Problem
In this chapter we describe methods for forming and solving nite-dimensional
approximations to the controller design problem. A method based on the
parametrization described in chapter 7 yields an
approximation of the re-
inner
gion of achievable specications in performance space. For some problems, an
approximation of this region can be found by considering a dual problem.
outer
By forming both approximations, the controller design problem can be solved to
an arbitrary, and guaranteed, accuracy.
In chapter 3 we argued that many approaches to controller design could be described
in terms of a family of design speci cations that is parametrized by a performance
vector
L
,
a
2
R
satis es hard 1( )
1 ...
( )
(15.1)
H
D
H
a
H
a
:
L
L
Some of these speci cations are unachievable the designer must choose among
the speci cations that are achievable. In terms of the performance vectors, the
designer must choose an
, where denotes the set of performance vectors that
a
2
A
A
correspond to achievable speci cations of the form (15.1). We noted in chapter 3
that the actual controller design problem can take several speci c forms, e.g., a
constrained optimization problem with weighted-sum or weighted-max objective,
or a simple feasibility problem.
In chapters 7{11 we found that in many controller design problems, the hard
constraint hard is convex (or even a ne) and the functionals 1 ...
are convex
D
L
we refer to these as convex controller design problems. We refer to a controller
design problem in which one or more of the functionals is quasiconvex but not
convex as a quasiconvex controller design problem. These controller design problems
can be considered convex (or quasiconvex) optimization problems over
since
H
H
351
352
CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
has in nite dimension, the algorithms described in the previous chapter cannot be
directly applied.
15.1
Ritz Approximations
The Ritz method for solving in nite-dimensional optimization problems consists of
solving the problem over larger and larger nite-dimensional subsets. For the con-
troller design problem, the Ritz approximation method is determined by a sequence
of
transfer matrices
n
n
z
w
0
1
2 ...
(15.2)
R
R
R
2
H
:
We let
8
9
= <
=
0 + X
1
H
R
x
R
x
2
R
i
N
N
i
i
i
:
1 i N
denote the nite-dimensional a ne subset of that is determined by 0 and the
H
R
next
transfer matrices in the sequence. The th Ritz approximation to the
N
N
family of design speci cations (15.1) is then
satis es hard 1( )
1 ...
( )
(15.3)
H
D
H
a
H
a
H
2
H
:
L
L
N
The Ritz approximation yields a convex (or quasiconvex) controller design problem,
if the original controller problem is convex (or quasiconvex), since it is the original
problem with the a ne speci cation
adjoined.
H
2
H
N
The th Ritz approximation to the controller design problem can be considered
N
a nite-dimensional optimization problem, so the algorithms described in chapter 14
can be applied. With each
N
we associate the transfer matrix
x
2
R
( ) = 0 + X
(15.4)
H
x
R
x
R
N
i
i
1 i N
with each functional we associate the function ( )
N
: N
given by
R
!
R
i
i
( )
N
( ) = ( ( ))
(15.5)
x
H
x
i
N
i
and we de ne
( )
N
=
( ) satis es hard
(15.6)
D
fx
j
H
x
D
g
:
N
Since the mapping from
N
into given by (15.4) is a ne, the functions
x
2
R
H
( )
N
given by (15.5) are convex (or quasiconvex) if the functionals are similarly
i
i
the subsets ( )
N
are convex (or a ne) if the hard constraint
N
hard is. In
D
R
D
15.1 RITZ APPROXIMATIONS
353
section 13.5 we showed how to compute subgradients of the functions ( )
N
, given
i
subgradients of the functionals .i
Let
denote the set of performance vectors that correspond to achievable
A
N
speci cations for the th Ritz approximation (15.3). Then we have
N
1
A
A
A
N
i.e., the Ritz approximations yield inner or conservative approximations of the
region of achievable speci cations in performance space.
If the sequence (15.2) is chosen well, and the family of speci cations (15.1) is
well behaved, then the approximations
should in some sense converge to as
A
A
N
. There are many conditions known that guarantee this convergence see
N
!
1
the Notes and References at the end of this chapter.
We note that the speci cation slice of chapter 11 corresponds to the = 2
H
N
Ritz approximation:
0 = (c)
1 = (a)
(c)
2 = (b)
(c)
R
H
R
H
;
H
R
H
;
H
:
15.1.1
A Specific Ritz Approximation Method
A speci c method for forming Ritz approximations is based on the parametriza-
tion of closed-loop transfer matrices achievable by stabilizing controllers (see sec-
tion 7.2.6):
stable = 1 + 2 3
stable
(15.7)
H
fT
T
QT
j
Q
g
:
We choose a sequence of stable
transfer matrices 1 2 ... and form
n
n
Q
Q
u
y
0 = 1
= 2
3
= 1 2 ...
(15.8)
R
T
R
T
Q
T
k
k
k
as our Ritz sequence. Then we have
stable, i.e., we have automatically
H
H
N
taken care of the speci cation stable.
H
To each
N
there corresponds the controller
( ) that achieves the closed-
x
2
R
K
x
N
loop transfer matrix
( )
: in the th Ritz approximation, we search over
H
x
2
H
N
N
N
a set of controllers that is parametrized by
N
, in the same way that the family
x
2
R
of PID controllers is parametrized by the vector of gains, which is in 3. But the
R
parametrization
( ) has a very special property: it preserves the geometry of the
K
x
N
underlying controller design problem. If a design speci cation or functional is closed-
loop convex or a ne, so is the resulting constraint on or function of
N
. This
x
2
R
is not true of more general parametrizations of controllers, e.g., the PID controllers.
The controller architecture that corresponds to the parametrization
( ) is shown
K
x
N
in gure 15.1.
354
CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
u
y
nom
K
v
e
+
r
1
q
x
1
Q
+ +
r
+
q
:
:
:
:
+
:
:
r
+
q
:
:
x
Q
N
N
Q
( )
K
x
N
The Ritz approximation (15.7{15.8) corresponds to a
Figure
15.1
parametrized controller
( ) that consists of two parts: a nominal con-
K
x
N
troller nom, and a stable transfer matrix that is a linear combination of
K
Q
the xed transfer matrices 1 ...
. See also section 7.3 and gure 7.5.
Q
Q
N
15.2
An Example with an Analytic Solution
In this section we demonstrate the Ritz method on a problem that has an analytic
solution. This allows us to see how closely the solutions of the approximations agree
with the exact, known solution.
15.2.1
The Problem and Solution
The example we will study is the standard plant from section 2.4. We consider the
RMS actuator e ort and RMS regulation functionals described in sections 11.2.3
and 11.3.2:
RMS(
1 2
p) = rms yp( ) = ; 12 sensor 22 +
13 proc 22 =
y
H
kH
W
k
kH
W
k
RMS( ) =
1 2
rms u( ) = ; 22 sensor 22 +
23 proc 22 =
u
H
kH
W
k
kH
W
k
:
15.3 AN EXAMPLE WITH NO ANALYTIC SOLUTION
355
We consider the speci c problem:
min
rms u( )
(15.9)
H
:
rms yp( ) 0 1
H
:
The solution, rms u = 0 0397, can be found by solving an LQG problem with
:
weights determined by the algorithm given in section 14.5.
15.2.2
Four Ritz Approximations
We will demonstrate four di erent Ritz approximations, by considering two di erent
parametrizations (i.e., 1, 2, and 3 in (15.7)) and two di erent sequences of stable
T
T
T
's.
Q
The parametrizations are given by the formulas in section 7.4 using the two
estimated-state-feedback controllers (a) and (d) from section 2.4 (see the Notes
K
K
and References for more details). The sequences of stable transfer matrices we
consider are
=
1 i
~ = 4 i
= 1 ...
(15.10)
Q
Q
i
i
+ 1
i
+ 4
s
s
We will denote the four resulting Ritz approximations as
( (a) )
( (a) ~)
( (d) )
( (d) ~)
(15.11)
K
Q
K
Q
K
Q
K
Q
:
The resulting nite-dimensional Ritz approximations of the problem (15.9) turn
out to have a simple form: both the objective and the constraint function are con-
vex quadratic (with linear and constant terms) in . These problems were solved
x
exactly using a special algorithm for such problems see the Notes and References
at the end of this chapter. The performance of these four approximations is plot-
ted in gure 15.2 along with a dotted line that shows the exact optimum, 0 0397.
:
Figure 15.3 shows the same data on a more detailed scale.
15.3
An Example with no Analytic Solution
We now consider a simple modi cation to the problem (15.9) considered in the
previous section: we add a constraint on the overshoot of the step response from
the reference input to the plant output p, i.e.,
r
y
min
rms u( )
(15.12)
H
:
rms yp( ) 0 1
H
:
os( 13) 0 1
H
:
Unlike (15.9), no analytic solution to (15.12) is known. For comparison, the optimal
design for the problem (15.9) has a step response overshoot of 39 7%.
:
356
CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
0:2
0:18
0:16
0:14
) 0:12
u
0:1
( (a) ~)
RMS(
K
Q
0:08
(d) ~
;
(
)
K
Q
;
;
0:06
( (d) )
K
Q
;
;
( (a) )
K
Q
;
;
;
;
0:04
@
I
@
0:02
exact
0
0
2
4
6
8
10
12
14
16
18
20
N
The optimum value of the nite-dimensional inner approxi-
Figure
15.2
mation of the optimization problem (15.9) versus the number of terms N
for the four dierent Ritz approximations (15.11). The dotted line shows
the exact solution, RMS( ) = 0 0397.
u
:
0:045
0:044
) 0:043
u
0:042
RMS(
( (a) )
K
Q
0:041
(d)
;
~
;
(
)
K
Q
;
;
0:04
;
@
I
@
(d)
@
I
(
)
@
K
Q
(a) ~
@
I
(
)
@
exact
K
Q
0:039
0
2
4
6
8
10
12
14
16
18
20
N
Figure 15.2 is re-plotted to show the convergence of the nite-
Figure
15.3
dimensional inner approximations to the exact solution, RMS( ) = 0 0397.
u
:
15.3 AN EXAMPLE WITH NO ANALYTIC SOLUTION
357
The same four Ritz approximations (15.11) were formed for the problem (15.12),
and the ellipsoid algorithm was used to solve them. The performance of the approx-
imations is plotted in gure 15.4, which the reader should compare to gure 15.2.
The minimum objective values for the Ritz approximations appear to be converging
to 0 058, whereas without the step response overshoot speci cation, the minimum
:
objective is 0 0397. We can interpret the di erence between these two numbers as
:
the cost of reducing the step response overshoot from 39 7% to 10%.
:
0:2
0:18
0:16
( (d) ~)
K
Q
0:14
;
)
;
0:12
( (a) ~)
K
Q
( (d) )
;
K
Q
;
u
;
0:1
;
(a)
RMS(
(
)
K
Q
;
0:08
;
0:06
0:04
@
I
@
0:02
without step spec.
0
0
2
4
6
8
10
12
14
16
18
20
N
The optimum value of the nite-dimensional inner approxi-
Figure
15.4
mation of the optimization problem (15.12) versus the number of terms N
for the four dierent Ritz approximations (15.11). No analytic solution to
this problem is known. The dotted line shows the optimum value of RMS( )
u
without the step response overshoot specication.
15.3.1
Ellipsoid Algorithm Performance
The nite-dimensional optimization problems produced by the Ritz approximations
of (15.12) are much more substantial than any of the numerical example problems
we