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Nigeria power generation is circa 4000 MW as at October 2016, this is far less than the national required value.Jebba Hydroelectric Power Station (JHEPS) is one of the three major hydropower stations in the country which its maximumaverage generation value is still less than the rated value of 578.4 MW. This paper presented some basic concepts of analysisof failure and repair. An algorithm was set up to extract the contiguous set of up-times and down-times to determine time-tofail (TTR) and time-to-repair (TTR) events, these events were analyzed for stochastic study. It was observed that the longestup time was 262 days and seen in turbo-alternator unit five, while the shortest was zero day and seen in the entire units exceptthat of unit two. The longest downtime was 133 days and seen in the turbo-alternator of unit one, the least time-to-repair is aday and common to all the units. The frequency distributions for both TTF and TTR showed that 79% of TTF events and 57%of TTR events are distributed in the first 2 classes

Low-density parity-check (LDPC) codes are capable of achieving excellent performance and provide a useful alternative for the high-performance applications. However, at medium to high signal-to-noise ratios, an observable error floor arises from the loss of independence of messages passed under iterative graph-based decoding. In this paper, the error floor performance of the short block length codes is improved by the use of a novel candidate selection metric in code graph construction. The proposed multipath extrinsic message degree (EMD) approach avoids harmful structures in the graph by evaluating certain properties of the cycles introduced in each edge placement. We present multipath EMD-based designs for several structured LDPC codes, including quasi-cyclic and the irregular repeat accumulate codes. In addition, an extended class of the diversity-achieving codes on the challenging block fading channel is proposed and considered with the multipath EMD design. This combined approach is demonstrated to provide the gains in decoder convergence and error rate performance. A simulation study evaluates the performance of the proposed and existing state-of-the-art methods.

A novel construction for irregular low-density parity-check (LDPC) codes based on a modification of the Progressive Edge Growth (PEG) algorithm is presented. Edge placement of the PEG algorithm is enhanced by use of the Sum-Product algorithm in the design of the parity-check matrix. The proposed algorithm is highly flexible in block length and rate. The codes constructed by the proposed methods are tested in the AWGN channel and significant performance improvements are achieved. The flexibility of the proposed decoder optimisation operation is then shown by its use in modifying the Improved PEG (IPEG) algorithm to achieve further performance gains.

A novel algorithm to design Root-Check LDPC codes based on Progressive Edge Growth (PEG) techniques for block-fading channels is proposed. The performance of the new codes is investigated in terms of the Frame Error Rate (FER) and the Bit Error Rate (BER). Numerical results show that the codes constructed by the proposed algorithm outperform codes constructed by the existing methods by 0.5dB.

A novel algorithm to construct structured LDPC codes based on graph connectivity and cycle quality is proposed. The performance of the constructed QC-LDPC and IRA codes is investigated and a performance improvement in the error floor region is demonstrated. The code constructions provided are flexible in rate and length and demonstrate that the enhanced EMD-driven PEG-based approach proposed may be applied to a number of other structured LDPC code classes, given consideration of correct constraints of the construction.

A novel construction for quasi-cyclic (QC) regular and irregular low-density parity-check (LDPC) codes based on a modification of the QC Progressive Edge Growth (PEG) algorithm is presented. Edge placement of the PEG-based algorithm is enhanced by use of the sum-product algorithm in the design of the parity-check matrix. The proposed algorithm is highly flexible in block length and rate, in particular when compared with algebraic constructions. The codes constructed by the proposed methods are tested in the AWGN channel and performance improvements are achieved. The proposed QC-LDPC codes provide an inherent trade-off between code performance and encoding/decoding complexity.

Important modifications to the theory of evidential reasoning (Dempster-Shafer theory) are presented. By relaxing the requirement that basic probability mass assignments have unity norm and by omitting the normalization step in Dempster′s rule of combination the algorithm is improved. A detailed comparison with the original theory is made and formal results are proved. Applicability of the modifications is examined and some cautionary notes are given.

A novel construction for irregular low density parity check (LDPC) codes based on a modification of the Progressive Edge Growth (PEG) algorithm is presented. The edge placement procedure of the PEG algorithm is enhanced by the use of the Sum-Product decoder in the design of the parity-check matrix. The proposed algorithm, like the PEG algorithm, is highly flexible in block length and rate. The codes constructed by the methods presented are tested in the AWGN channel and significant performance improvements are achieved. The flexibility of the proposed decoder optimisation operation in its application to PEG-based construction methods is then demonstrated by its use in modifying the Improved PEG (IPEG) extension to the PEG algorithm to achieve further performance gains.

Many digital signal processing applications require linear phase filtering. For applications that require narrow-band linear phase filtering, frequency sampling filters can implement linear phase filters more efficiently than the commonly used direct convolution filter. In this paper, a technique is developed for designing linear phase frequency sampling filters. A frequency sampling filter approximates a desired frequency response by interpolating a frequency response through a set of frequency samples taken from the desired frequency response. Although the frequency response of a frequency sampling filter passes through the frequency samples, the frequency response may not be well behaved between the specific samples. Linear programming is commonly used to control the interpolation errors between frequency samples. The design method developed in this paper controls the interpolation errors between frequency samples by minimizing the mean square error between the desired and actual frequency responses in the stopband and passband. This design method describes the frequency sampling filter design problem as a constrained optimization problem which is solved using the Lagrange multiplier optimization method. This results in a set of linear equations which when solved determine the filter's coefficients.

Many digital signal processing applications require linear phase filtering. Under certain conditions, a frequency sampling filter can implement a linear phase filter more efficiently than an equivalent filter implemented by a direct convolution structure. However, frequency sampling filter structures require pole-zero cancellations on the unit circle. For practical implementations of frequency sampling filters, finite word length effects usually prevent exact pole-zero cancellation. An uncancelled pole on the unit circle causes the filter to be unstable. Therefore, designers move the filter's poles and zeros off the unit circle and onto a circle of radius r where r < 1, by replacing z−1 with rz−1 in the filter's system function. As a result of this modification, the frequency response of the new filter using rz−1 where r < 1 is different from the frequency response of the original filterwhere r = 1. As r decreases, the frequency response of the modified frequency sampling filter increasingly differs from the frequency response of the original filter, and thus values of r close to 1 are usually chosen. However, it has been shown that the filter's output roundoff noise decreases as r decreases. Thus, there is a need for a design technique which chooses a value of r which renders an acceptable roundoff noise level while satisfying various frequency response design constraints. This paper develops an optimization method for designing a frequency sampling filter with r < 1 such that a linear combination of the mean square error between the desired and actual frequency responses in the passband and stopband and the sum of the square error of the impulse response symmetry is minimized while the stopband frequency samples are constrained to zero.

A technique is developed for designing linear phase frequency sampling filters where the interpolation errors between frequency samples are controlled by minimizing the mean square error between the desired and actual frequency responses in the stopband subject to constraints on the passband frequency response. The frequency sampling filter design problem is defined as a constrained optimization problem, which is solved using the Lagrange multiplier optimization method. The Lagrange multiplier optimization method results in a set of linear equations, the solution of which determines the filter's coefficients. >

A convexity proof of the function λ max ( e D A e −D + ( e D A e −D ) ∗ ) with respect to the D matrix is presented. This function is applicable to the analysis of the multivariable control system stability margin with one-sided structured uncertainty. The multivariable stability margin (MSM) of a control system can be defined as the ‘size’ of the smallest structured perturbation in component dynamics (e.g., aerodynamic stability or control derivatives) that will destabilize the system. Convexity of this function will guarantee the reliable convergence of an optimization algorithm towards a stationary point which is certain to be the global minimum. This function improves the existing MSM bound when the sign of the perturbation is known a priori. Using this property, a reliable stability margin program has been developed and used to analyze the robustness of several multivariable flight control system designs.

Under certain conditions, frequency sampling filters can implement linear phase filters more efficiently than direct convolution filters. Unlike a direct convolution filter, a frequency sampling filter uses a recursive structure which requires exact pole zero cancellation on the unit circle. When implemented with special purpose digital hardware or as a program on a digital computer, errors due to finite length registers and finite precision arithmetic prevent exact pole zero cancellation making the filter unstable. To stabilize the filter, the poles and zeros on the unit circle are moved to a circle of radius r where 0 r 1. The errors due to finite length registers and finite precision arithmetic also degrade the performance of frequency sampling filters. In this chapter, the effects that finite precision fixed point arithmetic have on frequency sampling filters are analyzed. These effects, represented as output noise levels, are determined as a function of register length, time and the value of r.

Under certain conditions, a frequency sampling filter can implement a linear phase filter more efficiently than an equivalent filter implemented by a direct convolution structure. However, the system function of a frequency sampling filter requires pole-zero cancellations on the unit circle. For practical implementations, finite word length effects usually prevent exact pole-zero cancellation which can result in filter instability. To prevent instability, the poles and zeros on the unit circle can be moved to a circle of radius r where 0 The resulting filter is guaranteed to be stable; however, the filter’s magnitude and phase characteristics are affected by the choice of r. This chapter develops a method for determining optimal coefficients which minimizes a linear combination of the mean square error in the stopband and passband, and the sum of square error of the impulse response symmetry subject to passband and stopband constraints and a fixed value of r

A discrete time transport processor is a discrete time system that is comprised only of delays, adds and subtracts. For example, a linear time invariant digital filter whose implementation uses only coefficients of + 1, − 1 and 0 is a discrete time transport processor. This chapter develops a technique for the design and implementation of frequency selective linear phase discrete time transport processors. This technique determines an optimal integer valued finite impulse response and determines a linear time invariant transport processor structure that can realize this impulse response.

How should the minimizing player choose the values of certain parameters, if he wants to further optimize his payoff at the maximizing player’s expense? Hence what would be the greatest lower bound for the maximizing player’s payoff? To answer these questions, necessary conditions for parameter optimization for linear quadratic differential games will be derived. Iterative numerical techniques for determining optimal parameters will be developed. Search techniques which will locate a “small” region of uncertainty in which the optimal parameter must lie will also be discussed.

A robust tracking control algorithm based mainly on the theory of variable structure systems and Lyapunov's direct method is developed in this paper, to treat the tracking control problem of a class of non-linear time-varying systems with uncertain dynamics. A nominal model is used instead of an exact one. together with a description of the bounds of plant uncertainty. The objective is to design a control law such that all the plants lying within a specified band of uncertainly will be driven to reach the goal. The modelling error, time-varying system dynamics, system uncertainties, and disturbances are all treated as general disturbances to the nominal system. This algorithm greatly eases the difficulties present in requiring a precise model of the physical system for the control purpose and for handling the non-linear time-varying system. The non-linear time-varying system is first linearized to a linear time-varying system (with time-varying coefficients completely defined) via the method of variable s...

In the past, a major objection to application of the well known solution to the linear stochastic optimal control problem has been that the complexity, or dimension, of the solution exceeds that which is necessary for satisfactory feedback control. For a system described by n first order linear differential equations excited by white noise, the solution requires a filter of dimension n. If non white noise sources excite the system, an even higher dimensional filter is required. This paper presents a technique which alleviates the above objection. A recursive algorithm similar to the Kalman filter algorithm is presented which permits design of a reduced order linear estimator to replace the well known Kalman filter. The new estimator, called an observer, is stochastically optimal subject to its reduced order dimensionality constraint, but its performance is not as good as a full Kalman filter. The observer algorithm is general in that it applies to time variable, multivariable systems. A minimum order = n - m"1 exists for the observer. Here n = state dimension and m"1 = dimension of the non white measurement. Non white noise sources of any order q may exist and need not contribute to the dimension of the optimal observer. Optimal observers of all orders between n - m"1 and n - m"1 + q may be designed, the latter case being a Kalman filter. Detailed examples are given to illustrate the theory.

An observer theory for estimating the state variables of deterministic linear time-invariant dynamic forced system is presented. The variation in the cost performance when some of the state variables are estimated by using a linear observer system is evaluated.

A formula is derived for interpolation between output samples of a fast Fourier transform (FFT), i.e., in the frequency domain. Such a formula is useful for obtaining greater frequency resolution when two coarse FFT outputs are available. Consideration is also given to the effect of such interpolation on a weighted FFT.

A combination of pure colored and pure white measurement noise may occur in several classes of linear discrete time dynamic stochastic systems of interest. Accurate state estimation in the presence of this nonwhite measurement noise is achieved through the use of a low order least squares filter. Previously the statistical nature of this nonwhite noise has been restricted, and computationally burdensome algorithms have been developed to handle this particular type of measurement noise. In overcoming these difficulties a filter is developed, similar in form to a Kalman filter, whose order is no higher than the order of the state.

One of the most important objectives of a radar angle-tracking loop is to keep the target within the beamwidth of the radar antenna. Thus, the behavior of the antenna pointing error is of vital interest in determination of tracking performance. For a tracker with a general polynomial linearity (representing nonlinear receiver characteristics), subjected to constant line-of-sight rate inputs, random initial antenna pointing errors, and white Gaussian receiver noise, a method to obtain approximations to the transient mean and variance of the antenna pointing error as explicit functions of time is presented.

A minimal-order observer for obtaining state estimates in linear stochastic time-delay systems is presented. The system model considered contains white noise disturbance in both the process and the output measurements. An observer gain matrix is derived which minimizes the estimation error in the mean-square sense. The error variance of the observer has been compared to that of an optimal Kalman filter using a second-order time-delay system as an example

The Prigogine–Defay ratio, Π≡ΔCp Δκ/TV (Δα)2, has been derived for systems whose states need to be specified in terms of a number of order parameters z by considering the conditions of tangency of the equilibrium free energy surface and the free energy surface for constant values of the order parameters. It is shown that Π?1. The equality applies for the case of a single order parameter or for the case of more than one order parameter which satisfies the constraint (∂V/∂zi)/ (∂S/∂zi) = (∂V/∂zk)/(∂S/∂zk). The inequality applies for the case of more than one order parameter which does not satisfy this constraint. The relevance of the Prigogine–Defay ratio to the kinetics of the structural relaxation in glass is discussed.

The applicability of sensitivity functions to parameter estimation of nonlinear, time variant systems was investigated. The method showed the ability to move the parameter vector toward the direction of optimality, but the nonlinearity of the system created problems of stability and rate of convergence of the algorithm. These problems were corrected by the introduction of artificial variable step sizes to improve the rate of convergence at start and to dampen the system near optimality. Thus modified, the method yielded a satisfactory solution of the estimation problem.

A combination of bilinear transformation, the Routh stability criterion and the integral squared error (ISE) criterion is proposed for the linear model reduction of high-order discrete-time systems. The approach consists of a four-step computational scheme. In the first three steps, by the use of the bilinear transformation and Routh approximation methods, the denominator of the reduced model is obtained. In the fourth step, a recursive algorithm is used to minimize the variance of the spectral density between the original and the reduced model to obtain optimal coefficients for the numerator of the reduced model.

In this paper we define and solve the multiple linear quadratic gaussian (LQG) problem for discrete-time systems using Salukvadze's ideal point method which treats dynamic, but deterministic, multicriteria optimization problems. The solution consists in designing a scalar regulator where the weighting matrices are replaced by the sums of their counterparts in the individual criteria. The resulting control is optimal, in the sense that it minimizes the difference between the cost of the implemented policy from the cost that would occur if we optimized each criterion separately and added the outcomes. All the nice properties of the scalar LQG regulator are enjoyed by the multicriteria regulator as well. Extension to the continuous-time case is straightforward, yielding identical results.

Least squares estimation techniques are employed to overcome previous difficulties encountered in accurately estimating the state and measurement noise covariance parameters in linear stochastic systems. In the past accurate and rapidly converging covariance parameter estimates have been achieved with complex estimation algorithms only after specifying the statistical nature of the noise in the system and constraining the time variation of the covariance parameters. Weighted least squares estimation allows these restrictions to be removed while achieving near optimal accuracy using a filter on the same order of complexity as a Kalman filter. Allowing the covariance parameters to vary in as general a manner in time as the state in a linear discrete time stochastic system, and assuming that a Kalman filter is applied to this system using incorrect knowledge of the a priori statistics, it is shown how a covariance system is developed similar to the original system. Unbiased least squares estimates of the covariance parameters and of the original state are obtained without the necessity of specifying the distribution on the noise in either system. The accuracy of these estimates approaches optimal accuracy with increasing measurements when adaptive Kalman filters are applied to each system.

A new smoothing algorithm is presented for discrete models with independent observation and disturbance noise. This algorithm first reduces the smoothing problem to a multiple hypothesis testing problem, and then estimates the states of the models by using a stack sequential decoding algorithm. Simulation results have shown that the algorithm performs well.

A suboptimum smoothing algorithm is presented for dynamic systems with or without an arbitrary random interference. For the disturbance noise, observation noise and interference, only independency is assumed. The model functions are any defined functions. This algorithm estimates the states by first representing the state model by a trellis diagram and then using a suboptimum decoding algorithm. It requires a constant memory for its implementation. Numerical results have shown that for either linear or non-linear discrete models with interference, the new algorithm performs very well. Further, in the absence of interference, it performs better than the extended Kalman filter algorithm for some non-linear discrete models.

The mathematical basis for the approximation technique involving truncation of the mixed Cauer form is presented and its superiority through the entire frequency range of the system response is established.

The sensitivity of the discrete-time Kalman filter to errors in the state transition matrix is examined. The approach makes use of the matrix Taylor expansion and estimates the deviation from the ‘nominal’ computed performance. Although only the first term of the expansion is retained, satisfactory accuracy is achieved. The method is considerably simple and constitutes a quick way to diagnose or establish limits for divergence. Direct application to stochastic observers is also possible

Global positioning by means of satellites requires simultaneous observation by at least four satellites. The problem is to determine the minimum number of satellites and the corresponding orbital geometry necessary to satisfy this requirement on a continuous basis. To model the problem, a fixed number of users are assumed uniformly distributed in a known manner over the surface of the earth, and the satellites are restricted to exist in either three or four orbital planes. However, the orbit radius and inclination angle are left as variables. Under these assumptions, and starting with a small number of satellites which will be increased afterwards, an algorithm is developed to determine the visibility of satellites at each surface location. In this way it is possible to specify the minimum number of satellites needed by any desired orbital geometry. It is found that the number of satellites required for three-dimensional continuous worldwide coverage decreases as the orbit radius is increased. There appears to be no general trend regarding the effect of the inclination angle on the minimum number of satellites.

The optimal control solution of a dynamical system with time delays is generally very difficult to obtain. In this paper, a uniformly convergent iterative algorithm is developed to obtain a partially closed-loop suboptimal control function for the time-varying linear tracking system with multiple time delays in both state variables and control variables such that a proposed quadratic cost functional was minimized. The essential feature of this method was to treat the time-delayed functions evaluated from the (K–1)th iteration step as perturbations to the kth optimizing iteration of the non–delayed system. The convergency rate of suboptimal solution was very fast.

In this work, we consider a class of multistage stochastic linear differential game with quadratic cost function subject to a post initial measurement decision constraint, Secure strategies, in the sense of minimax and maximin, are derived for each player by applying dynamic programming. Both of the minimax and maximin controls are shown to be of linear feedback form. It is demonstrated that both the certainty equivalence principle and the separation theorem from stochastic control theory hold. The derived algorithm, even for the multistage and stochastic nature of the problem, is of closed form and can be employed in real-time applications

The problem of finite time controllability of linear time invariant systems by piecewise constant output feedback is considered. It is shown that complete controllability and complete observability are both necessary and sufficient conditions for output feedback controllability by piecewise constant gain. Dead-beat output feedback controllers that drives the system to the origin in at most (μd × vd) steps, are also presented, where μd and vg are the indices of controllability and observability.

In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for discrete-time pole assignment by output feedback for linear time-invariant systems. The resultant output feedback gain is a periodic function of time with a period equal to the sampling interval. It is also shown that periodic discrete-output feedback gains can be used in stabilizing a linear time-invariant system if and only if its unstable modes are controllable and observable.

A stack sequential decoding algorithm is used to estimate the states of dynamic systems with an arbitrary random interference in the observation model. First, the models are approximated so that the motion model is represented by a trellis diagram. Second, some metrics are assigned to the nodes, branches, and paths of this trellis diagram. Finally, using a stack sequential decoding algorithm, the states are estimated. The simulation results have shown that the new smoothing algorithm performs well.

Linear discrete stochastic control systems containing unknown multiple time delays, plant parameters and noise variances are considered. An algorithm is established which uses the maximum-likelihood technique to identify the unknown parameters. An estimated likelihood function is evaluated based on the previous parameter estimates, which in turn generates a new descent direction vector to update the unknown parameters. The delays and plant parameters are identified in their respective parameter spaces. An example of a second-order stochastic system has been implemented by digital simulation to demonstrate the applicability of the algorithm.

Lower bounds for a quadratic cost functional, applicable to certain optimal control problems, arc derived. These bounds are tight, and are found to bo useful in determining the quality of suboptimal controls. The use of a cost functional lower bound in the snboptimal control of a linear distributed parameter system is included.

The necessary and sufficient conditions for the completion of a class of pursuit— evasion games are given. The state variables of the pursuer are assumed to be defined by a set of linear differential equations and the evader's state variables are given by a vector valued non-linear differential equation. The game is defined to be completed if certain (or all) state variables of the pursuer ; i.e. position, velocity, etc. are equal to the corresponding state variables of the evader. The Krein's L-method is used to determine the existence of the optimal completion time and the optimal player strategies. The general results are specialized to the case in which the evader's state vector is also governed by a set of linear differential equations. A computational technique—steepest descent method has been developed to obtain the numerical solution of the game. Three examples are presented to illustrate the method. The first one deals with a linear, second order, pursuit-evasion game with constant coeff...

In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for the complete controllability of linear time-invariant systems by discrete output feedback. In the case where the desired final state is the origin, the resultant, output feedback gain is periodic and the transfer is accomplished in at most vd periods, where vd is the discrete-time index of observability. In the more general case where the desired final state is non-zero, the transfer is accomplished using at most vd output measurements.

Luenberger's observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. An interesting new solution to the problem of constructing optimal and suboptimal reduced-order observers is presented. The solution contains as special cases both Kalman's optimal filter and the optimal minimal-order observer of Leondes and Novak. Also, the Tse and Athans observer is obtained as a special case of the reduced-order observer solution.

The decentralized control of discrete time stochastic linear systems is considered in this paper. The decentralized control laws are obtained by determining a sequential Nash equilibrium point for a set of quadratic cost functions. In the absence of disturbances, the Nash equilibrium set has the property of driving the state of the closed-loop system to the origin in finite time or asymptotically. In addition, the control law is a linear function of the instantaneous output measurements and can be computed on-line using relatively simple algorithms.

In the present paper it is shown that the optimal solution to the stochastic adaptive control of linear dynamic systems with uncertainties depends on the unconditional means and covariances of the uncertainties. The random Riccati-like systems of equations generated as a result of the presence of multiplicative and additive noise leads to the necessity of constraint on the means and covariances of the uncertainties. Furthermore, the existence and uniqueness of an optimal solution of the given system depends upon the above constraints : steady-state solutions exist only under restrictive statistical conditions of the system parameters

This paper presents a new smoothing algorithm for discrete models with arbitrary random interference. For the disturbance noise, observation noise and interference, only independency is required. Moreover, the motion and observation models are not restricted to be linear functions of the disturbance noise and interference. This algorithm estimates the states by reducing the smoothing problem to a multiple composite hypothesis testing problem, and then using the Viterbi decoding algorithm. Simulation results have shown that the new algorithm performs very well.

For an n-dimensional linear non-stationary stochastic discrete system with multiplicative and additive noise, the problem of estimation of the plant phase coordinate vector from partially noisy observation is considered. An algorithm is developed which is of lower dimension than the Kalman-Bucy filtering algorithm, namely (n-l)

Practical filters for non-linear system, in the sense of having- finite dimensional equations, are treated through a new approach which combines the model simplification and filtering aspects of the problem. Three filters are derived, the first extended filter (E1F), the second extended filter (E2F), and the second extended non-linear filter (E2NF). The EIF enjoys the accuracy of the extended Kalman filter (EKF) and the computational simplicity of the Kalman filter (KF). The E2F and the E2NF have structures similar to the KF and the EKF respectively, with with new formulae for the filter gain. The gain formula for the E2F and the E2NF provide an explanation as to the effect of the system non-linearities. Numerical results comparing the performance of the developed filters to those of other known filters like KF, EKF, and the modified second order filter (M2F) using the van der Pol equations are given.

Background information on the Kalman filter is given first. A discussion of the filter parameters and their a priori determination follows. The discussion points out the need for adaptive determination of the process noise statistics. The filter innovations are presented as a means for developing the adaptive criteria. The criteria center around the estimation of the true mean and covariance of the filter innovations. A method for the numerical approximation of the mean and covariance of a locally stationary random process is presented. The definition of a local stationarity is presented. Local stationarity allows for the separation of the process statistics into a stationary component and a time-varying component. The separation method is discussed. A method for estimating the stationary and time-varying components is presented. As an example of its application to real problems, the algorithm is applied to the problem to the problem of reentry trajectory estimation for the Space Shuttle. Both the adaptive algorithm and the steady-state Kalman filter are applied to the problem. The results of the reconstructions are presented. The adaptive algorithm exhibits superior performance.