# Detection, Estimation modulation theory part 1 - Vantress H.

ISBN 0-471-09517-6

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/ñ = 1 n

= - 2 Pk-lxk(t) + b0u(t).

(191)

(191)

Denoting the set of xt(t) by a column matrix, we see that the following first-order «-dimensional vector equation is equivalent to the wth-order scalar equation.

dx(t) ä

dt

\(t) = Fx(/) + G u(t),

(192)

where

and

" 0 1

0 i 0

0 ³

F = 0 ••

0 1

-~Po -Pl -Pi -Pa ¦¦¦ -Pn-1.

(193)

G =

0

0

-bo_

(194)

The vector x(t) is called the state vector for this linear system and (192) is called the state equation of the system. Note that the state vector x(/) we selected is not the only choice. Any nonsingular linear transformation of x(r) gives another state vector. The output y(t) is related to the state vector by the equation

X0 = Cx(0, (195)

Differential Equation Representation of Linear Systems 521

where Ñ is a 1 x n matrix

Ñ = [1 ! 0 ! 0 ! 0 • • • 0]. (196)

Equation (195) is called the output equation of the system. The two equations (192) and (195) completely characterize the system.

Just as in the first example we can generate both nonstationary and stationary random processes using the system described by (192) and (195). For stationary processes it is clear (190) that we can generate any process with a rational spectrum in the form of

Sy{cu) = d2na>2n + </2„-2«>2n-2 + • • • + d0 (197)

by letting u(t) be a white noise process and t0 = — oo. In this case the state vector x(0 is a sample function from a vector random process and y(t) is one component of this process.

The next more general differential equation is

y{n\t) + />„_! Óâ_1>(0 + • • • + p0y{t)

= + ••• + b0u(t). (198)

The first step is to find an analog computer-type realization that corresponds to this differential equation. We illustrate one possible technique by looking at a simple example.

Example 2A. Consider the case in which n — 2 and the initial conditions are zero. Then (198) is

ßÎ + Pi AO + ÐîÓ²Î = u(t) + b0 u(t). (199)

Our first observation is that we want to avoid actually differentiating u(t) because in many cases of interest it is a white noise process. Comparing the order of the highest derivatives on the two sides of (199), we see that this is possible. An easy approach is to assume that u(t) exists as part of the input to the first integrator in Fig. 6.28 and

examine the consequences. To do this we rearrange terms as shown in (200):

[ß0 “ bi m(01 + Pi AO + Po AO = bo u(t). (200)

The result is shown in Fig. 6.28. Defining the state variables as the integrator outputs, we obtain

*i(0 = Ó²Î (201a)

and

*a(0 = AO ~ bx u(t). (201 b)

Using (200) and (201), we have

ii(0 = *2(0 + bi u(t) (202a)

MO = -Po *i(0 - Pi (*2(0 + bx «(0) + bo u(t)

= ~Po *i(0 - Pi *2(0 + (b0 - bxpx) u{t). (202b)

We can write (202) as a vector state equation by defining

f° Ï

L-/>o -p J

(203a)

522 6.3 Kalman-Bucy Filters u(t) u(t)

and

Then

The output equation is

G-f 1.

Lb0 - pibd

(203b)

x(t) = Fx(/) + Gu(t). (204a)

y(t) = [1 I 0]x(O é Ñ x(0. (204b)

Equations 204a and 204b plus the initial condition x(r0) = 0 characterize the system.

It is straightforward to extend this particular technique to the wth order

(see Problem 6.3.1). We refer to it as canonical realization No. 1. Our

choice of state variables was somewhat arbitrary. To demonstrate this, we reconsider Example 2A and develop a different state representation.

Example 2Â. Once again

ËÎ + Pi y(t) + Po y(t) = bx u(t) + bo u(t). (205)

As a first step we draw the two integrators and the two paths caused by bx and b0. This partial system is shown in Fig. 6.29a. We now want to introduce feedback paths and identify state variables in such a way that the elements in F and G will be one of the coefficients in the original differential equation, unity, or zero. Looking at Fig. 6.29ä, we see that an easy way to do this is to feed back a weighted version of *i(0 (= y(t)) into each summing point as shown in Fig. 6.29b. The equations for the state variables are

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