LEWIS,F.L.Optimal Estimation: With an Introduction to Stochastic Control Theory.John Wiley and Sons, Inc., 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the first two are actually bilinear in either X or U and dP for = 1 or 2, respectively. This is necessary so that a modification of the LQG analysis Work supported in part by the National Science Foundation Grants DMS93 01107, DMS 96 26692 and CDA 94 13948. see Bryson and Ho [5] or Lewis [11]) will work from the dynamic programming point of view. It is assumed that the SDE (1) is interpreted in the sense of It o. The LQGP problem with linear dynamics in the form of (1) has its origins in an early set of lecture notes by Wonham [13] on random differential equations. A similar ....

....equation (PDE) 9) is known as the HamiltonJacobi Bellman equation and is subject to the final condition, v(x; t f ) 2 x S(t f )x. 3.1. Formal LQGP Solution To solve (9) assume a modification solution of the form for a LQG problem (for the usual LQG, see Bryson and Ho [5] or Lewis [11]) v(x; t) S(t)x D (t)x E(t) T Delta ( S( d: 10) The modification of the assumed solution form takes into the account that the Poisson mark distribution is not centered, i.e. non zero mean amplitudes, so a linear term with time dependent coefficient D (t) and a ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

....model and the nonlinear measurement model: where w k N (0,Q) and v k N (0, R) We assume model covariance matrices equal to: where, a1, and Q and R are constant matrices. For a 1, as time k increases, the R k and Q k decrease, so that the most recent measurement is given higher weighting [5]. If a=1, it is a regular EKF. Corrected position, velocity,etc Predicted measurements Estimated INS errors Pseudo range Residuals a Figure 2. Fuzzy adaptive extended Kalman filter. Computation of the Kalman gain and covariance matrices has been described in [6] Here we concentrate only on ....

Lewis F.L. 1986. Optimal Estimation with Introduction to Stochastic Control Theory. John Wiley and Sons, New York.

....be modeled as AWGN [1] 7] As discussed in Section I, it is practically impossible to exactly model the packet jitter as is. In [4] a maximum likelihood estimator (MLE) approach was employed instead, which could be a good choice of estimator for the jitter of unknown statistical characteristics [8], but cannot be an optimal estimator. In addition, it does not take advantage of the a priori information on the source clock frequency and . C. Estimation of Source Clock Frequency The prefilter in the KALP is intended to transform the packet jitter into a low pass signal, which then can be ....

F. Lewis, Optimal Estimation With an Introduction to Stochastic Control Theory. New York: Wiley, 1986.

.... detectors and the scattering coefficients (matrix S) we are asked to calculate the absorption coefficients at every position (matrix P ) From system s point of view, the inverse problem can be treated as a parameter identification problem, which can be solved using nonlinear filtering techniques [1,2,5, 6, 7, 8, 10]. The extended Kalman filtering method is a standard approach to nonlinear filtering problem and has particular applications in parameter estimations [7] To enhance the convergence properties of the extended Kalman filter, higher order terms of the power series are also considered in the model ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

....Also, the brackets around the coefficients, H 1 (t)x] and [H 2 (t)u] indicate nonstandard linear algebra or matrix forms. Analogous terms can be used to generalize the Gaussian coefficient, G(t) This is necessary so that a modification of the LQG analysis (see Bryson and Ho [2] or Lewis [7]) will work from the dynamic programming point of view. The LQGP problem with linear dynamics in the form of (1) has its origins in an early set of lecture notes by Wonham [11] on random differential equations. A similar formulation,but including Work supported in part by the National Science ....

....to the final condition: v(x; t f ) 1 2 x T S(t f )x: 15) 4 The argument of the minimum function is called the Hamiltonian of the system. 4. Formal Solution To solve (14) assume a modification solution of the form for a LQG problem (for the usual LQG, see Bryson and Ho [2] or Lewis [7]) v(x; t) 1 2 x T S(t)x D T (t)x E(t) 1 2 Z t f t d(GG T ) S( 16) The modification of the assumed solution form takes into the account that the Poisson mark distribution are not centered, i.e. not zero mean amplitudes, so a linear term with time dependent coefficient D T ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

....interactions of the locations of the features are modeled, in that the movement of each feature plays a role in updating the state estimates x k , as described below. 5 Extended Kalman Filter This system utilizes the Extended Kalman Filter (EKF) an extension of the standard Kalman Filter [23] [16] to the case of nonlinear observation and dynamic functions. Many tracking researchers use KF or EKF to track individual features or pixels, thus estimating the motion of the feature in the image plane or the workspace [22] 21] 25] After tracking, feature location and movement information is ....

F. L. Lewis. Optimal Estimation with an introduction to stochastic control theory. John Wiley & Sons, New York, NY, 1986.

....: 19) 2.2 The Extended Kalman Filter The problem of tracking an articulated object, such as a robot, is not a linear problem. Therefore the derivation above does not exactly capture the dynamics of the problem. Update equations for the exact nonlinear system dynamics can be derived (see [8] for an example) but they have been shown to be intractable. The Extended Kalman Filter (EKF) provides an approximation to the nonlinear system dynamics by linearizing them about the current state estimate. Consider a nonlinear model of the form x k 1 = f (x k ) G 0 k (x k )w k (20) z k = h ....

F. L. Lewis. Optimal Estimation with an introduction to stochastic control theory. John Wiley & Sons, New York, NY, 1986.

....for matrices A and A, of dimension n Theta m, vec(A A) vec(A) vec( A) show that the continuous time Lyapunov equation (3.21) can be written as vec( P u ) F(t) Omega I n I n Omega F(t) vec(P u ) G(t) Omega G(t) vec(Q) equivalent to Problem 3. 1 1 in Lewis [94]. d) Analogously, show that the discrete time Lyapunov equation (3.40) can be written as vec(P u (k 1) Psi(k 1; k) Omega Psi(k 1; k) vec(P u (k) Gamma(k) Omega Gamma(k) vec(Q k ) equivalent to Problem 2.2 1 in Lewis [94] 1 Leibnitz integration rule is d dt Z g(t) ....

....vec(Q) equivalent to Problem 3.1 1 in Lewis [94] d) Analogously, show that the discrete time Lyapunov equation (3.40) can be written as vec(P u (k 1) Psi(k 1; k) Omega Psi(k 1; k) vec(P u (k) Gamma(k) Omega Gamma(k) vec(Q k ) equivalent to Problem 2. 2 1 in Lewis [94]. 1 Leibnitz integration rule is d dt Z g(t) h(t) f(t; d = Z g(t) h(t) f(t; t d f [t; g(t) dg(t) dt Gamma f [t; h(t) dh(t) dt 51 4. Maybeck [101] Problem 2.15) a) Show that, for all t 0 , t 1 , and t, Phi(t; t 0 ) Phi(t; t 1 ) Phi(t 1 ; t 0 ) by showing that both ....

Lewis, F.L., 1986: Optimal Estimation with an Introduction to Stochastic Control Theory. John Wiley & Sons, 376 pp.

....= x kjk Gamma1 K k (z k Gamma D k u k Gamma H k x kjk Gamma1 ) k = 1; 2; 3.16) 3.2 Extended Kalman Filter Often, the systems we are interested in are not linear, and therefore, the derivation above does not capture their dynamics exactly. Exact update equations can be derived (see [87]) but they are intractable. Fortunately, the robustness of the standard Kalman filter is often enough to compensate for nonlinearities in the system. However, sometimes a more precise approximation to the nonlinear system dynamics is needed. In the extended Kalman filter (EKF) we take the first ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory. New York: John Wiley & Sons, 1986.

....optimal control is known as the regular control, reg MG29 . To solve (20) subject to the final condition, for the LQGP problem a modification of the formal state decomposition of the solution for the usual LQG problem (for the usual LQG, see Bryson and Ho [3] Dorato et al. 4] or Lewis [9]) is assumed: 3 MG24 1 M 0 54 0 76 u98 8 7 7 ; 4 ; 5 : f (23) 3 The final condition is satisfied, provided that 8 2=4 14600 6 2 f (24) The ansatz (23) would not, in general, be true for the state dependent case, but would be ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

.... Independence To solve (26) subject to the final condition (25) for the LQGP problem (for further details see Westman and Hanson [10] a modification of the formal state decomposition of the solution for the usual LQG problem (for the usual LQG, see Bryson and Ho [1] Dorato et al. 2] or Lewis [8]) is assumed: v(x; t) 1 2 x T S(t)x D T (t)x E(t) 32) 1 2 Z t f t Gamma GG T Delta ( S( d: The final condition (25) is satisfied, provided that S(tf ) Sf ; D(tf ) 0; and E(tf ) 0: 33) The ansatz (32) would not, in general, be true for the state dependent case, ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

....we want an efficient online method of estimation. A basic introduction to the Kalman filter can be found in Chapter 1 of [17] while a more complete introductory discussion can be found in [20] which also contains some interesting historical narrative. More extensive references can be found in [7, 12, 14, 16, 17, 30]. The Kalman filter has been used previously to address similar or related problems. See for example [2, 3, 9, 10, 18, 23] and most recently [11] The SCAAT approach in particular is described in great detail in [28, 29] The benefits of using this approach, as opposed to a multiple constraint ....

Lewis, Richard. 1986. Optimal Estimation with an Introduction to Stochastic Control Theory, John Wiley & Sons, Inc.

....nonlinear. The SDE (1) is interpreted in the sense of Ito (see Gihman and Skorohod (1972) or see Hanson 1996 for applicable exposition) The LQGP U problem advances the model of the LQGP problem (Wonham 1970, Westman and Hanson 1997) which is an extension of the LQG analysis (Bryson and Ho (1975) Lewis 1986, Dorato et al. 1995) In contrast to the LQGP problem, the LQGP U problem does not exhibit a formal closed form solution in terms of a state independent, nonlinear system of ordinary differential equations. Numerical methods for partial differential equations are necessary in order to solve the ....

....for boundary points. The delay terms associated with the Poisson noise, HV0 j;k are computed using linear interpolation having the same order of accuracy as that used for the derivatives. In the case of linear dynamics, quadratic costs and Gaussian noise (the LQG problem, see Bryson and Ho 1975, Lewis 1986, Dorato et al. 1995) formal solutions exist that require numerical solution for a state independent matrix Riccati equation. Further, if there are Poisson noise terms (the LQGP problem, see Wonham 1970, Westman and Hanson 1997) then formal solutions exist that are quadratic state space ....

Lewis, F. L., 1986, Optimal Estimation with an Introduction to Stochastic Control Theory (New York: Wiley).

....Open loop and closed loop simulations for a Van Der Pol oscillator are used to illustrate the results. Introduction In the case of linear systems with known parameters, there exists vast literature on estimation theory that allows asymptotic tracking of the actual state by its estimate, e.g. [1, 2]. At the opposite end of the spectrum one can find approaches for nonlinear plants with uncertain parameters. Many publications have been devoted to the design of adaptive observers for nonlinear systems that are linear with respect to unknown parameters, e.g. 3, 4] Parameter update laws are ....

Lewis F. Optimal estimation: with an introduction to stochastic control. A Wiley Interscience Publication, New York. 1986.

.... words: neural networks, nonlinear observer, adaptive control, output feedback problem 3 Introduction In the case of linear systems with known parameters, there exists vast literature on estimation theory that allows asymptotic tracking of the actual state by its estimate, e.g. see References [1, 2]. At the opposite end of the spectrum one can find approaches for nonlinear plants with uncertain parameters. Many publications have been devoted to the design of adaptive observers for nonlinear systems that are linear with respect to unknown parameters, e.g. see References [3, 4] Parameter ....

F. Lewis, Optimal Estimation: With an Introduction to Stochastic Control. A Wiley Interscience Publication, New York. 1986.

....position and orientation tracking systems. A very friendly introduction to the Kalman filter can be found in Chapter 1 of [30] while a more complete introductory discussion can be found in [31] which also contains some interesting historical narrative. More extensive references can be found in [1,30,32,33,34,35]. In this section we attempt to describe the method in a manner that does not imply a specific tracking system. In section 4 we will present simulations of a specific implementation, a one step at atime wide area optoelectronic tracking system. In section 3.1 we describe the method for tracking, ....

Lewis, Richard. 1986. Optimal Estimation with an Introduction to Stochastic Control Theory, John Wiley & Sons, Inc.

....is extremely difficult to solve and so the theory has not (to our knowledge) been used in applications. Due to the success of Kalman filters for linear systems, the extended Kalman filter (EKF) which is based on the linearization of the nonlinear system) was proposed for nonlinear estimation [8, 9]. Although the EKF is effective in numerous applications, there are limitations. A key assumption underlying the EKF is that deviations from the actual state trajectory are small. When these deviations are large, the EKF often performs poorly and can become unstable. The EKF also needs an accurate ....

F. L. Lewis, Optimal Estimation: With an Introduction to Stochastic Control Theory. New York: John Wiley & Sons, 1986.

....the dP 1 is bilinear in the control vector U and a nonlinear function of the state vector X. It is assumed that the SDE (1) is interpreted in the sense of Ito. The NLQGP problem advances the model of the LQGP problem [19] which is an extension of the LQG analysis (see Bryson and Ho [5] or Lewis [15]) In contrast to the LQGP problem, the NLQGP problem does not exhibit a closed form solution and numerical methods are necessary in order to solve the partial Work supported in part by the National Science Foundation Grants DMS 96 26692 and CDA 94 13948. differential equation of stochastic ....

.... This system can not be solved exactly, and numerical methods must be employed in order to determine the optimal, expected performance v(x; t) and optimal control vector u (x; t) In the case of linear dynamics, quadratic costs and Gaussian noise (the LQG problem, see Bryson and Ho [5] or Lewis [15]) formal solutions exist that require numerical solution for a matrix Ricatti equation, additionally if there is Poisson noise (the LQGP problem, see Westman and Hanson [19] formal solutions exist that require the numerical solution for a system on nonlinear differential equations. In order to ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

....in Special issue on Breakthrough in the Control of Nonlinear Systems in International Journal of Control, 15 January 1998. The LQGP U problem advances the model of the LQGP problem (see Wonham 1970 or Westman and Hanson 1997) which is an extension of the LQG analysis (see Bryson and Ho 1975 or Lewis 1986 or Dorato et al. 1995) In contrast to the LQGP problem, the LQGP U problem does not exhibit a formal closed form solution in terms of a state independent, nonlinear system of ordinary differential equations. Numerical methods for partial differential equations are necessary in order to solve the ....

....for boundary points. The delay terms associated with the Poisson noise, HV0 j;k are computed using linear interpolation having the same order of accuracy as that used for the derivatives. In the case of linear dynamics, quadratic costs and Gaussian noise (the LQG problem, see Bryson and Ho 1975, Lewis 1986 or Dorato et al. 1995) formal solutions exist that require numerical solution for a state independent matrix Riccati equation. Further, if there are Poisson noise terms (the LQGP problem, see Wonham 1970 or Westman and Hanson 1997) then formal solutions exist that are quadratic state space ....

LEWIS, F. L. 1986, Optimal Estimation with an Introduction to Stochastic Control Theory (New York: Wiley).

....Poisson random perturbations. The dynamics are governed by a stochastic differential equation (SDE) 1) which describes fairly general Markov processes in continuous time. The LQGP U problem generalizes the model of the LQGP problem (see [15, 11] which is an extension of the LQG analysis (see [1, 8, 2]) In contrast to the LQGP problem, the LQGP U problem does not exhibit, Work supported in part by the National Science Foundation Grant DMS 96 26692. in general, a formal closed form solution. Numerical methods for partial differential equations are necessary in order to solve the ....

....boundary points. The delay terms associated with the Poisson noise processes, HV0 jk and HV1 jk , are computed using linear interpolation that has the same order of accuracy as that used for the derivatives. In the case of linear dynamics, quadratic costs and Gaussian noise (the LQG problem, see [1, 2, 8]) formal solutions exist that require numerical solution for a state independent matrix Riccati equation. Further, if there are Poisson noise terms (the LQGP problem, see [15, 11] then formal solutions exist that are quadratic state space decompositions, that require the numerical solution for a ....

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, Wiley, New York, 1986.

No context found.

LEWIS,F.L.Optimal Estimation: With an Introduction to Stochastic Control Theory.John Wiley and Sons, Inc., 1986.

No context found.

Lewis, F. L., Optimal Estimation with an Introduction to Stochastic Control Theory, John Wiley, New York, NY, 1986.

No context found.

, 713-720. 238 Lewis86 Lewis, Frank L. Optimal Estimation with an Introduction to Stochastic Control Theory. John Wiley & Sons, Inc.

No context found.

Lewis86 Lewis, Frank L. Optimal Estimation with an Introduction to Stochastic Control Theory. John Wiley and Sons (1986). ISBN 0471 -83741-5.

No context found.

Lewis, F.L., Optimal Estimation with an Introduction to Stochastic Control Theory, John Wiley and Sons, New York, 1986.

No context found.

Lewis86 Lewis, Richard. 1986. Optimal Estimation with an Introduction to Stochastic Control Theory, John Wiley & Sons, Inc.

No context found.

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory, New York, NY: John Wiley and Sons, 1986.

No context found.

F.L. Lewis, Optimal Estimation: With an Introduction to Stochastic Control Theory, New York: John Wiley & Sons, Inc., 1986.

No context found.

Lewis, F. L. (1986). Optimal Estimation with an Introduction to Stochastic Control Theory. John Wiley, New York.

No context found.

F. L. Lewis, Optimal Estimation: With an Introduction to Stochastic Control Theory. New York, New York: John Wiley & Sons, 1986.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC