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Showing papers in "IEEE Transactions on Automatic Control in 1967"



Journal ArticleDOI
TL;DR: A recursive algorithm is developed which calculates a time-varying ellipsoid in state space that always contains the system's true state and is motivated by the problem of tracking an evasive target, but the results have wider applications.
Abstract: A method is discussed for estimating the state of a linear dynamic system using noisy observations, when the input to the dynamic system and the observation errors are completely unknown except for bounds on their magnitude or energy. The state estimate is actually a set in state space rather than a single vector. The optimum estimate is the smallest calculable set which contains the unknown system state, but it is usually impractical to calculate this set. A recursive algorithm is developed which calculates a time-varying ellipsoid in state space that always contains the system's true state. Unfortunately the algorithm is still unproven in the sense that its performance has not yet been evaluated. The algorithm is closely related in structure but not in performance to the algorithm obtained when the system inputs and observation errors are white Gaussian processes. The algorithm development is motivated by the problem of tracking an evasive target, but the results have wider applications.

905 citations


Journal ArticleDOI
W. M. Wonham1
TL;DR: In this paper, it was shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed loop system, formed by means of suitable linear feedback of the state.
Abstract: It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable linear feedback of the state. As an application of this result, it is shown that an open-loop system can be stabilized by linear feedback if and only if the unstable modes of its system matrix are controllable. A dual of this criterion is shown to be equivalent to the existence of an observer of Luenberger's type for asymptotic state identification.

853 citations


Journal ArticleDOI
TL;DR: The characterization of φ of all feedback matrices which decouple the system is characterized to determine the number of closed-loop poles which can be specified for the decoupled system and to develop a synthesis technique for the realization of desired closed- loop pole configurations.
Abstract: Necessary and sufficient conditions for the "decoupling" of an m -input, m -output time-invariant linear system using state variable feedback are determined. Given a system which satisfies these conditions, i.e., which can be decoupled by state variable feedback, the class φ of all feedback matrices which decouple the system is characterized. The characterization of φ is used to determine the number of closed-loop poles which can be specified for the decoupled system and to develop a synthesis technique for the realization of desired closed-loop pole configurations. Transfer matrix consequences of decoupling are examined and practice implications discussed through numerical examples.

774 citations


Journal ArticleDOI
TL;DR: In this article, a class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems, and it is shown that, unlike the single variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer.
Abstract: A class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems. It is shown that, unlike the single-variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer. A major result of the paper is that a multi-input system can be transformed to a set of coupled single-input subsystems.

667 citations


Journal ArticleDOI
TL;DR: In this article, a qualitative theory of stability for non-autonomous systems operating over finite time intervals has been developed, and sufficient conditions are given for various types of finite time stability of a system under the influence of perturbing forces which enter the system equations linearly.
Abstract: This paper continues the development of a qualitative theory of stability, recently initiated by the authors, for systems operating over finite time intervals. The theory is motivated by 1) the need for a more practical concept of stability than is provided by the classical theory; and 2) the search for methods for investigating stability of a system trajectory (either analytically or numerically given) without the necessity of performing complicated transformations of the differential equations involved. The systems studied in this paper are nonautonomous, i.e., they are under the influence of external forces, and the concept of finite time stability (precisely defined in the paper) in this case involves the bounding of trajectories within specified regions of the state space during a given finite time interval. (The input is assumed to be bounded by a known quantity during this time interval.) Sufficient conditions are given for various types of finite time stability of a system under the influence of perturbing forces which enter the system equations linearly. These conditions take the form of existence of "Liapunov-like" functions whose properties differ significantly from those of classical Liapunov functions. In particular, there is no requirement of definiteness on such functions or their derivative. The remainder of the paper deals with the problem of determining finite time stability properties of a system from knowledge of the finite time stability properties of lower-order subsystems which, when appropriately coupled, form the original system. An example is given which illustrates some of the concepts discussed in the paper.

571 citations


Journal ArticleDOI
J. Nagumo1, A. Noda1
TL;DR: In this paper, a method for system identification is proposed which is based on the error-correcting training procedure in learning machines, and is referred to as learning identification, which is applicable to cases where the input signal is random and nonstationary, and can be completed within a short time, so that it may be used to identify linear quasi-time-invariant systems in which some parameters vary slowly in comparison with the time required for identification.
Abstract: A method for system identification is proposed which is based on the error-correcting training procedure in learning machines, and is referred to as "learning identification." This learning identification is nondisturbing, is applicable to cases where the input signal is random and nonstationary, and can be completed within a short time, so that it may be used to identify linear quasi-time-invariant systems in which some parameters vary slowly in comparison with the time required for identification. This merit also makes it possible to eliminate noise disturbances by means of the moving average method. Computer simulation of the learning identification was carried out and the times required for identification were obtained for various cases. Some modifications of the learning identification were also investigated together with their computer simulations.

523 citations


Journal ArticleDOI
TL;DR: In this article, the authors extended the conjugate gradient minimization method of Fletcher and Reeves to optimal control problems, which is directly applicable only to unconstrained problems; if terminal conditions and inequality constraints are present, the problem must be converted to a non-constrained form; e.g., by penalty functions.
Abstract: This paper extends the conjugate gradient minimization method of Fletcher and Reeves to optimal control problems. The technique is directly applicable only to unconstrained problems; if terminal conditions and inequality constraints are present, the problem must be converted to an unconstrained form; e.g., by penalty functions. Only the gradient trajectory, its norm, and one additional trajectory, the actual direction of search, need be stored. These search directions are generated from past and present values of the objective and its gradient. Successive points are determined by linear minimization down these directions, which are always directions of descent. Thus, the method tends to converge, even from poor approximations to the minimum. Since, near its minimum, a general nonlinear problem can be approximated by one with a linear system and quadratic objective, the rate of convergence is studied by considering this case. Here, the directions of search are conjugate and hence the objective is minimized over an expanding sequence of sets. Also, the distance from the current point to the miminum is reduced at each step. Three examples are presented to compare the method with the method of steepest descent. Convergence of the proposed method is much more rapid in all cases. A comparison with a second variational technique is also given in Example 3.

401 citations


Journal ArticleDOI
TL;DR: In this article, the signal and noise processes are given as solutions to nonlinear stochastic differential equations and several methods of obtaining possibly useful finite dimensional approximations are considered, and some of the special problems of simulation are discussed.
Abstract: Let the signal and noise processes be given as solutions to nonlinear stochastic differential equations. The optimal filter for the problem, derived elsewhere, is usually infinite dimensional. Several methods of obtaining possibly useful finite dimensional approximations are considered here, and some of the special problems of simulation are discussed. The numerical results indicate a number of useful features of the approximating filters and suggest methods of improvement. The paper is concerned with problems where the noise and nonlinear effects are much too large for the use of "linearization" methods, which for the simulated problem, at least, were useless.

336 citations



Journal ArticleDOI
TL;DR: In this paper, the authors explore the possibility of using the instrumental variable method to estimate the parameters of linear time-invariant discrete-time systems and propose an online identification scheme based on recursive computation.
Abstract: This paper explores the possibility of using the instrumental variable method to estimate the parameters of linear time-invariant discrete-time systems. The existence of optimal estimates is established, methods for their approximate computation are given, and an on-line identification scheme based on recursive computation is proposed. Experimental results are included.

Journal ArticleDOI
TL;DR: This paper considers an important class of problems known as measurement adaptive problems, in which control is available over not only the plant but also the measurement subsystem, and shows that the optimization of plant control can be carried out independently of the measurement control optimization.
Abstract: This paper considers an important class of problems known as measurement adaptive problems, in which control is available over not only the plant (i.e., the state equation contains a control variable) but also the measurement subsystem (i.e., the measurement equation contains a control variable). In the general situation the problem is shown to be a generalization of the combined optimization problem. In the special situation of linear systems, quadratic cost, and Gaussian random processes, it is shown that the optimization of plant control can be carried out independently of the measurement control optimization and, furthermore, that optimization of the measurement control can be done a priori. Two examples illustrating this latter situation are presented.

Journal ArticleDOI
TL;DR: In this paper, the problem of finding a filter of fixed order to estimate a time-invariant random process from a related time invariant process is considered, and necessary conditions for a solution are developed and stated in terms of standard Wiener filter theory notation.
Abstract: In this paper two problems are considered, the problem of modeling a given constant linear system by a constant linear system of fixed lower order, and the problem of finding a filter of fixed order to estimate a time-invariant random process from a related time-invariant random process. A quadratic criterion is used to select the optimum system in both cases. It is shown that the filtering problem reduces to the problem of modeling the corresponding Wiener filter. Necessary conditions for a solution are developed and stated in terms of standard Wiener filter theory notation. Numerical solution of the equations embodying the necessary conditions is considered and several examples are presented.

Journal ArticleDOI
TL;DR: In this paper, the optimal linear filtering theory of Kalman and Bucy is extended to include linear systems with multiple time delays as well as the smoothing problem, and an explicit solution is given for the smoothhing problem for systems without time delays.
Abstract: The optimal linear filtering theory of Kalman and Bucy is extended to include linear systems with multiple time delays as well as the smoothing problem. The (ordinary) filter differential equation and variance equation of the Kalman-Bucy theory are replaced by partial differential equations. An explicit solution is given of the smoothing problem for systems without time delays.

Journal Article
G. Axelby1
TL;DR: Questions about future selection of correspondence items, including those containing new technical developments, are being considered and will influence the content, publication speed, and subsequent usefulness of correspondence.
Abstract: The following questions about future selection of correspondence items are being considered. Should those containing new technical developments be placed in the short paper category and thus be subjected to the complete review and selection procedures? Should all or only some be subjected to one or more reviews? Should they be published in the Neuisletter instead of the Transactions to save publication costs? Should the length of each, if discussing new technical developments, be so limited, or should it be labeled a "technical note" instead? Answers to these questions should be of interest to all Transactions readers and will influence the content, publication speed, and subsequent usefulness of correspondence. Your reactions to the status of past, present, and future correspondence items will be greatly appreciated and may create new publication policies to be described in a future editorial.

Journal ArticleDOI
TL;DR: In this paper, the problem of spectral factorization of a class of matrices arising in Wiener filtering theory and network synthesis is tackled via an algebraic procedure, where a quadratic matrix equation involving only constant matrices is shown to possess solutions which directly define a solution to the spectral factorisation problem.
Abstract: The problem of giving a spectral factorization of a class of matrices arising in Wiener filtering theory and network synthesis is tackled via an algebraic procedure. A quadratic matrix equation involving only constant matrices is shown to possess solutions which directly define a solution to the spectral factorization problem. A spectral factor with a stable inverse is defined by that unique solution to the quadratic equation which also satisfies a certain eigenvalue inequality. Solution of the quadratic matrix equation and incorporation of the eigenvalue inequality constraint are made possible through determination of a transformation which reduces to Jordan form a matrix formed from the coefficient matrices of the quadratic equation.

Journal ArticleDOI
TL;DR: In this paper, an alternate derivation of optimal linear filters is presented, using a matrix version of the maximum principle of Pontryagin coupled with the use of gradient matrices to derive the optimal values of the filter coefficients under the requirement that the estimates be unbiased.
Abstract: The purpose of this paper is to present an alternate derivation of optimal linear filters. The basic technique is the use of a matrix version of the maximum principle of Pontryagin coupled with the use of gradient matrices to derive the optimal values of the filter coefficients for minimum variance estimation under the requirement that the estimates be unbiased. The optimal filter which is derived turns out to be identical to the well-known Kalman-Bucy filter.

Journal ArticleDOI
TL;DR: In this paper, a penalty function approach is presented to the solution of inequality constrained optimal control problems, where a point interior to the constraint set is approached from within, by solving a sequence of problems with only terminal conditions as constraints.
Abstract: This paper presents a penalty function approach to the solution of inequality constrained optimal control problems. The method begins with a point interior to the constraint set and approaches the optimum from within, by solving a sequence of problems with only terminal conditions as constraints. Thus, all intermediate solutions satisfy the inequality constraints. Conditions are given which guarantee that the un "constrained" problems have solutions interior to the constraint set and that in the limit these solutions converge to the constrained optimum. For linear systems with convex objective and concave inequalities, the unconstrained problems have the property that any local minimum is global. Further, under these conditions, upper and lower bounds in the optimum are easily available. Three test problems are solved and the results presented.

Journal ArticleDOI
TL;DR: The investigation incorporates basic coding concepts into the currently emerging common basis for automata and continuous systems, and it gives explicit examples of the resulting benefits accruing to each of these areas from the others.
Abstract: Close relationships are established between convolutional codes and zero-state automata and between cyclic codes and zero-input automata. Furthermore, techniques of automata theory and continuous system theory are used to elaborate on the coding problem; and approaches from coding and automata are used to establish and interpret typical structural conditions in continuous systems. The investigation incorporates basic coding concepts into the currently emerging common basis for automata and continuous systems, and it gives explicit examples of the resulting benefits accruing to each of these areas from the others.

Journal ArticleDOI
TL;DR: A review of the theory of dynamic programming and the standard computational algorithm is included and several applications of the new techniques are discussed.
Abstract: Although dynamic programming has long provided a powerful approach to optimization problems, its applicability has been somewhat limited because of the large computational requirements of the standard computational algorithm. In recent years a number of new procedures with greatly reduced computational requirements have been developed. The purpose of this paper is to survey a number of the more promising of those techniques. A review of the theory of dynamic programming and the standard computational algorithm is included. Several applications of the new techniques are discussed.

Journal ArticleDOI
TL;DR: In this paper, a qualitative theory of stability for non-autonomous systems operating over finite time intervals has been developed, and sufficient conditions are given for various types of finite time stability of a system under the influence of perturbing forces which enter the system equations linearly.
Abstract: This paper continues the development of a qualitative theory of stability, recently initiated by the authors, for systems operating over finite time intervals. The theory is motivated by 1) the need for a more practical concept of stability than is provided by the classical theory; and 2) the search for methods for investigating stability of a system trajectory (either analytically or numerically given) without the necessity of performing complicated transformations of the differential equations involved. The systems studied in this paper are nonautonomous, i.e., they are under the influence of external forces, and the concept of finite time stability (precisely defined in the paper) in this case involves the bounding of trajectories within specified regions of the state space during a given finite time interval. (The input is assumed to be bounded by a known quantity during this time interval.) Sufficient conditions are given for various types of finite time stability of a system under the influence of perturbing forces which enter the system equations linearly. These conditions take the form of existence of "Liapunov-like" functions whose properties differ significantly from those of classical Liapunov functions. In particular, there is no requirement of definiteness on such functions or their derivative. The remainder of the paper deals with the problem of determining finite time stability properties of a system from knowledge of the finite time stability properties of lower-order subsystems which, when appropriately coupled, form the original system. An example is given which illustrates some of the concepts discussed in the paper.

Journal ArticleDOI
TL;DR: In this paper, the controllability and observability of the parallel and the tandem connection of two linear time-invariant differential systems were studied using the Jordan form representation.
Abstract: This paper studies the controllability and observability of the parallel and the tandem connection of two linear time-invariant differential systems; it uses the Jordan form representation; it does not assume that the eigenvalues of each representation are simple nor that the two sets of the eigenvalues are disjoint. The controllability and observability of the composite representations require only testing the linear independence of some constant vectors. Some sufficient conditions require just the transfer function matrices.

Proceedings ArticleDOI
TL;DR: A generalized algorithm for on-line identification of a stochastic linear discrete-time system using noisy input and output measurements is presented and shown to converge in the mean-square sense.
Abstract: The parameter identification problem in the theory of adaptive control systems is considered from the point of view of stochastic approximation. A generalized algorithm for on-line identification of a stochastic linear discrete-time system using noisy input and output measurements is presented and shown to converge in the mean-square sense. The algorithm requires knowledge of the noise variances involved. It is shown that this requirement is a disadvantage associated with on-line identification schemes based on minimum mean-square-error criteria. The paper also presents two off-line identification schemes which utilize measurements obtained from repeated runs of the system's transient response and do not require explicit knowledge of the noise variances. These algorithms converge with probability one to the true parameter values.

Journal ArticleDOI
F. Waltz1
TL;DR: In this article, the authors present an approach appropriate to problems in which two or more performance measures can be ranked as "most important", "next most important", etc., with the constraint set at a given stage in the process being chosen on the basis of the results obtained in the previous stages.
Abstract: An area of difficulty in the effective application of modern optimization techniques is often the choice of a performance criterion which adequately reflects the various factors of importance in the proper proportions. This short paper presents an approach appropriate to problems in which two or more performance measures can be ranked as "most important," "next most important," etc. It involves the successive application of the performance measures, with the constraint set at a given stage in the process being chosen on the basis of the results obtained in the previous stages. The resulting optimal control may be a much better compromise choice between the competing criteria than might be obtained by blind reliance on some preselected criterion.

Journal ArticleDOI
TL;DR: In this paper, the effect of errors due to incorrect a priori information on initial states as well as on noise models in continuous Kalman-Bucy filters has been investigated, and a convenient formula for computing error bounds has been derived which will allow parametric studies of the error effect.
Abstract: The effect of errors due to incorrect a priori information on initial states as well as on noise models in continuous Kalman-Bucy filters has been investigated in this paper. A conservative design criterion has been established, and a convenient formula for computing error bounds has been derived which will allow parametric studies of the error effect. The results are applied to typical orbit determination problems of the spacecraft subject to random acceleration having noise contaminated Doppler or range data.

Journal ArticleDOI
T. Pavlidis1
TL;DR: In this article, an extension of Liapunov's second method is presented which can be used for the investigation of the stability of such systems, which is obtained by introducing a positive definite function V(x), which decreases during the occurrence of an impulse and remains constant or decreases during free motion of the system.
Abstract: The class of systems described by differential equations containing impulses is of interest because most models for biological neural nets belong in that category as well as most pulse frequency modulation systems. In this paper an extension of Liapunov's second method is presented which can be used for the investigation of the stability of such systems. This is obtained by introducing a positive definite function V(x) , which decreases during the occurrence of an impulse and remains constant or decreases during the free motion of the system.

Journal ArticleDOI
TL;DR: In this article, a constructive design procedure for the problem of estimating the state vector of a discrete-time linear stochastic system with time-invariant dynamics when certain constraints are imposed on the number of memory elements of the estimator is presented.
Abstract: The paper presents a constructive design procedure for the problem of estimating the state vector of a discrete-time linear stochastic system with time-invariant dynamics when certain constraints are imposed on the number of memory elements of the estimator. The estimator reconstructs the state vector exactly for deterministic systems while the steady-state performance in the stochastic case may be comparable to that obtained by the optimal (unconstrained) Wiener-Kalman filter.

Journal ArticleDOI
TL;DR: In this article, the exact dynamical equation for the mode of the conditional density of a signal x t is derived and discussed, and the exact dynamics of the signal are discussed.
Abstract: The signal x t is a stochastic process satisfying the stochastic differential equation dx = f(x)dt+dz . Observations \dot{y} =g(x) +\xi are taken, where \xi is white noise. The exact dynamical equation for the mode of the conditional density of x t is derived and discussed.

Journal ArticleDOI
TL;DR: Algorithms for explicit calculation of the identification and control of linear discrete systems with Gauss-Markov random parameters and conditioned quadratic cost function are given for systems with perfect state measurement.
Abstract: Real-time identification and control of linear discrete systems with Gauss-Markov random parameters and conditioned quadratic cost function are considered. Algorithms for explicit calculation of the identification and control are given for systems with perfect state measurement and are illustrated by example applications. The control strategy generated by the cost function is an identitication-adaptive controller which is continually revised as new data are received. The identification equations develop an explicit model of the system which is available for other purposes if desired, and can be used apart from the control context.

Journal ArticleDOI
TL;DR: It is shown that under certain conditions an overall optimum system design is obtained by first optimizing the system with all quantizers removed and then applying the procedure for the static open-loop case mentioned above.
Abstract: In this paper the problem of optimally designing a quantizer imbedded in a closed-loop dynamic system is considered. The criterion for the design is that the overall system performance as expressed by a variational criterion is optimized. The function of quantization is thus related to the functions of control and estimation that are performed in the system. First, a procedure is described for optimally designing a quantizer in a static open-loop system, where the design criterion is the expected value of a function of the instantaneous error between the input and output of the quantizer. This procedure reduces the search over all quantizer parameters to an iterative search over a single parameter. Next, the existing methods for finding the optimal design of a quantizer imbedded in a dynamic system are reviewed. The most general method found in the literature involves a combination of dynamic programming with an exhaustive search for all quantizer parameters. The computational requirements of this procedure are quite large even for low-order systems with few quantizer parameters. Finally, a new result is presented that leads to greatly reduced computational requirements for the dynamic system case. It is shown that under certain conditions an overall optimum system design is obtained by first optimizing the system with all quantizers removed and then applying the procedure for the static open-loop case mentioned above. This result is analogous to the separation of the functions of estimation and control that occurs under similar conditions. The computational savings over the existing procedures are very extensive, and the new procedure is computationally feasible for a large class of practical systems.