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Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions
Systems & Control North-Holland
Letters
7 (1986)
165-172
June 1986
Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions Y.S. HUNG Departmenl University
a/ Electronic and Electrical of Surrey, Guildford, Surrey,
Engineering, GU2 SXH,
[G(s)]- = G(s) - [G(s)]+, the unstable projection of G(s). U.K.
K. GLOVER Engineering Trumpington
1. Introduction
Deparfmenr, University 01 Cambridge, Street, Cambridge, CB2 I PZ, U.K.
Received 5 November 1985 Revised 22 January 1986 Abstract: Given a stable rational t’ransfer function G(s) and weighting function W(S), the problem of finding G(s) of MacMillan degree k so as to minimise 11W(G - G 11u is considered. This problem is solved for W(s) = (s - /3)/(s - a) with no assumptions on the signs of a and /I. This gives rise to approximations where 11W(G - G) II m can be accurately bounded from above and below in terms of the Hankel singular values of WG (when a, B > 0). Keywords: Order approximation.
reduction,
Frequency
weighting,
Hankel-norm
Notation
I]G( S) 11”: Hankel-norm of (the stable part of) a rational transfer fuction matrix G(s). ]]G(s) I]-: Loo-norm of G(s). a(G(s)): The set of Hankel singular values of G(s). u,(G(s)): The i-th Hankel singular value (in descending order of magnitude) of (the stable part 00 G(s). , &(G(s)): McMillan degree of G(s). Hy, H? Hardy spaces of functions bounded and analytic in the right (resp. left) half plane. MT: Transpose of a matrix M. G*(s) = G( -s)~, the para-hermitian conjugate of G(s). [G(s)]+: The stable projection of G(s), i.e., the stable part of a partial fraction expansion of G(S). 0167-6911/86/$3.50
8 1986,
Elsevier
Science
Publishers
Over the past few years, considerable interest has been generated by the use of optimal Hankelnorm approximation techniques in the model reduction problem for a transfer-function matrix (see [l-4] and references therein). Given a stable transfer function matrix G(s) of McMillan degree n, an optimal Hankel-norm approximation is a stable transfer-function matrix G(s) of some prescribed degree k which minimises the Hankelnorm ]]G(s) - G(s) ]I,+ In the paper [2], Glover has derived in a state-space setting a closed-form characterisation of all optimal Hankel-norm solutions to the above minimisation problem, together with some L--error bounds. More recently, Latham and Anderson [4] have considered introducing frequency weighting into the optimal Hankel-norm approximation problem. They proposed that one may shape the approximation error by minimising
IIW)(Gb> - e(s)) 11~ where W( -s) is stable and of minimal phase. The reason for imposing such a condition on the weighting matrix is a technical one but it was argued that given any rational weighting matrix, one can always ‘reflect’ its poles and zeros with respect to the imaginary axis without changing the magnitude characteristics along the imaginary axis. It is however interesting to consider the following variations of the weighted optimal Hankel-norm approximation problem. Let us define four scalar
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W,(s)=$g,
G&(k)
(2.3)
%(s)=$$, K(s)= -#,
where U= (H?‘)pxnl
where a(s), b(s) are Hurwitz polynomials of the same degree and * denotes para-hermitian conjugation (i.e. a*(s) = a( --s)), and let us consider the rninimisation problems: (Pi>
In this section, we state some preliminary results which will be needed in the development of Section 3. Let G(s) be a stable, rational p x m transfer function matrix with Hankel singular values
- G(s) Ih 2 OL-+I-
see [1,2]) that for any of McMillan degree k, approximation error is + l)-th Hanlcel singular
(2.2)
Further, the Hankel singular values of G(s) can 166
(C?E(H~)~~“~~S(~)<~).
The next result states that the lower bound given in (2.2) is in fact achievable. G(s) E S(k) and F(s) E U (whose McMillun degree cun be chosen to be < n - k - 1) such that
2. Preliminaries
It is well established-(e.g. stable approximant G(s) the Hankel norm of the bounded below by the (k value,
S(k)=
and
Lemma 1 [1,2]. In the notation of (2.1), Ihere exist
finllW)(G(s) - G(s)) llH9 d stable of degree k
(i = 1, 2, 3, 4). Clearly, all four problems become identical should the Hankel norm be replaced by the Loo-norm. Nevertheless, it is conceivable that the optimal Hankel-norm solutions will be different and will produce different Lm-errors. The question then is which weighting function will be more appropriate if we want to keep the L--error small. The purpose of this paper is to present some preliminary results related to this question. Problem (P2) was solved in [4] and Lemma 3 (see Section 2) will make the minor modification to solve (P4). In Section 3, we will show how to solve (Pl) and (P3) in the case where a(s) and b(s) are of degree 1 and we will also provide and Lm-error bound for the achieved minimum of (PI).
Il’W
1986
be characterised as
weighting functions:
w,(s)=$y,
June
LEl-l’ERS
G(s)-&)-F(s)=a,+,E(s)
(2.4)
where E(s) is an all-puss matrix (or cun be uugmented to form an all-puss matrix in the care when G(s) is non-square). Furthermore, G(s) uchieues the lower bound of (2.2), llG(S) - ‘%S) Ih =f’k+,
(2.5)
and the constant term of d(s) can always be chosen to ensure that IIf’
lb,
s
uk+2
+
*.
. +a,,
IIG(s)-~‘(s)~~,,
**-
+‘J,.
A proof ofALemma 1 and explicit algorithms for determining G(s) can be found in [2]. The following lemma is a direct consequence of (2.3) and its proof is given below for completeness. Lemma 2. Let G(s) be u stable rational trunsferfunction matrix and let A(s) be a stable, all-puss rational matrix. Then
oi(G(s)) fori=l,2
2 oi(A*(s)G(s)) ,..., s([A*G]+).
(2.6)
Proof. Let G(s) and F(s) attain the infimum of (2.3) with k = i - 1. Then
oi(G(s)) = IIG(s) - G(s) - F(s) IIm = 11 A*(s)@(s) - &s) - f’(s)) Ilm =IIA*(s)Gb) - [A*(+%)] + -[Abel -- A*(s)F(s) 11,x, >,O/(A*(s)G(S))
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where the last step follows from (2.3), noting that the McMillan degree of [A(s)G(s)]+ is not greater than ~(G(s)) Q i - 1. 0 The next lemma is conncerned with minimising the Hankel norm of a weighted error system where the weighting matrix is totally unstable. Lemma 3 [4]. Let G(s) be a stable rational transfer function matrix and let W(s) be a totally unstable, biproper rational matrix and let W,(s) and G,(s) be the unstable and stable projections of W(s)G(s) respectively,
W(s)G(s)
= W,(s) + G,(s).
(2.7)
Let 6,(s) be a k-th order optimal Hankel-norm approximant of G,(s). Assume that W(s)-’ and 6,(s) have no common poles. Then there are rational matrices ME S(k) and c,(s)= U such that W(s)d(s)
= bv,(s) + d,(s)
(2.8)
and G(s) defined by (2.8) is a solution to (2.9)
and the minimum achieved is o&+,( W(s)G(s)).
Under the condition that G,(s) and W(s)-’ have no common poles, we can decompose W(s)-%,(s) as Proof.
w(s)-%,(s)
= d(s) - W(s)
- b(s))
=II(w,b) =llG,b>
+ G,(s))
June 1986
stable G(s) of McMi11a.n degree k, we can work through the same steps as given above except that the third equality has to be replaced by >, . This shows that a,+,(W(s)G(s)) is ? lower bound to (2.9) and is in fact attained by G(s). 0 Lemma 3 is essentially the continuous version of the approximation scheme proposed in [4]. It should be noted, however, that we do not require W( -s) to be of minimal phase. Such a condition is not necessary as far as minimising the Hankel norm is concerned, although it appears from the examples we have worked out that putting the zeros of W(s) in the right half plane as in [4] tends to produce a singificantly smaller Lm-error in comparison with other weightings having lefthalf-plane zeros but having identical magnitude characteristics along the imaginary axis. 3. Hankel-norm weighting
approximation
with
first-order
Let G(s) be a stable rational transfer function matrix and let w(s) be a first order scalar weighting function of the form w(s)=
S-P s+a’
a>O.
We will now consider the problem
(2.10)
where the poles of G(s) and w(s) are contained in the poles of G,(s) and W(s)-’ respectively. (2.8) then follows from (2.10) if we define l@,(s) = W(s)W(s). To prove (2.9), we first note that G(s) E S(k). Making use of (2.7) and (2.8), we have IIW(s)(G(s)
LETTERS
11~ - @‘I(S) + &b>)ll~
- ‘%b)lh
=%+,(6(S))
It is perhaps surprising that the solution to this problem is very closely related to the solution to
In fact, as will be shown below, the attainable minima for (Pl) and (P2) are equal, and if G(s) solves (P2), then we can always find a constant matrix b such that (Pl) is solved by G(s) = G(s) + b. Let us start with the problem (P2). Following the steps outlined in Lemma 3, we write SC(s)
where the third equality follows since G, is an k-th order approximant to G,. Now if G(s) were to be replaced by any other
= G,(s) + &
where G,(s)ES(k) constant matrix.
and R=(a-B)G(a)
(3.1)
is a
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Let a,, -->(I~>u~+~, ..aa,, be the Hankel singular values of G,(s). By Lemma 1, there are G,(S) E S(k) and F(s) E U such that
G,(s) - &b)
-f’(s)
= u,+,E(s)
(3.4
where E(s) is an all-pass rational matrix. We also assume that the constant term of C?,(S) has been chosen such that IIwII,-Jk+2+
(3.3)
*-* +%v
IIG,(s) - 6(S) 1103 Q uk+l
+ uk+2
June
(3.4)
Next, we decompose
1986
can be derived if the zero of the weighting function lies in the right half plane. Theorem 1. (a) In rhe notation of this section, ij we ser B=2nF(-a)-(R-A) (a+P)
,
then
(3.10)
G(s) = C(s) + h
* * * +a,.
[5s-1S,(s) =G(s) --& [s 1C,(s)
LElTERS
is a solution to the problem (Pl) and the minimum achieved is equal IO ok + , . (b) Furthermore, if p > 0, then
as
G (I+ d)uk+l
A
Note (see (3.1) may
2
.
[G(s) - &)I
1
uk+,E(d*
P-7)
Now we ‘re-adjust’ the constant term of the approximant by introducing a constant matrix D into the last equation,
2
[G(s) - G(s) - b]
+a,)
(3.11)
d-
b-PI a+P
(3.12)
(3.13)
Remark 1. In (3.9), the constant matrix h is chosen to introduce a zero at -a into the second term on the-left-hand side .of the equation (3.8). For such a D, we have that ,. +-F(s)-= (3.14)
for some X(S) E U. Thit can be viewed as a particular way to choose D in the method of [4]. Proof of Theorem 1. (a) By Lemma 2, we have
,.
1=
uk+lE(s)*
(3.8)
The-next theorem shfws that by a suitable choice of D, we can turn G(s) into a solution for the problem (Pl). An upper bound for the Lm-error 168
***
and
that G(s) is a solution to the problem (Pi) Lemma 3 and compare (2.7) and (2.8) with and (3.6)). Making use of (3.1) and (3.6). we rewrite (3.2) as
=
+d+m)(U,+,+
where
where G(S) is stable of McMillan degree k and A = (a - /3)Gi(/3) is a constant matrix. It follows from (3.5) that
Si;(s)=E,(s)+&
+(l
(3.5)
=
uk+l.
Hence the achievable minimum
for problem (Pl)
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is bounded below by that of (P2). In particular, II+(s)
-e(s)]
LETTERS
June
1986
where m is given by (3.13) and we have used (3.3) in the last step. Clearly
(3.15)
llH-x+,.
(3.20)
Now from (3.8) and (-3.14), we have Finally, from (3.1) and (3.6), we have 2
[G(s) - 8(s)]
+%X(s)
= ~,+1Jw. Since G,(s)- &i(s) is analytic in the right half plane, applying the maximum-modulus principle and then making use of (3.4), we get
It follows that
+x(s) =
uk+l
(3.16)
EE(s).
This implies that
<
d(
uk+i
+
ok+2
+
(3.21)
’ * ’ +a,,)
where d is given by (3.12). It follows (3.18)-(3.21) that which together with (3.15) show that G(s) achieves the lower bound ek+i for the problem (Pl). (b) To prove the L--error bound, we first deduce from (3.16) that
“X(S)“m~duk+l+(1+m+d)(uk+~+
from
*** +a,,>
Putting this into (3.17) yields the desired error bound (3.11). 0 Remark 2. Let us now consider the specific case k = n - 1, when F(s) in (3.2) can be shown to be
It remains to bound the Lm-norm of X(s). Now from (3.14),
.
X(s)= Sb-
ZF(s)-
zero, and X(s) in (3.18) is given by . X(s) = - s
=-+,(B)-&(B)]
gf.
Substituting for D from (3.9) gives 24
-PI
x(s)= (a+/3)(s+a)
(3.22)
q+ (3.18)
Making use of the maximum-modulus principle and noting that F(s) is anaiytic in the left half plane, we can bound the La-norm of the.first term on the right-hand side of (3.18) as
(Note that this upper bound may b,” weak, particularly if -/3 is near a pole of G. or G,, since E(s) is ah-pass; see the example in Section 4.) Denote the degree-(n - 1) approximation by @‘-l(s), we obtain
112 [G(s)- @-l(~)]11 02 <(l+d)u,.
(3.23)
Further, the Hankel singular values of (3.19) 169
SYSTEMS & CONTROL LETTERS
Volume 7. Number 3
are given by
q +-ys)] [
constant term is (assume p z=0)
= q(C,)
b = G(a)
- C?(a)
= G,(P)-‘%(P),
= q(G,), i=l,2 ,***, n - 1, by [2, Lemma 9.11. Now if we apply the above procedure sequentially, reducing the order one state at a time to obtain a degree k approximant G&(s), then
so that the pole at s = a is cancelled in the weighted error,
2
[G(s) - d(s) = G,(s) - 6,(s)
< (1+ d)(a,+,
+ * *. +a,).
(3.24)
Comparing (3.11) and (3.24) we see that the latter provides a smaller upper bound on the Lm-error, although in general @(s) will not be an optimal solution to (Pl). Remark 3. Yet another method for choosing the
Fig. 1. Frequency 170
June 1986
of G, 6 and e.
- I>] -b
and
(3.25)
by the maximum principle and (3.4). This upper bound will be inferior to that in (3.24) when d is small.
SYSTEMS & CONTROL LE’l-l-ERS
Volume 7, Number 3 4. An example
Figure 1 shows the frequency responses of G(s), G(s) and the unweighted error system
We illustrate the results given in Theorem I by an example. Consider the system G(s) =
June 1986
-s+4 s3 + 4s2+ 6s + 4
(4.1)
e(s) = G(s) -C?(s). Using Theorem l(b) (or (3.23)), we deduce that the weighted Lm-error is bounded as
which has Hankel singular values
11w(s)e(s)
a(G(s))
This is confirmed by the magnitude plot of e(s) given in Figure 2, which also shows that the weighting function has been very effective in shaping the frequency response of the error system in the desired way. In fact, from (3.22) and Figure 2,
= {0.7797,0.3275,0.04773}.
Now consider the stable weighting function w(s)=
S-l s+5’
In the notation of Section 3, the Hankel singular values of G,(s) (= stable projection of ((s - l)/ (s - 5)G(s)) are u(G,(s))
= h 029 ~31 = {0.2044,0.1321,0.02615}.
(4.3)
Since a3 is smaller than u2 by a factor of 5, it seems sensible to approximate G,(s) by a G,(s) of degree 2, which can be readily obtained by an altorithm of [2]. Calculating G(s) according as (3.5), (3.9) and (3.10) we get
C(s) = - 0.3719s + 0.8995 + o 02578 s2+1.068s+1.064
le(jw>l 4
’
’
(4-4
Ilm d ;a3 = 0.04358.
u3 Q 1)w(s)e(s)
)looz 0.0265
au3(1+d]E(/3)])=1.014u3 and it is seen that the resulting error is 1.4% from the unachievable lower bound. The method of Remark 3 gives a weighted error of about 1.02~~ and is almost identical. As a matter of interest, we repeat the above calculations with the weighting function (4.2) replaced by b(S)
s+l = s+5
which differs from w(s) in that it is minimal phase but otherwise has the same magnitude characteristics as w(s) along the imaginary axis. The relevant Hankel singular values are u[ sG(s)]
= {0.1576,0.04915,0.006077} (4.5)
and the degree-2 approximant
&(s) =
- 1.808s - 1.741 s2+1.835s+2.574
turns out to be _ o 41g1 * .
(4.6)
Although the Hankel singular values given in (4.5) compare favourably with those given in (4.3) and that IIw,,&)(G(s)
Fig. 2. Unweighted error magnitude.
- &(s))
IlH = 0.~6077,
this is not to be inferred as an indication that G,,,(s) is superior to G(s). To the contrary, Theorem l(b) does not hold in this case (since /3 = - 1 < 0) and a frequency-response plot (see Figure 3) shows that G,(s) is far from being a reasonable approximation to G(s). It is however interesting to 171
Volume 7. Number 3
SYSTEMS & CONTROL LETTERS
June 1986
IIIAG I .0
t
Fig. 3. Frequency response of G and &,.
note that the frequency response of w,,,(s)(G(s) G,,,(S)) lies on a small circle of radius 0.006077 centred around 0.4251 (= 0.4191 + 0.006077). The problem is that the Hankel norm is only a seminorm when the function is not strictly proper (as in this case), or not casual. Finally, the frequency-weighted balanced realisation method developed in [5] has been applied to this example. The weighted Loo-error was found to be approximately 0.07, worse than 0.0265 achieved above. However, this method does not use a ‘b-term’.
References [l] V.M. Adamjan, D.Z. Arov and M.G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the
172
[2] [3] [4] [S]
generalised Schur-Takagi problem, Math. USSR Sbornik 15 (1971) 31-73 (English translation: Amer. Marh. Sot. Trod. 111 (1978) 87-112). K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their Lm-error bounds, Internal. J. Control 39 (6) (1984) 1115-1193. S.Y. Kung and D.W. Lin, Optimal Hankel-norm model reductions: Multivariable systems, IEEE Trans. Auromar. Control 26 (4) (1981) 832-852. G.A. Latham and B.D.O. Anderson, Frequency-weighted optimal Hankel-norm approximation of stable transfer functions, Sysrems Conrrol Len. 5 (4) (1985) 229-236. D. Enns, Model reduction for control systems, Ph.D. dissertation, Dept. Aeronautics & Astronautics, Standford University (June 1984).
Letters
7 (1986)
165-172
June 1986
Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions Y.S. HUNG Departmenl University
a/ Electronic and Electrical of Surrey, Guildford, Surrey,
Engineering, GU2 SXH,
[G(s)]- = G(s) - [G(s)]+, the unstable projection of G(s). U.K.
K. GLOVER Engineering Trumpington
1. Introduction
Deparfmenr, University 01 Cambridge, Street, Cambridge, CB2 I PZ, U.K.
Received 5 November 1985 Revised 22 January 1986 Abstract: Given a stable rational t’ransfer function G(s) and weighting function W(S), the problem of finding G(s) of MacMillan degree k so as to minimise 11W(G - G 11u is considered. This problem is solved for W(s) = (s - /3)/(s - a) with no assumptions on the signs of a and /I. This gives rise to approximations where 11W(G - G) II m can be accurately bounded from above and below in terms of the Hankel singular values of WG (when a, B > 0). Keywords: Order approximation.
reduction,
Frequency
weighting,
Hankel-norm
Notation
I]G( S) 11”: Hankel-norm of (the stable part of) a rational transfer fuction matrix G(s). ]]G(s) I]-: Loo-norm of G(s). a(G(s)): The set of Hankel singular values of G(s). u,(G(s)): The i-th Hankel singular value (in descending order of magnitude) of (the stable part 00 G(s). , &(G(s)): McMillan degree of G(s). Hy, H? Hardy spaces of functions bounded and analytic in the right (resp. left) half plane. MT: Transpose of a matrix M. G*(s) = G( -s)~, the para-hermitian conjugate of G(s). [G(s)]+: The stable projection of G(s), i.e., the stable part of a partial fraction expansion of G(S). 0167-6911/86/$3.50
8 1986,
Elsevier
Science
Publishers
Over the past few years, considerable interest has been generated by the use of optimal Hankelnorm approximation techniques in the model reduction problem for a transfer-function matrix (see [l-4] and references therein). Given a stable transfer function matrix G(s) of McMillan degree n, an optimal Hankel-norm approximation is a stable transfer-function matrix G(s) of some prescribed degree k which minimises the Hankelnorm ]]G(s) - G(s) ]I,+ In the paper [2], Glover has derived in a state-space setting a closed-form characterisation of all optimal Hankel-norm solutions to the above minimisation problem, together with some L--error bounds. More recently, Latham and Anderson [4] have considered introducing frequency weighting into the optimal Hankel-norm approximation problem. They proposed that one may shape the approximation error by minimising
IIW)(Gb> - e(s)) 11~ where W( -s) is stable and of minimal phase. The reason for imposing such a condition on the weighting matrix is a technical one but it was argued that given any rational weighting matrix, one can always ‘reflect’ its poles and zeros with respect to the imaginary axis without changing the magnitude characteristics along the imaginary axis. It is however interesting to consider the following variations of the weighted optimal Hankel-norm approximation problem. Let us define four scalar
B.V. (North-Holland)
165
Volume
7, Number
SYSTEMS
3
& CONTROL
W,(s)=$g,
G&(k)
(2.3)
%(s)=$$, K(s)= -#,
where U= (H?‘)pxnl
where a(s), b(s) are Hurwitz polynomials of the same degree and * denotes para-hermitian conjugation (i.e. a*(s) = a( --s)), and let us consider the rninimisation problems: (Pi>
In this section, we state some preliminary results which will be needed in the development of Section 3. Let G(s) be a stable, rational p x m transfer function matrix with Hankel singular values
- G(s) Ih 2 OL-+I-
see [1,2]) that for any of McMillan degree k, approximation error is + l)-th Hanlcel singular
(2.2)
Further, the Hankel singular values of G(s) can 166
(C?E(H~)~~“~~S(~)<~).
The next result states that the lower bound given in (2.2) is in fact achievable. G(s) E S(k) and F(s) E U (whose McMillun degree cun be chosen to be < n - k - 1) such that
2. Preliminaries
It is well established-(e.g. stable approximant G(s) the Hankel norm of the bounded below by the (k value,
S(k)=
and
Lemma 1 [1,2]. In the notation of (2.1), Ihere exist
finllW)(G(s) - G(s)) llH9 d stable of degree k
(i = 1, 2, 3, 4). Clearly, all four problems become identical should the Hankel norm be replaced by the Loo-norm. Nevertheless, it is conceivable that the optimal Hankel-norm solutions will be different and will produce different Lm-errors. The question then is which weighting function will be more appropriate if we want to keep the L--error small. The purpose of this paper is to present some preliminary results related to this question. Problem (P2) was solved in [4] and Lemma 3 (see Section 2) will make the minor modification to solve (P4). In Section 3, we will show how to solve (Pl) and (P3) in the case where a(s) and b(s) are of degree 1 and we will also provide and Lm-error bound for the achieved minimum of (PI).
Il’W
1986
be characterised as
weighting functions:
w,(s)=$y,
June
LEl-l’ERS
G(s)-&)-F(s)=a,+,E(s)
(2.4)
where E(s) is an all-puss matrix (or cun be uugmented to form an all-puss matrix in the care when G(s) is non-square). Furthermore, G(s) uchieues the lower bound of (2.2), llG(S) - ‘%S) Ih =f’k+,
(2.5)
and the constant term of d(s) can always be chosen to ensure that IIf’
lb,
s
uk+2
+
*.
. +a,,
IIG(s)-~‘(s)~~,,
**-
+‘J,.
A proof ofALemma 1 and explicit algorithms for determining G(s) can be found in [2]. The following lemma is a direct consequence of (2.3) and its proof is given below for completeness. Lemma 2. Let G(s) be u stable rational trunsferfunction matrix and let A(s) be a stable, all-puss rational matrix. Then
oi(G(s)) fori=l,2
2 oi(A*(s)G(s)) ,..., s([A*G]+).
(2.6)
Proof. Let G(s) and F(s) attain the infimum of (2.3) with k = i - 1. Then
oi(G(s)) = IIG(s) - G(s) - F(s) IIm = 11 A*(s)@(s) - &s) - f’(s)) Ilm =IIA*(s)Gb) - [A*(+%)] + -[Abel -- A*(s)F(s) 11,x, >,O/(A*(s)G(S))
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where the last step follows from (2.3), noting that the McMillan degree of [A(s)G(s)]+ is not greater than ~(G(s)) Q i - 1. 0 The next lemma is conncerned with minimising the Hankel norm of a weighted error system where the weighting matrix is totally unstable. Lemma 3 [4]. Let G(s) be a stable rational transfer function matrix and let W(s) be a totally unstable, biproper rational matrix and let W,(s) and G,(s) be the unstable and stable projections of W(s)G(s) respectively,
W(s)G(s)
= W,(s) + G,(s).
(2.7)
Let 6,(s) be a k-th order optimal Hankel-norm approximant of G,(s). Assume that W(s)-’ and 6,(s) have no common poles. Then there are rational matrices ME S(k) and c,(s)= U such that W(s)d(s)
= bv,(s) + d,(s)
(2.8)
and G(s) defined by (2.8) is a solution to (2.9)
and the minimum achieved is o&+,( W(s)G(s)).
Under the condition that G,(s) and W(s)-’ have no common poles, we can decompose W(s)-%,(s) as Proof.
w(s)-%,(s)
= d(s) - W(s)
- b(s))
=II(w,b) =llG,b>
+ G,(s))
June 1986
stable G(s) of McMi11a.n degree k, we can work through the same steps as given above except that the third equality has to be replaced by >, . This shows that a,+,(W(s)G(s)) is ? lower bound to (2.9) and is in fact attained by G(s). 0 Lemma 3 is essentially the continuous version of the approximation scheme proposed in [4]. It should be noted, however, that we do not require W( -s) to be of minimal phase. Such a condition is not necessary as far as minimising the Hankel norm is concerned, although it appears from the examples we have worked out that putting the zeros of W(s) in the right half plane as in [4] tends to produce a singificantly smaller Lm-error in comparison with other weightings having lefthalf-plane zeros but having identical magnitude characteristics along the imaginary axis. 3. Hankel-norm weighting
approximation
with
first-order
Let G(s) be a stable rational transfer function matrix and let w(s) be a first order scalar weighting function of the form w(s)=
S-P s+a’
a>O.
We will now consider the problem
(2.10)
where the poles of G(s) and w(s) are contained in the poles of G,(s) and W(s)-’ respectively. (2.8) then follows from (2.10) if we define l@,(s) = W(s)W(s). To prove (2.9), we first note that G(s) E S(k). Making use of (2.7) and (2.8), we have IIW(s)(G(s)
LETTERS
11~ - @‘I(S) + &b>)ll~
- ‘%b)lh
=%+,(6(S))
It is perhaps surprising that the solution to this problem is very closely related to the solution to
In fact, as will be shown below, the attainable minima for (Pl) and (P2) are equal, and if G(s) solves (P2), then we can always find a constant matrix b such that (Pl) is solved by G(s) = G(s) + b. Let us start with the problem (P2). Following the steps outlined in Lemma 3, we write SC(s)
where the third equality follows since G, is an k-th order approximant to G,. Now if G(s) were to be replaced by any other
= G,(s) + &
where G,(s)ES(k) constant matrix.
and R=(a-B)G(a)
(3.1)
is a
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7. Number
SYSTEMS
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Let a,, -->(I~>u~+~, ..aa,, be the Hankel singular values of G,(s). By Lemma 1, there are G,(S) E S(k) and F(s) E U such that
G,(s) - &b)
-f’(s)
= u,+,E(s)
(3.4
where E(s) is an all-pass rational matrix. We also assume that the constant term of C?,(S) has been chosen such that IIwII,-Jk+2+
(3.3)
*-* +%v
IIG,(s) - 6(S) 1103 Q uk+l
+ uk+2
June
(3.4)
Next, we decompose
1986
can be derived if the zero of the weighting function lies in the right half plane. Theorem 1. (a) In rhe notation of this section, ij we ser B=2nF(-a)-(R-A) (a+P)
,
then
(3.10)
G(s) = C(s) + h
* * * +a,.
[5s-1S,(s) =G(s) --& [s 1C,(s)
LElTERS
is a solution to the problem (Pl) and the minimum achieved is equal IO ok + , . (b) Furthermore, if p > 0, then
as
G (I+ d)uk+l
A
Note (see (3.1) may
2
.
[G(s) - &)I
1
uk+,E(d*
P-7)
Now we ‘re-adjust’ the constant term of the approximant by introducing a constant matrix D into the last equation,
2
[G(s) - G(s) - b]
+a,)
(3.11)
d-
b-PI a+P
(3.12)
(3.13)
Remark 1. In (3.9), the constant matrix h is chosen to introduce a zero at -a into the second term on the-left-hand side .of the equation (3.8). For such a D, we have that ,. +-F(s)-= (3.14)
for some X(S) E U. Thit can be viewed as a particular way to choose D in the method of [4]. Proof of Theorem 1. (a) By Lemma 2, we have
,.
1=
uk+lE(s)*
(3.8)
The-next theorem shfws that by a suitable choice of D, we can turn G(s) into a solution for the problem (Pl). An upper bound for the Lm-error 168
***
and
that G(s) is a solution to the problem (Pi) Lemma 3 and compare (2.7) and (2.8) with and (3.6)). Making use of (3.1) and (3.6). we rewrite (3.2) as
=
+d+m)(U,+,+
where
where G(S) is stable of McMillan degree k and A = (a - /3)Gi(/3) is a constant matrix. It follows from (3.5) that
Si;(s)=E,(s)+&
+(l
(3.5)
=
uk+l.
Hence the achievable minimum
for problem (Pl)
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7, Number
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is bounded below by that of (P2). In particular, II+(s)
-e(s)]
LETTERS
June
1986
where m is given by (3.13) and we have used (3.3) in the last step. Clearly
(3.15)
llH-x+,.
(3.20)
Now from (3.8) and (-3.14), we have Finally, from (3.1) and (3.6), we have 2
[G(s) - 8(s)]
+%X(s)
= ~,+1Jw. Since G,(s)- &i(s) is analytic in the right half plane, applying the maximum-modulus principle and then making use of (3.4), we get
It follows that
+x(s) =
uk+l
(3.16)
EE(s).
This implies that
<
d(
uk+i
+
ok+2
+
(3.21)
’ * ’ +a,,)
where d is given by (3.12). It follows (3.18)-(3.21) that which together with (3.15) show that G(s) achieves the lower bound ek+i for the problem (Pl). (b) To prove the L--error bound, we first deduce from (3.16) that
“X(S)“m~duk+l+(1+m+d)(uk+~+
from
*** +a,,>
Putting this into (3.17) yields the desired error bound (3.11). 0 Remark 2. Let us now consider the specific case k = n - 1, when F(s) in (3.2) can be shown to be
It remains to bound the Lm-norm of X(s). Now from (3.14),
.
X(s)= Sb-
ZF(s)-
zero, and X(s) in (3.18) is given by . X(s) = - s
=-+,(B)-&(B)]
gf.
Substituting for D from (3.9) gives 24
-PI
x(s)= (a+/3)(s+a)
(3.22)
q+ (3.18)
Making use of the maximum-modulus principle and noting that F(s) is anaiytic in the left half plane, we can bound the La-norm of the.first term on the right-hand side of (3.18) as
(Note that this upper bound may b,” weak, particularly if -/3 is near a pole of G. or G,, since E(s) is ah-pass; see the example in Section 4.) Denote the degree-(n - 1) approximation by @‘-l(s), we obtain
112 [G(s)- @-l(~)]11 02 <(l+d)u,.
(3.23)
Further, the Hankel singular values of (3.19) 169
SYSTEMS & CONTROL LETTERS
Volume 7. Number 3
are given by
q +-ys)] [
constant term is (assume p z=0)
= q(C,)
b = G(a)
- C?(a)
= G,(P)-‘%(P),
= q(G,), i=l,2 ,***, n - 1, by [2, Lemma 9.11. Now if we apply the above procedure sequentially, reducing the order one state at a time to obtain a degree k approximant G&(s), then
so that the pole at s = a is cancelled in the weighted error,
2
[G(s) - d(s) = G,(s) - 6,(s)
< (1+ d)(a,+,
+ * *. +a,).
(3.24)
Comparing (3.11) and (3.24) we see that the latter provides a smaller upper bound on the Lm-error, although in general @(s) will not be an optimal solution to (Pl). Remark 3. Yet another method for choosing the
Fig. 1. Frequency 170
June 1986
of G, 6 and e.
- I>] -b
and
(3.25)
by the maximum principle and (3.4). This upper bound will be inferior to that in (3.24) when d is small.
SYSTEMS & CONTROL LE’l-l-ERS
Volume 7, Number 3 4. An example
Figure 1 shows the frequency responses of G(s), G(s) and the unweighted error system
We illustrate the results given in Theorem I by an example. Consider the system G(s) =
June 1986
-s+4 s3 + 4s2+ 6s + 4
(4.1)
e(s) = G(s) -C?(s). Using Theorem l(b) (or (3.23)), we deduce that the weighted Lm-error is bounded as
which has Hankel singular values
11w(s)e(s)
a(G(s))
This is confirmed by the magnitude plot of e(s) given in Figure 2, which also shows that the weighting function has been very effective in shaping the frequency response of the error system in the desired way. In fact, from (3.22) and Figure 2,
= {0.7797,0.3275,0.04773}.
Now consider the stable weighting function w(s)=
S-l s+5’
In the notation of Section 3, the Hankel singular values of G,(s) (= stable projection of ((s - l)/ (s - 5)G(s)) are u(G,(s))
= h 029 ~31 = {0.2044,0.1321,0.02615}.
(4.3)
Since a3 is smaller than u2 by a factor of 5, it seems sensible to approximate G,(s) by a G,(s) of degree 2, which can be readily obtained by an altorithm of [2]. Calculating G(s) according as (3.5), (3.9) and (3.10) we get
C(s) = - 0.3719s + 0.8995 + o 02578 s2+1.068s+1.064
le(jw>l 4
’
’
(4-4
Ilm d ;a3 = 0.04358.
u3 Q 1)w(s)e(s)
)looz 0.0265
au3(1+d]E(/3)])=1.014u3 and it is seen that the resulting error is 1.4% from the unachievable lower bound. The method of Remark 3 gives a weighted error of about 1.02~~ and is almost identical. As a matter of interest, we repeat the above calculations with the weighting function (4.2) replaced by b(S)
s+l = s+5
which differs from w(s) in that it is minimal phase but otherwise has the same magnitude characteristics as w(s) along the imaginary axis. The relevant Hankel singular values are u[ sG(s)]
= {0.1576,0.04915,0.006077} (4.5)
and the degree-2 approximant
&(s) =
- 1.808s - 1.741 s2+1.835s+2.574
turns out to be _ o 41g1 * .
(4.6)
Although the Hankel singular values given in (4.5) compare favourably with those given in (4.3) and that IIw,,&)(G(s)
Fig. 2. Unweighted error magnitude.
- &(s))
IlH = 0.~6077,
this is not to be inferred as an indication that G,,,(s) is superior to G(s). To the contrary, Theorem l(b) does not hold in this case (since /3 = - 1 < 0) and a frequency-response plot (see Figure 3) shows that G,(s) is far from being a reasonable approximation to G(s). It is however interesting to 171
Volume 7. Number 3
SYSTEMS & CONTROL LETTERS
June 1986
IIIAG I .0
t
Fig. 3. Frequency response of G and &,.
note that the frequency response of w,,,(s)(G(s) G,,,(S)) lies on a small circle of radius 0.006077 centred around 0.4251 (= 0.4191 + 0.006077). The problem is that the Hankel norm is only a seminorm when the function is not strictly proper (as in this case), or not casual. Finally, the frequency-weighted balanced realisation method developed in [5] has been applied to this example. The weighted Loo-error was found to be approximately 0.07, worse than 0.0265 achieved above. However, this method does not use a ‘b-term’.
References [l] V.M. Adamjan, D.Z. Arov and M.G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the
172
[2] [3] [4] [S]
generalised Schur-Takagi problem, Math. USSR Sbornik 15 (1971) 31-73 (English translation: Amer. Marh. Sot. Trod. 111 (1978) 87-112). K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their Lm-error bounds, Internal. J. Control 39 (6) (1984) 1115-1193. S.Y. Kung and D.W. Lin, Optimal Hankel-norm model reductions: Multivariable systems, IEEE Trans. Auromar. Control 26 (4) (1981) 832-852. G.A. Latham and B.D.O. Anderson, Frequency-weighted optimal Hankel-norm approximation of stable transfer functions, Sysrems Conrrol Len. 5 (4) (1985) 229-236. D. Enns, Model reduction for control systems, Ph.D. dissertation, Dept. Aeronautics & Astronautics, Standford University (June 1984).