=
h
1
1
3 + 2p2
)
k
k
1 + 2Ttrk
i.e., any h that satis es the constraints in (12.47) has an objective that exceeds the
objective of our candidate solution (12.52). This proves that our guess is correct.
(The origin of this mysterious is explained in the Notes and References.)
From our solution (12.52) of the optimization problem (12.47), we conclude that
the speci cations corresponding to Emax and Ttrk are achievable if and only if
Emax(1 + 2Ttrk) > 3 + 2p2:
(12.56)
(We leave the construction of a controller that meets the speci cations for Emax
and Ttrk satisfying (12.56) to the reader.) This region of achievable speci cations
is shown in gure 12.6.
5
4:5
4
3:5
3
Emax 2:5
2
1:5
1
0:5
0
0
0:5
1
1:5
2
T2:5 3 3:5 4 4:5 5
trk
The tradeo between peak tracking error and tracking band-
Figure
12.6
width specications.
Note that to guarantee that the worst case peak tracking error does not exceed
10%, the weighting lter smoothing time constant must be at least Ttrk 28:64,
which is much greater than the time constants in the dynamics of P0, which are on
the order of one second. In classical terminology, the tracking bandwidth is consid-
erably smaller than the open-loop bandwidth. The necessarily poor performance
implied by the tradeo curve (12.56) is a quantitative expression that this plant is
\hard to control".
NOTES AND REFERENCES
291
Notes and References
LQR and LQG-Optimal Controllers
Standard references on LQR and LQG-optimal controllers are the books by Anderson
and Moore AM90], Kwakernaak and Sivan KS72], Bryson and Ho BH75], and the
special issue edited by Athans Ath71]. Astrom and Wittenmark treat minimum variance
regulators in AW90]. The same techniques are readily extended to solve problems that
involve an exponentially weighted 2 norm see, e.g., Anderson and Moore AM69].
H
Multicriterion LQG
The articles by Toivonen Toi84] and Toivonen and Makila TM89] discuss the multicri-
terion LQG problem the latter article has extensive references to other articles on this
topic. See also Koussoulas and Leondes KL86].
Controllers that Satisfy an
Norm-Bound
H
1
In Zam81], Zames proposed that the
norm of some appropriate closed-loop trans-
H
1
fer matrix be minimized, although control design specications that limit the magnitude
of closed-loop transfer functions appeared much earlier. The state-space solution of sec-
tion 12.3 is recent, and is due to Doyle, Glover, Khargonekar, and Francis DGK89, GD88].
Previous solutions to the feasibility problem with an
norm-bound on were consid-
H
H
1
erably more complex.
We noted above that the controller me of section 12.3 not only satises the speci-
K
cation (12.26) it minimizes the -entropy of . This is discussed in Mustafa and
H
Glover Mus89, MG90, GM89]. The minimum entropy controller was developed inde-
pendently by Whittle Whi90], who calls it the linear exponential quadratic Gaussian
(LEQG) optimal controller.
Some Other Analytic Solutions
In OF85, OF86], O'Young and Francis use Nevanlinna-Pick theory to deduce exact trade-
o curves that limit the maximum magnitude of the sensitivity transfer function in two
dierent frequency bands.
Some analytic solutions to discrete-time problems involving the peak gain have been found
by Dahleh and Pearson see Vid86, DP87b, DP88b, DP87a, DP88a].
About Figure 12.4
The step response shown in gure 12.4 was found as follows. We let
20
( ) =
10
X
i
(12.57)
T
s
x
i
=1
+ 10
s
i
where
20 is to be determined. (See chapter 15 for an explanation of this Ritz
x
2
R
approximation.) ( ) must satisfy the condition (12.34). The constraint ( ) = 0 is
T
s
T
1
automatically satised the interpolation condition (1) = 0 yields the equality constraint
T
on ,x
T
= 0
(12.58)
c
x
292