Linear State Estimation Research Articles

Minimax state estimation for linear stochastic systems with noise uncertainty

The problem of minimax linear state estimation for linear stochastic systems driven and observed in noises whose second-order properties are unknown is considered. Two general aspects of this problem are treated: the single-variable problem with uncertain noise spectra and the multivariable problem with uncertain componentwise noise correlation. General minimax results are presented for each of these situations involving characterizations of the minimax filters in terms of least favorable second-order properties. Explicit solutions are given for the spectral-band uncertainty model in the single-variable cases treated and for a matrix-norm neighborhood model in the multivariable case. Characterization of saddlepoints in terms of the extremal properties of the noise uncertainty classes is also discussed.

On-Line Computer Control of a Nuclear Reactor Using Optimal Control and State Estimation Methods

The paper discusses an on-line computer control system for a nuclear reactor using optimal state feedback based on the linear quadratic regulator and state estimation using a Kalman filter. Practical methods are presented for selecting the gain of the state estimator. It is shown that when pure time delays occur in the output states it is possible to use the Kalman filter to virtually eliminate the effects of the time delay on the control loop performance. The experimental results obtained using the control scheme show that the control scheme designed using modern control techniques has improved performance compared with conventional reactor control methods having greater speed of response, wider stability margins and significantly improved noise filtering properties

Multipartitioning linear estimation algorithms: Continuous systems

The partitioned estimation algorithms of Lainiotis for the linear continuous-time state estimation problem have been generalized in this paper in two important ways. First, the initial condition of the estimation problem can, using the results of this paper, be partitioned into the sum of an arbitrary number of jointly Gaussian random variables; and second, these jointly Gaussian random variables may be statistically dependent. The form of the resulting algorithm consists of an imbedded Kalman filter with partial initial conditions and one correction term for each other partition or subdivision of the initial state vector. Emphasis in this paper is on ways in which this approach, called multipartitioning, can be used to provide added insight into the estimation problem. One significant application is in the parameter identification problem where identification algorithms can be formulated in which the inversion of the information matrix of the parameters is replaced by simple division by scalars. A second use of multipartitioning is to show the specific effects on the filtered state estimate of off-diagonal terms in the initial-state covariance matrix.

On decentralized estimation and control with application to freeway ramp metering

This paper presents an approach to the suboptimal design of time invariant, infinite time horizon, L-Q-G regulator in which decision making and state estimation are decentralized into local subsystems. For a prespecified information pattern (set of admissible intercommunications) the problem of finding a best linear control law and state estimator is converted to a parameter optimization which can be solved off line. A modified Davidon-Fletcher-Powell algorithm is used to obtain numerical solutions to the static optimization. A design is illustrated by means of a 12 state freeway corridor ramp metering problem; behavior of centralized and decentralized designs in a nonlinear macroscopic simulation of the freeway are provided.

Hybrid Estimators for Discrete-Time Stochastic Systems

The problem of estimating the state of a linear discrete-time system from input-output data in the presence of input disturbance noise when some of the outputs are not obscured by output measurement noise is considered. The proposed hybrid estimator gives the linear least squares state estimate and has order less than the order of the system whose state is being estimated by the number of outputs which are not obscured by noise.

A note on the design freedom in singular state estimation

A singular case of linear state estimation with exact observations is analyzed with the object of obtaining an explicit characterization of the design freedom that exists in this case.

The use of pseudo and oblique pseudo-inverse matrices in power system state estimation algorithms

The applications of generalized, pseudo and oblique pseudo-inverse matrices to both linear and non-linear state estimation algorithms are studied. Examples showing the solutions to power system network state estimation problems are given along with computational schemes for the matrices. The oblique pseudo-inverse matrix is shown to be particularly useful when parts of the system are effectively isolated because of lost data measurements.

On Optimal Control of Linear Systems in the Presence of Multiplicative Noise

This correspondence considers the problem of optimal regulator design for discrete time linear systems subjected to white state-dependent and control-dependent noise in addition to additive white noise in the input and the observations. A pseudo-deterministic problem is first defined in which multiplicative and additive input disturbances are present, but noise-free measurements of the complete state vector are available. This problem is solved via discrete dynamic programing. Next is formulated the problem in which the number of measurements is less than that of the state variables and the measurements are contaminated with state-dependent noise. The inseparability of control and estimation is brought into focus, and an enforced separation solution is obtained via heuristic reasoning in which the control gains are shown to be the same as those in the pseudo-deterministic problem. An optimal linear state estimator is given in order to implement the controller.

Measurement designs to improve the computational efficiency of linear system state estimators

Abstract A design procedure for the measurement process is proposed to decrease the computational burden of state estimation in linear discrete-time systems. System order reduction is involved, which is achieved by a derived transformation, yielding an alternative system and measurement process representation that includes time delays in both the system and the measurement process. Without loss of filter optimality, measurement designs yielding reductions in the computational requirements of the Kalman filter applied to the reduced order system are derived. A computational comparison is included to show that the proposed method, as opposed to the direct application of the Kalman filter, can yield large reductions in the filter computation time.

Estimation and Feedback in Linear Time-Varying Systems: A Deterministic Theory

The problems of (i) state-variable feedback, (ii) deterministic state estimation, and (iii) state estimation and feedback are considered for a finite-dimensional linear time-varying system. The following are the questions; they can be regarded as natural and logical extensions to a time-varying case of the pole assignment problem [2]–[6] for a time-invariant system.(i) What is the relation between controllability and reachability of an open loop system and the possibility of constructing a linear state-variable feedback law which realizes the prescribed stability and instability degrees of the closed loop system?(ii) What is the relation between observability and reconstructibility of the system to be estimated and the possibility of constructing a linear state estimator which realizes the prescribed stability and instability degrees of the estimate error?(iii) What is the relation between controllability, reachability, observability and reconstructibility of an open loop system and the possibility of con...

Canonical equations for boundary feedback control of stochastic distributed parameter systems

The plant considered is described by a set of time-invariant linear partial differential equations with spatially generalized Wiener process disturbance of the state and point measurement. Boundary control is used to minimize the expected value of a functional that is quadratic in the state and control. The well-known separation into linear optimal state estimator and linear optimal deterministic controller is heuristically shown using an extension of the Caratheodory lemma to obtain sufficient conditions. Each resulting Hamilton-Jacobi equation is shown equivalent to a pair of linear canonical partial differential equations. Use of the canonical equations leads to exact analytical solutions under certain conditions by means of closed loop modes. This permits a paper and pencil analysis of the engineering properties of the control of a large class of practical distributed systems with boundary controls and point measurements. An example is given of the boundary control of a one space dimension diffusion equation with input noise at the other boundary and noisy measurement taken at an internal point. Because an analytical solution is obtained, the effect of the location of this internal measurement point can be analyzed.

Empirical Bayes state estimation in discrete time linear systems

Linear dynamic systems state estimation, using empirical Bayes decision theory to develop filter set

Estimation of multi-input systems with ‘noiseless’ outputs

A procedure is presented for linear discrete system state estimation in which the observation does not contain additive white noise. The order of the system for estimation is reduced according to the number of noiseless outputs and noiseless output differences (here noiseless implies the absence of white noise). The advantage of this approach is its general application. Unlike earlier methods, it is not necessary for the noise states to be separable or even distinguishable from other system states. A systematic approach to the formulation of the required estimation model from a general system model is presented. The procedure is straightforward but somewhat lengthy to describe for the general problem. However, once the basic scheme is understood, it can be applied without undue difficulty to a large number of problems. For the modified estimation model, required filtering algorithms are derived which are of essentially the same complexity as the usual white noise algorithms. Direct estimation of the original system states, using the reduced order system for calculation of gains and the required error variances, is described. Modelling errors often occur in representing physical systems, and error analysis algorithms are derived. The use of the estimation and error analysis algorithms is demonstrated by examples and the results compared with the usual discrete (augmented states) Kalman filter approach.

On the stability of fixed-lag smoothing algorithms

The stability of fixed-lag smoothers for linear system state estimation is considered. The main result proved is that the smoother is unstable when the usual set of conditions hold which guarantee asymptotic stability of the optimum filter.

An algebraic solution of the state estimation problem.

The general linear state estimation problem is considered here from a purely algebraic viewpoint without resorting to the concepts of probability theory. A least-squares parameter estimation formula is first established and then extended to permit estimation on a sequential basis. By combining the relations derived with the properties of the state transition matrix, the solutions to the problem of filtering, smoothing, and prediction for systems with both forcing and measurement noise are found.

Application of curvature methods to parameter and state estimation

Parameter and state estimation may be formulated as a nonlinear minimisation problem, and many iterative techniques are available to perform the minimisation. However, the class of techniques using curvature has received little attention in the past, largely a result of excessive computational difficulties and convergence problems. In this paper, it is shown that the curvature methods are readily adaptable to parameter and state estimation, and it is also shown that the curvature methods are more rapidly convergent than other iterative techniques. Examples in linear and nonlinear parameter and state estimation serve to illustrate the advantages of the curvature methods.