Reduced-order Estimation Research Articles

Separate-bias estimation with reduced-order Kalman filters

This paper presents the optimal two-stage Kalman filter for systems that involve noise-free observations and constant but unknown bias. Like the full-order separate-bias Kalman filter, this new filter provides an alternative to state vector augmentation and offers the same potential for improved numerical accuracy and reduced computational burden. When dealing with systems involving accurate, essentially noise-free measurements, this new filter offers an additional advantage, a reduction in filter order. The optimal separate-bias reduced order estimator involves a reduced order filter for estimating the state, the order equalling the number of states less the number of observations.

Reduced-order estimator for closed-loop online estimation of cross-directional parameters in a plastics extrusion process

The paper addresses the problem of estimating the ‘effective’ actuator centre locations in a plastics extrusion process while the process is operating in closed loop. It shows that it is possible to use a lower-order model estimator and hence considerably reduce the complexity of the procedure. The estimator is described as a radial basis function network, where the basis functions in the hidden layer are defined using discrete Chebyshev polynomials. The problem is considered as a nonlinear optimisation problem and solved online using a recursive prediction error method.

On-Line Estimation of CD Actuator Locations in a Paper Process

This paper addresses the problem of estimating the effective actuator centre locations in a paper machine whilst the process is operating in closed loop. A reduced order estimator employing a Radial Basis Function Network is described, where the basis functions in the hidden layer are defined using Discrete Chebyshev Polynomials. The problem is considered as a nonlinear optimisation problem and solved online using a recursive prediction error method.

H2 optimal controllers with measurement feedback for discrete-time systems: Flexibility in closed-loop pole placement

For a general H 2 optimal control problem, first all H 2 optimal measurement-feedback controllers are characterized and parameterized, and then attention is focused on controllers with estimator-based architecture. The H 2 optimal control problem with strictly proper controllers and the H 2 optimal control problem with proper controllers are essentially different and hence clearly delineated. Next, estimator-based H 2 optimal controllers are characterized and parameterized. Systematic methods of designing them are also presented. Three different estimator structures, namely prediction, current and reduced-order estimators, are considered. Since in general there exist many H 2 optimal measurement-feedback controllers, utilizing such flexibility and freedom, one can place the closed-loop poles at more desirable locations while still preserving H 2 optimality. All the design algorithms developed here are easily computer-implementable.

REDUCED-ORDER ESTIMATION AND POLE PLACEMENT FOR SPEED CONTROL OF A CENTRIFUGAL SPRAYER PUMP

A servovalve-controlled, hydraulic motor-driven centrifugal pump of the type used on liquid fertilizer andpesticide applicators was linearly modeled using classical control theory in the continuous time domain. A controller wasdesigned according to modern control theory using reduced-order estimation and pole placement in the discrete timedomain. Experimental tests of the closed-loop controller indicated sufficient performance which approximated simulationresults.

Reduced-Order Estimation Technique Using Covariance Data of Observed Value in Linear Discrete-Time Systems

This paper proposes reduced-order estimation technique by the recursive least-squares filter and fixed-point smoother in linear discrete-time systems, given output measurement data. The estimators require the information of the system matrix, the observation vector of the signal generating model and the variances of the state and white Gaussian observation noise processes. This paper shows that these necessary quantities are calculated in terms of the output measurement data for the signal process modeled by the AR (autoregressive) process of order n. As a consequence, we can estimate the signal via the estimators in terms of output measurement data. Furthermore, to shorten the computer implementation time, a reduced-order estimation technique is proposed by decreasing the model order in comparison with the optimum order of the AR model. A numerical simulation example is examined to ascertain that the estimation accuracy based on the reduced-order AR model is almost equivalent to that based on the AR model of optimum order.

Estimation and control of tethered satellite systems

This paper describes the analysis of a tethered system during the station-keeping phase. The system is composed of two small satellites connected by a long tether. The satellites are modeled as particles, and the tether is represented by eight rigid, massive rods. The angles between adjacent elements can vary with time. Only measurements at the end bodies are considered, and the angles between the tether segments are estimated by using a reduced-order estimator. The compensator design is based on the linear quadratic regulator algorithm and a steady-state Kalman filter. It is shown that all motion modes are observable with the available measurements and controllable with thrusts on one of the satellites.

State-Optimization Method for Order Reduction of Linear Models and of State Estimators

Solution of the problem of optimal model reduction and that of optimal reduced-order state estimation are obtained by adopting a proposed state-optimization method. The method is based on the concept of nonsimilarity transformation on states. Factorization of the transformation in terms of a partial isometry enables the controllability and observability conditions of the optimal reduced versions to be expressed in optimal projection equations. The effect of the factorization is interpreted to be that of an additional constrained-condition to the reduction problems. The method is found applicable to systems of any nature with the possibility of preserving the physical significances of original modelled-states in the reduced ones

From Pole Placement Feedback to Estimator Design Using Analog Computers

From pole placement feedback to estimator design using analog computers This paper describes analog-based control lab. simulations. Simulations deal with pole placement feedback and asymptotic observers. For asymptotic state observers, the transfer function approach is just outlined. All simulations are implemented using the state variable approach. Simulations are extended to analyze heuristic pole-placement selection and to evaluate reduced-order estimators.

A Reduced Order Observer for Robust Servomechanism Applications

A reduced order estimator for estimating the state of a linear system driven by constant unknown disturbances is proposed in this article. It is assumed that the unknown disturbances can effect the plant, and as well they can contaminate the measurements. The paper outlines a systematic design approach for designing the estimator for multivariable systems. Necessary and sufficient conditions for existence of the PIO is given. Finally, the performance of such an estimator is illustrated through a numerical example.

Concurrent hyperthermia estimation schemes based on extended Kalman filtering and reduced-order modelling.

The success of treating cancerous tissue with heat depends on the temperature elevation, the amount of tissue elevated to that temperature, and the length of time that the tissue temperature is elevated. In clinical situations the temperature of most of the treated tissue volume is unknown, because only a small number of temperature sensors can be inserted into the tissue. A state space model based on a finite difference approximation of the bioheat transfer equation (BHTE) is developed for identification purposes. A full-order extended Kalman filter (EKF) is designed to estimate both the unknown blood perfusion parameters and the temperature at unmeasured locations. Two reduced-order estimators are designed as computationally less intensive alternatives to the full-order EKF. Simulation results show that the success of the estimation scheme depends strongly on the number and location of the temperature sensors. Superior results occur when a temperature sensor exists in each unknown blood perfusion zone, and the number of sensors is at least as large as the number of unknown perfusion zones. Unacceptable results occur when there are more unknown perfusion parameters than temperature sensors, or when the sensors are placed in locations that do not sample the unknown perfusion information.

Optimal control of a single zone environmental space

The problem of finding an optimal control scheme for space heating is explored. A bilinear model for the plant is linearized, and an optimal control law is found by solving the Riccati equation. The output response of the system to impulse and step disturbances is investigated. A simplified design leading to a suboptimal control law, where only the dominant states of the plant are considered is given. The system performance with optimal and suboptimal control is compared. A reduced order estimator is designed and system performance with the estimator is tested under simulated step disturbances. From the view point of zone air temperature control, the system performance with suboptimal controller is shown to be acceptable.

Decentralized state estimation in large-scale interconnected dynamical systems

A simple approach for decentralized state estimation in large-scale control systems is proposed. Given that a large-scale dynamical system can be decomposed into a number of interconnected input-output decentralized subsystems, a method for designing totally decentralized local reduced order estimators is proposed. It is shown that totally decentralized state estimation may be possible as long as the number of the subsystem interaction inputs are less than or equal to the locally measured outputs. Here the interconnection effects among the subsystems are treated as totally unknown inputs to each subsystem. The method builds upon a novel approach in designing reduced order unknown input observers (UIO) for dynamical systems, and it therefore results in totally decentralized estimators' structure. The pros and cons of the estimation scheme are pointed out and necessary and sufficient conditions for designing local estimators are stated. Finally, the design approach is verified by a numerical example.

Fixed-order sampled-data estimation

For the Kalman filter-type sampled-data estimation problem utilizing an averaging A/D device, the equivalent discrete-time problem is shown to be of increased order. The fixed-structure optimal projection approach for reduced-order, discrete-time estimation is applied to the equivalent discrete-time problem in order to characterize reduced-order estimators.

Robust Fault Detection based on Low Order Models

In this paper we consider the problem of comparing 2 SISO noise corrupted input-output data records and trying to decide if they were produced by the same linear dynamic system or not. We propose to use a likelihood ratio test statistic based on the BLUE’s of the parameters of the dynamic systems for the resultant hypothesis testing problem. For the case of Gaussian distributed additive noise disturbances on the output data, we examine the power of the critical region for this statistic and relate it to the signal to noise ratio for the data. We then show how a reduced order estimator may be intelligently chosen and applied to one of the data records in order to derive a test statistic with more powerful critical region, so that for a given false alarm rate, less true faults will be missed. A simulation example illustrating the efficacy of these approaches concludes the paper.

Optimal reduced-order observer-estimators

A unified approach to designing reduced-order observer-estimators is presented. Specifically, an attempt is made to design a reduced-order estimator satisfying an observation constraint which involves a prespecified, possibly unstable subspace of the system dynamics and which also yields reduced-order estimates of the remaining subspace. The results are obtained by merging the optimal projection approach to reduced-order estimation of D.S. Bernstein and D.C. Hyland (IEEE Trans. Autom. Control, vol.AC-30, p.583-5, 1985) with the subspace-observer results of the authors (Proc. IEEE Conf. on Decision and Control, p.2364-6, Dec. 1988). A salient feature of this theory is the treatment of unstable dynamics within reduced-order stable-estimation theory. In contrast to the standard full-order estimation problem involving a single algebraic Riccati equation, the solution to the reduced-order observer-estimator problem involves an algebraic system of four equations consisting of one modified Riccati equation and three modified Lyapunov equations coupled by two distinct oblique projections. >

Optimal reduced-order estimators in the frequency domain: the discrete-time case

The discrete-time stationary Kalman filter problem is solved in the z-domain for nth-order systems with >m outputs, where 0 ≤ >k ≤ >m measurements are noise-free. The design equations are very similar to the continuous-time cage, and they can be solved by spectral factorization, giving the polynomial matrix D¯∼( z) which parametrizes the reduced-order optimal estimator in the frequency domain. By solving a single linear equation the equivalent time-domain representation for the optimal filter can also be obtained. A simple example demonstrates the filter design in the frequency domain.

Comments on ‘Design of reduced-order optimal estimators directly in the frequency domain’

The derivation by Hippe (1989) of a frequency-domain Riccati equation for a reduced-order Kalman filter was somewhat lengthy. It is shown that this derivation can be considerably streamlined by simple arguments.

Design of reduced-order optimal estimators directly in the frequency domain

The frequency-domain design of optimal reduced-order estimators ((Caiman filters) for continuous-time linear time-invariant systems containing 0 ≤ k ≤ m noise-free output measurements is considered. In a direct frequency-domain approach to state feedback and estimation the ‘states’ of a system do not appear explicitly, and therefore we here focus our attention mainly on the frequency-domain characteristics of the optimal filter. By suitably modifying known time-domain results, a simple polynomial matrix equation can be derived from which the optimal filter parameters are obtained by spectral factorization. The equivalence between the time- and the frequency-domain results is established with the aid of a non-minimal representation of reduced-order observers.

Optimal reduced-order state estimation for unstable plants

The problem of optimal reduced-order steady-state state estimation is considered for the case in which the plant has unstable poles. In contrast to the standard full-order estimation problem involving a single algebraic Riccati equation, the solution to the reduced-order problem involves one modified Riccati equation and one Lyapunov equation coupled by a projection matrix. This projection is completely distinct from the projection obtained by Bernstein and Hyland (1985) for stable plants.