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Note that the noise process and linear system must be speci ed jointly, since there are more degrees of freedom than design constraints Also, the selection of G(s) from the two factors of eqn (452) is not arbitrary The transfer function G(s) is selected so that its poles and zeros lie in the strict left half of the complex s-plane This results in a causal, stable linear system model Such a choice is always possible, when the spectral density

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is the ratio of polynomials containing only even powers of s The interested reader is referred to the subject of spectral factorization When the driving noise is white and Gaussian, the process speci ed in eqn (454) is an example of a scalar Gauss-Markov process Higher order vector Gauss-Markov processes, such as the system of eqn (455), result when the correlation function is more complex Additional examples of simple Gauss-Markov process that are often useful in navigation system error model are discussed in Section 463

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As motivation for the study of stochastic systems, consider the error dynamics for a single channel of a tangent plane INS, as depicted in Figure 44 and the following equation x 0 1 0 x 0 0 v = 0 0 g v + 1 0 a (456) g 1 0 1 0 0 R where [x, v, ] represents error in the navigation state The error model is driven by two sources of error: accelerometer error a and gyro error g These errors originate in the sensors, but appear in the error model as process noise The error at each source may take several forms: white noise, correlated (colored) noise, bias error, scale factor error, etc After any deterministic error sources have been compensated or modeled, the remaining instrument errors will be modeled as random inputs The use of random variables to model the error sources is reasonable, since although the analyst may not be able to specify the value of the various error sources at some future time, it is usually possible to specify the dynamic nature and statistics of the error sources as a function of time It is also reasonable to restrict our attention to linear stochastic systems, since the primary focus of the stochastic analysis in navigation systems will be on linearized error dynamics

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The primary objectives for this section are to present: (1) how to account for stochastic sources of error in state space dynamic system modeling; and, (2) how the mean and covariance of system errors propagate in linear stochastic systems This section presents various results for linear, state space systems with random inputs that will be used throughout the remainder of the book

services Instructing the library clientele in the use of the catalog, special reference tools, and indexes In order to become a medical librarian, you need four years of college, plus a master of library science degree from a school of library science Some approved library schools offer courses in medical librarianship Scholarships are also available The Medical Library Association includes membership for individuals and institutions

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The model for a nite-dimensional linear continuous-time system with stochastic inputs can be represented as x(t) = F(t)x(t) + G(t)w(t) y(t) = H(t)x(t) + v(t) (457)

where w(t) and v(t) are random variables The random variable w is called the process noise The random variable v is called the measurement noise At each time, x(t) and y(t) are also random variables The designation random variable implies that although the value of the random variable at any time is not completely predictable, the statistics of the random variable may be known Assuming that the model is constructed so that v(t) and w(t) are white, the mean and covariance of the random variables w(t) and v(t) will be denoted w (t) = E w(t) = 0 and cov (w(t), w( )) = Q(t) (t ), v (t) = E v(t) = 0 and cov (v(t), v( )) = R(t) (t ) (458) (459) (460) (461)

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In the analysis that follows, it will often be accurate (and convenient) to assume that the process and measurement noise are independent of the current and previous state cov (w(t), x( )) = 0 for t cov (v(t), x( )) = 0 for t and independent of each other cov (w(t), v( )) = 0 for all t, 0 (464) (462) (463)

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