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y(t) = C(0 x(f), (2206)
as the basic representation^ The matrices F(f) and G(f) may be functions of time. By using a white noise input
E[u[t) u(r)] = r), (221)
we have the ability to generate some nonstationary random processes. It is worthwhile to observe that a nonstationary process can result even when F and G are constant and x(f0) is deterministic. The Wiener process,
defined on p. 195 of Chapter 3, is a good example.
Ul(t) ^ Single yi(t) ^
U2(t) Single Ó2(0
u (t) Vector y(t)
Fig. 6.32 Generation of two messages.
t The canonic realizations in Figs. 6.28 and 6.30 may still be used. It is important to observe that they do not correspond to the same «th-order differential equation as in the time-invariant case. See Problem 6.3.14.
528 6.3 Kalman-Bucy Filters
Example 3. Here F(/) = 0, G(0 = <j, C(t) = 1, and (220) becomes
= a u(t).
Assuming that x(0) = 0, this gives the Wiener process.
Other specific examples of time-varying systems are discussed in later sections and in the problems.
The motivation for studying multiple input-multiple output systems follows directly from our discussions in Chapters 3, 4, and 5. Consider the simple system in Fig. 6.32 in which we generate two outputs yx(t) and Ó2²0- We assume that the state representation of system 1 is
*i(0 = Fx(0 xx(0 + Gx(0 wx(0, (223)
*(0 = Cx(0 xx(0, (224)
where xx(0 is an ÿ-dimensional state vector. Similarly, the state representation of system 2 is
^2(0 = ^2(0 ^2(0 4" ^2(0 ^2(0? C^25)
72(0 = Ca(0 x2(0, (226)
where x2(0 is an m-dimensional state vector. A more convenient way to describe these two systems is as a single vector system with an (n + m)-dimensional state vector (Fig. 6.32b).
x(0 = F(/) = G (0 = u(0 = C(0 =
Fi(0 : 0 ³
. 0 ³ W.
*Gi(/) ³ î
. 0 ! g2(0
'Q(0 j 0
. 0 ³ ñ2(f).
>i(0" ¦ MO.
Differential Equation Representation of Linear Systems 529 The resulting differential equations are
x(r) = F(0 x(0 + G(0 u(0, y(0 = C(t) x(t).
The driving function is a vector. For the message generation problem we assume that the driving function is a white process with a matrix covariance function
where Q is a nonnegative definite matrix. The block diagram of the generation process is shown in Fig. 6.33.
Observe that in general the initial conditions may be random variables. Then, to specify the second-moment characteristics we must know the covariance at the initial time
Kx(f0,*o)^ E[x(t0)xT(t0)] (236)
and the mean value 2?[x(f0)]. We can also generate coupled processes by replacing the 0 matrices in (228), (229), or (231) with nonzero matrices.
The next step in our discussion is to consider the solution to (233). We begin our discussion with the homogeneous time-invariant case. Then (233) reduces to
x(0 = Fx(0, (237)
with initial condition x(/0). If x(f) and F are scalars, the solution is familiar,
x(t) = e^-^xito). (238)
For the vector case we can show that (e.g. , , , or )
x(0 = eF(t-*o>x(;0), (239)
Matrix time-varying gain
Fig. 6.33 Message generation process.
530 6.3 Kalman-Bucy Filters
where eFt is defined by the infinite series
åø A I + Ft + ^ +•••. (240)
The function åð<(-‘î> is denoted by ô(/ — t0) é. ô(ò). The function
Ô(* - to) is called the state transition matrix of the system. Two properties
can easily be verified for the time-invariant case.
Property ll.f The state transition matrix satisfies the equation
d%~- = W - to) (241)