Showing papers on "State vector published in 1972"
TL;DR: In this article, the formulation of the optimal, linear, time-invariant, multivariable control problem with quadratic performance index is extended so that the solution includes multivariability integral feedback and model-following capabilities in addition to the normal proportional state feedback.
43 citations
TL;DR: An exact method is given for the temperature determination of devices in circuits analysed by nonlinear-circuit-analysis programs by introducing the ‘thermal-node’ temperatures as well as the node voltages.
Abstract: An exact method is given for the temperature determination of devices in circuits analysed by nonlinear-circuit-analysis programs. The method is based on the simultaneous solution of the electronic and thermal network equations by introducing the ‘thermal-node’ temperatures as well as the node voltages. These voltages and temperatures are arranged in a joint state vector.
41 citations
TL;DR: In this paper, the estimation of a vector of unknown spatially varying parameters in non-linear partial differential equations from noisy observations is considered, and a method of steepest descent based on consideration of the unknown parameter vector as a control vector is presented.
Abstract: The estimation of a vector of unknown spatially varying parameters in non-linear partial differential equations from noisy observations is considered. Two algorithms are presented. The first is a method of steepest descent based on consideration of the unknown parameter vector as a control vector. The second is baaed on treating the parameter as an additional state vector and employing least-square filtering. Computational results are presented on the estimation of the diffusivity in the heat equation
35 citations
TL;DR: This paper extends the application of QP to non-linear problems through quasi-linearization and the solution of a sequence of linear-quadratic sub-problems whose solutions converge to the solutions of the original non- linear problem.
Abstract: Quadratic programming (QP) has previously been applied to the computation of optimal controls for linear systems with quadratic cost criteria. This paper extends the application of QP to non-linear problems through quasi-linearization and the solution of a sequence of linear-quadratic sub-problems whose solutions converge to the solution of the original non-linear problem. The method is called quasi-linearization-quadratic programming or Q-QP. The principal advantage of the Q-QP method lies in the ease with which optimal controls can be computed when saturation constraints are imposed on the control signals and terminal constraints are imposed on the state vector. Use of a bounded-variable QP algorithm permits solution of constrained problems with a negligible increase in computing time over the corresponding unconstrained problems. Numerical examples show how the method can be applied to certain problems with non-analytic objective functions and illustrate the facility of the method on problems ...
23 citations
TL;DR: In this article, the authors describe the application of minimum-variance estimation techniques for in-flight alignment and calibration of an inertial measurement unit (IMU) relative to another IMU and/or some other reference.
Abstract: This is the second part of a two-part paper which summarizes work pursued by the author in 1967 [2]. The paper describes the application of minimum-variance estimation techniques for in-flight alignment and calibration of an inertial measurement unit (IMU) relative to another IMU and/or some other reference. The first paper [1] formulates the problem, and this paper reports numerical results and analyses. The approach taken is to cast the problem into the framework of Kalman-Bucy estimation theory, where velocity and position differences between the two IMU's are used as observations and the IMU parameters of interest become part of the state vector. Instrument quantization and computer roundoff errors are considered as measurement noise, and environmental induced random accelerations are considered as state noise. In this paper, numerical results for three important IMU error parameter configurations are presented and discussed. The main results of the paper determine the effects of state and observation noise levels and the nominal trajectory on the identifications of the errors for these configurations. A discussion of the minimum number of trajectory maneuvers and of the optimal trajectory maneuvering is given.
22 citations
TL;DR: In this paper, it was shown that the quantum-mechanical state vector correctly describes not only the probabilities for the outcomes of measurements, but also the correlations between the outcome of successive measurements.
Abstract: It is shown that the quantum-mechanical state vector correctly describes not only the probabilities for the outcomes of measurements, but also the correlations between the outcomes of successive measurements. In particular, von Neumann's axiom $M$ is shown to be redundant. Consequently, no extra---quantum-mechanical "reduction" of the joint object-apparatus state vector is required for a full statistical description of a sequence of measurements. It is also shown that any attempt to determine experimentally whether or not a reduction of the joint state vector has taken place during a measurement is incompatible with the preservation of the outcome of that measurement.
20 citations
TL;DR: In this article, a compatible dynamic observer of the Luenberger type is designed to reconstruct the entire state vector for a synchronous machine-infinite bus system, and the performance of the system with the observer cascaded to the optimal controller is compared with the performance when all the state variables are available for feedback.
Abstract: The closed-loop optimal control law obtained for a linear time-invariant system always requires the entire state vector to be available for direct measurement It is seldom that all the state variables of a physical system are available for measurement A compatible dynamic observer of the Luenberger type is designed to reconstruct the entire state vector for a synchronous machine-infinite bus system The performance of the system with the observer cascaded to the optimal controller is compared with the performance of the system when all the state variables are available for feedback The transfer matrix relating the output and input is derived
19 citations
TL;DR: This paper extends the approach to include all desired handling qualities by using the model x = Lx + 7V8, where 8 is the vector of pilot commands to computes both optimal feedback and optimal feedforward from 8 to the controls.
Abstract: The technique known as implicit model following allows desired closed-loop characteristics of a system to be included in an optimal control algorithm. Previous authors have used a model of the form x = Lx9 where x is the state vector. The optimal control algorithm then computes the feedback which makes the system response optimally close to the model. This paper extends this approach to include all desired handling qualities by using the model x = Lx + 7V8, where 8 is the vector of pilot commands. The algorithm which is derived computes both optimal feedback and optimal feedforward from 8 to the controls. Algorithms are given for both sampled-data and continuous control. General guidelines for choosing L and TV are presented. An example is given for the design of the landing approach control for a short takeoff and landing (STOL) airplane.
17 citations
TL;DR: In this article, a two-level control structure is proposed to consider all known properties of a plant within an open-loop, feed-forward, upper level control scheme containing a model of the plant to be controlled and a controller which feeds back the model's state vector.
16 citations
TL;DR: In this article, the random vibration statistics of linear or perturbed linear dynamic systems with variable parameters, in particular of lifting rotors with feedback controls, are analyzed using shaping filters to solve the response variance matrix directly from another set of linear equations.
15 citations
TL;DR: In this article, an optimal control policy for a discrete-time linear system with interrupted observations and an expected quadratic cost is proposed, which is realized by cascading a nonlinear estimator, which computes the conditional mean of the state vector, with the optimal feedback gain matrix in which all uncertainties are removed.
TL;DR: In this article, the problem with undetermined final time is transformed to a problem with fixed final time by augmenting the control vector with one additional component, and a fixed number of steps can then be used for the determination of unknown constant system parameters or unknown initial conditions of the state vector.
Abstract: For solving non-linear control problems several methods exist, the conjugate-gradient method being one of the most successful Gradient and conjugate-gradient methods have also been extended to treat control problems with undetermined final time Two basic approaches have been suggested; one in which the final time in each iteration is determined by a search and the other in which the final time for the next iteration is computed In both approaches the number of steps in the numerical integration procedure is variable In this paper an alternative approach is suggested By augmenting the control vector with one additional component the problem with undetermined final time is transformed to a problem with fixed final time A fixed number of steps can then be used The same approach can be applied to the determination of unknown constant system parameters or unknown initial conditions of the state vector and for the computation of the on-off control
TL;DR: In this article, it is suggested that instead of totally neglecting the affected state variables, a minimal sensitivity estimate of these be obtained on the basis of the nominal values of the imperfectly known parameters.
Abstract: The problem of divergence that often arises when the filter mechanization is based on lower order process dynamics is examined. This situation may be caused, for example, by the impossibility of completely identifying some of the process parameters. It is suggested that instead of totally neglecting the affected state variables a minimal sensitivity estimate of these be obtained on the basis of the nominal values of the imperfectly known parameters. The result of applying this scheme to a system, part of whose state vector is thus affected, is also examined.
TL;DR: The receiver is shown to effectively eliminate intersymbol interference and perform as well as a matched-filter receiver that uses only the signal energy received during one message interval.
Abstract: The problem of eliminating intersymbol interference is studied from the viewpoint of channel-state estimation. The channel state vector, at the beginning of the present message baud, contains the information required to eliminate the effects of past channel inputs. A receiver is proposed that makes a maximum-likelihood estimate of the channel state vector conditioned upon all previous digits having been received correctly. The channel-state estimate is used to control the motion of a variable decision threshold. Correct adjustment of the threshold removes the effects of past symbols on the present decision. The receiver performance is studied via certain approximation techniques used in the study of threshold learning systems. In particular, the performance of one- and two-pole channels is evaluated. In these cases, the receiver is shown to effectively eliminate intersymbol interference and perform as well as a matched-filter receiver that uses only the signal energy received during one message interval.
TL;DR: In this paper, it is shown that Renninger's negative result does constitute an actual measurement process, which is a process termed by Schrodinger disentanglement, and it is proposed to regard a state vector as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.
Abstract: The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrodinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrodinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.
01 Jan 1972
TL;DR: In this paper, the authors compare Kalman filtering and quasilinearization for the identification of system parameters and show that the corresponding estimates will be identical provided that the Kalman estimation is performed with the following restrictions: the state vector about which the process equations are linearized is the solution to the state equations containing the smoothed parameter and initial condition estimates computed at the end of the last data cycle.
Abstract: : Two techniques which have had wide application in the identification of system parameters are Kalman filtering and quasilinearization. This paper discusses the quantitative differences between these two procedures and shows that corresponding estimates will be identical provided that the Kalman estimation is performed with the following restrictions: The state vector about which the process equations are linearized is the solution to the state equations containing the smoothed parameter and initial condition estimates computed at the end of the last data cycle. No updating of this trajectory should be performed as the data is serially processed. At some time after the start of the data record the recursive equations should be initialized with the parameter and state estimates which correspond to a least squares curve fit of the data up to that time. (Author)
TL;DR: In this article, it is shown that the expected value of the performance index may be written as a sum of two terms under the assumption of first-order variations of the system state.
Abstract: Development of a design method for including the effects of parameter uncertainties in the design of linear control systems. The approach taken to this problem may be classified as a special case of the stochastic control problem. Thus the formulation is based on the minimization of the expected value of a quadratic performance index defined in terms of the system state vector. The uncertainty in the value of the performance index is the result of the statistical nature of the system parameters rather than a random input signal. It is shown that the expected value of the performance index may be written as a sum of two terms under the assumption of first-order variations of the system state. The first of these terms expresses the nominal performance of the system when the system parameters assume their mean values. The second term represents the effect of the uncertainties on the expected value of the performance index, and is interpreted as an index of system sensitivity.
TL;DR: In this article, the problem of observing the states of two tandem-connected linear stationary systems is considered, and it is shown that if the tandem systems are observable then the tandem system observer can be composed of two separate systems, each the same order as the corresponding system being observed.
11 Sep 1972
TL;DR: In this paper, the authors derived N-burn analytic solutions for trajectories of a vehicle in a vacuum between arbitrary boundary conditions, in general, in terms of variational changes in the control vector and in the initial state vector.
Abstract: Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.
TL;DR: An appropriate iterative algorithm for computing the control law is developed using an augmented state vector which includes the plant, model and command signal, and the results of a digital simulation are presented.
Abstract: A suitable I.P. is developed for use in a discrete implicit model-following problem. An appropriate iterative algorithm For computing the control law is developed using an augmented state vector which includes the plant, model and command signal. The algorithm was tested on an aeroplane lateral axis stability augmentation system problem, and the results of a digital simulation are presented.
22 Aug 1972
TL;DR: An approach is presented based on the state-variable representation for random processes and makes use of the Markovian nature of the state vectors for simultaneous detection of random signals in additive white Gaussian noise, estimation of the signals and identification of the systems that generate the signals.
Abstract: : A unified approach is presented for the problem of simultaneous detection of random signals in additive white Gaussian noise, estimation of the signals and identification of the systems that generate the signals. The approach is based on the state-variable representation for random processes and makes use of the Markovian nature of the state vectors. Optimal solutions are derived for the cases where both the dynamical system for the signal state and the observation data are continuous, and both the system and the observation data are discrete. Optimal solutions are presented in the form of optimal nonlinear filtering for the sufficient statistic for Bayes decision, the signal state-vector and the parameter vector that characterize the system. This problem is treated for both single shot and multishot observations of various signal sequences. In addition, the optimal state vector estimation for the linear systems where the initial state vector is non-Gaussian is presented. (Author Modified Abstract)
TL;DR: In this article, the optimal solution to a cooperative game which is described by a common quadratic cost functional subject to linear ordinary differential equation constraints is presented, and the optimal controller is constructed for both players when all components of the state vector can be measured.
Abstract: The optimal solution is presented to a cooperative game which is described by a common quadratic cost functional subject to linear ordinary differential equation constraints. The optimal feedback controller is constructed for both players when all components of the state vector can be measured. For the case that only some components of the state vector are available, the optimal feedback controller is designed using a dynamic observer. As a specific application, the approach is used in determining the optimal feedback controllers for two satellites which are performing a rendezvous on a circular orbit. Simulation results are presented to illustrate the applicability of the approach.
01 Mar 1972
TL;DR: In this article, the problem of minimum mean-square state estimation for linear stationary systems is considered, and a quantitative measure for the degree of optimality for this estimate is derived.
Abstract: The problem of minimum mean-square state estimation for linear stationary systems is considered. State-vector partitioning is employed to arrive at a computationally efficient estimate, and a quantitative measure for the degree of optimality for this estimate is derived. This quantitative measure can then be used to relate the degree of optimality to the particular state-vector partition employed, thus providing an ordering over all admissible partitions of the system. A method of selecting the best partition (i.e. the one that maximises the degree of optimality) is outlined.
TL;DR: In this paper, the optimally sensitive control is extended to the problem of state dynamics with unknown time-varying inputs, where the state dynamics contain a vector of stochastic inputs which can be represented as Martingale processes.
Abstract: The implementation of control systems capable of identifying and adapting to time-varying unknown parameters has become increasingly important in air-traffic control and other applications. In the recent literature the control problem, in which both the initial state vector as well as a vector of constant plant parameters are unknown, has been treated utilizing sensitivity techniques referred to as optimally sensitive control. The concepts of optimally sensitive control as developed by Kokotovic, Perkins, Cruz, and others are extended to the problem in which the state dynamics contain a vector of stochastic inputs which can be represented as Martingale processes. The resulting optimally sensitive system is shown to be an effective and realistic adaptive controller for systems containing unknown time-varying parameters. A numerical example is presented to demonstrate the effectiveness of the resulting control system at identifying and adapting to the levels of the unknown time-varying inputs.
01 Jun 1972
TL;DR: The Precision State and Filter Weighting Matrix Extrapolation Routine (PSME) as mentioned in this paper provides the capability to extrapolate any spacecraft geocentric state vector either backwards or forwards in time through a force field consisting of the earth's primary central-force gravitational attraction and a superimposed perturbing acceleration.
Abstract: The Precision State and Filter Weighting Matrix Extrapolation Routine is described which provides the capability to extrapolate any spacecraft geocentric state vector either backwards or forwards in time through a force field consisting of the earth's primary central-force gravitational attraction and a superimposed perturbing acceleration. The routine also provides the capability of extrapolating the filter-weighting matrix along the precision trajectory. This matrix is a square root form of the error covariance matrix and contains statistical information relative to the accuracies of the state vectors and certain other optionally estimated quantities. The routine is a cooled algorithm for the numerical solution of modified forms of the basic differential equations which are satisfied by the geocentric state vector of the spacecraft's center of mass and by the filter-weighting matrix.
TL;DR: In this article, the question is asked: When does there exist a family of coordinate transformations defined by a set of nonsingular matrices T(\lambda), continuously dependent on \lambda, with T(0) = I and with mapping the state vector associated with one minimal realization into the one associated with the other?
Abstract: Suppose one is given two minimal realizations of the same transfer function matrix. The question is asked: When does there exist a family of coordinate transformations defined by a set of nonsingular matrices T(\lambda) , continuously dependent on \lambda , with T(0) = I and with (1) mapping the state vector associated with one minimal realization into the state vector associated with the other? The quesion is answered, and a procedure is given for constructing the family when it exists.
TL;DR: In this article, a compatible controller theory is proposed for optimizing time-varying linear plants, which can often be implemented with less complexity than that required of observers or Kalman filter type designs.
Abstract: Problems relating to the optimal control of linear systems with quadratic cost invariably lead to analytic solutions requiring complete state feedback for their implementation. It is often the case in actual practice, however, that only a subset of the entire state vector can be measured. In such cases, one approach is to construct an estimator of the complete state via dynamic feedback elements, whose output is then used to construct a control that asymptotically approaches the optimum. A more general compatible controller theory, described here for optimizing time-varying linear plants, can often be implemented with less complexity than that required of observers or Kalman filter type designs.
01 Oct 1972
TL;DR: The matrix extrapolation routine presented in this article is a coded algorithm for the numerical solution of modified forms of the basic differential equations which are satisfied by the geocentric state vector of a spacecraft's center of mass and by the filter-weighting matrix.
Abstract: The matrix extrapolation routine presented is a coded algorithm for the numerical solution of modified forms of the basic differential equations which are satisfied by the geocentric state vector of a spacecraft's center of mass and by the filter-weighting matrix. The equations are described in detail and the various input and output variables designated. The capability is thus established to extrapolate any spacecraft's geocentric state vector, either backwards or forwards in time, through a force field consisting of the earth's primary central-force gravitational attraction, and a superimposed perturbing acceleration.