Data processing: structural design – modeling – simulation – and em – Modeling by mathematical expression
Reexamination Certificate
1999-08-10
2001-09-04
Teska, Kevin J. (Department: 2123)
Data processing: structural design, modeling, simulation, and em
Modeling by mathematical expression
C700S044000, C700S038000, C708S300000
Reexamination Certificate
active
06285971
ABSTRACT:
FIELD OF INVENTION
The invention pertains to the field of nonlinear system state estimation and control in which the nonlinear system is describable by a set of nonlinear differential equations. More specifically the invention relates to the use of extended Kalman filtering (EKF) type techniques for state estimation and system control.
BACKGROUND TO THE INVENTION
Observation of a physical system for understanding its behavior requires obtaining access to key parameters (state variables) within the system. Often, these state variables are not directly available for observation so that they must be inferred from indirect and noisy measurements. Optimal linear estimation theory has been developed for estimating these state variables by producing a minimal error estimate from their contaminated images. The need for optimal estimation technology is widespread and includes such diverse applications as monitoring system behavior in hostile environments, estimating system parameters for system modeling, estimating (detecting) messages in communication systems, remote measuring systems, and controlling of physical systems.
Optimal estimation techniques are based on statistical signal processing (filtering) techniques for extracting a statistical estimate of the desired state variables from inaccurate and distorted measurements by minimizing a prescribed error function. The form of the error function determines the nature of the estimator optimality.
Kalman filtering (Kalman, R. E., “A New Approach to Linear Filtering and Prediction Problems”, Trans. ASME, J. Basic Eng., Vol. 82D, pp. 34-45, March 1960) is an optimal filtering technique commonly used for estimating state variables of a linear system. Kalman filtering is a time domain operation that is suitable for use in estimating the state variables of linear time-varying systems that can be described by a set of linear differential equations with time-varying coefficients (linear differential equations with constant coefficients being a special case). Although Kalman filtering has found application in state estimation in many systems that may be approximately described as linear, the basic Kalman filter technique can not adequately accommodate general nonlinear systems. Because nonlinear systems are common, attempts have been made to adapt the Kalman filter to estimation of states in nonlinear systems by quasi-linearization techniques. These adaptations, when restricted to computationally feasible methods, result in sub-optimal estimators which do not yield a minimal (optimal) error estimation.
Because it is desirable to use digital computers for applying Kalman filter techniques, discrete-time (time-sampled) adaptations have been developed. Discrete Kalman filtering is ideally suited for estimation of states in discrete-time systems that can be properly described by a set of finite difference equations with discrete time-varying coefficients. However, because of the strong incentives for incorporating digital signal processing techniques for continuous-time Kalman filter estimation, extensive use has been made of discrete Kalman filters in continuous-time system state estimation.
Because Kalman filters (and other state variable estimators, such as the Luenberger observer (Luenberger, D. G., “Observing the State of a Linear System”, IEEE Trans. On Military Electronics, pp. 74-80, April 1964)) are based on a model of a system whose states are to be estimated, the use of a discrete Kalman filter for estimating the states of a continuous system implies modeling the continuous system as a discrete time-sampled system in which integrators are replaced by accumulators and continuous time-varying coefficients are replaced by discrete time-varying coefficients. In addition to propagating the state estimates, estimators may propagate state estimate error covariance using a model that may be a different approximation to the original continuous system model. As the time interval between sampled data points is increased, the approximation error for each model increases and may result in model behaviors that departs drastically from the actual system behavior.
When the effects of modeling continuous time systems by using discrete-time models are combined with the inaccuracies of quasi-linear model approximations to nonlinear systems, estimator error stability can be severely impaired. Loss of stability means severe departure between the actual state values and the estimates. Increasing the sampling rate of the model produces smaller finite difference intervals with improved performance but often at an unacceptable cost for the increased computational burden.
Optimal state estimators for linear systems form the basis for optimal control in the so-called Linear Quadratic Gaussian (LQG) control problem (Reference: Applied Optimal Control, A. E. Bryson and Y. C. Ho, John Wiley & Sons, 1975). System state variables are estimated using optimal estimation techniques and then a quadratic objective performance criterion is applied to establish a control design strategy. Kalman filtering is commonly used as a means for estimating the state variables of a system. When the estimated state variables are combined, using the control law based on the objective performance criterion (or performance index), optimal LQG control system results.
The objective performance criterion summarizes the objectives of the system by the performance index, J, which is a mathematical expression that must be minimized in order to meet the objectives of the system. For example, the performance index can represent the final error, total time, or the energy consumed in meeting the system objective. The various possible performance indices may be used individually or in combination so that more than one objective is satisfied.
FIG. 1
is a block diagram of a basic state variable control system
10
in which the physical plant
11
that is to be controlled has a vector (multichannel) input, u(t), and a state vector, x(t), with vector elements corresponding to the set of state variables representing the attributes of plant
11
that are to be controlled. State vector x(t) is generally not directly accessible but through the measurement vector, z(t), which are available from a set of appropriate sensors in sensor unit
12
. The measurement vector may have more or less elements than the state vector. These measurements can also be contaminated by measurement noise due to the system environment and/or due to sensor noise. Typically, the sensors are transducers that convert the various physical elements of the state vector representing diverse physical activity (e.g. heat, light, mass, velocity, etc.) into electrical signal representations suitable for additional signal processing. State observer
13
accepts vector z(t) as a noisy and distorted representation of the state vector x(t) from which an estimated state vector, {circumflex over (x)}(t), of the state vector is made. The estimated state vector, {circumflex over (x)}(t), is a statistical estimate of necessity because of the stochastic nature of the random noise introduced into the measurement process and into the dynamics. The estimated state vector, {circumflex over (x)}(t), is then used by controller (−C(t))
14
to form a suitable input vector, u(t), to drive plant
11
. Because of the stochastic nature of real control systems, the performance index, J, must be expressed as an average value, {overscore (J)}=E{J}, where E{.} is the expectation operator. In order to accommodate time-varying plants, i.e. plants that change physical attributes as a function of time, controller
14
may also be a time-varying controller based upon a time-varying performance index, {overscore (J)}(t), and a time varying state observer
13
.
FIG. 2
is a signal flow block diagram representing a plant such as plant
11
of FIG.
1
. Summing junction
111
provides at its output vector {dot over (x)}(t), the time derivative of state vector x(t) that is the sum of three inputs to summing junction
111
: the output vector
Pandey Pradeep
Shah Sunil C.
Blakely & Sokoloff, Taylor & Zafman
Phan Thai
Teska Kevin J.
Voyan Technology
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