12.3
Minimum Entropy Regulator
The LQG solution method described in section 12.2 was recently modi ed to nd
the controller that minimizes the -entropy of H, de ned in section 5.3.5. Since
the -entropy of H is nite if and only if its
norm is less than , this analytic
H
1
solution method can be used to solve the feasibility problem with the inequality
speci cation H < .
k
k
1
The plant is identical to the one considered for the LQG problem, given by (12.7{
12.15) we also make the same standard assumptions for the plant that we made
for the LQG case. The design speci cations are realizability and the
norm
H
1
inequality speci cation
H <
(12.26)
k
k
1
(which are stronger than internal stability under the standard assumptions). We
will show how to solve the feasibility problem for this one-dimensional family of
design speci cations.
It turns out that if the design speci cation (12.26) (along with realizability) is
achievable, then it is achievable by a controller that is, except for a scale factor,
an estimated-state-feedback controller. This controller can be found as follows. If
is such that the speci cation (12.26) is feasible, then the two algebraic Riccati
equations
ATXme + XmeA Xme(BR 1BT
2W
;
;
)Xme + Q = 0
(12.27)
;
;
(c.f. (12.17)), and
AYme + YmeAT Yme(CTV 1C
2Q
;
;
)Yme + W = 0
(12.28)
;
;
(c.f. (12.19)) have unique positive de nite solutions Xme and Yme, respectively. (The
mnemonic subscript \me" stands for \minimum entropy".) These solutions can be
found by the method described in section 12.1, using the associated Hamiltonian
matrices A (BR 1BT 2W)
A
W
;
;
;
;
Q
AT
;
(CTV 1C
2Q) AT
;
;
;
;
;
;
;
if either of these matrices has imaginary eigenvalues, then the corresponding ARE
does not have a positive de nite solution, and the speci cation (12.26) is not feasible.
From Xme and Yme we form the matrix
Xme(I
2Y
1
(12.29)
;
meXme);
;
which can be shown to be symmetric. If this matrix is not positive de nite (or the
inverse fails to exist), then the speci cation (12.26) (along with realizability) is not
feasible.
12.4 A SIMPLE RISE TIME, UNDERSHOOT EXAMPLE
283
If, on the other hand, the positive de nite solutions Xme and Yme exist, and the
matrix (12.29) exists and is positive de nite, then the speci cation (12.26) (along
with realizability) is feasible. Let
Ksfb = R 1BTX
2Y
1
(12.30)
;
me(I
;
meXme);
;
and
Lest = YmeCTV 1
(12.31)
;
(c.f. (12.16) and (12.18)). A controller that achieves the design speci cations is
given by
K
1
me(s) = Ksfb ;sI A + BKsfb + LestC
2Y
L
;
meQ ; est
;
;
;
(c.f. the LQG-optimal controller (12.20)).
12.4
A Simple Rise Time, Undershoot Example
In this section and the next we show how to nd explicit solutions for two speci c
plants and families of design speci cations.
We consider the classical 1-DOF system of section 2.3.2 with
P
1
0(s) = s ;
(s + 1)2:
It is well-known in classical control that since P0 has a real unstable zero at s = 1,
the step response from the reference input r to the system output yp, s13(t), must
exhibit some undershoot. We will study exactly how much it must undershoot, when
we require that a stabilizing controller also meet a minimum rise-time speci cation.
Our design speci cations are internal stability, a limit on undershoot,
us(H13) Umax
(12.32)
and a limit on rise time,
rise(H13) Tmax:
(12.33)
Thus we have a two-parameter family of design speci cations, indexed by Umax and
Tmax.These design speci cations are simple enough that we can readily solve the
feasibility problem for each Umax and Tmax. We will see, however, that these design
speci cations are not complete enough to guarantee reasonable controller designs
for example, they include no limit on actuator e ort. We will return to this point
later.
284