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(5.23 6)
(5.237)
Zi = Pi Ax + Pi Bu (5.238)
Because this transformed equation is suppased to be in phase-variable form, we
must have zi - 22. This gives 22 - PiAx and PiB - 0. From Eq. (5.234),
we have 22 - P2X. Therefore, P2 - PiA. Continuing this further, we find that
22 - Pl A2X, Pi AB - 0, P3 - Pl A2,...,2n_l = Pi An-lX, PiAl-2B = O,
and Pn-I = PiAn-2.Finally,we have
Zn = Pi A Ix + Pi An -1Bu (5.239)
Comparing this with Eq. (5.235), we find that PiAI-lB = 1. With this, we can
construct the following matrix:
[PiB PiAB PiA2B . PiAn-iB]=[O O 0 . 1] (5.240)
Pi-[0 O 0 . 1][B AB A2B . A"-lB]-l
-[o o 0 . 1lQcl
(5.241)
(5.242)
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 517
Because we have assumed that the given systemis controllable, the controllability
matnix Qc is nonsingular and Qc 1 exists. Once Pi is known, then P2, P3,..., Pn
can be calculated using the relations derived above. Then, the phase-variable form
ofthe given systemis .
z = (PAP-1)z + (PB)u (5.243)
5.70.8 ' Conversion of Transfer Function Form to
Phasl-Variable Form
Suppose the relation between the input and output of a system is given in the
form of a transfer function; we can convert this to state-space phase-variable repre-
sentation using a number of different approaches. Here, we will discuss a method
based on decomposition ofthe transfer function. To illustrate the method, consider
the system shown in Fig. 5.41a
Let the open-loop transfer function of a system be given by
G(s) = _ k(S2 + ais + a2) .
s3+b~s2+b2s+b3 (5.244)
The first step is to decompose the grven system into two blocks, one for the
denominator with transfer function Gi(s) and the other for the numerator with
transfer function G2(S) as shown in Fig. 5.41b. Let the output of the :first block be
denoted as xi(s). Then, for the first block,
so that
Gi(s) = xl(s) k
= r~s~ = s3+b~s2+~2s+b3 (5-245)
Xl(S)(S3 + biS2 + b2s +b3) = kr (s) (5.246)
Taking the inverse Laplace transforms,
dd +b,ddt +b2ddt+b3x,=kr(t) (5-247)
┏━━━┳━━━━━━━━━━━━━━┳━━━┓
┃f (s) ┃G(s. : k (82 +aIS + ck) ┃y(8) ┃
┃ ┃U\8) - 83 + bjs2 + b28 + b3 ┃ ┃
┣━━━╋━━━━━━━━━━━━━━╋━━━┫
┃ ┃ ┃ ┃
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