state. It is shown that such control is usually not feasible, and if feasible can have serious defects.
TL;DR: In this article, the authors deal with stochastic control of linear systems in which the information available to distinct controllers is different, and they obtain necessary conditions for optimality of the controller structure parameters.
Abstract: This paper deals with the stochastic control of linear systems in which the information available to distinct controllers is different. Constraints on the control structure are imposed. Necessary conditions for optimality of the controller structure parameters are obtained. The results show that the separation theorem does not hold in this case.
TL;DR: In this article, an algorithm for the design of asymptotic state estimators (observers) for index-invariant uniformly observable time-varying linear finite-dimensional multivariable systems is presented.
Abstract: This paper presents an algorithm for the design of asymptotic state estimators (observers) for index-invariant uniformly observable time-varying linear finite-dimensional multivariable systems. The results obtained indicate that asymptotic estimators can be employed in optimally designed regulators provided an increase from the optimal cost is tolerable. It is also shown that any uniformly observable and uniformly controllable plant with index-invariant observability and controllability matrices can be stabilized with an observer.