LIDS-P-2025
Controller Reduction by 7-4-Balanced
Truncation
Denis Mustafa*
Keith Glover
Laboratory for Information
and Decision Systems
Massachusetts Institute of
Technology
Cambridge MA 02139, USA
Department of Engineering
Cambridge University
Cambridge CB2 1PZ, UK
(0223) 332711
(617) 253-2156
February 5. 1990
Revised July 25, 1990
To appear in the IEEE Transactions on Automatic Control
Abstract
7-F,0-balanced truncation may be used to obtain reduced-order plants or controllers. The plant (possibly unstable) is compensated using a particular robustly
stabilizing controller. The two Riccati equations involved are then used to define a
values. That
set of closed-loop input-output invariants called the 71, -characteristic
0
part of the plant or controller corresponding to 'small' %7'/-characteristicvalues is
discarded to give a reduced-order plant or controller. By exploiting an intimate
connection with coprime factorization, a simple a prioritest is derived for the ability of such a reduced-order controller to stabilize the full-order plant. Furthermore,
the performance of the resulting closed-loop may also be bounded a priori i.e., in
terms of the prespecified level of robustness and the discarded 7-/,-characteristic
values.
1
Introduction
Controllers of low complexity are often desirable in practice. Unfortunately, modern
controller design techniques frequently lead to high complexity controllers. So there is
'Financial support by the Commonwealth Fund under its Harkness Fellowships program, and by
grant AFOSR-89-0276
a real need for reliable model reduction methods which allow a low order controller to
be extracted from a high order controller without incurring too much error.
One way of obtaining a low order controller is to work with a low order plant.
For example, one well-established way of obtaining a low order plant is the so-called
balanced truncation method initiated in [24]. The state-space description of a stable
system is transformed to balanced coordinates, where the observability and controllability Gramians are equal and diagonal ('balanced'). The diagonal elements of the
balanced Gramian in fact form a set of closed-loop input-output invariants (the Hankel
singular values) which quantify the contribution of each state to the input-output map
of the system. States contributing only weakly to the input-output map are deleted.
The method has some appealing properties: it generically leads to a stable reducedorder model and an error bound exists in terms of the truncated Hankel singular values
(see [8, 6J and Lemma 2.3).
Intuitively, however, there are objections to reducing the order of the plant without
regard for the controller to be designed for that reduced-order plant. As pointed out
in [1], approximation early in a design process may lead to the undesirable propagation of errors as the design progresses. Furthermore, as argued in [6, 33], satisfactory
approximation of the plant should take account of the presence of the controller. Openloop balanced truncation suffers from these criticisms, and if the plant happens to be
unstable, compensation is needed anyway. Closed-loop model reduction, where there is
some compatibility between the model reduction and the control strategy, would not
suffer from these objections.
One closed-loop model reduction procedure was introduced in [161. The open-loop
system (which may be unstable) is first compensated with a standard Linear Quadratic
Gaussian controller (the so-called Normalized LQG Controller). Two algebraic Riccati
equations are needed-one for filtering and one for control. Balancing the solutions to
these two Riccati equations, so that they are equal and diagonal, exposes the difficulty
of filtering and controlling each state. To be exact, the diagonal elements .i of the
solution to the 'LQG-balanced' Riccati equations form a set of closed-loop input-output
invariants, known as the LQG-characteristic values. States corresponding to small /i
are easy to filter and control in an LQG sense; these states may be discarded to give a
reduced-order plant or reduced-order LQG-controller.
More recently there has been much research in the design of 7',,O controllers, which
are robust to system uncertainty (see [7] for background). In [25], it was shown how
LQG-balancing ideas can be carried over to the corresponding 74,, case. That is,
first compensate a plant with a standard 7H,o controller (called the Normalized 7',t
controller), then balance the solutions to the Riccati equations to define a set of inputoutput invariants (called the -,oo-characteristic values, vi). States corresponding to
small vi are easy to filter and control in an 3,o sense: these states may be discarded
to give a reduced-order plant or a reduced-order HOO controller. Only a brief analysis
of this 1oo-balanced truncation method of model reduction was given in [25]. Two
important questions in particular were not answered:
. How 'small' do the truncated 7H,-characteristic values have to be to guarantee
that a reduced-order controller stabilizes the full-order plant?
2
* What is the degradation in performance caused by using a reduced-order controller
in place of the full-order controller?
As well as providing a complete description of the 7ioo-balancing approach and of 7-ocharacteristic values, we will also answer the above two questions.
A different approach to obtaining reduced-order LQG or 7/,' controllers may be
found in [15] and f3], respectively. (See also [13] for an LQG/'H,o plant reduction
problem to which similar methods are applied.) There a fixed (reduced-order) structure
for the controller is assumed and optimization theory is applied to derive four coupled
matrix equations which the controller state-space matrices must satisfy (two Riccati
equations and two Lyapunov equations in the LQG case, four Riccati equations in the
7'tm case). Our approach, in contrast, whilst not known to be optimal, only requires the
pair of decoupled Riccati equations needed for the full-order Normalized 14' Controller.
The layout of this paper is as follows. Sections 2 and 3 are preliminary sections,
included for completeness: Section 2 summarizes the balanced truncation method of [24]
whereas Section 3 summarises the LQG-balancing method of [16]. Some readers may
already be very familiar with the material in Sections 2 and 3. In that case they may
wish to skip directly to Section 4 which begins the treatment of Ho-balanced truncation
with a discussion of XO-balancing and '7,o-characteristic values as introduced in [25].
Section 5 contains the main results of the paper. There the tCO-balanced truncation
method is analysed in detail. The analysis hinges on exploiting a close relationship
with coprime factorization. Stabilizing and performance properties of the reducedorder controllers are derived, and a numerical example is provided to illustrate the
utility of the results. Conclusions are given in Section 6, and Appendices A-C contain
proofs of the main results.
Notation All systems are taken to be linear, multivariable and time-invariant, with
real-rational transfer function matrices. In general, capital letters are used for matrices
and lower case letters are used for vectors. Often we will not distinguish between a
(continuous) time domain signal w(t) and the Laplace domain signal w(s); the context
will make it clear which is intended. A (proper) transfer function matrix is represented
in terms of state-space data by
or b ,C:= D + C(s-l -
)-iB,
or by (A, B, C, D), where A. B, C and D are real matrices of appropriate dimensions.
If D = 0, then the system is strictly proper and we write (A,B,C). The matrix
A is asymptotically stable if and only if each of the eigenvalues of A has a strictly
negative real part. The symbol IR denotes the real numbers; the prefix R denotes
'real-rational'. We will need the usual Hardy space 7 7 i{o consisting of real-rational
transfer matrices bounded on the imaginary axis with analytic continuation into the
right half plane. (See [7] for further details). If X(s) E 7ZR-, then the 7- 0o-norm of
X is defined by ]IXKl[ := sup{\ma,,(X*(jw )X(jw))}, where X*(.s) := XT(-s). If
.(s) := (A,B, C. D) then X E 7Z7oo if .A is asymptotically stable. Suppose M = MT
solves the algebraic Riccati equation
ATM+ MA-
MRM+ Q = 0
3
where R = RT and Q = QT. Then M is said to be the stabilizing solution if A - RM
is asymptotically stable. Note that if a stabilizing solution exists, it is unique (see [17,
Lemma 3.4.1]).
2
Balanced Truncation
For completeness and later reference, in this section we give a brief summary of the
balanced truncation method of model reduction proposed in [24].
Consider an n state, minimal and asymptotically stable system G = (A, B, C).
Its controllability Gramian P = pT E IRnXn is given by the unique positive definite
solution to the Lyapunov equation
PAT + AP + BBT = 0.
(1)
Its observability Gramian Q = QT E IR" Xn is given by the unique positive definite
solution to the Lyapunov equation
QA + ATQ + CTC = 0.
(2)
9
Under a nonsingular state transformation S, (A, B, C) s (SAS-'.SB, C'S-), so Q
S-TQS -1 and P s SPST. Hence QP s S-TQpST. Since QP and S-TQpST are
similar matrices, the eigenvalues of QP are similarity invariants. These invariants
are the squares of the Hankel singular values of the system G (see e.g., [8]); a formal
definition and some basic properties are given next. It is well-known that the largest
singular value is the Hankel norm of G.
Proposition 2.1 (Balancing and Hankel singular values [24, 8])
Let the system G = (A, B, C) be minimal and asymptotically stable with n states,
and with (positive definite) controllability Gramian P solving (1) and (positive definite)
observability Gramian Q solving (2). Then the eigenvalues of QP are real. strictly
positive similarity invariants, as are their positive square roots which are called the
Hankel singular values of G. Let arl > a2 > ... > a2 > 0 denote the n eigenvalues of
QP arranged in decreasing order, then there exists a similarity transformation which
The system is then
,
transforms both Q and P to the form E := diag(-al,a2,...on).
said to be in balanced coordinates with balanced Gramian E.
For a given system a balancing transformation can be calculated using the method
in [8, Section 4].
In [24] it was argued that, in a balanced realization. the ith Hankel singular value
quantifies how observable and controllable the ith state is. Those states with 'small'
Hankel singular values may be truncated to leave a reduced-order plant, with only a
'small' error. The details are given next.
Procedure 2.2 (Reduced-order plant by balanced truncation [24])
Let G = (A,B, C) be minimal and asymptotically stable with n states, and in
balanced coordinates with Hankel singular values orl - o2 > ... > o > 0. That
4
is.
:= cliag(a<,cr2,...,cr,) = P = Q is the unique positive definite solution of both (1)
and (2). Pick k < n and partition E accordingly into
=diag(l,...
) anddiagk+l,
,)
where 2d = diag(ol,... k) and 22 = diag(ok+l*..on).
conformably with the partitioning of 2:
A2
=
A,
1A
2
1
BA1[B1
B2
an]d
C
Parttion .4,
and C
Partition A, B and C
[C
C2 ]
(3)
A k-state reduced-order plant is then G, = (All, B 1, C1 ).
Lemma 2.3 Let G be asymptotically stable and minimal with n states. Let k < n and
G, be a k-state reduced-order plant obtained by balanced truncation (Procedure 2.2).
Then
(i) [32] G, is in balanced coordinates with balanced Gramian t1.
(ii) [32]
IfO k > crk+l
(iii) [8, 6] IlG-
then G, is asymptotically stable and minimal.
GIlo <- 2 trace[Z 2].
Note carefully that item (iii) above gives an error. bound in terms of the truncated
Hankel singular values. The existence of such an a priori error bound is one of the
attractions of model reduction by balanced truncation.
3
LQG-Balancing and LQG-Characteristic Values
Although balanced truncation has some attractive features it is an open-loop model
reduction procedure; no account is taken of the presence of a controller, which may
not be realistic (especially if the plant is unstable). Instead, consider a closed-loop
procedure where the plant is first stabilized with a standard ('normalized') controller,
and then balanced truncation ideas are applied to the closed-loop system. For example,
one might connect the plant G to a standard LQG controller and balance that configuration, or one might do the same with a standard lHo controller. The former idea was
introduced in [16] and is described in this section; the latter idea was introduced in [25]
and is described and analysed in the next and subsequent sections.
The LQG-balancing approach is based on the Normalized LQG Problemof [16]. This
problem is concerned with a system
x = Ax + Bwl 1 + Bu
z-1 = Cx,
z2 = U
(4)
y = C'X + w.2,
where wlt and w 2 are zero mean Gaussian white noise signals. each with a spectrum
equal to the identity, and the given plant G = (A, B, C) is assumed to be minimal.
Put Z
:= and
[Wz w :_ [WrTfTAT
W]T. Some simple manipulations show that the
closed-loop transfer function from w to z is
(G, hK) =
(I - GK)-'G
(I - GK)- 1 G
(I- GK
-GK
)
(I - GK)- 1
(5)
See Figure 1 for a block diagram of the closed-loop system. We recognise this system
to be the one used to analyse internal stability [34, p101].
The LQG cost is then defined to be
C(X(G, K)) = lim E{ 1t
=lim E
t -oo
/ z-(t)z(t) dt}
2tf
zT(t)CTC(t) + U(t)U(t)
j
t!
where E denotes expectation.
Problem 3.1 (The Normalized LQG Control Problem) Find the controller K
which minimizes the LQG cost C( H(G, K)) over all stabilized closed-loops 7H(G, K).
Remark 3.2 It is easily seen that the Normalized LQG Problem has a 'standard
plant' P (in the sense of e.g., [7]) given by
P21 P2 2
A
BO]
B
0
[C]
0 OO
[0
[0]
[
]
(6)
That is, [zT yT]T = p[wT uT]T where u = Ky. It is well-known [7] that a controller K
stabilizes P if and only if it stabilizes P 22, or since P22 = G, K stabilizes P if and only
if it stabilizes G. (By 'stabilizes' we as usual mean 'internally stabilizes'. That is ([5]),
the states of G and K go to zero from every initial condition when w = 0.)
The solution to the Normalized LQG Problem follows from standard references [2,
18, 161.
Proposition 3.3 (Solution to the Normalized LQG Problem)
Let G = (A, B, C) be minimal. Then there exists a unique positive definite stabilizing
solution X 2 = XT E IRnX n of the control algebraic Riccati equation (CARE):
A4 T2
+ X 2 A - X 2 BBTX 2 + CTC' = 0
C'ARE
(7)
and there also exists a unique positive semidefinite stabilizing solution Y2 = yT E IRnxn
of the filter algebraic Riccati equation (FARE):
AY 2 + Y 2 AT - Y2CTrCy + BBT = O.
6
FARE
(8)
Z1
U=Z 2
K
Figure 1: Block diagram for the Normalized Problems
The Normalized LQG Controller KLQG = (A, B, C) takes the form of an optimal observer
(9)
= (A- Y 2 CTC - BBTX 2 )± + Y 2CT y
B
A
together with the optimal state-feedback
u = -BTY
2
:.
(10)
The minimum value of the LQG cost is
C(li(G, KLQG)) = trace[BT X 2 B + BrX2Y 2 X 2B].
(11)
Just as for ordinary balancing, under a nonsingular state transformation S, X 2 Y2 s
25 Since X 2Y and STX 2Y 2 ST are similar matrices, the eigenvalues of X 2Y2
are similarity invariants. These invariants are the squares of the LQG-characteristic
values of the system G, as defined in [161; a formal definition and some basic properties
are given next.
Proposition 3.4 (LQG-balancing and LQG-characteristic values [16])
Let the system G = (A, B, C) be minimal -with n states and let X 2 and Y2 be the
unique positive definite stabilizing solutions of the CARE and FARE respectively. Then
the eigenvalues of Y2 X'2 are real. strictly positive similarity invariants. as are their
positive square roots which are called the LQG-characteristic values of G. Let 12 >
..
> _/2 > 0 denote the n. eigenvalues of X 2'2 arrangedin decreasing order. then
JL
there exists a similarity transformation which transforms both X.N and 12 to the form
M := diag(pl,,2, 2 .. . ,In,). The system is then said to be in LQG-balanced coordinates,
and Al is the diagonal matrix of LQG'-characteristic values of G.
7
As argued in [161, in an LQG-balanced realization. small ui correspond to states
which are easy to filter and control in an LQG sense. This motivates the following
model reduction schemes.
Procedure 3.5 (Reduced-order plant by LQG-balanced truncation [16])
Let G = (A,B,C) be minimal with n states and in LQG-balanced coordinates
with LQG-characteristic values pl1 > p 2 > .
>._ P, > O. That is, M :=
diag(l,A2, ... , J,, ) = X2 = Y 2 is the stabilizing solution of the CARE and FARE
associated with G = (A, B,C). Pick k < n such that Alk > PIk+l and partition M
accordingly into
M=[
where M, = diag(Ilt,..., ,k)
M1
0 ]
and M2 = diag(&k+l,...
,n)).
Partition A. B and C
conformably with the partitioning of M:
A2,1
B =
A22
B2
and
C=
C1
C 2 ].
A k-state reduced-order plant is then G, = (All, B 1, C 1).
Procedure 3.6 (Reduced-order controller by LQG-balanced truncation [16])
Let G = (A, B, C) be minimal with n states and in LQG-balancedcoordinateswith LQGcharacteristicvalues 1 > It2 > ...
An > O. That is, M := diag( 1,/ 2 ,..., t,) =
I>
X2 = Y 2 is the stabilizing solution of the CARE and FARE associated with G =
(A, B, C). Pick k < n such that Ak > /tk+l and partition M accordingly into
M
=
where MI
= diag(tl,. .. ,uLk)
[M
02]
and Ml2 = diag(/.k+l,.. .,,,).
Let KLQG
=
(A,B,C)
be the Normalized LQG Controllerfor the plant G = (A, B, C) (as given in Proposition 3.3). PartitionA, B and C conformably with the partitioning of M:
[ A21
2
'
B=[
and C= [ C
,4 k-state reduced-order controller is then K, = (All,,,
C2]
B 1 , C 1 ).
Remark 3.7 The reduced-order controller hK, is the full-order Normalized LQG Controller for the reduced-order plant G,. This is an immediate consequence of the easily seen fact that AllMis the stabilizing solution to the CARE and FARE for C, =
(All, B1, C ).
Of course, it is important to know what happens to the stability and performance of
the closed-loop when a reduced-order Normalized LQG Controller is connected to the
full-order system (.1, B, C). Sufficient conditions are derived in [16] for stability but
these tend to be very conservative. since they give sufficient conditions for a dissipative
closed-loop (further they only apply to the single-input single-output case).
8
4
7-,-Balancing and 7,--Characteristic Values
In this section, we show how all the results of the previous section can be generalized
to the minimum entropy/'j-, case. See [251 for further details of the setup, the original
definition of the 7',t-characteristic values vi, and their basic properties. The Normalized 4,O Control Problem on which the ",c-balancing method is based is simply the
minimum entropy/7,t control problem associated with the system (4). The samle block
diagram is appropriate (see Figure 1); the closed-loop transfer function is 'l(G, K) as
before (see (5)). Some definitions and remarks are in order to begin with.
Definition 4.1 (.(G,y)-admissible controller) A controller K is said to be (G,Y)-
admissible if K stabilizes G and II'(G, K)I1, < 7.
Definition 4.2 ((G,-y)-admissible closed-loop) If K is a (G,y)-admissible controller, then 'H(G, K) is said to be a (G, -y)-admissible closed-loop.
Remark 4.3 Each element of N(G, K) has a robustness interpretation, as follows. (IGK)-1 G corresponds to 'additive' uncertainty All on the controller K; (I-GK)-1GK
corresponds to 'output multiplicative' uncertainty A 12 on the plant G; K(I - GK)-'G
corresponds to 'input multiplicative' uncertainty A 21 on G; and K(I - GK)- 1 corresponds to additive uncertainty A22 on G. Block diagrams illustrating these four
uncertainty types Aij, i,j = 1,2, are provided in Figure 2. The upper bound 7 on
II['(G, K)lt; implies robust stability guarantees for each of the four uncertainty types.
This follows from the simple observation that
II(r - GK)-'Gl11o < 7
(I- GK)- 1 G
K(I - GK)-' G
<
(I - GK)-<K
K( - GK)- '1
II(Z - GK)-'GIItl <
- GK)-GII
IIK(
<"
11K(I - GK)-1 tlI < 7
(12)
Using the Small Gain Theorem of [361 implies that closed-loop stability is maintained
for any one Aij E Ztoo satisfying
IlaijllI
< -1,
i,j = 1,2.
(Note that because K stabilizes G, we have (I - GK)-'G E .7-,,, (I - GK)-IGK E
1RZT , hK(I - GK)-'G E 47ZHo and K(I - GK)-1' 47',o.) If we use the Small Gain
Theorem on a frequency-by-frequency basis then a frequency-wise description of the
tolerable Aij is obtained: closed-loop stability is maintained for any one Aij E R.7-,
satisfying
al{All(j'c)}al{(I
- G(jw)hK(jw))- G(jw)} < 1
.)} '1{(I - G(jw)K(jw))-'G(jwo)K((jw)} < 1
71 {A21(jw) } a { ( jw )(I- G(jw)K(jw)')- G(j w)} < 1
({A21(j2
o'l{A 22(ju)}o'{K(jw)(I
-
G(jw)K( jw))-1} < 1
9
Vw E IR.U {oo};
w E IRU {oO};
vw,
IR U {oo};
'~~ E IRU {oO}.
G
Additive uncertainty on K
AG
Output multiplicative uncertainty on G
1
IK
Ka
Input multiplicative uncertainty on G
Additive uncertainty on G
Figure 2: Uncertainty types Aij covered by the Normalized X', Problem
Let %, denote the smallest 7 for which a (G,-y)-admissible controller exists. Finding this 7o is an 'H,o-optimal control problem (see [7]). Throughout this paper we
assume 7 > 0,. This permits the introduction of a minimum entropy criterion into the
framework.
Definition 4.4 (Entropy) Let H E 14,, with H(oo) = 0 and IIHIo. < 7y. Then the
entropy of H is defined to be
2
I(H,7) :=
2
0o
lnldet(I - In
2
H'(jw)H(jw))ldw
Note that if H is a (G, 7)-admissible closed-loop and G(oo) = 0 then we do have
H E R7ZO, H(oo) = 0 and IIHIJI
< y.
Entropy minimization in the context of 1,,O control theory has been explored elsewhere (see [28, 26, 121). There are strong relations between minimum entropy/7'X control and risk-sensitive LQG control (see [10, 9]), and also with the combined Ioo/LQG
control considered in [3] (see [27]). However, due to lack of space we shall not elaborate
further, except to quote the following result linking entropy with the LQG cost, which
follows essentially from [28].
Proposition 4.5 ([28]) Let H E 7Z7-ko with H(oo) = 0 and IIHIok
(i) I(H, y) = C(H) + O( 7 -2);
(ii) I(H, 7) > C(H);
10
< m. Then
(iii) I(H,oo):= lim_.{I(H,y)} = C(H).
Remark 4.6 It is clear from Remark 4.3 and Proposition 4.5 that the Normalized 7-,
Controller guarantees a prespecified level of robust stability and provides a guaranteed
LQG cost bound.
At last we may specify the problem of interest, which was introduced and discussed
in [26].
Problem 4.7 (The Normalized X7,O Control Problem) Find the controller K
which minimizes the entropy [I(7'(G, K), y) over all (G, )-admissible closedloops XH(G, K).
The solution to the Normalized H7/ Control Problem may now be stated, using the
results of [5] and [12]. (In fact ([12]), the Normalized X7,0 Controller is the central
member of the class of all (G, y)-admissible controllers.)
Proposition 4.8 (Solution to the Normalized 7-,0 Problem)
Let G = (A, B, C) be minimal, and let 7 > ,0. Then there ezists a unique positive
definite stabilizing solution X, = XT of the 7iH,o control algebraic Riccati equation
(HCARE):
<-)Xo.BBTXo.
ATX, + XoA - (1-
+ CTC = 0
HCARE
(13)
and there ezists a unique positive definite stabilizing solution YO = YT of the 73,o filter
algebraic Riccati equation (HFARE):
2 )YCTCY
AY, + Y, AT - (1 -
+ BBT = O.
HFARE
(14)
There ezists a (G, y)-admissible controller if and only if there ezists positive definite
X,Y, and Y,, as above, which satisfy Ama.(XoY,) < y2. Define
Z.o
-,- 2 YX¥o)'.
=(
(15)
The Normalized 714, Controller KME.o = (A, B, C) takes the form of an observer
z = (A -(1
-
-2)YooCTC
BBT
A
oZ);
+ YC
r
T
y
(16)
B
together with the state-feedback
u = -BTXZ' o.
(17)
The minimum value of the entropy is
I(71(G, hKMEoo),) = trace[BTrX oB + BT Z,,rXoZoDoB].
(18)
Although we are not concerned in this paper with 74o,-optimality, the following
remark is worth noting: Corollary 5.5 should also be consulted.
Remark 4.9 By definition the 7-/o-optimal version of the Normalized X7-, Control
Problem is to find
: K stabilizes G}.
7o = inf{ll7t(G, K)j11
Consider the closely related problem of finding
%f:= inf{jlli(G, K)llo : K stabilizes G},
where
wr(G,K)
:= (GK)+ [0 0
(19)
This latter problem has an exact (non-iterative) solution [19, Remark 4.211 in terms
of X 2 and Y2 , the stabilizing solutions to the CARE and FARE. respectively:
To =
I1+ An.1 {X 2Y 2 } > 1.
But the triangle inequality applied to the 7-oo-norm of (19) gives
II,(G,K)IIo - 1 < I'l(G, K)Ioo < IIl(G,K)llo + 1.
Taking the infimum over all K which stabilize G gives
0<
_< + 1,
-1 < -o <
which provides (non-iterative) upper and lower bounds on y,.
Just as in ordinary balancing, under a nonsingular state transformation S, ,oYYo
S-TJXoYoST, so the eigenvalues of XoYoo are similarity invariants, the positive square
roots of which we define to be the 7',o-characteristic values. All the arguments of
Section 3 carry across and we obtain the following minimum entropy/7-o,o generalization of Proposition 3.4. Note that since - > 70 by assumption, Proposition 4.8
gives Am,,a(XYoYo) < y 2 so each of the H,o-characteristic values is strictly less than y.
and X7,o-characteristic values [25])
Proposition 4.10 (H, o-balancing
0
Let the system G = (A,B,C) be minimal with n states, let 'y > -y, and let Xo
and Y} be the unique positive definite stabilizing solutions of the HCARE and HFARE
respectively. Then the eigenvalues of XoYYo are real, strictly positive similarity invariants, as are their positive square roots which are called the XoH,-characteristic values of
arrangedin decreasG. Let v1/ > V2 > ... > V2 > 0 denote the n eigenvalues of Xoo,ZO
ing order, then y > vi and there ezists a similarity transformation which transforms
both X-, and YO to the form N := diag(tv, V 2 , .. .,v,). The system is then said to be
in Hoo-balanced coordinates. and N is the diagonal matrix of 7H, -characteristicv'alues
of G.
Note that the ',oo-characteristic values are functions of y. Strictly speaking, we
should write 'i({y) and N(y). In the interests of notational simplicity, we will write vi
and N.
The following proposition states some properties of the vi, and is included for completeness.
12
Proposition 4.11 ([25]) For the vi as defined in Proposition4.10 we have
(i) - > vi > tLi.
(ii) Each vi is a monotonically decreasing function of y.
(iii) If the vi are distinct then dvl d-y < 0.
(iv) Each vi is a continuous function ofT.
(v) lim_{li} = Hi.
5
Model Reduction by 7,oo-Balanced Truncation
As proposed in [25], reduced-order plants or reduced-order 7-oo controllers may be
obtained by truncating those states in an 7t,o-balanced realization associated with small
h,O-characteristic values. This section gives a detailed analysis of this 7,H-balanced
truncation method of model reduction. It will be seen how -lo-balanced truncation is
an extension of LQG-balanced truncation to the minimum entropy/7Xo case. We will
find, in contrast to model reduction using LQG-balanced truncation, that the a priori
data 7 and the truncated vi are all that are needed to answer the questions posed in
the Introduction, repeated here for convenience.
* How 'small' do the truncated Rio-characteristic values have to be to guarantee
that a reduced-order controller stabilizes the full-order plant?
* What is the degradation in performance caused by using a reduced-order controller
in place of the full-order controller?
Answers to these questions are contained in Section 5.4.
5.1
7-H,-Balanced Truncation
Procedure 5.1 (Reduced-order plant by '7,--balanced truncation [25])
Let G = (A, B, C) be minimal with n states and in oHo-balanced coordinates for
given 7 >'2
> z,, with 'too-characteristic values y >
>_
Ž ... > v, > 0. That
is, N := diag(v l ,w 2,...,
= Xoo = Y,o is the stabilizing solution of the HCARE and
HFARE associated with G = (A,B,C) and -y. Pick k < n such that vk > vk.l and
L,)
partition N accordingly into
L O NV
where .V1 = diag(v 1,.... ,k)
and N 2 = diag(vk+l,....
conformably with the partitioning of N:
[
21
22
B=
B2
and C
.4 k-state reduced-order plant is then G = (All, B 1 , C').
13
v,n).
Partition .4. B and C'
C'1C2
Procedure 5.2 (Reduced-order control by X7-,-balanced truncation [25])
Let G = (A, B, C) be minimal with n states and in -t,,0-balanced coordinates for
given y, > 7,, with 'tI,-characteristic values 7 > vl > v,2 > ... > ,vn > O. That
v,,) = X', = Yo, is the stabilizing solution of the HCARE and
is, N := diag(vl, v,...
HFARE associated with G = (A,BC) and y. Pick k < n such that vk > Vk+l and
partition N accordingly into
N2
N=[l0
where N1 = diag(vl,...,vk) and N 2 = diag(vk+l,...v,). Let KMEOO = (A,B,C)
be the Normalized X7i' Controller for the plant G = (A, B, C) and 'y (as given in
Proposition4.8). PartitionA, B and C conformably with the partitioning of N:
A
A=
ll
l
[
A 12
2
1]
and
B
B=[
|
C=[C
C2 ]
.4 k-state reduced-order controller is then K, = (A 1 , B 1, C 1 ).
Remark 5.3 The reduced-order controller K, is the full-order Normalized 7',0. Controller for the reduced-order plant G,. This is an immediate consequence of the
easily seen fact that N 1 is the stabilizing solution to the HCARE and HFARE
for G, = (A 11, B 1 , C 1).
We can now relate the H,-characteristic values of G to the Hankel singular values
of G.
Proposition 5.4 ([25]) Suppose G = (A, B, C) is asymptotically stable and minimal,
with Hankel singular values Oal >
oL,,-characteristic values 7 > 1 v>
(i) If
> a, > O. Let - > 70 and let G have
2 > ...
2 > ...
>
, > O. Then
> 1 then cri > vi.
(ii) If 7 = 1 then
ai
= vi.
(iii) If y < 1 then ai < vi.
Proof The following proof is taken from [25], and is included for completeness. Note
that (A, B, C) is not necessarily a balanced realization of any type. The controllability
Gramian P = pT > 0 and observability Gramian Q = QT > 0 are given by
AP + PAT + BBT = 0
(20)
0.
(21)
ATQ + QA + CTC
and then
j
= ¾QP}.
Subtract the HFARE from (20), and subtract the HCARE from (21):
)Y-:C.TCYo =0
A(P - Y o) + (P - Y.)AT + (1 AT(Q - \:O) + (Q - X.O)A + (1 - ?- 2).XBBT_.Y0= 0.
14
(22)
(23)
Part (i) By assumption, A is asymptotically stable and 1 - y- 2 > 0. A standard
result on Lyapunov equations (see [8, Theorem 3.3(7)]) applied to (22) and (23) then
gives P > Y,, and Q > XS. Hence, by the argument in the proof of Proposition 4.11(i),
Ai{QP} > Ai{XYo} }. Recalling cr = \i{QP} and v 2 = AI{XY }, the result follows.
Part (ii) Put 7 = 1 in (22) and (23):
A(P - ',) + (P - Y )AT = 0
AT(Q _ X o) + (Q-XoX,)A = 0.
Since A is asymptotically stable by assumption, it follows from [31] that P = Y,. and
Q = Xoo. Hence the result.
Part (iii) The same argument as used to prove Part (i) may be applied, except
now 1 - a- 2 < O0.It follows that P < Yo and Q < XY,, which implies the result. a3
Corollary 5.5 Suppose G = (A, B, C) is minimal. Then y, < 1 if and only if G is
asymptotically stable and has Hankel norm cr < 1.
Proof If Suppose G is asymptotically stable and has Hankel norm cl < 1. Then
the controllability Grainian P = pT > 0 and the observability Gramian Q = QT > 0
satisfy (20) and (21) respectively, and A,,az(QP) = al < 1. Setting y = 1, from
the proof of Part (ii) of Proposition 5.4 we have P = YO and Q = Xo,.
Hence
Proposition 4.8 implies that there exists a (G, 1)-admissible controller. So -y, < y = 1.
Only if Suppose y, < 7 < 1. Then (14) can be written as a Lyapunov equation:
0 = YoAT + AYo + [aYoCT BI[aYoCT BIT,
2 := 7 - 2 - 1 > 0. The pair (A, [aY.CT B]) is certainly stabilizable
where Ca
because,
by assumption, (A,B) is controllable. This together with Y,, > 0 implies that A is
asymptotically stable, using a standard result on Lyapunov equations [35, Lemma 12.21.
Finally, combining Parts (ii) and (iii) of Proposition 5.4 gives Oal _ v < 1.
[]
Remark 5.6 From Proposition 4.11(v) we see that if the 7H,O-norm bound 7 is relaxed completely i.e., Y - oo, the '7,o-characteristic values become precisely the
LQG-characteristic values. Indeed, Proposition 4.5(iii) implies that in this case the
Normalized H,,O Control Problem becomes the Normalized LQG Control Problem. So
the results of Section 3 on LQG-balanced truncation could all be viewed as limiting
cases of the results on 1o,-balanced truncation.
5.2
Coprime Factorization via the Normalized 7-, Problem
A key step in the analysis of 2,o-balanced truncation will be to link it to balanced
truncation of the normalized coprime factors of a scaled plant. To be specific. consider
the scaled plant 3G where
15
For /3 to be well-defined we assume -y > max{1,yo} so that 0 < 3 < 1. This is not a
severe restriction; Corollary 5.5 shows that yo < - < 1 is possible if and only if G is
asymptotically stable and has Hankel norm less than one.
Balanced truncation of normalized coprime factors has been considered in [21]
= M1-'3V be a normalized
and [19, Chapter 5]. The idea is as follows. Let L3G
left-coprime factorization. That is.
(i) ON E ~7Z,
and M E 17Xo;
(ii) 7Mis non-singular;
(iii) /3NV and M are left-coprime (i.e., 3BX,Y
Y E 77-I, s.t. ,3UNf + MYf'= 1);
(iv) 3 2 VNN* + MM*= I;
(v) /G= M- 1,/3N.
In the above, item (iv) is the normalizing condition.
The matrix ([{N .I] is asymptotically stable and hence balanced truncation may be
applied to obtain reduced order coprime factors [3N, 37I,. The detail will be provided
in the next subsection. Before that we must show how to construct [/3N 17f] and write
down its Hankel singular values, all in terms of quantities we already have available
from the Normalized iO Problem.
The first step is to use [22] to construct the normalized left-coprime factorization
of 3G using one of the 74,O algebraic Riccati equations of the Normalized Noo Problem.
Lemma 5.7 Let G = (A,B,C) be minimal and let 7 > max{1,y 0 }. Let /3 = (1 ,- 2 )1/2 . Let Ye be the unique positive definite stabilizing solution of the associated
HFA RE. Define N and ill by
[o/3~' g=
,}}
[- A
0
CB
'=
(24)
Then
(i) /G = M-'IN is a normalized left-coprime factorization of/3G.
-
(ii) The controllability Gramian of [p3N Ai) is
p = /32yoo,
and the observability Gramian of [3iN A11] is
Q = (o-- 1
d2+)-I
(iii) Let l > &2 > ... > an > 0 be the Hankel singular values of [/3N AI].
y > vl > v2 >_... > v, > 0 be the X',-characteristic values of G. Then
-
=
2"
32
1 +32V
16
'
Let
Proof Appendix A.
a
Remark 5.8 From [29, 231, the realization (24) of [3fi3V ]f is minimal if and only if
the realization G = (A, BC) is minimal. Minimeality of G = (A.B.C') is a standing
assumption in this paper.
Remark 5.9 Suppose [03N JM] is in balanced coordinates. Then its controllability Gramian P and its observability Gramian Q satisfy P = Q = v where ~ =
diag(a 1 , &2, ... , an) is the balanced Gramian. But then from Lemma 5.7(ii), Y,2 = 3-2:
and XO = (S-1 _ )-', which are both diagonal, with product
oYo ='
-:(g-1
_)-1l
= diag(/3-2(1 _ a2)-l, 3- 2 2(1 _ &2)-1,.
-diag(v21,
-N
2
,-2&2(1
_ 2)-)
)
v2
,
using Lemma 5.7(iii). Thus. G may be put into Xo-balanced coordinates by applying
the diagonal state similarity transform
i31/2(1 _- f2)-(1/4)
(which was derived in a straightforward way by applying the method of [8, Section 4]).
Remark 5.10 Suppose G is in 74Ho-balanced coordinates. Then Xo = Y, = N where
N = diag(vj,v 2 ,..., V,). But then from Lemma 5.7(ii), P = ,3 2 N and Q = (N-' +
/2N)-', which are both diagonal, with product
PQ
,3 2 N(N-1 + 3 2N)-'
= diag(3 2v2( 1 +/3 2v1)-1
2
2 v2) - 1
v ,2(1 +
.-3
3 2 v2(1 + o2
2
)-1
S- 2
using Lemma 5.7(iii). Thus, [3N t1IV]
may be put into balanced coordinates by applying
the diagonal state similarity transform
/3-(1/2)(
+ -
2
N
2
)-(1/4),
(which was derived in a straightforward way by applying the method of [8. Section 4]).
5.3
Model Reduction via the Coprime Factors
We are now in a position to carry out balanced truncation of the normalized left-coprime
factors [/3N Mf] of the scaled plant3 G, to obtain a reduced-order plant. The analysis
here and in the next subsection is reminiscent of [19. Chapter 5].
17
Procedure 5.11 (Reduced-order plant by coprime factorization of L3G)
Let G = (A,B. C) be minimal with n states and let y >
3 = (1 -
max{ 1,y 0 }.
Let
Let 3G = if-13l3N be the normalized left-coprime factorization
-2)1/2.
Vifi = (AB, , D) be as given in
of fiG based on the HFARE-that is, let [,3N
Lemma 5.7. Let the state-space realization of [iN Mf] be in balanced coordinates
> O. Pick k < n such that Ok > &k+l
> .2.. >
with Hankel singular values 1 Ž &2 >
and let [i3N, Mil] := (Al 1, B 1, Cl, D) be obtained by balanced truncation of [(3N fM]
;'lN
to k states (Procedure 2.2). Then a k-state reduced-order plant is G, :=
and [23] G0, = M 3r-liNr, is a normalized left-coprime factorization.
How does this reduced-order plant compare with the one obtained via 7-,"-balanced
truncation in Procedure 5.1? The next result answers this question.
Theorem 5.12 The plant model reduction schemes described in Procedure 5.1 and
Procedure 5.11 yield identical reduced-orderplants. To be precise, let G, be the k-state
reduced-orderplant obtained by performing 7-',,-balanced truncation (Procedure 5.1) on
the full-order plant G for y > max{1, 7}. Let G, be the k-state reduced-orderplant
obtained by performing balanced truncation of the coprime factors [OfN MfI]of fiG
(Procedure 5.11) where /3 = (1 - 7- 2 ) 1/ 2. Then
G, =G,.
Proof Suppose G = (A,B, C) is 7,4-balanced, and we form G, by X7-/-balanced
truncation. Thus, according to Procedure 5.1, we have the partitioning
X,=Y =NY. =[ZN
0N]
IV2
0
where N, = diag(vl,..., vk,) and VN
2 = diag(vk+l,..., v,,), together with
=[
= B2
A122
A
and
C
[ C1 C2 ]
The k-state reduced-order plant G, is then
G, =A[
(25)
B1
Using Lemma 5.7, we can use N to construct the normalized left-coprime factorization 3G = .I'-'/3N (with the partitioning corresponding to that of N made explicit):
11 -3
[031V 1i] =
1
-
2
NVC' CI
2
3 CV2C
C,
-'12 - .32
Cf' 2
A22 - 32N 22TC2
C
CC
2
3B,
B2
0
-.32 VC'
,32N 2CT
I
, (26)
with Gramians P = 3 2 N and Q = (N-' + da2 N) - 1 . But from Remark 5.10, this
may be put into balanced coordinates by a diagonal state-transformation. Denote
this diagonal balancing transformation by L = diag(L1 , L 2 ), partitioned conformably
18
with iV = diag(NV
1, V 2 ). (An explicit expression for the balancing transformation L
is given in Remark 5.10 but we shall not need it.) Applying this diagonal balancing
transformation to (26) gives
L1AllL -
[0V IV] =
1
LiAi2L
I
1
L1B
jL 2B 2
L 2 A 2 2 L'2
L2A21L-1
CL
1
C 2 Lll
-_2LIVICT
-d2L
0
2 CT
2 1N
I
where Aij := Aij- 32 NiC Cj, i,j = 1,2, which is a balanced realization with (Remark 5.10) balanced Gramian E. Applying Procedure 5.11, the k-state reduced-order
where [iN, MI,]
'f'r-l3N,
plant 0, has a normalized left-coprime factorization 30, =
is obtained by balanced truncation of [AIN M]. Thus, truncating the above balanced
realization to k states,
[/3N,.
ll] =
[L
=A[AI
C,.
z= Ml7l:%r'
=
K
2N
1'
1 (AL-3
CL-
-
CTC1)L'
3 2 N 1 Cl C
C1
3B 1
0
I
/3LB 1
0
N
]1
-3 2 LiNCT
I
-3 2 NjC=T
IC
Ah
a
a
But this is precisely G, in (25).
The following corollary is immediate on taking note of Remark 5.3.
Corollary 5.13 Let K, be the k-state reduced-order controller obtained by performing
7to,-balanced truncation (Procedure 5.2) on KMEO,, where KMEoo is the full-order Nor-
malized .,O Controller for the full-order plant G. Let A, be the k-state Normalized
iH, Controller for the k-state reduced-order plant G, (Procedure 5.1 or equivalently
Procedure 5.11). Then
K,=
-,.
It is therefore irrelevant whether we consider doing model reduction by 7H,-balanced
truncation before or after the control; there is a compatibility between the closed-loop
objectives of Normalized 7-0 0, Control and model reduction by 7,o-balanced truncation.
5.4
Stability and Performance with the Reduced-Order Controller
We can now piece together results from the previous sections to consider the stability
and performance of the closed-loop consisting of the reduced-order Normalized 7Ho.
controller K, with the full-order plant G (as illustrated in Figure 3). It is an advantage
of using Xi,-balanced truncation that the results may be expressed in terms of a priori
quantities only (i.e., 7/ and the neglected vi). This comes about because the model
reduction error e defined by
Oi3:=-11[/3,
X.H]{,
19
(27)
where
AN := N?-
:N,
A~Q := M - Air,
may be bounded above using y and the neglected vi only. To be precise, the balanced
truncation error bound of Lemma 2.3(iii) applied to (27) gives an upper bound on ,3 in
terms of the neglected Hankel singular values of [,/N ITJ.Hence, from Lemma 5.7(iii),
d3 may be bounded in terms of the 7O4,-characteristic values of G:
< 2 trace[
= 2
2
:
+
i=k+I/1
v2
(28)
Therefore,
<2Z
2a
i=k+l 1
+ /32 v 2
(29)
and by inspection, a weaker but simpler bound is
< 2
!
E
vi = 2 trace[lV
2].
(30)
i=k+l
This last upper bound is 'twice the sum of the tail,' just as in ordinary balanced
truncation (Lemma 2.3(iii)). Using either of the above upper bounds in place of e
in the following results gives a (conservative) statement of closed-loop stability and
performance using the reduced-order controller without having to calculate the reducedorder controller in advance.
Proposition 5.14 Let K, be the k-state reduced-order controller obtained by performing '7',-balancedtruncation (Procedure 5.2) on the full-order Normalized 74, Controller
for the full- order plant G with 7 > max{1,y 0 }. Let iG = iM-1/3N be the normalized
left-coprime factorization of /3G given in Lemma 5.7. Let flG, = MTIf-7/3Nr be the normalized left-coprime factorization (Procedure 5.11) of the k-state reduced-orderplant G,
obtained by performing 74, -balanced truncation (Procedure 5.1). Let
Kr =Ur.
-
be any right-coprime factorization of K,. Define
r,:= A, ;Vr RU.
l
Let
e
be the model reduction error as defined in (27) above. Then the condition
i/13
]R-1
<1
ensures closed-loop stability of the system consisting of the reduced-ordercontroller IK.
and the full-order plant G.
20
Figure 3: Reduced-order Normalized 'H,O Controller with full-order plant
Proof The proof is basically an application of the Small Gain Theorem. Rather
than repeat the argument here, we will for brevity quote a result [19, Corollary 3.7]
which, for our case, states that K, stabilizes G if K, stabilizes G, (which it does by
construction) and
where S, := (I - G,K,) - '. The claim of the proposition follows easily on making the
necessary substitutions to obtain
S,, = ~V R;-.Mqf
(31)
and then
/r S,,
-
1
=
U
rr
Corollary 5.13 shows that K, is in fact the normalized 17o controller for G,. Define
7-(Gr,.Kr)11f,
:=
(32)
to be the actual ',o-norm of 7'-(G,,hKr). Then
< '¥
is immediate, and the following corollary to Proposition to .5.14 is obtained.
21
(33)
Corollary 5.15 With definitions as above, we have that
R-[1~
U]R
<S+f,
(34)
so the condition
(0 + i) < 1
is sufficient to ensure closed-loop stability of the system consisting of the reduced-order
controller K, and the full-order plant G.
Proof Appendix B.
0
Remark 5.16 An a priori test for the stability of the system consisting of K, with G
is immediate. Just replace -y with its upper bound 7 (equations (32) and (33)), and
replace e with e, where e is an upper bound on c, such as that given in (29). It is simple
to verify that
( +)
< 1
.
(+)
<
.
Hence to test if K, stabilizes G it suffices to check if e(/3 + y) < 1, a simple test which
depends only on - and the truncated vi.
Now assume that i(,/3+j) < 1 so that, by Corollary 5.15, the closed-loop consisting of
the reduced-order controller K, and the full-order plant G is stable. From Remark 4.3
we know that the bound II'H(G, KMEO.)l[.O
< 7 inherent in Normalized X7,
Problem
gives robust stability guarantees. The next proposition tells us how much the bound
II7/(G, KhME.O )IIo < 7 is degraded (increased) by using the reduced-order controller K,
in place of the full-order Normalized 7'H0 Controller KMEoo.
Proposition 5.17 Definitions as above. Assume i(/3+f)< 1 so that the reduced-order
controller Kh; stabilizes the full-order plant G. Then the associated closed-loop transfer
function satisfies
II /(G,KE,)1lo.
Proof
__<+ ~(l + j)(1
7)^(p+
)
(35)
Appendix C.
[
Remark 5.18 An a prioriupper bound on II7-(G, K,)l[oo is immediate from (35). Just
replace - with its upper bound 7 (equations (32) and (33)), and replace e with e, where
e is an upper bound on e such as that given in (29).
Provided that e(3 + Y) < 1,
Remark 5.16 predicts that K, stabilizes C. Under this condition. it is simple to verify
that the right-hand side of (35) can only increase on replacing e with e and with -y.
'
That is,
( ,K)
<+E(1+ y)(l+ 3 + )(36)
(1 - e(/3 + 7))
which depends only on 7 and the truncated vi. So in this context. the n - k smallest
vi are 'small enough' to be discarded if e(/3 + y) < 1. Beyond this. smaller discarded
vl lead to a better reduced-order controller; 'better' in that the upper bound (35) is
smaller (closer to -y).
22
(a)
(b)
(c)
(d)
Upper
bound
e from (29)
Does Remark 5.16
B VI>2on
=
K. stabilizes G?
K-
(e)
Upper
bound
on ,H(G,K.)JI
from Remark
2.3600
0.9058
2.3544
0.0100
0.0200
Yes
2.667
3.0000 )
0.9428
2.2030
0.0100
0.0200
Yes
3.4293
7.0000
0.9897
2.0339
0.0100
0.0200
Yes
8.7119
10.0000
0.9950
2.0167
0.0100
0.0200
Yes
13.3828
40.0000
0.9997
2.0010
0.0100
0.0200
Yes
231.3253
100.0000
0.9999
2.0002
0.0100
0.0200
No
2.0000
0.0100
0.0200
No
.00.00
}
Table 1: 7-,o-balanced truncation example: a priori numerical results
Remark 5.19 Note that as 7 increases. the a posterioriresult of Proposition 5.17 remains viable, whereas the a priori result of Remark 5.18 becomes increasingly weak.
In the limit as 7 -x oo (the LQG case), Proposition 5.17 is still viable, whereas Remark 5.18 becomes vacuous.
Remark 5.20 Note that if the model reduction error
0, the upper bound (35)
recovers the definition ~ = ll1(G,,K,)llj,. Similarly, the upper bound (36) recovers
the full-order bound 1ItI(G, KMEo)oI < -' if the model reduction error bound E -- 0.
-i
5.5
A Numerical Example
The results derived above are illustrated here using a numerical example. Consider the
system
G= A A3/4B
-
[
-98/201
-9999/200
98/201
[ 61
=
1
11
0
This system was constructed in [16, Section VIII to be in LQG-balanced coordinates
with LQG-characteristic values tl, = 2.0000 and JL2 = 0.0100. To verify this, one merely
has to check that M = diag(pl,/A 2) is the stabilizing solution of the CARE and FARE.
The system has a left-half plane zero at -24.1349 and poles at 0.7547 and -49.9997.
Using y-iteration gives 7o = 2.3559. We chose seven representative values of 7 > y, at
which to perform X7,-balanced truncation: ' = 2.36. 3, 7, 10, 40, 100 and oo.
The results are tabulated in Tables 1 and 2: the 'predicted' results in Table 1 and
the 'actual' results in Table 2. We know that when -- . the R7-,-balancing method
recovers the LQG-balancing method (see Remark 5.6). This is confirmed in Tables 1
and 2, where the last row (corresponding to y, - oo) of Columns (a), (b) and (i) agrees
with the results of [16. Section VIII.
For each of the chosen values of 7, the HCARE and HFARE are solved and the
'Lo-characteristic values are calculated using Proposition 4.10. This gives Columns (a)
23
(f)
I
I
(g)(h)
(i)
Does
A
v i~
2.3600
2.3600
I
K
(k)
(j)
()
Upper
on
bound
Actual
Actual
value
of
JI((G,K,) jKMjoo
Actual
~of i
from (27)
Cor.
5.15
Kvaluestabils
Does K,
stabilize
G?
0.0182
Yes
Yes
2.6373
2.4273
2.3600
3.0249
2.9087
II7/(G,K,)IIo
from
5.17
Prop.
value of
3.0000
2.9076
0.0189
Yes
Yes
3.2939
7.0000
4.1396
0.0198
Yes
Yes
4.8339
4.4268
4.1524
10.0000
4.3451
0.0199
Yes
Yes
5.0997
4.6688
4.3612
40.0000
4.5472
0.0200
Yes
Yes
5.3642
4.9092
4.5669
100.0000
4.5590
0.0200
Yes
Yes
5.3794
4.9233
4.5790
oo
4.5612
0.0200
Yes
Yes
5.3823
4.9260
4.5812
Table 2: 7X,-balanced truncation example: exact and a posteriorinumerical results
and (b) of Table 1. For each choice of - we have vl > v2 , which prompts us to consider obtaining single-state reduced-order controllers by discarding v 2 via 'X,-balanced
truncation (Procedure 5.2). But before calculating the reduced-order controllers, we
can predict their success using the a priori data - and v 2. Using (29), an upper bound
on the model reduction error e may be calculated from the neglected v 2 , giving Column (c). As explained in Remark 5.16, this upper bound on e may then be used in
place of e in Corollary 5.15 to predict stability of each reduced-order controller hK, with
the full-order plant G, Column (d). Notice that stability of K, with G is predicted
for all but the two largest choices of 7, when nothing can be deduced. In the cases
where stability of K, with G is predicted, the performance of that configuration can be
bounded using Remark 5.18, as shown in Column (e).
Having predicted the satisfactory behaviour of the reduced-order controllers we can
go ahead and calculate them using Procedure 5.2, from which Table 2 is constructed.
The model reduction error e may be calculated exactly from (27) using the algorithm
in [4], to give Column (g). Stability of K, with G may be tested for explicitly by
calculating the closed-loop poles, to give Column (i). The actual 7t,-norm of the
closed-loop of K, with G is also calculated explicitly using the algorithm in [4], to give
Column (k). Likewise, A, the actual 7X,-norm of the closed-loop of IK, with G,, is
calculated explicitly to give Column (f). For reference, in Column (1) we give the fullorder results, that is, the actual 7',O-norm of the closed-loop 7'(G, KME.o) consisting
of the full-order controller KMEoo with G.
Using the a posterioridata j and i, Corollary .5.15 may be used to check if stability
of fK. with G is predicted, see Column (h). Proposition .5.17 then allows calculation of
an upper bound on the XO,-norm of the closed-loop of K, with G (Column (j)). Notice
that it is predicted that K, stabilizes G for all the values of y, including infinity. Also,
the upper bound in Column (j) is within 10% of the actual value in Column (k), again
even for large 7.
Comparing the two tables of results it is clear that the 7',o-balanced truncation
method can give a good reduced-order controller. Furthermore. when y is small (of the
24
order of o), indicating a preference for robustness over LQG performance, the 70balanced truncation method gives accurate a priori prediction of the performance of IK,
with G. This is particularly clear on comparing the first three rows of the two tables.
Using the a posterioridata 7 and I, even tighter results are possible: they indicate that
a good reduced-order controller can be obtained even when y is large (in which case the
a priori predictions may be weak, as indicated in Remark 5.19). This is particularly
clear on comparing the final three rows of the two tables.
6
Conclusions
',o.-balanced truncation has been analysed here as a method for deriving reducedorder plants or reduced-order 7,/ controllers. An easily tested criterion for the ability
of such a reduced-order 7/,, controller to stabilize the full-order plant was derived. The
criterion required knowledge only of the prespecified X7o-norm bound and the discarded
7o,-characteristic values. Then we derived an upper bound on the Xoo-norm of the
closed-loop consisting of the reduced-order 7'/00 controller and the full-order plant. That
too relied on the same a priori data. ,o-balanced
truncation is one of the few known
0
methods where such a prioriresults exist for the reduced-order controller. The results
arise because of the compatibility between the reduction procedure and the closed-loop
objectives. Another problem where a priori results are possible is the coprime factor
problem in [20].
Computationally the 7oo-balanced truncation method is very simple, since the 7/.characteristic values are easily calculated from the solutions to the design Riccati equations in advance of doing any model reduction. This was seen in the numerical example.
In [30], it was shown that the LQG-balancing approach is particularly applicable to
symmetric passive systems, such as large space structures with colocated rate sensors
and actuators. We would expect this to be the case for the 7H,-balancing approach too,
and it would be instructive to compare results. However, as pointed out in [30, p831, for
systems with no symmetry or passivity properties, a non-normalized approach may be
more suitable. One possible way round this would be to apply the normalized approach
not to the actual plant, but to a 'shaped' plant which has been (open-loop) pre- and
post-compensated. The success of such 'loop-shaping' ideas has been demonstrated
in [19], using controller synthesis via the robust stabilization of the normalized coprime
factors of a plant. It would be interesting to apply and analyse such loop-shaping ideas
in the context to -oo-balanced truncation.
A
Proof of Lemma 5.7
Part (i)
Multiply the HFARE by d2 and define }
WAYS Y~Ar
Y-YCTrCi
:= 3 2 Y.
Then
+ (dB)(13B)T = 0.
Using this equation the required normalized left-coprime factorization of d3G
(A, 3B, C) may be written down immediately using [11. Lemma 2.1].
25
Part (ii) The controllability Gramian P = pT is the unique positive definite solution
to the Lyapunov equation
P(A - IO2yooCTC)T + (A _ 32YCTC)P +
2BBT + 3 4 yoCTCyo = 0.
It is easy to show. using the HFARE, that P = 3 2 Yo solves this equation.
The observability Gramian Q = QT is the unique positive definite solution to the
Lyapunov equation
Q(A - 3 2YooCC)
+ (A - 32YooCTC)TQ + CrC = 0.
Pre- and post-multiply by Q-l and substitute Q-' = X 1 +f 2 Yo, (XYl exists because
XO > 0 from Proposition 4.8.) It is then straightforward to use the HFARE and the
HCARE to show that the equation is indeed solved by this choice of Q.
Part (iii) Firstly recall the definitions 2 = A 1 {QP} and v2 = Ai{Xo,'
using Lemma 5.7(ii) to express P and Q in terms of S, and Yo, we have
o}. Then,
. = Ai(QP}
= A1 {((xl +
2
y, )-132 Y
= A;{(I + :3 XY0,)-13
2
_=
2
}
2 X,
¥o}
Ai{X(oYoo }
1 + /32 Ai{XoYo}
1 + /3 2 V
B
Proof of Corollary 5.15
We only have to prove that (34) holds: the corollary follows immediately from that
and Proposition 5.14. Now, K, is the Normalized 7'H Controller for the plant G,. The
closed loop transfer function is
7(G,,K,) =
KS, G,
K, S,,
where S,, := (I - G,K,)-l. We need to rewrite this closed-loop transfer function in
terms of the coprime factorizations G, = ll Y,.r and K, = U,1j'- '. We have
R.., = I,I'; - *Yr,
by definition and
S,, = Vr R;' .lr
26
from (31). Straightforward manipulations then give
H(G.,)= [
S Gr
]
[
Srr
- VR-l V, VrfRs - 1
= U, Rr'N-,
V,
OI
LR -lM,
L
rr
0 I
[U.]R;'[Nr M ] t
O]|
[go
Hence
[Ur ]R-1l. [
(37)
+ -H(G, ;r)
M=
and
[U, j R [ Nr
r ]
[
I]
(3[ ° ° ]jGKr))
(38)
Similarly,
[U]R.
]
[3 r
I
] +H(G,K))[
r]=([O
]
(39)
and also
Ur ]
[
[0 ]
1I]([ oo]
0)
[ °v ° ,
O
[
I ] (G:t
·
[
I*
(40)
Take the lH, 0 -norm of (37)-(40) and use the triangle inequality and the submultiplicative property of the 7,O-norm. Also use that j[iH(Gr, K,)1[oJ = 5 by definition
and that 0 </ < 1. Then
[.
]R;[Nt
M ]
< 1+
(41)
a[],
<3+.
(42)
and
3[VrI
RrI[r
Also.
[
];r]r[ '
=4[
]
M
jRrr|
< 1+,
(43)
and finally
[I j]Rrr [3N.
Mt
V
[sIr3]
]
Rrr1
< ±.3
+
(44)
The first equality in (43) and in (44) arises because [13Nr, fI,] is normalized. Equation (44) is (34) as claimed; equations (41)-(43) will be needed in Appendix C.
O]
27
Proof of Proposition 5.17
C
We shall stick to the notation and definitions used in the above proof. Introduce
MV, -iV U,
R =f
then
e7(G,K,)
[V=
]R
[N M]
(45)
[
Noting that
R. = R, +- ( 3-'aq)( v3,.) - a.
-3-'a, ]
[a
= R.-
[V],
an application of the well-known matrix inversion lemma [14, p191 gives
- 3-'-
R, = R:1 + R-1 [ a
T-L [T
(46)
R-'
where
p .[ |[R
T:=I-
- 3-1,
A
].
Substitute (46) into the expression for '7(G,K,;) in (45) and use the identity [IV M =
1M]
to obtain
[N. + ax M, +
[Ur ]
I[
I 0
R.[
+ [V]
]
-R'z
+ [U]R;[zi
]
-,-1/\
A
Taking 'HX-norms, and using the triangle inequality and sub-multiplicative property of
the XH--norm, leads to
II7H(G.,K,)IIo <
I=L(Gr,,K,K)
u,]R-1
+
[ur] Rro
[
+
[Ir
1
X [ aV
+||.
n].|
3. A ] o
R,1
VI R{
]
3-1,
8-,,
n']I2
28
Mr
1
orr[3]
'l
o2IT]dl1M
[
] ILIT-'11
To complete the derivation, use (41)-(44), and recall that 1II7(G,,Kr)Ioo =
nition and that
I1[ A.
3-'.
] l. =
-.
by defi-
e(47)
by definition. Also, using the inequality [14, p3011
(I- A)-'1o <_(1 - ItAIlo)-1
for
IAllo, < 1,
(48)
we get
IIt(G,)Il[o<
1
+(1+
-( i
f)+R(1 +[)(i+A)(1
< (1 + )(1 + d +j)
(1
+
+ ))- +
(3
))
+
c
as claimed.
References
[11 B. D. O. Anderson and Y. Liu. Controller reduction: concepts and approaches.
IEEE Transactions on Automatic Control, 34(8):802-812, 1989.
[2] B. D. O. Anderson and J. B. Moore. Linear Optimal Control. Prentice-Hall, 1971.
[3] D. S. Bernstein and W. M. Haddad. LQG control with an Hoo performance bound:
a Riccati equation approach. IEEE Transactionson Automatic Control, 34(3):293305, 1989.
[4] S. Boyd. V. Balakrishnan. and P. Kabamba. A bisection method for computing
the 7X, norm of a transfer matrix and related problems. Mathematics of Control,
Signals. and Systems, 2(3):207-219, 1989.
[5] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis. State-space solutions to standard X72 and X7, control problems. IEEE Transactions on Automatic
Control. 34(8):831-847, 1989.
lModel Reduction for Control System Design. PhD thesis, Stanford
[6] D. F. Enns.
University, 1984.
29
[7] B. A. Francis. A Course in 7' C'ontrol Theory. Volume 88 of Lecture Notes in
Control and Information Sciences, Springer Verlag, 1987.
[81 K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their £,--error bounds. International Journal of Control. 39(6):11151193, 1984.
[9] K. Glover. Minimum entropy and risk-sensitive control: the continuous time case.
In Proceedings of the Conference on Decision and Control, Tampa, Fl, 1989.
[10] K. Glover and J. C. Doyle. State-space formulae for all stabilizing controllers that
satisfy an 7tb,-norm bound and relations to risk sensitivity. Systems and Control
Letters, 11:167-172, 1988.
[11] K. Glover and D. C. McFarlane. Robust stabilization of normalized coprime factor
plant descriptions with 7',o.-bounded uncertainty. IEEE Transactions on Automatic Control. 34(8):821-830, 1989.
[12] K. Glover and D. Mustafa. Derivation of the maximum entropy J-,o-controller and
a state-space formula for its entropy. InternationalJournal of Control, 50(3):899916, 1989.
[131 W. M. Haddad and D. S. Bernstein. Combined L2/17o, model reduction. International Journal of Control, 49(5):1523-1535, 1989.
[141 R. A. Horn and C. R. Johnson. Matriz Analysis. Cambridge University Press,
1985.
[151 D. C. Hyland and D. S. Bernstein. The optimal projection equations for fixed-order
dynamic compensation. IEEE Transactions on Automatic Control, 29(11):10341037, 1984.
[161 E. A. Jonckheere and L. M. Silverman. A new set of invariants for linear systemsapplication to reduced order compensator design. IEEE Transactionson Automatic
Control, 28(10):953-964, 1983.
[17] T. Kailath. Linear Systems. Prentice-Hall, 1980.
[181 H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley, 1972.
[191 D. C. McFarlane and K. Glover. Robust Controller Design Using Normalized Coprime Factor Plant Descriptions. Volume 138 of Lecture Notes in Control and
Information Sciences, Springer Verlag, 1990.
[201 D. C. McFarlane, K. Glover, and M. Vidyasagar. Reduced order controller design
using coprime factor model reduction. IEEE Transactions on Automatic Control,
35(3):369-373, 1990.
[21] D. G. Meyer. Model Reduction via FractionalRepresentation. PhD thesis, Stanford
University, 1987.
30
[22] D. G. Meyer and G. F. Franklin. A connection between Linear Quadratic Regulator
theory and normalized coprime factorizations. IEEE Transactions on Automatic
Control. 32(3):227-228, 1987.
[231 D.G. Meyer. A fractional approach to model reduction.
American Control Conference, Atlanta, GA, 1988.
In Proceedings of the
[24] B. C. Moore. Principal component analysis in linear systems: controllability,
observability and model reduction. IEEE Transactions on Automatic Control,
26(1):17-32, 1981.
[25] D. Mustafa. 7-o-characteristic values. In Proceedingsof the Conference on Decision
and Control, Tampa, Fl, 1989.
[26] D. Mustafa. On '7,, control, LQG control and maximum entropy. In Proceedings of the International Symposium on the Mathematical Theory of Networks and
Systems, Amsterdam, 1989.
[27] D. Mustafa. Relations between maximum entropy/7-H, control and combined
'Noo/LQG control. Systems and Control Letters, 12:193-203, 1989.
[281 D. Mustafa and K. Glover. Controllers which satisfy a closed-loop '-,o-norm bound
and maximize an entropy integral. In Proceedings of the Conference on Decision
and Control, Austin, TX, 1988.
[291 R. J. Ober and D. C. McFarlane. Balanced canonical forms for minimal systems:
a normalized coprime factor approach. Linear Algebra and Its Applications, 122124:23-64, 1989.
[30] Ph. Opdenacker and E. A. Jonckheere. LQG balancing and reduced LQG compensation of symmetric passive systems. InternationalJournal of Control, 41(1):73109, 1985.
[31] A. Ostrowski and H. Schneider. Some theorems on the inertia of general matrices.
Journal of MathematicalAnalysis and Applications, 4:72-84, 1962.
[321 L. Pernebo and L. M. Silverman. Model reduction via balanced state-space representations. IEEE Transactions on Automatic Control, 27(2):382-387, 1982.
[331 R. E. Skelton. Model error concepts in control design. International Journal of
Control, 49(5):1725-1753, 1989.
[34] M. Vidyasagar. Control Systems Synthesis: A FactorizationApproach. MIT Press,
1985.
[351 W. M. Wonham. Linear Multivariable Control: A Geometric Approach. Springer
Verlag, 1979.
[36] G. Zames. On the input-output stability of time-varying nonlinear feedback systems, Part I. IEEE Transactions on Automatic Control, 11(2):228-238, 1966.
31