Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00004176/00001
## Material Information- Title:
- Kalman filtering as applied in tracking
- Creator:
- Tsoulkas, Vasilis N
- Publication Date:
- 1988
- Language:
- English
- Physical Description:
- 77 leaves : illustrations ; 28 cm
## Subjects- Subjects / Keywords:
- Kalman filtering ( lcsh )
Tracking radar ( lcsh ) Kalman filtering ( fast ) Tracking radar ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 69-70).
- General Note:
- Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering.
- Statement of Responsibility:
- by Vasilis N. Tsoulkas.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- |Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 19817802 ( OCLC )
ocm19817802 - Classification:
- LD1190.E54 1988m .T76 ( lcc )
## Auraria Membership |

Full Text |

KALMAN FILTERING AS APPLIED IN TRACKING
by Vasilis N. Tsoulkas B.S., University of Colorado, 1986 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering 1988 This thesis for the Master of Science degree by Vasilis N. Tsoulka^ has been approved for the Department of Electrical Engineering by Edward T. Wall Date iii Tsoulkas, Vasilis Nicholas (M.S. Electrical Engineering) Kalman Filtering as Applied in Tracking Thesis directed by Professor Edward T. Wall The applicability of the Kalman filter as applied in target tracking is analyzed. The mathematical treatment of the overall dynamic process is developed, using the generalized state space approach. The tracking behavior of the filter is examined, within a certain range of signal to noise ratios for two different types of target trajectories. The tracking matrix of the state estimator is mathematically formulated, and its use, as applied in the tracking problem, is demonstrated. The effectiveness of the so-called Kalman tracking filter is presented and compared for a parabolic and sinusoidal target trajectory. Further concepts of tracking matrices are discussed for further use in stochastic control theory. The form and content of this abstract are approved. I recommend its publication. Signed Faculty member in charge of thesis IV ACKNOWLEDGEMENTS I wish to express my gratitude to Dr. Alfred Fermellia from Hughes Aircraft Co. for his continuing support and direction during the course of thesis work. Gratitude is extended to Dr. Edward T, Wall for many useful discussions and suggestions and to Dr. Arum Majumdar for serving in the graduate committee for the defense of this thesis. CONTENTS TABLES...................................................vii FIGURES..................................................viii CHAPTER 1. INTRODUCTION...........................................1 1.1 General System Description......................1 2. THE STATE SPACE.......................................3 2.1 Methodology of the State Space..................3 2.2 Problem Description (Application)..............10 3. THE KALMAN FILTER ALGORITHM...........................12 3.1 Kalman Filter Equations........................12 3.2 Mathematical Justifications....................17 4. THE TRACKING MATRIX...................................30 4.1 Mathematics of the Tracking Matrix.............30 4.2 Harmonic Oscillations..........................35 5. STATE ESTIMATION-APPLICATION VIA THE KALMAN TRACKING FILTER................................44 5.1 Parabolic-Trajectory......................... 44 5.2 Sinusoidal-Trajectory..........................54 5.3 Comparisons....................................65 6. CONCLUSION............................................67 BIBLIOGRAPHY.....................................69 APPENDIX A.......................................71 APPENDIX B.......................................76 APPENDIX C.......................................77 LIST OF TABLES Table 5.1 Summary of Noise Deviations and SNR's.........45 Table 5.2 Summary of Noise Deviations and SNR's.........55 LIST OF FIGURES Figure 2.1 Configuration of an I/O System in the State Space...............5 Figure 2.2 Classification Set..............................................8 Figure 2.3 Classification Set of Continuous Equations......................9 Figure 2.4 Classification Set of Discreet Equations........................9 Figure 3.1 Kalman Filter Diagram............................... .........29 Figure 4.1 Harmonic Oscillations in One Plane.............................35 Figure 5.1 Target Trajectory with SDP = 0.001, SDM = 0.01, T = 0.5...46 Figure 5.2 Target Trajectory with SDP = 0.001, SDM = 0.09, T = 0.5...47 Figure 5.3 Parabolic Trajectory with Noise SDP = 0.001, SDM = 0.2, T= 0.5, SNR 250.............................................48 Figure 5.4 Innovation Process with SDP = 0.001, SDM = 0.2, T = 0.5, Parabolic Case................................................49 Figure 5.5 Target Trajectory with SDP = 0.001, SDM = 1.5,T = 0.5.........50 Figure 5.6 Target Trajectory (dotted) with SDP = 0.01, SDM = 2, T = 0.1, SNR = 5..............................................51 Figure 5.7 Target Trajectory with SDP = 2, SDM = 0.2, T = 0.5.............52 Figure 5.8 Target Trajectory (dotted) with SDP = 5, SDM = 0.01, T = 0.5.......................................................53 Figure 5.9 Target Trajectory with SDP = 0.01, SDM = 0.001, T = 0.5.......56 Figure 5.10 Target Trajectory with SDP = 0.01, SDM = 0.1, T = 0.5.........57 Figure 5.11 Tracking Trajectory (dotted) with SDP = 0.01, SDM = 1.5, T = 0.7, SNR = 3.5............................................58 Figure 5.12 Innovation Process SDP = 0.01, SDM = 1.5, T = 0.5, SNR = 3.5.....................................................59 Figure 5.13 Target Trajectory with SDP = 0.1, SDM = 0.01, T = 0.5.........60 Figure 5.14 Innovation Process with SDP = 0.1, SDM = 0.01, T = 0.5, Sinusoidal Case..........................................61 Figure 5.15 Target Trajectory with SDP = 1.0, SDM = 0.01, T = 0.5......62 Figure 5.16 Target Trajectory (dotted) with SDP = 2, SDM = 0.1, T = 0.5, SNR = 2..................................................63 Figure 5.17 Noisy Signal z(k+1) End Estimated State SDP = 0.1, SDM = 0.3, T = 0.5................................. 64 CHAPTER 1 INTRODUCTION 1.1 General System Description System description is the most important step leading to the solution of a general control problem. To be successfully conpleted, a basic understanding of the physical laws which occupy the system is required; then these laws can be described through a wide variety of mathematical equations that will govern the overall process. Their solution will reveal the basic properties of the system. It is essential that the nature of these equations be identified properly since failure to do so will result in an incorrect modelling and an erroneous result. In general, control systems may be described using different types of equations. For example, digital equations, ordinary differential equations, partial differential equations and delay differential equations. Moreover, the above formulations may exhibit linear or non- linear behavior; consequently other factors may need to be considered, such as system stability, "operation in the small," "operation in the large" and other related concepts. All of the above questions, including the order of the system, can be determined when the governing mathematics and physical laws have been carefully analyzed and understood. Another important issue in control theory is whether the representation of the system should be formulated deterministically or stochastically; failure to realize the correct nature of the input-output control signals will produce incomplete results which will lead to inadequate design and synthesis. To successfully include these considerations as well as numerous design specifications in the system description, a generalized higher- level mathematical framework known as the state-space-approach will be employed. It is a purpose of this research to expose the reader to some of the power of the mathematics of state-space-theory and to demonstrate it's use and applicability to a specific stochastic control problem. CHAPTER 2 THE STATE SPACE 2.1 Methodology of the State Space Formulation In recent years, a major effort has been made to investigate and apply state variable techniques in many different areas of science and engineering. Work done by Belman, Pontryagin, Liapunov, Kalman and many others has shown that the state-space formulation offers many advantages over the classical methods. The main reason for the wide interest in state space modeling, in contrast to the classical approaches (root locus methods, frequency response, etc.) is that the latter, which are basically transformational methods, present certain difficulties when applied to the analysis of non- linear time varying and multivariable sytems. In other words, classical control theory is generally applicable only to sinlge-input-single-output (SISO) and linear time invariant systems. Since classical control is essentially a complex frequency domain approach which relies heavily on trial and error techiques, it is usually not possible to optimize the design of a system when given specified performance measures. On the other hand, optimization is possible with the state variable approach. In 4 fact, the state equations are best suited for this type of problem since the quantities required in control system optimization are represented by state variables, control variables and the system parameters. Another advantage of modern control theory is that the equations are formulated in the time domain, allowing the designer to work with a whole class of input functions, including the initial conditions of the problem. Hence, state equation formulation is a powerful tool with which the designer is able to completely describe complex multiple-input-multiple-output (MIMO) systems. Consider a high level formulation which represents the nonlinear and the time varying dynamics of a plant in the general discreet form: x(tk+i) = f(x(tk),u(tk),tk) y(tk) = S(x(tk),li(tk),tk) where: x(tk) is a vector of state-variables y(tk) is an input-control-vector y(tk) is the output vector and f, g. describe the non-linear functional relationships of the system dynamics. It is possible to linearize the above model description and obtain the following discreet form: x(tk+i) = F x(tk) + B u(tk) 2.1.1 5 y(tk) = C x(tk)) + D u(tk) 2.1.2 where F, B, C, D are the corresponding dynamic coefficient matrices of the variables of the system. It is important to point out that the linearized equations (2.1.1,2.1.2) show how the state-space-approach relates not only to the input-output (I/O) behavior of the system but also to its internal state behavior. Ui(tk) rxi(tkn u2(tk) *2(tk) X(tk) = Un(tk) _ x/tk) _ yi(tk) y2(*k) ym(tk) FIGURE 2.1 CONFIGURATION OF AN I/O SYSTEM IN THE STATE SPACE For dynamic systems in general, there are two basic equations that typically represent such processes; the state equation and the measurement equation. The latter describes, in a linear fashion, which states are measured (Equation 2.1.2) and the former relates to the effects of forcing functions and to the systems state transition matrix from one time frame to another with a possible existing forcing function (Equation 2.1.1). 6 Another important aspect of system representation is the identification of its type; as has been already mentioned, the question to be resolved is whether the system is deterministic or stochastic. A system is of deterministic nature if there are no uncertainties present, or are negligible with respect to the overall process; consequently, the properties of the system propagate in a predictable manner. On the other hand, when there is a sufficient amount of disturbances, inherent to the system or introduced during the observations, the nature of the process becomes random, and realization of its performance can only be completed if the noise statistics are well understood. It should be emphasized at this point that the state space approach to system description is particularly useful in providing a statistical description of the system behavior. The dynamics of linear lumped-parameter systems with random coefficients can be modeled by the generalized first order continuous time vector matrix differential equation: Â£(t) = F(t) *(t) + B(t) m + G(t) vy(t) 2.1.3 or x(t) = F (t) *(t) + G (t) w(t) 2.1.4 where li(t) is some deterministic forcing function x(t) is the system's state-vector, \&(t) is a random forcing function and F(t), G(t) are time varying dimensioned matrices. The measurement equation in the continuous time domain is of the form: z(t) = H(t) x(t) + y(t) 2.1.5 where x(t) is the state vector that is measured H(t) is the dynamics measurement coefficient matrix y(t) is a random noise vector introduced during measurements The previous models are non-unique, which means that there are many different sets of the coefficient matrices {F(t), G(t), H(t)} which will yeild the same overall input -output behavior. Equation 2.1.3 is the state vector equation of a dynamic system which is composed of a minimum set of variables (state variables) such that knowledge of these variables at time t = to together with the input for t > tO, is sufficient to completely describe the unforced motion of the system for every t > tO. for linear-time-invariant-systems, t0 is usually zero (t0 = 0). in order to gain a better understanding of the dynamic representation of a system, a classification set may be defined in such a way that it combines the independent and dependent variables, the system coefficients and the sensor output. These four mathematical quantities are necessary and sufficient to model a given process. Additionally, realization of the dependent and independent variables leads to the solution of the estimation problem and the realization of the system coefficients, results in a solution of the identification part. C A INDEPENDENT VARIABLES DEPENDENT COEFFICIENTS VARIABLES OF EQUATIONS SENSOR OUTPUT ESTIMATION IDENTIFICATION CONTROL FIGURE 2.2 CLASSIFICATION SET Solution of both subsets (estimation, identification), together with the sensor output (Figure 2.2), yields a complete solution to the control problem. Hence the proper estimation and identification with the estimated terms will result in the correct modeling of the system. The classification set (Figure 2.2) can be modified to a more explicit continuous form as follows: 9 C = IND VAR DEP VAR COEFFICIENTS U X F,B,H,W,V OUTPUT] Z FIGURE 2.3 CLASSIFICATION SET OF CONTINUOUS EQUATIONS or, in the discrete form: CA flND VAR U DEP VAR X COEFFICIENTS OUTPUT] Z ESTIMATION IDENTIFICATION CONTROL FIGURE 2.4 CLASSIFICATION SET OF DISCRETE EQUATIONS From Figure 2.4 the corresponding discrete state equations are: X(tk+1) = Wk+iM X(tk) + 0(tk+1,tk) U(tk) + S(tk) Z(tk+i) = H(tk+1)X(tk+1) + V(tk+1) From now on, the discrete state equations for notational convenience will be written as: Xi<+1 = ())k+i ik X|< + 0k+1 ik Uk + Zk+i = Hk+1 Xk+1 Vk+1 If the control input (independent variables) is absent from the state space formulation, the following state equations result: ^K+^ = ())k+1 k Xk + 2.1.6 Z|<+1 = Hk+1 Xi<+1 Vk+1 2.1.7 2.2 Problem Description (Application) The problem application to illustrate the power of this approach consists of two cases. In the first case, the Kalman tracking filter will be used in a system consisting of a moving vehicle (such as an aircraft) following a parabolic trajectory. In the second case, which is perhaps more vivid, the same filter algorithm will be applied to a moving vehicle describing a sinusoidal path with time varying acceleration. In both cases the Kalman tracking filter is attempting to simultaneously estimate and track the position of the target point from the simulated incoming noisy data. In both situations, the target is perturbed by zero mean white Gaussian noise (which takes into account the maneuvering of the target and the possible effects of other random plant parameters, such as wind gusts, etc.). Both results of the Kalman tracking filter are modeled within a practical range of noise variances, for both the measurement noise and the plant noise. This case illustrates the power of the overall process. Finally, the two simulations are compared and limitations are discussed. CHAPTER 3 THE KALMAN FILTER ALGORITHM In order to proceed to the Kalman filter formations, it is appropriate to mention at this point the Gauss-Markov state random process. Since the Kalman filter is a special case of a Gauss-Markov sequence. 3.1 Kalman Filter Equations Definition: "A stochastic process is Gauss-Markov if it assumes a Gaussian distribution and if it is statistically dependent only on its previous past value." The non-unique formation of the Kalman filter is centered about the following. Given the discrete model of a system and the discrete measurements in the following form: Xk+1 = <|> k+i,k*k + Wk 3.1.1 zk+1 = Hk+1 xk+1 + ^k+1 3.1.2 Determine the optimal estimate of the state-vector xk+i (at t = tk+1), (system state) where wk : random process noise white sequence vj^-i : measurement noise white sequence with the following statistics: E {wk} = 0 E{vk+1} = 0 where: E {} is the expectations operator 8kj- is the Knonecker-delta Qk Rk+i are constant covariance matrices and: xk+i A nx1 state vector ^k+l.k 4 nxn state transition matrix wk A nx1 noise state vector Zk+1 4 lx1 measurement vector Hk+i A Ixn measurement-dynamics matrix vk+i A 1x1 noise measurement vector Since wk is a white noise sequence (uncorellated from one-step to the next) and since the system is driven by a noise process, the state vector xk (xk+1) becomes itself a noise process and the statistical properties of the states can be described through the mean of xk and the associated covariance matrix (to be analyzed later); so that determination of the optimal estimator is feasible. It is also assumed that the random variables have zero ensemble average values. i.e. E{Xk} = (2 => E{xk+1} = 0 (unbiased) The estimation of the state vector, which is created from discrete measurement data, corrupted by noise, can be optimized after minimization of the estimator error between the true-incoming-data and the one-step predicted estimated data. In other words, the Kalman filter can be derived via an optimal-recursive state estimator in the generalized sense that given a prior estimate at time tk an updated estimate at time tk+i is created based on the use of noisy measurements zk at tk, along with the associated dynamics matrix (STM) of the filter. Moreover, this estimator is a linear-least square minimum variance unbiased estimator. The unbiased estimate has an expected value which is the same as that of the quantity which is being estimated, that is E {x} = x. The minimum variance implies that the estimation error variance is less than or equal to the error variance of any other unbiased estimate. Since the observation process combines, linearly, the measurement data with the states of the system, along with some additional white Gaussian noise, the updated estimation error can be defined as: Â£k+1/k+1 4 ^k+1 ^k+1/k+1 3.1.3 where the indices k+1/k+i imply an estimate at t^+i given the observations attk+i. Equation 3.1.3 implies that, during the state-estimation of a dynamic system, an error vector exits, which is the deviation of the estimated state from the actual value. In summary, the filtering process involves the determination of that estimator which minimizes in the least-squares sense. (Equation 3.1.3) The previously defined estimation error is, ideally, equal to zero. To accomplish the minimization process, a scalar cost functional needs to be constructed, which is the inner product of xk+1/k+-|. ^k+1 Â§ ^-pk+l/k+1 Â£k+1/k+lj 3.1.4 The value of xk+1/k+i that minimizes Equation 3.1.4 is optimum for the actual state xk+1. In the following section, the well-known Kalman gain will be derived using Equation 3.1.4. The Kalman filter which is a generalization of the Wiener filter in the state space domain was first developed in the 1960's by Kalman- Bucy. Since then, due to its consistent results, especially in aerospace applications, it has gained a significant popularity in control systems engineering and other scientific research. The complete state space representation of the process has the following form: Xk+1/k+i = *k+l/k + Gk+l[^k+1 ik+1/k] 3-1-5 As it can be seen, Equation 3.1.5 is an a posteriori estimate of the state vector, and it is calculated from its state value using the state transitions matrix ( term of the R.H.S). It should be noted that the second term of the R.H.S of Equation 3.1.5 needs some special attention. At first, the quantity Gk+i (Kalman gain) is a weighting factor to the bracketed expression, often termed the residual or innovation process. Since this quantity is the error difference between the new incoming measurements signal Zk+i and the one-step predicted estimated signal ik+i/k, the residual quantity, in some sense, with the optimum gain, corrects at every iteration the updated estimate of the state (Equation 3.1.5). It is very important to emphasize that the Kalman gain forces the residual { Zk+i Zk+i/k} to seek a minimum level. The derivation, as 17 mentioned in an earlier section, is accomplished from the defined loss- function or performance measure, Equation 3.1.4. Note also that: ^+1/k = Then Equation 3.1.5 can be modified to a more explicit form as: ^k+1/k+1 = ^k+1 ,kv ^k/k + ^k+l[^k+1 ^k+1 ^k+l.k ^k/k] 3.1.6 3.2 Mathematical Justifications In order to gain a better understanding of the Kalman filter algorithm, it is desirable to go through certain mathematical derivations of certain important expressions. At first, note that the covariance matrix of an n-state vector is an (nxn) symmetric matrix. The diagonal elements are the mean square errors of the state variables. If the off-diagonal-elements are non-zero, they represent the cross-correlation terms of the elements of xk+1/k+1. Stochastically, the covariance matrix expresses the uncertainty involved in the estimation process of the state of a dynamic system, which is defined as: P AE{xxf}. 1 8 Now, define the a posteriori error covariance-matrix as: ~ ~T Pk+1/k+1 4 E {*k+1/k+1 Â£k+1/k+lJ 3.2.1 where: ^k+i/k+1 = ^k+1 ^k+l/k+1 3.2.2 Similarly, going one-step backwards, Pk/k 4 ^ {4k/k *kyk} 3.2.3 where =xk- 3.2.4 Also, define the one-step-prediction estimation error as: Pk+1/k 4 ^ {2k+1/k ^k+1/k} 3.2.5 where xk+1/k = Xj<+1 Vi/k 3.2.6 Moreover, let the state-model be described as: Xk+1 = and the one-step prediction estimation state as: A i /V ^k+1/k ~ 9k+1,k ^k/k 3.2.8 From Equation 3.2.3, Equation 3.2.4: pk/k ~ E {^k/k^k/k} ~ E {(4k 4k/k)(4k 4k/k )"*"} and from Equation 3.2.5, Equation 3.2.6 pk+1/k = E (4k+1/k 4k+1/k }= E {(4k+1 4k+1/k)( 4k+l 4k+1/k)Tl since, Xk+1 Xk+i/k = <|>k+i,k 4k + Wk ^k+1 ,k 4k/k or, Xk+1 Xj<+i/k = $k+1 ,k 4k ^k+1 ,k 4k/k+ ^k = = ^k+1 ,k(4k 4k/k) + ^k =Xk -4k+1/k = (l)k+1,k4k^ + wk 3.2.9 Combining Equation 3.2.5 and Equation 3.2.9 the covariance outer product is: pk+1/k = E {(^k+1 ,k 4k/k + ^k )( = E {(^k+1 ,k 4k/k + ^k )(4k/k ^k+1 ,k + ^k)} = = E {^k+1 ,k4k/k 4k/k ^k+1 ,k + ^k+1 ,k 4k/k ^k+ ^k 4k/k ^k+1 ,k+ + WkW1} 20 = E {(j)k+i,k *k/k 2k/k ^k+1,k } + E (^k+l.k^k/k ^k) + + E{Wkxk/k c|>k+1 k) + E {wkwk} = = ^k+1 ,k E {2k/k 2k/k Mk+I.k + ^k+l.k E {Â£k/k^k) + + E {WkxJyk} => pk+1/k = $k+1,k pk/k $ T k+1,k + Qk 3:2.10 Since, E {x^ WkT} = E{Wk } = 0 where Qk is an nxn matrix. Due to the properties of uncorreleatedness and orthogonality between x^ and WkT (error estimate and process noise), and since at least one of the above quantities has zero mean value, E {^} = 0, white sequence. Note that if *k+1 = the derivations are identical, and pk+1/k = ^k+l.k Â£k/k ^k+l.k + ^kW^k1 Equation 3.2.10 is the a priori error covariance matrix of the one- step prediction estimation error of the state vector. In addition, from Equation 3.1.5, which is repeated here for convenience: 21 *k+1/k+1 = *k+1/k + Gk+lfek+1 Hk+1 *k+1/kl and Equation 3.2.2: xk+1/k+1 = *k+i x^/k+i notice that: ** A Â£k+1/k+1 = ^k+1 Â£k+l/k+1 = = xk+1 {*k+1/k + Gk+lUk+1 l<+1/k]} = = Xk+1 {Xk+1/k + Gk+1 [2k+1 Hk+1 ^k+1/kl}= = > = Xk+1 *k+1/k Gk+1Hk+1 *k+l Gk+lÂ¥k+1 + + Gk+1Hk+1 *k+1/k = = ^k+1/k Gk+1^k+1 Gk+1 ^k+1 [ ^k+1 ^k+1/kl = = 2k+1/k Gk+1^k+1 Â£k+1/k Gk+1^k+1 ^ Â£k+1/k+1 = [I Gk+1^k+l] 2k+i/k Gk+1^k+1 3.2.11 Now the updated error covariance can be derived in the following manner. Since Pk+l/k+1 = E { Â£k+i/k+1 2k+-|/k+-|} and from Equation 3.2.11 Ek+1/k+1 = E {[(I Gk+1 ^k+1 ) ^k+1/k Gk+1^k+1 ] [(I Gk+1Hk+1) xk+1/k Gk+i^+i]T} = = E {[(I Gk+iHk+i)xk+i/k Gk+iVk+-|] [ 2k+1/k(l Gk+i Hk+1 )T vk+1Gk+1]} = = E {(I Gk+i Hk+-| ) xk+1/k xk+1/k (I Gk+-|Hk+i)T - - (I Gk+1 Hk+i )xk+1/k vk+1 Gj+1 - ' Gk+1^k+1 S+1/k(*' Gk+1Hk+l)T + Gk+lÂ¥k+lÂ¥k+1 Gk+1 } = 0 Gk+1 ^k+1 ) E { xk+1/k xk+1/k}(l Gk+iHk+1)T - - (I Gk+i Hk+1 ) E { xk+1/kT yk+1 }Gk+1 - ' Gk+1 E &k+1 *k+l/k} 0 Gk+1 Hk+1 )T + + Gk+1 E {^k+1 ^k+1 }Qk+i => 3.2.12 pk+1/k+1 = 0 Gk+1 Hk+l^k+l/kO Gk+1Hk+l)T + + Gk+1 Rk+lGk+l ~T where: E { xk+1/k xk+1/k} = Pk+1/k E {xk+1/k vk+i } = E {^+1 xk+1/k } = 0 E {Yk+1 vk+1 } = Rk+1 Recall Equation 3.1.4. From stochastic geometric theory the diagonal elements of the cost functions represent the geometric length (in an Euclidean space) of the stochastic error which is also the trace of the updated error covariance matrix, namely: ^k+1 = E tek+i/k+i Â£k+i/k+i} = Trace[Pk+i/k+i] 3.2.13 Realizing from Equation 3.2.13 that ^k+1 = E {2k+i/k+i I Â£k+l/k+i) where I is the identity matrix, and using Equation 3.2.12, proceed in the following results for the derivation of the optimum Kalman gain: pk+1/k+1 = 0 Gk+1 Hk+l)Pk+1/k(! Gk+1Hk+l)T + Gk+1Rk+lGk+1 - 0 Gk+1 H|<+i)P|<+i/|<(l Hk+i Gk+i )+ Gk+iRk+1^k+1 * = 0 Gk+1 Hk+l)(pk+1/k pk+1/k Hk+1 Gk+1 )+ Gk+1Rk+1Gk+1 - pk+1/k pk+1/k Rk+1 Gk+1 Gk+1Rk+1Pk+1/k + + Gk+1Rk+1pk+1/k Rk+1 Gk+1 + Gk+1Rk+1Gk+1 Taking the partial derivative of Jk+1 with respect to Gk+1 and setting equal to zero, dJk+1 9Gk+i = ^GkTT(F>k+1/k) 'aG^7(Pk+l/k Hk+1 Gk+l) 3G^7 + + (Gk+1 Hk+1 pk+1/k) + 3G^'+1' (Gk+1 Hk+1 pk+1/k Hk+1 Gk+l) + 3Gk+1 (Gk+1 Rk+1 Gk+1) 3Jk. 1 T 0Gk+1 = '2(' Gk+1 Hk+l) pk+1/k Hk+1 + 2Gk+1Rk+1 3.2.14 Recalling from linear algebra that 5Gk+i (pk+l/k) = a. 25 0Gk+1 ("Pk+1/k Hk+1 Gk+i) Rk+1/k Rk+1 b. 9Gk+i (^k+1 Rk+1 Rk+1/k) = 'Rk+1Rk+1/k c- Recall that Pk+i/k is a symmetric matrix, meaning that (Pk+i/k)T=pk+i/k So from c. taking the transpose of the whole quantity ('Rk+lPk+1/k)T = Pk+1/kRk+1 > 0Gk+i ^k+1 Rk+1 Pk+1/k Rk+1 Gk+l) = 2 Gk+1 Rk+1 Pk+1/k Rk+1 d. From linear algebra: since (XAXT) = 2Ax; A,X: matrices, with A symmetric, 3 T 9Gk+i" (Gk+1 Rk+1 Gk+l) = 2Gk+1 Rk+1 From Equation 3.2.14 and 3Jk+i 5Gk+l 26 there results: -2 (I Gk+i Hk+1) Pk+i/k Hj[+1 + 2Gk+i Rk+i = 0 => -2(l Gk+i Hk+1) Pk+i/k Hj[+1 = -2Gk+i Rk+i ^ pk+1/k Rk+1 Gk+1 Rk+1pk+1/k Rk+1 Gk+1 Rk+1 = 3 ^ Gk+1 (Rk+1 Pk+1/k Rk+1 + Rk+l) = Pk+1/k Rk+1 => Gk+1 = pk+1/k Hk+1 (Hk+1pk+1/k Hk+1 + Rk+l)'1 3-2- Equation 3.2.15 is the expression for the optimum Kalman gain. Substituting Equation 3.2.15 into 3.2.12 gives the updated error covariance in its final form as: pk+1/k+1 = ([ Gk+1 Hk+l)pk+1/k which is optimum in some sense. Summarizing, the complete computational algorithm of the Kalman filter consists of the following ordered operations: 27 T Pk+1/k ^k+l.k Pk/k ^k+l.k + Gk Gk+1 ~ Pk+1/k ^k+1 (^k+1 Pk+1/k ^k+1 + ^k+l) 1 Pk+1/k+1 O' Gk+-|Hk+l) Pk+1/k Â£ k+1/k+1 = ^k+1 ,k Â£ k/k + Gk+1 (Zk+1 H k+1 ^k+1 ,k k/k) IV. The initial conditions of the Kalman filter are: Pk/k is an arbitrarily large and diagonal matrix of the form (first iteration): (0/0) = tfx^O) * O 2 0 0 1 o ie 0 * %(0) 1 o 200_ For the initial state vector, Â£(0/0) = a For the general process noise covariance matrix: 28 I- 2 Qu- = cWl(0) 0 <%n(0) 3.2.18 For the general measurement noise covariance matrix: Rk+1 = ^ (0) 0 vn(0) 3.2.19 Note that the matrices P, Q, R must be always positive semidefinite. It is appropriate at this point to emphasize that Gk+i = f(Q,0,H,R). 3.2.20 The above relationship shows the strong coupling between the Kalman filter gain and the system's coefficient matrices; consequently the optimal conditions of the updated estimator Xk+i/k+i are also strongly affected by the same quantities. From what has been said so far, it is evident that the Kalman filter is a very sophisticated and useful algorithm in state estimation applications, due to its intrinsic properties in reflecting real world situations without the introduction of errors (under certain assumptions). Figure 3.1 (below) shows the corresponding Kalman filter diagram. Zk+1 = Hk+1 Â£k+1 + Vk+1 Zk+1/k = ^k+1 ^k+l.k^k/k FIGURE 3.1 KALMAN FILTER DIAGRAM CHAPTER 4 THE TRACKING MATRIX 4.1 Mathematical Formulation of the Tracking Matrix The tracking matrix is a method of discretizing and tracking (in the polynomiai sense) a smooth function via the Taylor series expansion. This methodology, unfortunately, as applied to tracking guidance systems, hasn't been communicated very well in the existing literature, for various reasons. In the next sections, this matrix will be used as the state transition matrix (dynamics matrix), of the Kalman filter formulations, thus solving in many respects, the identification problem of the assumed model. It is important that the identification part (recall from Chapter 2) is properly solved, or at least, if there are certain drawbacks, these must be realized a priori. The tracking matrix has the following generalized form: 4>(k+1,k)- T Â¥ o o 1 0 T 1. 4.1.1 31 The elements of this matrix are the generated sampling coefficients of the well-known Taylor series expansion. Consider the general Taylor series expansion of three variables; in this case a vector directional gradient can be formulated in the following way: f(x,y,z) A f(x0,y0,zo) lXo,yo,Zo + S (W) lXo,y0,z0 X x0' y yo Z Z0_ + V2f(x,y,z) 'x0.yo.zo '(x x0) 2-1 + Yn f(x,y,z) 'x0.yo.zo 2! (y yo)2 2! (z z0)2 L 2! J (x-xo)"- n! (y yo)n n! (z zp)n I n! I => f(x,y,z) Â£ f(x0,y0,zo) + Â§ lx0,y0>z0(x-xo) + Â§ ^^0) + 9f_. a^f, (x-x0)2 d2f (y-yp)2 + 8z lx0>y0,z0^ZZo) + 0X2 lx0ly0>z0 2! + 0y2 lx0,y0,z0 2! + 32f (z-z0)2 anf (x-xp)n + 5z2 x0.yo>zo 2! + *+ 3xn'x0,y0.z0 n! + 9nf (y-yp)n anf (z-z0)" + 3yn 'x0,yo,zo n! + 9zn'x0,yo.zo n! 4.1.2 32 The formidable expression of Equation 4.1.2 is actually not as complicated as it may seem, especially when the expansion is applied to a function of one variable. In the time domain, and about a center point to, the Taylor series assumes the following form: (t -10)2 (t t0)n f(t) A f(t0) + f(t0)(t to) + Ht0)- 2|~+ + ">(<<)) nj or, in closed form oo n=0 (t-t0)n 4.1.3 So for a general function x(t): x(t) = x(t0) +x(t0) (t -10) + x(t0) (t -10)2 2! + higher order terms. For the discretization process, let: * *k+i to-* *k t" *0 *k+1 tk = T where T is the sampling interval of the process. Then, as in Equation 4.1.3, EX(")(tk) nj (^k+1 tk) n=0 n-S n=0 XW(tk) n! 1 4.1.4 Consequently: x(tk+i) = x(tk) + x(tk) (tk+i tk) + x(tk> (tk+1 ~ tk)2 2! + remainder. or, xk+1 = xk + xk T + xk2|- + remainder. 4.1.5 Taking the derivative of x(tk+i) and expanding in the same way, we obtain : x(tk+i). = x(tk) + x(tk) (tk+1 tk) + remainder or, xk+i = xk + xk T + remainder. 4.1.6 Proceeding with the second derivative of X(tk+1), there results: x(tk+i) = x(tk) + x(tk) (tk+1 tk) + remainder or, xk+i = xk + xk T + remainder. 4.1.7 From Equations 4.1.5 4.1.7, the elements of the tracking matrix are easily recognized. In the physical sense: x(tk+i) denotes position x(tk+i) denotes velocity x(tk+i) denotes acceleration These quantities are the states of the assumed model. The tracking matrix is able to "follow", with very good accuracy, targets moving with constant acceleration, implying that the series expansion will be able to fit the polynomial trajectory of the target closely. Expanding in the neighborhood of the center point, tk(to). Considering the first three terms of the Taylor series, the tracking will be exact if the function can be regenerated after a few terms. For the case of a parabolic trajectory, the third term (second derivative) regenerates the parabolic function. then or, x(t) = t2lt0=o + 2t|t0=o(t to) + 2jto=0 2 => x(t) = t2 4.1.8 or, => x(t) =~2~ => *(t) =T2 4.1.9 where T is the sampling interval. In the extreme situation where the path of the target is a sinusoidal function, the tracking process becomes somewhat pathological. Even though, in general, additional derivative terms in the expansion give better results, this is not true for the sinusoidal case, since the expansion is about t0 = 0, and the sinusoid moves away from zero and also since this type of a function is infinitely differentiable. Both cases, parabolic and sinusoidal, will be examined in the next chapter, since it is of interest to demonstrate the behavior of the Kalman tracking filter while it tracks a noisy trajectory, especially in the sinusoidal case. 4.2 Harmonic Oscillation Consider the vibration of a spring with a weight, being attached to a support downwards. rS7777777777777777777777777777 FIGURE 4.1 HARMONIC OSCILLATIONS IN ONE PLANE If the block is moved away from its equilibrium point (0y, the motion of the block is described by a differential equation, with its associated initial conditions. This mechanical problem involves simple harmonic motion. Assuming that the motion of block A takes place entirely in a vertical plane, the velocity and acceleration are given by the second and third derivatives of x(ft) with respect to time t (seconds). Additionally, there is a restoring force F which is proportional to the distance from its equilibrium point, and which is directed towards point 0 (Fig. 4.2.1). In addition to the above, there will be, in general, a retarding force, which is created by the medium in which the motion occurs. Also a possible impressed force may be present, such as the motion of the support or the existence of some magnetic field. If the above parameters are to be considered in the model, the describing differential equation would assume the following form: jx(t) + Bx(t) + Kx(t) = ^F(t) 4.2.1 If the retarding force is proportional to the cube of the velocity, then Equation 4.2.1 becomes highly non-linear. For present purposes, both the retarding and the impressed forces are assumed to be zero. In such a case, Equation 4.2.1 becomes a second order linear homogeneous differential equation with constant coefficients of the form: 37 jx(t) + Kx(t) = 0 => => mx + Kx(t) = x(t) + ^x(t) = 0 => => x(t) + co2x(t) = 0 x + co2x = 0 or, x = -co2x where to = and with I.C.'s x(0) = x0 x(0) = v0 Solving Equation 4.2.3 there results x(t) = Asin(cot + <))) or x(t) = Acoscot + Bsjncot Using the given initial conditions with Equation 4.2.4: 4.2.2 4.2.3 4.2.4 x(0) = x0 x(0) = v0 the coefficients are obtained as follows: 38 x(t) = Acoscot + Bsincot fort0 = 0, x(t0) = A and x(t0) = -Acosincot + coBcoscot 4.2.5 x(t0) x(t0) = coB => B=-^ Now consider: x(t0) x(t) = x(t0)cosco(t-t0) + sinco(t-t0) CO 4.2.6 Let t *k+i lk t_t0 tk+i-tk = T. x(t) then x(tk+1) = x(tk)cosco(tk+1 tk) +-------sinco(tk+1-tk) CO x(tk) x(tk+i) = x(tk)coscoT +-------sincoT CO or xk Xk+1 = xkcoscoT + sincoT co 4.2.7 Equation. 4.2.7 can be described and formulated in the following manner: 39 ' Xk+1 " r t2 1 " xk " 1 T T Xk+1 d. Xk 0 1 T Xk + 1 0 0 1 Xk Recall matrix Equation. 4.1.1, realizing that: cos(x) = ^ n=0 2n! cos(coT) = I n=0 n/^r\2n (-1) (coT) 2n! sin(x) _y tin - Lu (2r n=0 x2n+1 (2n+1)! sin(coT) = I n=0 (-1 )n(coT)2n+1 (2n+1)l From Equation. 4.2.9 , . 0)2T2 o)4T4 03676 cos(coT) ~ 1 "2! + 4! "6! + . T co3T3 co5T5 (sPV sin(coT) ~ coT- 31 + 51 71 + Then from Equation. 4.2.11 C02f2 4.2.8 4.2.9 4.2.10 4.2.11 4.2.12 cos(coT) = 1 - 2! 4.2.13 and from Equation. 4.2.12 sin((oT) = coT 4.2.14 Substituting Equation. 4.2.13 and Equation. 4.2.14 into Equation. 4.2.7 there results: x(tk+i) = X(W 1 - 0)2T2 x(tk) _ +-------COT : co C02T2 . = x(tk) x(tk)-2+ x(tk)T: = x(tk)-co2X(tk)^+x(tk)T: or o T2 . xk+1 xk 032 xk 2 + xk^ but Xk = -0)2 Xk then . .. J2 Xk+1 = xk + xkT + xk ~2~ 4.2.15 Differentiating Equation. 4.2.7 and proceeding in the same manner: x(tk+i) = -cox(tk)sinco(tk+1 tk) + x(tk)cosco(tk+1 tk) = -cox(tk)sino)T + x(tk)coscoT = 41 = -cox(tk)[co7] + x(tk) 1 - C02T2 n . . co2T2 . 0 co2T2 CO2 x(tk)T + x(tk) x(tk)2 = X(tk) CO2 x(tk)T xpiJ-g = x(tk) + x(tk)T or xk+i = xk + xkT 4.2.16 Taking the first two terms and recalling that: x(tk) = -co2 x(tk) Taking the second derivative of Equation. 4.2.7: x(tk+i) = -o^x^coscoT cox(tk)sincoT = = -co2x(tk) 1 - co2T2 cox(tk)coT = = -co2x(tk) + co4x(tk)"2 co2x(tk)T choosing the first term of the previous expression: x(tk+i) = -co2 x(tk) => x(tk+1) = x(tk) or Xk+1 = Xk 4.2.17 42 Then the structure of Equation. 4.2.8 can be easily recognized from Equations 4.2.15-3.2.17. In the next chapter, the applicability of the Kalman tracking filter equation is demonstrated for a target, such as an aircraft, moving at first with constant acceleration (parabolic case) and then with time varying acceleration (sinusoidal case). The motion of the vehicle is perturbed by zero mean white Gaussian plant noise, which accounts for manoeuvres or other random factors such as wind gusts. Moreover, the measurement process is corrupted also by mean white Gaussian noise which accounts for the observation noise. Then the vehicle dynamics are assumed to be described by the following state vector equations. ^k+1 $k+i ,k + Fk wk 4.2.18 where: *k = [xk xk xk] 4.2.19 wj=[w(1) w{2) w(3)] I 4.2.20 rJ-[o o 1] 4.2.21 The measurement equation is: Z|<+1 = Hk+12k+1 + Vk+1 4.2.23 where: Hk+-| = [1 0 0] 4.2.24 The matrix equation 4.2.22 is used in the Kalman filter equation so that proper estimation and tracking is accomplished. CHAPTER 5 STATE ESTIMATION APPLICATION VIA THE KALMAN TRACKING FILTER 5.1 Parabolic Trajectory For the parabolic trajectory state, the general phase variable formulation is developed in the following way: Letting y = x2 5.1.1 X1 = x2l x-i = x2 x2 = 2xJ X2 = 2 5.1.2 5.1.3 Since X = A* + B yi X1 0 1 V 0 ' S + _x2. . o 0 . _x2_ 1 2 + )n 5.1.4 For the discretization process: = pAT (j) = e eAX dX B 5.1.5 5.1.6 where: r o i B = 1 The states are generated via formulation 5.1.4. Keeping in mind equations 5.1.4 5.1.6, then: *k+l ^k+uXk + TkWk 5.1.7 Zk+i = Hk+1 Xk+1 + Yk+i 5.1.8 where Hk+i = [ 1 0 ] The Kalman tracking filter is applied within a certain range of process and measurement noise deviations. SDP = 0.001 SDM = 0.01 SNR = 198E3 LO o ii 1- SDP = 0.001 SDM = 0.09 SNR 2.5E3 T = 0.5 SDP = 0.001 SDM = 0.2 SNR 500 to o II 1- SDP = 0.001 SDM = 1.5 SNR 8.8 T = 0.5 SDP = 0.01 SDM = 2 SNR = 1.8 H II p SDP = 2 SDM = 0.2 SNR 4.9 T = 0.5 SDP = 5 SDM = 0.01 SNR = 0.8 T = 0.5 TABLE 5.1 SUMMARY OF NOISE DEVIATIONS AND SNR'S. TRACKING TRAJECTORY FIGURE 5.1 TARGET TRAJECTORY WITH SDP = 0.001, SDM = 0.01, T = 0.5 TRACKING TRAJECTORY -P- -vl 5 3 5 2 5 1 5 0 5 ( FIGURE 5.4 INNOVATION PROCESS WITH SDP = 0.001, SDM = 0.2, T = 0.5, PARABOLIC CASE TRACKING TRAJECTORY tn o TRACKING TRAJECTORY FIGURE 5.6 TARGET TRAJECTORY (DOTTED) WITH SDP = 0.01, SDM 2, T = 0.1, SNR = 5 cn TRACKING TRAJECTORY FIGURE 5.7 TARGET TRAJECTORY WITH SDP = 2, SDM = 0.2, T = 0.5 TRACKING TRAJECTORY 0 10 20 30 40 50 60 70 00 90 100 FIGURE 5.8 TARGET TRAJECTORY (DOTTED) WITH SDP = 5, SDM = 0.01, T = 0.5 5.2 Sinusoidal Trajectory 54 For the sinusoidal case, which is of special interest, the phase variable representation develops in the following way. Consider, at first, the second order differential equation which is repeated here for convenience: x(t) + co2x(t) = 0 or x + co2x = 0 5.2.1 The sinusoidal trajectory is generated via the phase variable state model of 5.2.1. X1 = X] X, = x2 Let: .[ =* 1 * 5.2.2 X CM 3 1 II CM X >T n CM X 5.2.3 Since x = Ax + B X1 0 1 ' ' X1 0 S + . X2 . -CO 2 0 _ . X2 . 1 Similarly, for the discretization process: (j) A eAT 5.2.5 5.2.6 r = B then 5.2.4 becomes 5.2.7 *k+1 = $k+1,k + TkWk 5.2.8 55 and ^k+1 ^k+1 ^k+1 + Yk+1 5.2.9 where Hk+1 = [ 1 0] 5.2.10 ' *1 ' Note that Zk+1 [ 1 0 ] + Yk+1 *2 k+1 => Zk+I=xk+1 + Vk+1 5-2-11 Also the (-G)2) quantity of 5.25 is: .. 2k 2k Tsig 20Tsam 10Tsam and -co2 = -x 100T2 The Kalman tracking filter equation is applied to the sinusoidal case for the following range of process and measurement noise deviations. SDP = 0.01 SDM = 0.001 SNR = 39.6E3 H II o cn SDP = 0.01 SDM = 0.1 SNR 396 T = 0.5 SDP = 0.01 SDM = 1.5 SNR 1.7 H II p Vl SDP = 0.01 SDM = 0.01 SNR 396 H II p cn SDP = 1. SDM = 0.01 SNR-4 T = 0.5 SDP = 0.1 SDM = 0.1 SNR 1 T = 0.5 SDP = 0.1 SDM = 0.3 SNR 40 H II p cn TABLE 5.2 SUMMARY OF NOISE DEVIATIONS AND SNRS TRACKING TRAJECTORY tracking trajectory 6 T TARGET TRAJECTORY cn oo 5 A 3 2 1 0 1 2 3 A 5 5 A 3 2 1 0 1 2 3 A 5 i------1------1-------1------1----i-------1------1-------r __i------j___:___i_______i_______t_______i_______i_______i______i_______ 10 20 .. 30 40 50 60 70 80 90 100 FIGURE 5.13 TARGET TRAJECTORY WITH SDP = 0.1, SDM = 0.01, T = 0.5 CD O 15 ---------1--------1--------1--------1---------1--------1--------1--------1--------1 _i__________i__________i__________i----------1----------1---------i --------1----------1 10 20 30 40 50 60 70 00 -15 0 FIGURE 5.15 TARGET TRAJECTORY WITH SDP = 1.0, SOM = 0.01, T = 0.5 90 100 25 20 15 10 5 0 -5 10 15 20 25 10 20 30 40 50 60 70 80 90 100 FIGURE 5.16 TARGET TRAJECTORY (DOTTED) WITH SDP = 2, SDM = 0.1, T = 0.5.SNR = 2 CD co 5.3 Discussion From the simulations of 5.1 and 5.2, the applicability and usefulness of the Kalman tracking filter can be easily recognized. Comparing the two cases, it is possible to draw many different kinds of conclusions and explanations. Probably the most interesting ones are related to the consistency of the filter performance, for the parabolic case, and the expected error that is introduced in the sinusoidal case. The results verify and couple in a strong way the theory that was presented in Chapter 2, relating to the importance of the classification set. \ Recall that correct definitions and realization of the classification set is mandatory and unavoidable. The parabolic situation verifies the above notion since the overall tracking process behaves according to the laws of this set. The same observation can be made also for the sinusoidal case, where an error is introduced due to the approximating nature of the system dynamics matrix or, in other words, the tracking matrix. In addition to the above, the quantity that couples the results with the theoretical aspects of Kalman filtering is the innovation process. It is very important to realize that for the parabolic case the error is zero mean and unbiased, which is in agreement with the Kalman filter theory. On the other hand, in the sinusoidal case this weighted quantity, or residual, possesses some periodicity (Refer to Figure 5.4 and 5.12 or 5.14). This periodicity can be explained via the classification set. Since the error is a random convergent zero mean unbiased quantity for the sinusoidal case, it can be said that there are some trends in the identification part, due to the state transition matrix of the Kalman filter trying to approximate an infinitely differentiable function via the Taylor series expansion. CHAPTER 6 CONCLUSION The Kalman tracking filter was proved to be a good systematic state space approach for tracking moving targets which are perturbed by plant noise. Additionally, in this work, the capability of estimating the state (position) of a moving vehicle from noisy measurement data was demonstrated. Both applications presented some of the limitations of the filter for high measurement noise. Moreover, the extreme situation of a parabolic trajectory provided a verification for the importance of correct mathematical modeling. The most stimulating part is probably the ability of the Kalman filter to act as a two-fold algorithm; namely as a state estimator and as a tracker. The latter concept is of greater interest, from a controls point of view. It is necessary to emphasize that the algorithm, loosely speaking, doesn't know anything about the target dynamics; nevertheless, the tracking matrix is able to approximate the unknown system coefficients of the vehicle via the Taylor series expansion terms. An area of concern is the sensitivity of the tracking filter for low signal to noise ratios. Recalling the sinusoidal case, it seems that introduction of measurement noise in the process creates problems to the overall tracking behavior. A probable area of further research would be to identify the matrix parameters more correctly, especially when the problem of time varying acceleration occurs. Since time varying acceleration implies certain nonlinearities, a better matrix may be developed that would depend on different types of functional approximations; numerical mathematics offers a wide variety of functional relationships that could be used for data fitting, such as the Bessel expansion coefficients, the Hermite and Lagrange polynomials, the Laguerre and Legendre functions, and others. Perhaps a tracking matrix, with coefficients derived from the above set of functions, could provide, to the Kalman tracking filter, a higher robustness in the pressure of noise. Another important issue is the minimization of the real time execution of the Kalman tracking filter, equations, especially when it is extended to three dimensional coordinate systems. (Coupling between Cartesian and Polar coordinates) Finally, the great importance and the mechanics of state space mathematics as applied to estimation and tracking became apparent through this work. BIBLIOGRAPHY 1. Ogata, K., Modern Control Engineering. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1970. 2. Gear, William C., Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971. 3. Burden, Richard L., J. Douglas Faires, Albert C. Reynolds, Numerical Analysis (Second Edition). Prindle, Weber & Schmidt, Boston, MA, 1981. 4. Toltov, Georgi P., Fourier Series. Dover Publications, Inc., NY, 1962. 5. Gelb, A., Applied Optimal Estimation. The MIT Press, Cambridge, MA, 1974. 6. Roden, M.S., Introduction to Communication Theory. Pergamon Press, Inc., 1972. 7. Gagliardi, R., Introduction to Communications Engineering. John Wiley & Sons, 1978. 8. Priestley, M.B., Spectral Analysis and Time Series. Academic Press, Harcort Brace Jovanovich, 1981. 9. Papoulis, A., Probability. Random Variables and Stochastic Processes. McGraw-Hill Book Company, 1984. 10. 70 Beyer, W.H., CRC Standard Mathematical Tables (27th Edition^ CRC Press, Inc., 1984. 11. Houpis, C.H., J.J. D'Azzo, Linear Control System Analysis and Design (Conventional and Modern). McGraw-Hill Book Company, 1981. 12. Ramachandra, K.V., State Estimation of Manueuvenng Targets from Noisy Radar Measurements, IEEE Proceedings, F, Communications, Radar, and Signal Processing, Volume 135, Number 1, pp 82-84, February, 1988. 13. Ramachandra, K.V., Position, Velocity and Acceleration Estimates f rom the Noisy Radar Measurements, IEEE Proceedings, F, Communications, Radar, and Signal Processing, Volume 131, Number 2, pp 167-168, 1984. APPENDIX A Formal Derivation via Autocorrelation For the power of X(t) = Asin(cot + following: 7C P avg - dco A.1 Sxx() J Rxx(t) e-jcotdx A.2 Rxx(t) = 2% jSxx(C0) ejCOTdo) A.3 Sxx(co): even spectrum. oo oo Rxx(0) = ^ Jsxx(co) dco => Rxx(0) = ~Jsxx(co) dco A.4 RW = E{x(t) x(t + t)}=!|A | x(t) x(t + x) 6x A.5 -T/2 T/2 Since 72 From X(t) = Asin(cot + <|>) and Equation A.5, Rxx(x) = t^pT 1 Asin(cot + ()>) Asin(cot + cox + <(>) -T T lim A2 T->2T T rsT C 2f J sin(cot + <|>) sin(cot + cox + <()) d<{> A.6 -T From trigonometry tables, sin(cot + - g'COS^t + ()) + cot + cox + ^cos(o)t) ^-cos(2o)t + cox + 2 From Equations A.6 and A.7, RxxM- J [^cos(cox) 7jrcos(2cot + cox + 2$)] d<|> -T lim A2 t>oo 47 T J [cos(cox) cos(2cot + cox + 2 -T 73 lim A2 t-*~ 4T 2T cos(cox) jcos(2cot + cot + 2
-T |