Tài liệu Lọc Kalman - lý thuyết và thực hành bằng cách sử dụng MATLAB (P5) pdf

5.1.2 Main Points to Be Covered Many estimation problems that are of practical interest are nonlinear but ``smooth.'' That is, the functional dependences of the measurement or statedyna

Nonlinear Applications

The principal uses of linear ®ltering theory are for solving nonlinear problems

Harold W Sorenson, in a private conversation

5.1 CHAPTER FOCUS

5.1.1 Nonlinear Estimation Problems

Linear estimators for discrete and continuous systems were derived in Chapter 4.The combination of functional linearity, quadratic performance criteria, and Gaus-sian statistics is essential to this development The resulting optimal estimators aresimple in form and powerful in effect

Many dynamic systems and sensors are not absolutely linear, but they are not farfrom it Following the considerable success enjoyed by linear estimation methods onlinear problems, extensions of these methods were applied to such nonlinearproblems In this chapter, we investigate the model extensions and approximationmethods used for applying the methodology of Kalman ®ltering to these ``slightlynonlinear'' problems More formal derivations of these nonlinear ®lters andpredictors can be found in references [1, 21, 23, 30, 36, 75, 112]

5.1.2 Main Points to Be Covered

 Many estimation problems that are of practical interest are nonlinear but

``smooth.'' That is, the functional dependences of the measurement or statedynamics on the system state are nonlinear, but approximately linear for smallperturbations in the values of the state variables

 Methods of linear estimation theory can be applied to such nonlinearproblems by linear approximation of the effects of small perturbations inthe state of the nonlinear system from a ``nominal'' value

169

Kalman Filtering: Theory and Practice Using MATLAB, Second Edition,

Mohinder S Grewal, Angus P Andrews Copyright # 2001 John Wiley & Sons, Inc ISBNs: 0-471-39254-5 (Hardback); 0-471-26638-8 (Electronic)

 For some problems, the nominal values of the state variables are fairly wellknown beforehand These include guidance and control applications for whichoperational performance depends on staying close to an optimal trajectory Forthese applications, the estimation problem can often be effectively linearizedabout the nominal trajectory and the Kalman gains can be precomputed torelieve the real-time computational burden.

 The nominal trajectory can also be de®ned ``on the ¯y'' as the current bestestimate of the actual trajectory This approach is called extended Kalman

®ltering It has the advantage that the perturbations include only the stateestimation errors, which are generally smaller than the perturbations fromany prede®ned nominal trajectory and therefore better conditioned forlinear approximation The major disadvantage of extended Kalman ®ltering

is the added real-time computational cost of linearization about anunpredictable trajectory, for which the Kalman gains cannot be computedbeforehand

 Extensions of the linear model to include quadratic terms yield optimal ®lters

of greater applicability but increased computational complexity

5.2 PROBLEM STATEMENT

Suppose that a continuous or discrete stochastic system can be represented bynonlinear plant and measurement models as shown in Table 5.1, with dimensions ofthe vector and matrix quantities as shown in Table 5.2 and where the symbols

D…k `† stand for the Kronecker delta function and the symbols d…t s† stand forthe Dirac delta function (actually, a generalized function)

The function f is a continuously differentiable function of the state vector x, andthe function h is a continuously differentiable function of the state vector

Whereas af®ne (i.e., linear and additive) transformations of Gaussian RVs haveGaussian distributions, the same is not always true in the nonlinear case Conse-quently, it is not necessary that w and v be Gaussian They may be included asarguments of the nonlinear functions f and h, respectively However, the initial value

TABLE 5.1 Nonlinear Plant and Measurement Models

Ehw…t†w T …s†i ˆ d…t s†Q…t† Ehwkw T

i i ˆ D…k i†Qk

Ehv…t†v T …s†i ˆ d…t s†R…t† Ehv k v T

i i ˆ D…k i†R k

x0 may be assumed to be a Gaussian random variate with known mean and known

5.4 LINEARIZATION ABOUT A NOMINAL TRAJECTORY

5.4.1 Nominal Trajectory

A trajectory is a particular solution of a stochastic system, that is, with a particularinstantiation of the random variates involved The trajectory is a vector-valuedsequence fxkjk ˆ 0; 1; 2; 3; g for discrete-time systems and a vector-valuedfunction x…t†; 0  t, for continuous-time systems

The term ``nominal'' in this case refers to that trajectory obtained when therandom variates assume their expected values For example, the sequence fxnom

k gobtained as a solution of the equation

xnom

k ˆ f …xnom

with zero process noise and with the mean xnom

0 as the initial condition would be anominal trajectory for a discrete-time system

5.4.2 Perturbations about a Nominal Trajectory

The word ``perturbation'' has been used by astronomers to describe a minor change

in the trajectory of a planet (or any free-falling body) due to secondary forces, such

as those produced by other gravitational bodies Astronomers learned long ago thatthe actual trajectory can be accurately modeled as the sum of the solution of the two-body problem (which is available in closed form) and a linear dynamic model for the

TABLE 5.2 Dimensions of Vectors and Matrices in Nonlinear Model

perturbations due to the secondary forces This technique also works well for manyother nonlinear problems, including the problem at hand In this case, the perturba-tions are due to the presence of random process noise and errors in the assumedinitial conditions.

If the function f in the previous example is continuous, then the state vector

xk at any instant on the trajectory will vary smoothly with small perturbations ofthe state vector xk 1 at the previous instant These perturbations are the result of

``off-nominal'' (i.e., off-mean) values of the random variates involved Theserandom variates include the initial value of the state vector (x0), the processnoise (wk), and (in the case of the estimated trajectory) the measurement noise(vk)

If f is continuously differentiable in®nitely often, then the in¯uence of theperturbations on the trajectory can be represented by a Taylor series expansion aboutthe nominal trajectory The likely magnitudes of the perturbations are determined bythe variances of the variates involved If these perturbations are suf®ciently smallrelative to the higher order coef®cients of the expansion, then one can obtain a goodapproximation by ignoring terms beyond some order (However, one must usuallyevaluate the magnitudes of the higher order coef®cients before making such anassumption.)

Let the symbol d denote perturbations from the nominal,

If the higher order terms in dx can be neglected, then

dxk F‰1Šk 1dxk 1‡ wk 1; …5:7†where the ®rst-order approximation coef®cients are given by

3777777777

xˆx nom

an n  n constant matrix

5.4.3 Linearization of h about a Nominal Trajectory

If h is suf®ciently differentiable, then the measurement can be represented by aTaylor series:

xˆx nom k

dxk‡ higher order terms, …5:10†or

dzkˆ@h…x; k†@x

xˆx nom k

which is an `  n constant matrix

5.4.4 Summary of Perturbation Equations in the Discrete CaseFrom Equations 5.7 and 5.12, the linearized equations about nominal values are

Similar to the case of the discrete system, the linearized differential equations can

5.5 LINEARIZATION ABOUT THE ESTIMATED TRAJECTORY

The problem with linearization about the nominal trajectory is that the deviation ofthe actual trajectory from the nominal trajectory tends to increase with time As thedeviation increases, the signi®cance of the higher order terms in the Taylor seriesexpansion of the trajectory also increases

A simple but effective remedy for the deviation problem is to replace the nominaltrajectory with the estimated trajectory, that is, to evaluate the Taylor seriesexpansion about the estimated trajectory If the problem is suf®ciently observable(as evidenced by the covariance of estimation uncertainty), then the deviationsbetween the estimated trajectory (along which the expansion is made) and the actualtrajectory will remain suf®ciently small that the linearization assumption is valid[112, 113]

The principal drawback to this approach is that it tends to increase the real-timecomputational burden Whereas F, H, and K for linearization about a nominaltrajectory may have been precomputed off-line, they must be computed in real time

as functions of the estimate for linearization about the estimated trajectory

5.5.1 Matrix Evaluations for Discrete Systems

The only modi®cation required is to replace xnom by ^xk 1 and xnom

k by ^xk in theevaluations of partial derivatives Now the matrices of partial derivatives become

H‰1Š…^x; k† ˆ@h…x; k†@x

5.5.2 Matrix Evaluations for Continuous Systems

The matrices have the same general form as for linearization about a nominaltrajectory, except for the evaluations of the partial derivatives:

F‰1Š…t† ˆ@f …x…t†; t†

@x…t†

xˆ^x…t† …5:25†and

H‰1Š…t† ˆ@h…x…t†; t†@x…t†

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and Kalman gains This off-line computation is not possible for the extendedKalman ®lter, because these implementation parameters will be functions of thereal-time state...

5.7 DISCRETE EXTENDED KALMAN FILTER

The essential idea of the extended Kalman ®lter was proposed by Stanley F.Schmidt, and it has been called the ` `Kalman? ?Schmidt'''' ®lter [122,... the linearized Kalman ®lter inTable 5.3

EXAMPLE 5.1 Consider the discrete-time system

TABLE