Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, NJ.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....smoother instead of the Kalman filter, since smoothing involves additional data in the estimation procedure. As a consequence, the estimation error variance is further decreased, and with a large lag length (delay) the performance approaches that of a noncausal Wiener filter for stationary signals [3]. In order to obtain a Kalman fixed lag smoother we represent the AR signal model (2) in state space form s(n 1) A(n)s(n) u(n) 0; 0 : 4) In contrast to (2) the d 1 dimensional state vector s(n) now consists of the current signal sample s(n) and d delayed samples where d ....

G. C. Goodwin, "Adaptive filtering prediction and control", Chapter 7, Prentice-Hall, Inc., 1984.

....according to J for the so obtained augmented state space representation of the system. The resulting controller incorporates a self adjusting mechanism, in that it selects a control input that realizes an appropriate compromise between the control and the identification objectives (dual action, [1]) However, such an optimal dual control problem is doable only in a few simple cases where computing the solution to the optimization problem is actually feasible. A computationally feasible though sub optimal approach to the design of self adjusting controllers is the so called switching ....

G. C. Goodwin, K. S. Sin, Adaptive filtering prediction and control. Englewood Cli#s, Prentice-Hall, 1984.

....smoother instead of the Kalman filter, since smoothing involves additional data in the estimation procedure. As a consequence, the estimation error variance is further decreased, and with a large lag length (delay) the performance approaches that of a noncausal Wiener filter for stationary signals [4]. In order to obtain a Kalman fixed lag smoother we represent the AR signal model (1) in state space form s(n 1) A(n)s(n) u(n) 0; 0 Delta T : 3) In contrast to (1) the d 1 dimensional state vector s(n) now consists of the current signal sample s(n) and d delayed samples ....

G. C. Goodwin, "Adaptive filtering prediction and control", Chapter 7, Prentice-Hall, Inc., 1984.

.... not the required inner product of the parameter error vector ( 0 yy n QQand input vector ( nVnd y , but instead this inner product is filtered by the FIR filter is not strictly positive real (SPR) except for the case of 0 dB = i.e. convergence cannot be assured [48]. Therefore, the controller of Figure 2 is disqualified as viable explicit rate congestion controller. 10 ( yndnd y Q u(n d) ynndV Figure 2 A Direct Adaptive Controller System for Controlling MA Plant That MAY NOT CONVERGE. 3.2.2 An Unrealizable Controller Consider a ....

....Section 4.1 are lifted. In their place, Assumption 5 is made, as well as the minor Assumption 6 and Assumption 7. This leads to a cleaner proof with stronger global stability results. 4.2.2.1 Proof of Convergence and Global Stability The update equation (39) is identical to Equation (3.3. 19) of [48]. From (15) 39) and (14) en n n = Qy , and from Lemma 3.3.2 of [48] 1 2 lim 0 n T nn . yy , nk n = QQfor any finite k. 40) From (15) 13) and (14) T en ndV nndV n n ndV n nndV n nndV n = ....

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G. Goodwin and K. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984.

....on the so obtained augmented state space representation of the process. The resulting controller incorporates a self adjusting mechanism, in that it selects a control input that realizes an appropriate compromise between the control and the identification objectives (dual action, see, e.g. [1]) However, such an optimal dual control approach is generally di#cult, except in a few simple cases, where computing the solution to the optimization problem is actually feasible. A computationally feasible though sub optimal approach to the design of self adjusting controllers is the ....

....# #J # (#, #) 1 #J # (#, #) if #) is # stable 1, otherwise, 2.2) where J # (#, #) is some positive cost criterion (e.g. an H 2 or H# cost) and # is a positive constant. The criterion J thus combines both stability and performance. Note that J # is normalized so that J takes values in [0, 1]. This is done for technical reasons related to the implementation of the cautious switching logic. We can then formalize the required richness of the candidate controller set as follows. Assumption 2.3. J : sup ### inf ### J(#, #) 1. This means that for any admissible model, there is a ....

G.C. Goodwin and K.S. Sin. Adaptive filtering prediction and control. Prentice-Hall, 1984.

.... Delta Delta r 1 z r 0 : 3.6) Since in our application we want the system output y(k) to converge to a constant level, we specify the reference output signal as ym (k) c 6= 0, which is characterized in the zdomain as Qm (z)y m (z) 0; Qm (z) z Gamma 1: 3. 7) The adaptive controller [13] applied to our QoS proxy cache system (3.2) is u(k) a j (k)u(k Gamma j) b j (k) y m (k Gamma j) Gamma y(k Gamma j) 3.8) Web Cache R(z) P(z) y=H[0] H[1] u=S[0] S[1] Estimation Controller y m =Desired Ratio of Hit Rates Controller Design Controller Parameters Plant ....

....In the next two subsections we describe how the cache parameters and controller parameters are estimated and updated respectively. 3.2. 1 Parameter Estimation To estimate the web cache system parameters pn Gammaj and r n Gammaj , we use a standard gradient algorithm from control literature [13]. The input u(k) and output y(k) measurements are fit to the model described in equation (3.2) Define the vector OE(k) u(k Gamma n) Delta Delta Delta ; u(k Gamma 1) y(k Gamma n) Delta Delta Delta ; y(k Gamma 1) 3.9) Then the cache model(3.2) can be expressed as y(k) ....

Goodwin, G. C., and K. S. Sin. Adaptive Filtering Prediction and Control. Prentice Hall, 1984.

....Adaptive control, controller parametrization, dynamic modeling, high frequency gain matrix, LDU decomposition, multivariable systems, web cache. 1 Introduction Adaptive control of multivariable systems has been a major research topic with both theoretical and practical challenges [4] 8] [9], 15] 16] 18] Recently, continuing effort has been made to relax conditions on systems to be admissible to a stable adaptive control scheme [10] 11] 14] 20] especially, a certain positive definiteness condition on the system high frequency gain matrix in the continuous time case. ....

....any bounded initial conditions. An important concept used in designing multivariable model reference control schemes is the zero structure at infinity of the plant transfer matrix G(D) This structure is characterized by the Hermite normal form [15] or equivalently, the interactor matrix [4] [9], 21] or its modified version [18] a triangular polynomial matrix m (D) of the form: m (D) 6 6 6 6 6 6 p 1 (D) 0 Delta Delta Delta Delta Delta Delta 0 p 21 (D) p 2 (D) 0 Delta Delta Delta 0 p N Gamma1 1 (D) Delta Delta Delta p N Gamma1 N Gamma2 (D) p N Gamma1 ....

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Goodwin, G. C. and K. S. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, Englewood Cliffs, N. J., 1984.

....of an effective new approach of using adaptive control in web cache systems for performance guarantees in the presence of system uncertainties. 1 Introduction Adaptive control of multivariable systems has been a major research topic with both theoretical and practical challenges [4] 7] [8], 13] Recently, continuing effort has been made to relax conditions on systems to be admissible to a stable adaptive control scheme [9] 12] 15] especially, a certain positive definiteness condition on the system high frequency gain matrix in the continuous time case. Inspired by recent ....

....input signal, find the feedback control u(k) for the plant (3.1) with unknown G(D) such that y(k) tracks ym (k) asymptotically and all signals in the closed loop system are bounded. An important concept used in designing multivariable model reference control schemes is the interactor matrix [8] of the plant transfer matrix G(D) a triangular polynomial matrix m (D) which has a stable inverse and has the key property that lim D 1 m (D)G(D) K p , the high frequency gain matrix of G(D) is finite and nonsingular. For the system model (2.4) D 0 (3.3) if det[B] 6= 0, and for the ....

Goodwin, G. C. and K. S. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, 1984.

....and control of dynamical systems has been an active area of research for the last three decades. Although methods for controlling linear time invariant plants with unknown parameters had been pursued since the 1960s, it was not until the last decade that stable adaptive laws were established [1] [4] Recent advances in nonlinear control theory and, in particular, feedback linearization techniques [5, 6] have initiated activity aimed at developing adaptive control schemes for nonlinear plant models [7] 10] This area, which came to be known as adaptive totlitear cottrol, deals with ....

....of their rguments. This cn esily be chieved if the sigmoid used is smooth function. The logistic function nd the hyperbolic tngent re examples of popular sigmoids that lso stisfy the smoothness condition. By the Mean Value Theorem there exists f [0f, 0] i. e, f : 0f (1 ) 0 for some ) [0, 1], such that Now let where B(Mf) denotes a ball of radius Mf. Therefore, the higher order terms satisfy Using the same procedure we have oo where 0g : 0g 0j and 0o(X , 0v) satisfies and the constant 6g is defined as sup cs(yr, g[ 3.37) 3.39) Vx x, V (M) 3.40) From now ....

G.C. Goodwin and K.S. Sin, Adaptive Filtering Prediction and Control, Englewood Cliffs, N J, Prentice-Hall, 1984.

....data pairs ( xi, xi2 . Xiq] Yi) i = 1, 2 . p . Here W is the weight matrix and its element, wj, are defined by 1 d (d is the distance between the j th example and the corresponding cluster centre) j = 1, 2 . p. We can rewrite equations (4) with the use of a recursive LSE formula [8] as follows: P = A r A) b =PArWy, 5) In DENFIS, we use a weighted recursive LSE with a forgetting factor defined as follows. Let the k th row vector of a matrix A is denoted as ak r and the k th element ofy is denoted as yk. Then b can be calculated iteratively as follows: f b = bk w ....

Goodwin, G. C., Sin, K. S., "Adaptive Filtering Prediction and Control", Prentice-Hall, Englewood Cliffs, N.J., 1984.

....( eff yn N nun d = 2.4) Note that Fulton and Li do not use forward looking estimates of ( yn, therefore the notation ( yninstead of ( yndn is used. Assuming for now that the plant in (2. 4) is a valid model, a simple Minimum Prediction Error Adaptive Controller (Direct Approach) [63] can be created to determine, at time n , the control signal ( un that minimizes ( 2 Eyndyndn . As with the design of most adaptive controllers, for the purposes of analysis, it is assumed that the parameter oeff N = is constant within the time interval needed to ....

....knowledge of d is assumed, a Normalized Least Mean Squares (NLMS) 64] formulation is possible: 29 ( 2 1 eff eff eff N nNn undynundNn und = 2.5) 1 eff yndn un Nn = 2.6) Update equation (2. 5) converges to the desired value if 02 [63]. All poles and zeros of (2.4) are at the origin, thus within the unit circle, leading to the result of Lemma 2.1. ######### For the adaptive controller of (2.5) and (2.6) applied to the plant (2.4) 1. yn and ( un are bounded sequences, 2. lim 0 n yn y nn d = ....

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G. Goodwin and K. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984.

....are deterministic based on ordinary differential equations. However, to be usable in a Kalman filtering framework it is crucial that the model contain both deterministic and stochastic components stochastic differential equations. Such models can be learned effectively from training data [9, 5]. In this paper we develop two significant elaborations for stochastic dynamical models. The first concerns modelling object classes for objects in motion. The second addresses the efficient modelling of couplings between tracked objects. 1.1 Shape and Motion Variability The first problem ....

C.G. Goodwin and K.S. Sin. Adaptive filtering prediction and control. PrenticeHall, 1984.

....can describe signals of arbitrary spectral density. For a good discussion of disturbance modeling, the reader is referred to chapter 6 of the book by Astrom and Wittenmark (1990) The theory for output prediction is well developed (see for example the books by Astrom and Wittenmark (1990) and Goodwin and Sin (1984)) It is summarized in the following: x(kjk Gamma 1) Ax(k Gamma 1jk Gamma 1) Bu(k Gamma 1) 3) y(kjk Gamma 1) Cx(kjk Gamma 1) 4) Correction based on measurements: x(kjk) x(kjk Gamma 1) K(ym (k) Gamma y(kjk Gamma 1) 5) Prediction: x(k 1jk) Ax(kjk) Bu(k) 6) y(k 1jk) ....

Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering Prediction and Control, PrenticeHall, Englewood Cliffs, N.J.

....on the control of the output of the system, rather than the probability density function of the system output. Anumber of well known algorithms havethus been developed and successfully used in many practical industrial systems. Typical examples are minimum variance control, selftuning control ([1]) and stochastic linear quadratic control. In most existing approaches, it has been assumed that all the variables in the system obey a Gaussiantype distribution. This is based upon the fact that most input noise can be characterised as coloured noise which can be regarded as being generated by a ....

.... (2) can be further expressed as a d step ahead predictive form to give y k d = G(z ;1 )y k F (z ;1 )B(z ;1 )u k F (z ;1 )n k d (5) where F (z ;1 ) and G(z ;1 )areknown polynomialsof the orders d; 1 and n; 1, and are obtained by solving the following Diophantine equation ([1]) 1=F (z ;1 )A(z ;1 ) z ;d G(z ;1 ) 6) Denote k = F (z ;1 )n k (7) and assume that k is a bounded stochastic distribution whose continuous probability density function at sample time k is denoted by fl (x# k) which is defined onaknown interval x 2 [ff 1 #fi 1 ]as Pfff 1 k ....

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G.C.Goodwin, and K.S. Sin, "Adaptive Filtering, Prediction and Control", Prentice-Hall, Englewood Cliffs NJ, 1984.

....time for the subgroup are determined based on the sum of absolute coefficient values in each subgroup. They are redetermined every time all the subgroups have been selected once as the constrained region in the current hopping order. Coefficients are updated by the Normalized LMS (NLMS) algorithm [5] as in 1 # # 2 1 represents the channel index and is the number of iterations. is a fixed step size and # # is an absolute value operator. is defined as a partial input signal vector whose elements ....

....canceler has converged, A minimum number of coefficient adaptations are assigned to converged channels in order to keep track with echo path changes. 4. COMPUTATIONAL SAVING Assuming ADSP 21020 (Analog Devices) FIR filtering with sparse taps requires 4 D 10 multiply and add (MPA) operations [5]. When coefficient adaptation and tap position control are needed, the necessary MPA operations become 6.625 A 16. This is because a combination of FIR filtering, power calculation (POW) for the NLMS algorithm, and 75 of the absolute coefficient sum (SUM) is the heaviest load in a single ....

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G. C. Goodwin and K. S. Sin, "Adaptive Filtering, Prediction and Control," Prentice-Hall, Info.and Sys. Sci. Ser., 1985.

....Bernstein polynomial network basis, are formed by using an input vector s barycentric coordinates, which are obtained by using a new inverse de Casteljau procedure using backpropagation. The network weights are then trained by the least squares method used in other linear in the weights ANNs [4, 13]. The traditional backpropagation method has been extended in the de Casteljau inverse mapping algorithm, which can be viewed as a special network structuring algorithm. A simple matrix inversion for deriving barycentric coordinates from input data with respect to a triangle has been included in ....

....fuzzy membership function A i (x) Eq. 1) can be expressed as y(t) B T (x(t) 3) where denotes estimates. 1 ; p ] T 2 p , B T (k) is the p dimensional input basis(fuzzy) functions: x 2 n B(x) 2 p . Typical linear learning methods such as least squares [13] can be used, via the minimisation of cost P N t=1 (y(t) y(t) 2 , to obtain the weights following the determination of the membership function basis B(x) by some preprocessing procedure. 4 (a) b) Figure 1: An illustrative example of hidden nodes required in two modelling approaches; ....

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Goodwin G. C. and Sin, K. S. : `Adaptive Filtering Prediction and Control'. Information and System Science Series, (Prentice Hall, Englewood Clis, New Jersey, 1984). 17

....input output description satisfying (2) where and represent the k th input and output samples respectively. This can be rewritten as where the k th row of given by (3) and . 4) This example includes all systems admitting strongly uniformly observable bilinear state variable realizations [5]. This work is supported by the Belgian National Fund for Scientific Research (NFWO) the Flemish government (GOA IMMI) and the Belgian government as a part of the Belgian program on Interuniversity Poles of Attraction (IUAP50) initiated by the Belgian State, Prime Minister s Office, Science ....

G.C. Goodwin and K.S. Sin, "Adaptive Filtering, Prediction and Control", Englewood Cliffs, Prentice-Hall, 1984.

....the response: g(t) t k=L s(k)z T (k) and Q(t) t k=L z(k)z T (k) 3) Iteratively applying the above equations as new data become available is extremely expensive computationally. Instead, the estimate at time t 1 is used to compute the estimate at time t in the following manner [9] [10]: y t = y t 1 Q 1 (t)z(t) s T (t) z T (t) y t 1 ] 4) Q(t) Q(t 1) z(t)z T (t) 5) An example using the resulting algorithm is shown in Figure 1. s 1 s 2 s 3 s 4 r 1 r 2 r 3 vy 3 vy 2 vy 1 a 0 0.5 1 1.5 Time (sec) b 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (msec) c ....

G. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control, Prentice Hall, New Jersey, 1984.

....with broad applicability. Many of the ideas described here are developed more fully in standard textbooks in modern systems theory, particularly textbooks on discrete time systems ( Astrom Wittenmark, 1984) adaptive signal processing (Widrow Stearns, 1985) and adaptive control systems (Goodwin Sin, 1984; Astrom Wittenmark, 1989) These texts assume a substantial background in control theory and signal processing, however, and many of the basic ideas that they describe can be developed in special cases with a minimum of mathematical formalism. There are also issues of substantial relevance to ....

....converge to zero because it is not needed to represent the inverse model for this plant. The nonconvexity problem The direct inverse modeling approach is well behaved for linear systems and indeed can be shown to converge to correct parameter estimates for such systems under certain conditions (Goodwin Sin, 1984). For nonlinear systems, however, a di#culty arises that is related to the general degrees of freedom problem in motor control (Bernstein, 1967) The problem is due to a particular form of redundancy in nonlinear systems (Jordan, 1992) In such systems, the optimal parameter estimates (i.e. ....

Goodwin, G. C., & Sin, K. S. (1984). Adaptive filtering prediction and control. Englewood Cli#s, NJ: Prentice-Hall.

....and Nikolaou (1996) introduced the simultaneous Model Predictive Control and Identification (MPCI) paradigm. MPCI relies on on line optimization over a finite future horizon (Figure 20) Its main difference from standard MPC is that MPCI employs the well known persistent excitation (PE) condition (Goodwin and Sin, 1984) to create additional constraints on the process inputs in the following kind of on line optimization problem, solved at each sampling instant k: 169 ) minimize horizon on optimizati over objective control horizon control over values input process subject to ( 170 ) Standard MPC ....

Goodwin, G. C., and K. S. Sin, Adaptive Filtering: Prediction and Control, Prentice Hall (1984).

....can be integrated effectively. To help distinguish among a variey of alternative approaches, we first define some terminology. Barto, Sutton, and Watkins (1990) have introduced the term model based to describe what essentially correspond to indirect algorithms in the adaptive control field (Goodwin Sin. 1984). These algorithms explicitly estimate relevant parameters underlying the system to be controlled and then use this learned model of the system to compute the control actions. The corresponding notion for an immediate reinforcement learning system would be one that attempts to learn an explicit ....

Goodwin, G. C. & Sin, K. S. (1984). Adaptive Filtering Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall.

.... sum of matrix quadratic forms: 2 log P x 1 , x # , y 1 , y # = # X t=1 [ y t Cx t ) T R 1 (y t Cx t ) log R ] the state from a sequence of # consecutive observations is well defined as long k # #p (a notion related to observability in systems theory; Goodwin Sin, 1984). For this reason, in dynamic models it is sometimes useful to use state spaces of larger dimension than the observations, k p, in which case a single state vector provides a compact representation of a sequence of observations. 6 In other words the convolution of two gaussians is again a ....

Goodwin, G. C., & Sin, K. S. (1984). Adaptive filtering prediction and control. Englewood Cliffs, NJ: Prentice Hall.

....are Gaussian and the priors for the hidden states are Gaussian, the resulting posterior is also Gaussian. The special cases of the inference problem for state space models play a prominent role in the engineering literature: filtering, smoothing, and prediction (Anderson and Moore, 1979; Goodwin and Sin, 1984). The goal of filtering is to compute the probability of the current hidden state X t given the sequence of inputs and outputs up to time t P (X t jfY g t 1 ; fUg t 1 ) 3 The recursive algorithm used to perform this computation is known as the Kalman filter (Kalman and 2 One can also ....

....analysis problem to be well posed. For a statespace model, the equivalent constraint is that the dimensionality of X must be less than the product of the dimension of Y and the length of the observation sequence. This constraint derives from the notion of observability in linear system theory (Goodwin and Sin, 1984). 4 2.2 Hidden Markov models A hidden Markov models defines a probability distribution over sequences of observations fY t g. This distribution over sequences is obtained by specifying the probability over observations at each time step t given a discrete hidden state S t , and the probability ....

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Goodwin, G. and Sin, K. (1984). Adaptive filtering prediction and control. Prentice-Hall.

....following slightly modified form of the algorithm: #(k) #(k 1) q(k) q(k k 1) c # T (k 1)#(k 1)T #(k 1) 6) with c 0; 0 2. is the learning rate of the parameter vector #(k) Since the convergence properties of the projection algorithm (6) have been analyzed [7], detailed description is omitted here. The parameter vector # is utilized to predict the future queue length q(k 1) at the present time, and to control the source rates to be set at the desired value. The algorithm (5) can be motivated geometrically as follows. Given q(k) q(k 1) A(k ....

G.C. Goodwin and K.S. Sin, "Adaptive Filtering Prediction and Control," Prentice Hall, pp.50--58, 1984.

....widely used for various process controls of unknown systems. As long as the unknown systems are structured form whose priori knowledges are available, it is well known that the STC can behave well and maintain satisfactory performances irrespective of system variation anddisturbance e ect [1] [2]. However, in unstructured systems whose priori knowledges such as the degree, the number of parameter, the time delay and etc. are not available, the conventional approaches for the design of a STC cannot be applied. It is mainly due to the fact that most of the conventional approaches are based ....

G.C. Goodwin and K.K. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, 1984.

....dimensionality. In this work, we extend MPCI to deterministic auto regressive moving average (DARX) processes 4 in an effort to reduce the dimensionality of the on line optimization problem solved by MPCI. The difficulty lies in that the PE condition for DARX models depends on the process output (Goodwin and Sin, 1984), a signal on which the designer has no control. In the sequel, we develop a strong PE condition for DARX models that involves only process inputs. This PE condition is then used in the formulation of an on line optimization problem that purports to select optimal process inputs for simultaneous ....

....DARX models. We subsequently discuss the numerical solution of the resulting on line optimization problem. Finally, we illustrate the above ideas through simulations. 2. PERSISTENT EXCITATION The role of PE in system identification has been emphasized by several authors, including Ljung (1987) Goodwin and Sin (1984); Goodwin and Payne (1977) Anderson and Johnson (1982) Bitmead (1984) Lozano and Zhao (1994) Genceli and Nikolaou (1995) Parameter identification is feasible if the input to the process satisfies the PE property so as to excite all the modes of the system (Ljung, 1987) It has been shown ....

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Goodwin, G. C., and K. S. Sin (1984). Adaptive Filtering Prediction and Control, Prentice Hall.

....Reasonable default functions can be chosen for those densities. However, it is obviously more satisfactory to measure the actual densities or estimate them from data sequences (x 1 ; x 2 ; Algorithms to do this assuming Gaussian densities are known in the control theory literature [13] and have been applied in computer vision [6, 7, 4] 1.2 Sampling methods A standard problem in statistical pattern recognition is to find an object parameterised as x with prior p(x) using data z from a single image. This is a simplified, static form of the image sequence problem addressed in ....

C.G. Goodwin and K.S. Sin. Adaptive filtering prediction and control. PrenticeHall, 1984.

....(Terzopoulos and Metaxas, 1991; Blake et al. 1993b) Reasonable defaults can be chosen for those densities, however it is obviously more satisfactory to measure or estimate them from data sequences. Algorithms to do this, assuming Gaussian densities, are known in the control theory literature (Goodwin and Sin, 1984) and have been applied in computer vision (Blake and Isard, 1994; Baumberg and Hogg, 1995b) Blake and Isard have developed an active contour framework (Blake and Isard, 1998) which is used in this thesis and is more fully described in chapter 2. Contour models have also been extended to include ....

Goodwin, C. and Sin, K. (1984). Adaptive filtering prediction and control. Prentice-Hall.

....trajectories after transients die out. This is a remarkable result, since it shows that the system s state information can be recovered from a sufficiently long observation of the output time series, and should be contrasted with the conventional approach of Kalman observables in control theory [46]. In more mathematical terms this statement means that there is a one to one smooth map with a smooth inverse from the K th dimensional manifold M of the original system to the Euclidean reconstruction space R N . Such mapping is called an embedding and the theorem is known as Takens ....

....of the dynamical system, the problem is how to guarantee smoothness at the boundaries among the local models. Crutchfield [10] has experimentally shown that dynamic modeling fails if this condition is not imposed. The extended Kalman filter also develops a local linear model of the trajectory [46], but utilizes a formulation based on a n xn( time series local neighborhood Reconstruction Space x n ( 42 10 statistical data model which is the main stream of control theory. 3.3. State dependent prediction of nonlinear AR processes. The approach of locally linear ARMA fitting has ....

Goodwin, G.C., and K.S. Sin, "Adaptive filtering, prediction, and control", PrenticeHall, 1984.

....for the speech signal given only the noise source. Additional issues related to recurrent training, error coupling, 9 This optimization approach relates to work done by Nelson [41] for system identi cation, and to Matthews neural approach [35] to the recursive prediction error algorithm [17]. 20 Handbook of Neural Networks for Speech Processing x k 1 Measurement Update EKFx Measurement Update EKFw x k y k x k w k w k 1 Time Update EKFx Time Update EKFw (measurement) w k Figure 9: The Dual Extend Kalman Filter (Dual EKF) EKFx and EKFw represent the lters for ....

G. C. Goodwin and K. S. Sin. Adaptive Filtering Prediction and Control. Prentice-Hall, Inc., Englewood Clis, NJ, 1994.

....The problem of stochastic adaptive control of linear ARMAX systems has received considerable attention over the past decade. The notable pioneering contributions are due to Astr om and Wittenmark [1] and Ljung [2,3] Subsequently, Goodwin, Ramadge and Caines [4] and Goodwin and co workers [5] have proved the self optimality of some adaptive control algorithms for minimum variance regulation and tracking. By self optimality it is meant that the cost, the time average of the square of the tracking error, is minimal. Recently a stochastic gradient algorithm has been proved to be ....

G. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control," Prentice Hall, Englewood Clis, NJ, 1984.

....is also assumed that Phi u ( 0 and that Phi u ( has a finite dimensional spectral factorisation. At issue is the estimation of the (assumed unknown) time varying dynamics G k (q) by means of the observations fu k g and fy k g. There are many approaches to this problem, but a common theme [22, 5, 14] is to express the dependence (1) in a linear regression form y k = OE T k k k ; 2) where the regression vector OE k depends on measurements of fu t g and fy t g up until t = k and k 2 R p is a vector of p parameters in a model structure G(q; k ) that attempts to describe the true ....

....uncorrelated with fu k g in the sense that jE fu k w k Gamma g j 0 as 1. When employing any of these adaptive schemes, a central question is the accuracy of the estimate G(q; b k ) as a description of G k (q) The most common way of assessing this is to examine the accuracy of b k itself [22, 5]. This may be achieved by defining k as the true parameter vector that allows the model structure to exactly describe the underlying time varying dynamics as G(q; k ) G k (q) and by defining the estimation error e k as e k , k Gamma b k : 10) Of course, in general the model ....

G. Goodwin and K. Sin, Adaptive Filtering Prediction and Control, PrenticeHall, Inc., New Jersey, 1984.

....is also assumed that Phi u( 0 and that Phi u ( has a finite dimensional spectral factorisation. At issue is the estimation of the (assumed unknown) time varying dynamics G t (q) by means of the observations fu t g and fy t g. There are many approaches to this problem, but a common theme [2] is to express the dependence (4) in a linear regression form y t = OE T t t t where the regression vector OE t depends on measurements of fu t g and fy t g up until t = k and t 2 R n is a vector of n parameters in a model structure G(q; t ) that attempts to describe the true ....

....t Sigma (7) with Sigma 0 and symmetric is known as the Kalman Filter. When employing any of these adaptive schemes, a central question is the accuracy of the estimate G(q; b t ) as a description of G t (q) The most common way of assessing this is to examine the accuracy of b t itself [2]. This may be achieved by defining t as the true parameter vector that allows the model structure to exactly describe the underlying time varying dynamics as G(q; t ) G t (q) and by defining the estimation error e t as e t , t Gamma b t . The quality of an adaptive estimation ....

G.C. Goodwin and K.W. Sin. Adaptive Filtering Prediction and Control. Prentice-Hall, Inc., New Jersey, 1984.

....an on line version (see below) for systems with changing characteristics. Specifically, let the ith row vector of matrix A defined in equation (12) be a T i and the ith element of B be b T i , then X can be calculated iteratively using the sequential formulas widely adopted in the literature [1, 7, 25, 41]: X i 1 = X i S i 1 a i 1 (b T i 1 Gamma a T i 1 X i ) S i 1 = S i Gamma S i a i 1a T i 1 S i 1 a T i 1 S i a i 1 ; i = 0; 1; Delta Delta Delta ; P Gamma 1 9 = 14) where S i is often called the covariance matrix and the least squares estimate X is equal to XP . The ....

....formulas to account for the time varying characteristics of the incoming data, we need to decay the effects of old data pairs as new data pairs become available. Again, this problem is well studied in the adaptive control and system identification literature and a number of solutions are available [7]. One simple method is to formulate the squared error measure as a weighted version that gives higher weighting factors to more recent data pairs. This amounts to the addition of a forgetting factor to the original sequential formula: X i 1 = X i S i 1 a i 1 (b T i 1 Gamma a T i 1 X i ) ....

G. C. Goodwin and K. S. Sin. Adaptive filtering prediction and control. Prentice-Hall, ENglewood Cliffs, N.J., 1984.

....4200 CPU seconds whereas the asymptotic approximation required a mere 170 seconds. At the same time, the differences between the two estimates were statistically indistinguishable, despite the system being time varying which violates the strict existence criteria of the asymptotic error limit (Goodwin and Sin, 1984). However, the matrices in the static error estimate calculation have the same dimension the filter, and subsequently the computational requirements would still formidable for large models. For example, derivation of the asymptotic filter for the 100 x 100 x 10 example above would still require 90 ....

....and the and density, respectively, were set to (2, 0.5, 0. 5, 4, 2) The asymptotic steady state error is computed by the doubling algorithm (Anderson and Moore, 1979) Such limit can be shown to exist uniquely, if the unstable and neutral modes of the model are both observable and controllable (Goodwin and Sin, 1984). Observability is the ability to determine the state from observations in the absence of errors, and stochastic controllability is the ability to force the model from one arbitrary state to another by the model process noise (e.g. 1974) In particular, a mode can be shown to be observable when ....

Goodwin, G. C., and K. S. Sin, 1984. "Adaptive Filtering Prediction and Control", Prentice-Hall, Inc., Cliffs, N. J., 540 pp.

....on the basis of delayed CF training information, to switch when it sees PF patterns that code spring mass states that lie on curve proprioceptive input . 6 DISCUSSION The model we have presented is most closely related to adaptive control methods known as direct predictive adaptive controllers (Goodwin Sin, 1984). Feedback delays pose no particular difficulties despite the fact that no use is made of a forward model of the motor plant. Instead of producing predictions of proprioceptive feedback, the model uses its predictive capabilities to directly produce appropriately timed motor commands. Although the ....

Goodwin, GC & Sin, KS (1984). Adaptive Filtering Prediction and Control. Englewood Cliffs, N.J.: Prentice-Hall.

....error is smaller. To make the procedure faster and to avoid repeating for each model the parameter and the PRESS computation we adopt an incremental approach, based on recursive linear techniques. Recursive algorithms have been developed in model identification and adaptive control literature (Goodwin Sin, 1984) to identify a linear model when data are not available from the beginning but are observed sequentially. Here we employ these methods to obtain the parameters of the model fitted on n nearest neighbors by updating the parameters of the model with n Gamma 1 examples. Also, the leave one out ....

Goodwin G.C. & Sin K. S. 1984. Adaptive Filtering Prediction and Control. Prentice-Hall.

....system expressed in rational proper transfer function form using the differentiation operator p: y(t) u(t) A(p) G(p) A(p) pn an lp i . alp ao B(p) b,p bm lp m 1 . blp bo This can be converted to an equivalent state space form using a canonical representation [6]: p(t) A(t) Bu(t) 2) y(t) CT(t) bu(t) Note that if G(p) is strictly proper then b equivalent shift operator state space description is: 0. Assuming zero order hold sampling, the qk = Mk Nu (4) y = S Tu (5) Where M,N,S and T are given by [12] M = e A5 N = A i(eAS I)B =C T = ....

....front end anti aliasing filters can be relaxed. Since these filters have to be taken into account in system estimation this is a major advantage. There is a smoother progression in control input to the plant. If slow sampling is used then the control input can be a sequence of large step changes [6]. This can feed significant energy into high frequency mechanical resonances. Rapid sampling ensures a smooth sequence of smaller changes to achieve the same bandwidth. 3. The discrete time response is a better approximation to the desired continuous time response. 4. Higher closed loop ....

[Article contains additional citation context not shown here]

G.C. Goodwin and K.W. Sin. Adaptive Filtering Prediction and Control. Prentice- Hall, Inc., New Jersey, 1984.

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Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, NJ.

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Goodwin, G. C. and Sin, K. S. 1984. Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, New Jersey. 34

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G.C. Goodwin and K.S. Sin. Adaptive filtering prediction and control. PrenticeHall, 1984.

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G. C. Goodwin and K. S. Sin. Adaptive Filtering Prediction and Control. Prentice-Hall, 1984.

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G. Goodwin and K. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, Inc., New Jersey, 1984.

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G. GOODWIN AND K. SIN, Adaptive Filtering Prediction and Control, Prentice-Hall, Inc., New Jersey, 1984.

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G. GOODWIN AND K. SIN, Adaptive Filtering Prediction and Control, Prentice-Hall, Inc., New Jersey, 1984.

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G.C. Goodwin and K.W. Sin. Adaptive Filtering Prediction and Control. Prentice-Hall, Inc., New Jersey, 1984.

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G. C. Goodwin and K. S. Sin, "Adaptive Filtering, Prediction and Control," Prentice-Hall, Info.and Sys. Sci. Ser., 1985.

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Goodwin, G. C., Sin, K. S., "Adaptive Filtering Prediction and Control", Prentice-Hall, Englewood Cliffs, N. J., 1984 27

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G. C. Goodwin, "Adaptive filtering prediction and control", Chapter 7, Prentice-Hall, Inc., 1984.

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Goodwin, GC and Sin, KS. 1984. Adaptive Filtering Prediction and Control. Information and Systems Science, Prentice Hall. NJ.

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