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u + hot s f (xo , uo )
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F(t) x + G(t) u + hot s
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(332)
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f f where u = u uo , F(t) = x xo (t),uo (t) , and G(t) = u xo (t),uo (t) The resultant perturbation to the system output y = y yo is
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y(t) = = y(t) =
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h(x(t)) h(xo (t)) h(x) x(t) + hot s x xo (t),uo (t) H(t) x(t) + hot s (333)
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where H(t) = h xo (t),uo (t) By dropping the higher-order terms (hot s), x eqns (332) and (333) provide the time-varying linearization of the nonlinear system: x(t) = y(t) = F(t) x(t) + G(t) u(t) H(t) x(t) (334) (335)
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33 STATE SPACE LINEARIZATION
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which is accurate near the nominal trajectory (ie, for small x and u ) Applications of eqns (334) and (335) are common in navigation applications For example, the GPS range equations include the distance from the satellite broadcast antenna e ective position to the receiver antenna e ective position When the satellite position is known and the objective is to estimate the receiver location, the GPS measurement is nonlinear with the generic form of eqn (331), but frequently solved via linearization See Section 822 Also, the navigation system kinematic equations, discussed in s 9 12, are nonlinear and have the generic form of eqn (330) The following example illustrates the basic state space linearization process Example 39 Assume that there is a true system that follows the kine matic equation x = f (x, u) where x = [n, e, ] R3 , u = [u, ] R2 , and u cos( ) f (x, u) = u sin( ) (336) For this system, the variables [n, e] de ned the position vector, is the yaw angle of the vehicle relative to north, u is the body frame forward velocity, and is the yaw rate Also, assume that two sensors are available that provide measurements modeled as y= u + eu + e
A very simple dead-reckoning navigation system can be designed by inte gration of xo = f (xo , y) for t to from some initial xo (to ) where xo = [no , eo , o ] The resulting dead-reckoning system state, xo (t) for t 0, will be used as the reference trajectory for the linearization process The navigation system equation can be simpli ed as follows: xo = f (xo , y) u cos( o ) eu cos( o ) (u + eu ) cos( o ) = (u + eu ) sin( o ) = u sin( o ) + eu sin( o ) ( + e ) e eu cos( o ) = f (xo , u) + eu sin( o ) (337) e
The rst order (ie, linearized) dynamics of the error between the actual and navigation states are 0 0 sin( o ) u cos( o ) 0 eu u cos( o ) v sin( o ) 0 v= 0 0 (338) e 0 0 0 0 1
CHAPTER 3 DETERMINISTIC SYSTEMS
where u = y1 and v = [ n, e, ] = x xo To derive this linearized model, the function f in eqn (336) is expanded using Taylor series Then eqn (337) is subtracted from the Taylor series expansion of eqn (336) to yield eqn (338) For additional discussion related to this example, see 9 Error state dynamic equations enable quantitative analysis of the error state itself Later, the linearized error state equations will be used in the design of Kalman lters to estimate the error state Prior to that, we must discuss such issues as stochastic modeling of sensor errors which is done in 4
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