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RandomSignals and Applied Kalman Filtering with Matlab Exercise (0, 2014-08-22)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercise.pdf (5037894, 2014-08-19)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM (0, 2014-08-20)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO (0, 2014-08-20)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO\GPS.H (2188, 1996-06-02)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO\GPS.P0 (103, 1991-03-28)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO\GPS.PRM (72, 1996-08-06)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO\GPSQ.txt (103, 1991-03-28)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO\GPSR.txt (61, 1991-03-28)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\GPSDEMO\GPSSTM.txt (92, 1991-03-28)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS (0, 2014-08-20)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\ATT.BGI (6269, 1988-08-29)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\CGA.BGI (6253, 1988-08-29)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\DEFFONT.GLF (11750, 1989-11-03)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\EGAVGA.BGI (5363, 1988-08-29)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\EIGNPROC.EXE (44832, 1992-01-11)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\HERC.BGI (6125, 1988-08-29)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\IBM8514.BGI (6665, 1988-08-29)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\KALM.EXE (100160, 1995-11-04)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\N-LIES.EXE (163008, 1993-04-20)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\PC3270.BGI (6029, 1988-08-29)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\PRINTER.CFG (42, 1995-04-18)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\PROCESOR.EXE (98320, 1995-11-04)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\KALM\PROGS\STMQPROC.EXE (31520, 1993-04-04)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES (0, 2014-08-19)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\CHISQ1.M (190, 1996-08-12)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\CHISQ16.M (198, 1996-08-12)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\CHISQ2.M (164, 1996-08-12)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\CHISQ4.M (169, 1996-08-12)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\COVAR521.M (801, 1996-08-12)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\DRNCOS1.M (2047, 1996-06-04)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\DRNCOS2.M (2045, 1996-06-03)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\EX05_3.M (1181, 1996-06-01)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\EX06_1.M (894, 1996-04-04)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\EX06_5.M (929, 1996-04-04)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\EX10_2.M (4318, 1996-05-26)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\EX11_1.M (4590, 1996-08-13)
RandomSignals and Applied Kalman Filtering with Matlab Exercise\MFILES\FILCOVAR.M (1086, 1996-04-12)
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USER'S GUIDE FOR KALM 'N SMOOTH (VERSION 2.0) ============================================= Kalm 'N Smooth was the main tutorial tool used in conjunction with the previous edition of this book. It has now been supplanted by MATLAB for the versatility and power of the latter. Kalm 'N Smooth is still effective software for solving many Kalman filtering problems, so it is included for those who prefer a menu-oriented program. This user's guide in its text form is understandably limited in its typesetting qualities and hence must be read with caution and a little imagination. We suggest that the user's guide be printed out for easier perusal. Before describing the capabilities of the Kalm 'N Smooth software package, we wish to emphasize that the programs were written primarily for instructional purposes and in spite of its limitations, we invite its usage for solving practical problems as well. No extraordinary attempt was made to make the software especially efficient, either in terms of speed, program or memory capacity. Quite to the contrary, "user friendliness" was the prime consideration in the software development. All programs in the software package are menu oriented, and the user is prompted at each step as to what needs to be done to get to the next step. And the user will find that even on a personal computer with modest speed, most assigned tasks will be executed expediently enough for tutorial purposes. OVERVIEW OF THE SOFTWARE CAPABILITIES Covariance Analysis for Optimal Filtering and Smoothing: In covariance analysis, the random input (measurement) and output (estimate) processes are not generated. Rather, the program simply cycles through the error covariance equations recursively beginning with the initial covariance. In order to do this, of course, the user must specify the four key parameters phi, Q, H, and R, plus the initial P(0)-. The end result that the user would want to derive from this processing is a sequence of error covariance matrices. These data contain information about the estimation accuracy under optimal conditions. There are separate functions available for filtering and smoothing. The required parameters are identical for both, except that in the latter, the smoothing interval must be specified in addition. Suboptimal Analysis: It is not difficult to imagine a sequence of filter gains that are suboptimal. They could be the result of a mismodeled situation, or the suboptimal gain sequence might be specified completely arbitrarily with no connection to any model. Regardless, once the gain sequence is specified the structure of the filter dictates that the estimate update and projection be done in accordance with Eqs. 5.5.8 and 5.5.23. The theory allows us to compute the error covariance associated with this operation for any arbitrary gain K (with the proper dimensions, of course). The general recursive equations for the error covariance (optimal or suboptimal) can be found in Eqs. 5.5.18 and 5.5.25. The suboptimal analysis program simply implements these equations. Obviously, the user must specify the true phi, Q, H, and R which define the structure of the filter. This is sometimes referred to as the "truth" model. The user must also specify the gain sequence K. If this comes from an "untruth" model, then this system must be run first to generate the appropriate number of computational steps, the gains saved in a data file, and then these suboptimal gains used in the suboptimal component program. (Caveat: There are some restrictions to this elementary approach to suboptimal analysis. See Section 6.7 of the text for details). Monte Carlo Kalman Filter and Smoother Simulation: In a Monte Carlo filter simulation, a sample realization of the input (measurement) process must first be generated using the random process generation function described below. Separate Monte Carlo processing functions are available for filtering and smoothing. Therefore, in addition to parameters specified above for the covariance analysis, the user must specify the measurement sequence z as well as an initial state estimate x(0)-. The filter or smoother then yields a corresponding output (estimate) process that constitutes simply one sample realization for the specified number of computational steps. This can, of course, be repeated over and over again with different noisy measurement sequences made from the random process generation. In doing so, the user obtains an ensemble of experiments which may be analyzed statistically. This makes a vivid demonstration that the error covariance sequence associated with the experiment does, in fact, yield meaningful information about the estimation accuracy. Random Process Generation: In Monte Carlo simulation work we often have occasion to generate a discrete random sequence in state-space form (see Eqs. 5.5.1 and 5.5.2). This form allows the generation of any stationary Gaussian process with a given rational spectral function using the modeling methods of Chapter 5. Clearly, if the user wishes to generate a "noiseless" version of the process, the variance of the parameter v(k) can simply be set to zero. This program requires the user to specify the usual phi, Q, H, and R parameters, and also the initial conditions on x(k). The end results are files for both x(k) and z(k), which can then be used in subsequent programs. Schmidt Filtering (covariance analysis and Monte Carlo simulation): Schmidt filtering may be executed with Kalm 'N Smooth by setting up a state vector that consists of both the reduced-order states x and the "consider" states y (see Eq. 9.4.1). The Kalman filter parameters are to be set up according to Eqs. 9.4.2 and 9.4.3. When the Schmidt filter function is invoked, the user will be prompted for the number of reduced-order states (x). This number defines where the partition separating the x and y subvectors lie. In the subsequent processing, the Schmidt model is treated like a regular Kalman filter except the gains associated with the y states are zeroed on every iteration. Even though this is not the way a Schmidt filter is actually implemented, this way of processing performs the same exact function. There are separate functions to construct a Schmidt filter for covariance analysis and for Monte Carlo simulation. Steady-State Characteristic Roots Computation: It was mentioned in Chapter 6.9 that a Kalman filter behaves much the same as any other constant-parameter digital filter when (and if) the filter gain reaches a steady-state condition. The user only needs to specify the usual phi, Q, H, and R parameters and the program computes the steady-state gain. The end result of this program is the set of characteristic roots of the filter in the z-plane. State Transition Matrix (STM) and Q Matrix Computations: Calculation of the STM and Q matrices in higher-order systems is an onerous task with paper-and-pencil methods. This program relieves the user of this task. We begin with the continuous model for the system state vector x: dx/dt = F x + G u where F and G are constant matrices, and u is the input vector whose elements are independent unity white noises. The user specifies F and G in the model and the step size Dt, and the program does the rest. The end products are data files for both phi(k) and Q(k) that can be used in subsequent programs. Discrete Fourier Transform (DFT), Inverse DFT, and Periodogram: In signal analysis, it is important to be able to go back and forth between the time and frequency domain. Thus direct and indirect Fourier transform programs have been included in the software package. Note that the straightforward Discrete Fourier Transform (DFT) is implemented in this software (in contrast to the FFT), so the number of data points is not restricted to an integer power of 2. The DFT takes a little longer to execute than the FFT, but from a tutorial viewpoint it is better because it is less restrictive. In the included programs, the data stream may be of any reasonable length and may be either real or complex in both the time and frequency domains. It is worth noting that the DFT program transforms an N-point time function into an N-point complex function in the frequency domain. If the original time function is real, this of course leads to redundancy in the frequency domain, with points above the foldover frequency being complex conjugates of the corresponding lower-frequency points. We all know that this is excess baggage for the special case of real-valued time-domain functions, but it does no harm to do so and is correct mathematically. The periodogram of a time signal is also of special interest in signal analysis. Recall from Chapter 2 that the periodogram is proportional to the square of the Fourier transform, and that it has a close statistical connection to the autocorrelation function. Therefore, in the DFT program, an additional output is provided which is proportional to the square of the magnitude of the Fourier transform, i.e. the periodogram. Autocorrelation Function: The autocorrelation function is a basic descriptor of a stationary random process. There is, of course, no way that one can determine the true autocorrelation function from just one sample realization of a process. First, there is the matter of the sample length; and second there is the problem of ergodicity. However, if the data span is long relative to the pertinent correlation times, and if we assume that the sample realization at hand is representative of all possible samples (i.e., assume ergodicity), then we can compute something that approximates the true autocorrelation function. This is done simply by multiplying the time function by itself shifted t units in time, and then averaging. The autocorrelation function program included in this software package does just that. The approximation so obtained does have limitations, though, and these should be recognized, especially when the number of shifts (the "tau" parameter) is appreciable relative to the data span (see Chapter 2). Data Graphing: Who was it that said, "A picture is worth a thousand words"? How true this is, especially in analysis work. For this reason, data plotting capability has been built into this software package in two ways. First, in procedures where it is appropriate, the user is provided with a prompt such as "Press ENTER to view the data" after a procedure is executed. This provides a quick look at the data in bar graph form without formal plotting. Second, a more formal data graphing capability is provided in a separate program whereby several data files may be plotted together on the same X-Y or line (unspecified independent variable) graphs. Note on Learning to Use the Software: A User's Guide giving detailed instructions on how to use the accompanying software follows in the next section. After installation of the programs on your computer, we suggest that you first familiarize yourself with the various features provided in this software package by working simple examples of your own creation. For example, to check out the DFT procedure, you might try creating a data file for a simple 16-point unit impulse in the time domain. Then take the DFT of this function (which we know to have a constant magnitude); and, finally, form the inverse of the DFT to see if you get the unit impulse back again (within roundoff error, of course). In this way, you not only become familiar with the various procedures, but you gain a degree of confidence that the programs are doing the job and giving the correct answers. You are now ready to begin using the accompanying software package. Be forewarned that a modest amount of practice is necessary before you will begin to feel completely comfortable with the software. This is true with any new software package, and it is to be expected. We trust that the "learning curve" will be brief in this case. Once you get past the initial familiarization phase, we think you will find that Kalman filtering can be fun! HARDWARE REQUIREMENTS The programs provided will run on an IBM PC, PS/2, XT, or AT (or a 100 percent compatible computer) with ***0 KBytes of internal memory. Most of the standard display adapters and printers are supported. Color displays are recommended. DOS Version 3.2 or later should be used. INSTALLING PROGRAMS The programs can be executed from the floppy disk provided or from a hard disk. If it is desired to have the programs from your original diskette transferred to another diskette or installed onto a hard disk, simply copy all the programs over with any file copying utility. In DOS, the file copying utility COPY is suitable for such a task. Important: All programs and files furnished on the original diskette must be located on the same directory path. If installing on a hard disk, a (programs) subdirectory should be set up on the hard disk for this purpose of containing the furnished programs and files. For efficient management, it is recommended that a separate (work) subdirectory be set up on the hard disk for new data files created in the process of using these programs. Starting from this work subdirectory, the need program can be executed by properly specifying the proper path and the appropriate program name. The following example shows the execution of the program KALM which resides in the programs subdirectory named PROGS while all data files specified during execution will by default be stored or read from the work subdirectory named KALMDATA: C:\KALMDATA> \PROGS\KALM PROGRAMS AND FILES On the 1.44-MByte 3.5-inch diskette furnished with the book, there are a total of thirteen files and a subdirectory of files intended for a demonstration. Of those in the main directory, four are executable programs that work together to provide Kalman Filter-related functions: KALM.EXE, PROCESOR.EXE, EIGNPROC.EXE, STMQPROC.EXE. A fifth executable program N-LIES.EXE contains simple utilities intended to provide the user some additional analysis and graphing capabilities. In addition to these programs, the diskette also contains several display drivers for some of the more common graphics standards available today for IBM-compatible PC systems. These driver files are identified by the extension .BGI: CGA.BGI For IBM CGA or MCGA EGAVGA.BGI For IBM EGA or VGA HERC.BGI For Hercules Monochrome ATT.BGI For AT&T 6300 (400 line) PC3270.BGI For IBM 3270 PC IBM8514.BGI For IBM-8514 The capability for printing onscreen-generated graphics is also supported. The two files associated with this capability are named DEFFONT.GLF and PRINTER.CFG . PRINTER.CFG is an ASCII text file that contains user-defineable parameters pertaining to the desirable printer and printing configurations. Prior to doing any printing with these programs, PRINTER.CFG must be properly set up. See the section on Printer Configurations for instructions to do this, using the Data Editor program function provided with the N-LIES program. KALM The main program KALM.EXE handles input data files and runs in conjunction with several processing programs to provide Kalman filtering and other utility functions. They comprise the following: PROCESOR.EXE - Handles all the Kalman filtering functions including generation of random processes; EIGNPROC.EXE - Computes eigenvalues and eigenvectors of the state space model and also the steady- state covariance of a time-invariant state model; STMQPROC.EXE - Computes the state transition matrix and the process noise covariance matrix for a discrete-time model from a continuous-time equivalent. There is a limit to the size of the Kalman filter that can be modeled with the KALM 'N SMOOTH programs. The state dimensionality must be no larger than 20. The limit on the dimensionality on the measurement space is also 20. Therefore, the largest H matrix allowable is 20x20. The two principal data elements utilized here to manage the flow of information are the Parameter and Data files. They are described in detail in the discussion that follows. Parameter File: By convention, the full set of Kalman parameters that may be used consist of the following, although in certain cases, some are not necessary: 1. State transition matrix (phi) 2. Process noise covariance matrix (Q) 3. Measurement matrix (H) 4. Measurement noise covariance matrix (R) 5. Initial state (x0) 6. Initial error covariance (P0) 7. Measurement (z) 8. State output (x) In addition to the names of the Data Files for the Kalman parameters above, a Parameter File also records several other items which includes the mode of data entry for the respective Kalman parameter as well as the dimensions of the state and measurement spaces. Although these parameters are important for the data processing that follows, they are primarily intended for housekeeping functions by the programs that use them and should really be transparent to the user. Data File: A Data File is an ASCII text file which allows the user to input information for a particular Kalman parameter. There are three standard modes of data entry into the processing Kalman filter: 1) constant , 2) file, and 3) equation (see Mode of Data Entry). In each Data File, the mode of data entry must be specified along with the dimensions of the matrix representing the parameter. Both are located in the very first line of each Data File. The dimensions of the matrix must be delimited by brackets and separated (number of rows versus columns) by a colon. If the parameter represented is a vector instead, the dimension may be represented as a single number without use of the colon. The data that follow are read in sequence across the columns of the matrix and down the rows. Comments, which are ignored by the data processing, can be added to each line by placing them to the right of a colon. Equations used in the equation mode of data entry must be in the form of function of time, t or T (see Reference). The following are examples of various formats that can be used for a Data File: 1. A Data File with 2x3 matrices to be read at each iteration, with comments: [2:3] File 3.2 1.1 4.8 2.0 8.2 5.2E-2 : 0 3.1 1.1 5.0 2.5 8.4 5.2E-2 : 1 3.0 1.1 5.1 3.0 8.4 5.2E-2 : 2 . . . 2. A Data File with 2x3 constant matrix: [2:3] Constant 3.2 1.1 4.8 2.0 8.2 5.2E-2 3. A Data File with 2x2 equations matrix using semicolon separators: [2:2] Equation k1 = 2*t+3; k2 = cos (2*pi*t) $1$1 = k1*k2; $1$2 = k1+4 $2$1 = t ; $2$2 = 3 4. A Data File with 2x2 eq ... ...

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