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This version: 22/10/2004

Chapter 15

An introduction to H∞ control theory

The task of designing a controller which accomplishes a speciﬁed task is a challenging chore. When one wishes to add “robustness” to the mix, things become even more challenging. By robust design, what is meant is that one should design a controller which works not only for a given model, but for plants which are close to that model. In this way, one can have some degree of certainty that even if the model is imperfect, the controller will behave in a satisfactory manner. The development of systematic design procedures for robust control can be seen to have been initiated with the important paper of Francis and Zames [1984]. Since this time, there have been many developments and generalisations. The understanding of so-called H∞ methods has progressed to the point that a somewhat elementary treatment is possible. We shall essentially follow the approach of Doyle, Francis, and Tannenbaum [1990]. For a recent account of MIMO developments, see [Dullerud and Paganini 1999]. The book by Francis [1987] is also a useful reference.

Although all of the material in this chapter can be followed by any student who has come to grips with the more basic material in this book, much of what we do here is a signiﬁcant diversion from control theory, per se, and is really a development of the necessary mathematical tools. Since a complete understanding of the tools is not necessary in order to apply them, it is perhaps worthwhile to outline the bare bones guide to getting through this chapter. This might be as follows.

1.One should ﬁrst make sure that one understands the problem being solved: the robust performance problem. The ﬁrst thing done is to modify this problem so that it is tractable; this is the content of Problem 15.2.

2.The modiﬁed robust performance problem is ﬁrst converted to a model matching problem, which is stated in generality in Problem 15.3. The content of this conversion is contained in Algorithm 15.5.

3.The model matching problem is solved in this book in two ways. The ﬁrst method involves Nevanlinna-Pick interpolation, and to apply this method, follow the steps outlined in Algorithm 15.18. It is entirely possible that you will have to make some modiﬁcations to the algorithm to ensure that you arrive at a proper controller. The necessary machinations are discussed following the algorithm.

4.The second method for solving the model matching problem involves approximation of unstable rational functions by stable ones via Nehari’s Theorem. To apply this method, follow the steps in Algorithm 15.29. As with Nevanlinna-Pick interpolation, one should be prepared to make some modiﬁcations to the algorithm to ensure that things work out. The manner in which to carry out these modiﬁcations is discussed following the algorithm.

5.Other problems that can be solved in this manner are discussed in Section 15.5.

558

15 An introduction to H∞ control theory

22/10/2004

Readers wishing only to be able to apply the methods in this chapter should go through the chapter with the above skeleton as a guide. However, those wishing to see the details will be happy to see very little omitted. It should also be emphasised that Mathematica® and Maple® packages for solving these problems may be found at the URL http://mast.queensu.ca/~math332/.1

Contents

15.1	Reduction of robust performance problem to model matching problem . . . . . . . . . . .		558
	15.1.1	A modiﬁed robust performance problem . . . . . . . . . . . . . . . . . . . . . . . .	558
	15.1.2	Algorithm for reduction to model matching problem . . . . . . . . . . . . . . . . .	560
	15.1.3	Proof that reduction procedure works . . . . . . . . . . . . . . . . . . . . . . . . .	562
15.2	Optimal model matching I. Nevanlinna-Pick theory . . . . . . . . . . . . . . . . . . . . .		566
	15.2.1	Pick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	566
	15.2.2	An inductive algorithm for solving the interpolation problem . . . . . . . . . . . .	568
	15.2.3	Relationship to the model matching problem . . . . . . . . . . . . . . . . . . . . .	570
15.3	Optimal model matching II. Nehari’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .		570
	15.3.1	Hankel operators in the frequency-domain . . . . . . . . . . . . . . . . . . . . . .	571
	15.3.2	Hankel operators in the time-domain . . . . . . . . . . . . . . . . . . . . . . . . .	573
	15.3.3	Hankel singular values and Schmidt pairs . . . . . . . . . . . . . . . . . . . . . . .	576
	15.3.4	Nehari’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	578
	15.3.5	Relationship to the model matching problem . . . . . . . . . . . . . . . . . . . . .	580
15.4	A robust performance example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .		580
15.5	Other problems involving H∞ methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .		580

15.1 Reduction of robust performance problem to model matching problem

For SISO systems, the problem of designing for robust performance turns out to be reducible in a certain sense to two problems investigated by mathematicians coming under the broad heading of complex function approximation, or, more descriptively, model matching. In this section we shall discuss how this reduction takes place, as in itself it is not an entirely obvious step.

15.1.1 A modiﬁed robust performance problem				First recall the robust performance
				¯	+
problem, Problem 9.23. Given a proper nominal plant RP , a function Wu RH∞ given an
¯		¯			R(s), we seek
uncertainty set P×(RP , Wu) or	P+(RP , Wu), and a performance weight Wp				R(s), we seek
a controller RC that stabilises the nominal plant and satisﬁes either
\|WuT¯L\| + \|WpS¯L\| ∞		< 1 or	\|WuRC S¯L\| + \|WpS¯L\| ∞ < 1,

1These are not currently implemented. Hopefully they will be in the near future.

22/10/2004 15.1 Reduction of robust performance problem to model matching problem 559

depending on whether one is using multiplicative or additive uncertainty. The robust performance problem, it turns out, is quite di cult. For instance, useful necessary and su cient conditions for there to exist a solution to the problem are not known. Thus our ﬁrst step in this section is to come up with a simpler problem that is easier to solve. The simpler problem is based upon the following result.

15.1 Lemma Let R1, R2

R(s2) and

the

-valued

27→

(|R1(s)| + |R2(s)|) and by

|R1|

denote by

function

+ |R2|

the R-valued function

s 7→(|R1(s)| + |R2(s)| ).

|R1|2

|2 ∞ <

+ |R2

then

|R1| + |R2

| ∞ < 1.

Proof Deﬁne

S2 = (x, y) R

x, y > 0, x + y <

2 .

= (x, y)

x, y > 0,

|x + y|

< 1 ,

S2. However,

we note that S1 = [0, 1]

The result will follow if we can show that

[0, 1]

and S2 is the circle of radius √2

centred at the origin. Clearly S1 S2 (see Figure 15.1).

Figure 15.1 Interpretation of modiﬁed robust performance condition

This leads to a modiﬁcation of the robust performance problem, and we state this formally since it is this problem to which we devote the majority of our e ort in this chapter. Note that we make a few additional assumptions in the statement of the problem that are not present in the statement of Problem 9.23. Namely, we assume now the following.

1.Wp RH+∞: Thus we add the assumption that Wp be proper since, without loss of generality as we are only interested on the value of Wp on iR, we can suppose that all poles of Wp lie in C−.

2.Wu and Wp have no common imaginary axis zeros: This is an assumption that, if not satisﬁed, can be satisﬁed with minor tweaking of Wu and Wp.

560

15 An introduction to H∞ control theory

22/10/2004

3.RC is proper: We make this assumption mostly as a matter of convention. If we arrive at a controller that is improper but solves the problem, then it is often possible to modify the controller so that it is proper and still solves the problem.

Thus our problem becomes.

15.2 Modiﬁed robust performance problem Given
(i)		¯
(i)	a nominal proper plant RP ,
(ii)	a function Wu RH∞+ ,	¯	¯

(iii) an uncertainty model P×(RP , Wu) or P+(RP , Wu), and

(iv) a performance weight Wp RH+∞,

so that Wu and Wp have no common imaginary axis zeros, ﬁnd a proper controller RC that

(v)

stabilises the nominal system and

(vi)

satisﬁes either

∞

< 1 or

< 1, depending

|WuTL|

+|WpSL|

|WuRC SL|

+|WpSL|

∞

on whether

one is using multiplicative or additive uncertainty.

As should be clear from Figure 15.1, it is possible that for a given RP , Wu, and Wp it will not be possible to solve the modiﬁed robust performance problem even though a solution may exist to the robust performance problem. Thus we are sacriﬁcing something in so modifying the problem, but what we gain is a simpliﬁed problem that can be solved.

15.1.2 Algorithm for reduction to model matching problem The objective of this section is to convert the modiﬁed robust performance problem into the model matching problem. We shall concentrate in this section on multiplicative uncertainty, with the reader ﬁlling in the details for additive uncertainty in Exercise E15.3.

First let us state the model matching problem.

15.3 A model matching problem Let T1, T2 RH∞+ .	Find θ RH∞+ (s) so that kT1 − θT2k∞
is minimised.

The model matching problem may not have a solution. In fact, it will often be the case in applications that it does not have a solution. However, as we shall see as we get into our development, even when the problem has no solution, it can be used as a guide to solve the problem that is actually of interest to us, namely the modiﬁed robust performance problem, Problem 15.2. Some issues concerning existence of solutions to the model matching problem are the topic of Exercise E15.2

15.4 Remark Note that since T1, T2 RH+∞, if θ is to be a solution of the model matching problem, then it can have no imaginary axis poles. Also, since the model matching problem only cares about the value of θ on the imaginary axis, we can without loss of generality (by multiplying a solution θ to the model matching problem by an inner function that cancels all poles in θ in C+) suppose that θ has no poles in C+.

Let us outline the steps in performing the reduction of Problem 15.2 to Problem 15.3. After we have said how to perform the reduction, we will actually prove that everything works. The reader will wish to recall the notion of spectral factorisation for rational functions (Proposition 14.17) and the notion of a coprime factorisation for a pair of rational functions (Theorem 10.33).

22/10/2004 15.1 Reduction of robust performance problem to model matching problem 561

			¯
15.5 Algorithm for obtaining model matching problem for multiplicative uncertainty Given RP ,
Wu, and Wp as in Problem 15.2.
1.	Deﬁne
		WpWp WuWu
	U3 =			.
		WpWp + WuWu
2.	If kU3k∞ ≥ 21 , then Problem 15.2 has no solution.		¯
3.	Let (P1, P2) be a coprime fractional representative for RP .
4.	Let (ρ1, ρ2) be a coprime factorisation for P1 and P2:
	ρ1P1 + ρ2P2 = 1.
5.	Deﬁne
	R1 = Wpρ2P2,		S1 = Wuρ1P1,
	R2 = WpP1P2,		S2 = − WuP1P2.

6.Deﬁne Q = [R2R2 + S2S2 ]+.

7.Let V be an inner function with the property that

					R1R2 + S1S2				V
						Q
has no poles in		+.
has no poles in	C	+.
8. Deﬁne				R		+ S	S
			R	R		+ S	S
		U1 =	1	2		1	2	V, U2 = QV.
		U1 =			Q			V, U2 = QV.
					Q

9.Deﬁne U4 = [12 − U3]+.

10.Deﬁne

T1	=	U1	, T2	=	U2	.

		U4			U4

11.Let θ be a solution to Problem 15.3, and by Remark 15.4 suppose that it has no poles in C+.

12.If kT1 − θT2k∞ ≥ 1 then Problem 15.2 has no solution.

13.The controller

RC = ρ1 + θP2 , ρ2 − θP1

is a solution to Problem 15.2.

The above procedure provides a way to produce a controller satisfying the modiﬁed robust performance problem, provided one can ﬁnd θ in Step 11. That is to say, we have reduced the ﬁnding of a solution to the modiﬁed robust performance problem to that of solving the model matching problem. It remains to show that all constructions made in Algorithm 15.5 are sensible, and that all claims made are true. In the next section we will do this formally. However, before we get into all the details, it is helpful to give a glimpse into how Algorithm 15.5 comes about.

562 15 An introduction to H∞ control theory 22/10/2004

As we are working with multiplicative uncertainty (see Exercise E15.3 for additive uncertainty), the problem we start out with, of course, is to ﬁnd a proper RC R(s) that

satisﬁes

∞

¯ u

| p

+ W

for a given nominal proper

plant R

, an uncertainty model W

, and a performance

∞

and an

weight Wp RH∞. We choose a coprime fractional representative (P1, P2) for RP

associated coprime factorisation (ρ1, ρ2). By Theorem 10.37, any proper stabilising controller

is then of the form				ρ1 + θP2
				ρ1 + θP2
		RC =				.
A simple computation then gives		RC =		ρ2 − θP1		.
A simple computation then gives
¯			¯		= −(P1P2θ − ρ2P2).
TL = P1P2		θ + ρ1P1, SL			= −(P1P2θ − ρ2P2).
Deﬁning
R1 = Wpρ2P2,						S1 = Wuρ1P1,
R2 = WpP1P2,						S2 = − WuP1P2,
we then obtain
\|WuT¯L\|2 + \|WpS¯L\|2 ∞ = \|R1 − θR2\|2 + \|S1 − θS2\|2 ∞.
Up to this point, everything is	simple enough. Now we claim that there exists functions
Up to this point, everything is

U1, U2 RH+∞ and U3 R(s), deﬁned in terms of R1, R2, S1, and S2, and having the property that

	\|R1 − θR2\|2 + \|S1 − θS2\|2 ∞ = \|U1 − θU2\|2 + U3	∞.	(15.1)
That these	functions exist, and are as stated in Steps 1 and 8 of Algorithm 15.5, will be
That these

proved in the subsequent section. Finally, with U4 as deﬁned in Step 9, in the next section we shall show that

\|U1 − θU2\|2	+ U3 ∞ <	1	kU4−1U1 − θU4−1U2k∞ < 1.
		2
rough justiﬁcation behind us, let us turn to formal proofs of the validity of
With this

Algorithm 15.5. Readers not interested in this sort of detail can actually skip to Section 15.4.

15.1.3 Proof that reduction procedure works Throughout this section, we let RP ,

Wp, and Wp are as stated in Problem 15.2.

In Step 6 of Algorithm 15.5, we are asked to compute the spectral factorisation of R2R2 + S2S2 . Let us verify that this spectral factorisation exists.

15.6 Lemma R2R2 + S2S2 admits a spectral factorisation.

Proof We have

R2R2 + S2S2 = P1P1 P2P2 (WpWp + WuWu ).

Since P1, P2 RH+∞ we may ﬁnd an inner-outer factorisation

P1 = P1,inP1,out, P2 = P2,inP2,out

22/10/2004 15.1 Reduction of robust performance problem to model matching problem 563

by Proposition 14.10. Therefore

P1P1 = P1,inP1,outP1,inP1,out = P1,outP1,out,

since P1,in is inner. Since P1,out is outer, it follows that P1,out is a left spectral factor for P1P1 . Similarly P2P2 admits a spectral factorisation by the outer factor P2,out of P2. Thus P1P1 and P2P2 admit a spectral factorisation, and so too then does P1P1 P2P2 . Now let (Np, Dp) and (Nu, Du) be the c.f.r.’s of Wp and Wu. We then have

			N	N D	D	+ N	N D	D
W	W + W	W =	p	p u	u	u	u p	p	.
W	W + W	W =							.
p	p u	u		DpDpDuDu
				DpDpDuDu

Since Wp, Wu RH+∞ by hypothesis, Dp and Du, and therefore Dp and Du, have no imaginary axis roots. Also by assumption, Np and Nu have no common roots on iR. One can then show (along the lines of Exercise E14.2) that this infers that NpNp DuDu + NuNu DpDp has constant sign on iR. Since it is clearly even, one may infer from Proposition 14.12 that NpNp DuDu + NuNu DpDp admits a spectral factorisation. Therefore, by Proposition 14.17, so too does WpWp +WuWu . Finally, by Exercise E14.3 we conclude that R2R2 +S2S2 admits a spectral factorisation.

Our next result declares that U1, U2, and U3 are as they should be, meaning that they satisfy the relation (15.1).

15.7 Lemma If U1, U2, and U3 as deﬁned in Steps 8 and 1 satisfy								∞.
			\|R1 − θR2\|2 + \|S1 − θS2		\|2 ∞	= \|U1 − θU2\|2 + U3
Proof	It is su
			cient that the relation				U1 − θ U2 + U3 (15.2)
R1 − θR2		R1 − θ R2 + S1 − θS2		S1 − θ S2 = U1 − θU2

holds for all θ and for s = iω. Doing the manipulation shows that if the three relations

	R2R2 + S2S2				= U2U2		(15.3)
	1	2	1	2	1	2
R		R	+ S	S = U		U

R1R1 + S1S1 = U1U1 + U3.

hold for s = iω, then (15.2) will indeed hold for all θ. We let Q = [R2R2 + S2S2 ]+. If V is an inner function with the property that

R1R2 + S1S2 V

has no poles in C+, then if U2 = QV we have

U2U2 = QV Q V = QQ = R2R2 + S2S2 .

Thus U2 satisﬁes the ﬁrst of equations (15.3). Also,

	R	R	+ S	S
U1 =	1	2	1	2	V
U1 =		Q			V
		Q

564

15 An introduction to H∞ control theory

22/10/2004

clearly satisﬁes the second of equations (15.3). To verify the last of equations (15.3), we may directly compute

						(R R2	+ S S2)(R1R		+ S1S )
						1	1	2	2
R1R1	+ S1S1	− U1U1	= R1R1	+ S1S1	−					,	(15.4)
							R2R2 + S2S2

using our solution for U1 and the fact that V is inner. A straightforward substitution of the

deﬁnitions of R1, R2, S1, and S2 now gives U3 as in Step 1.
Our next lemma veriﬁes Step 2 of Algorithm 15.5.
15.8 Lemma If kU3k∞ ≥ 21 then Problem 15.2 has no solution.
Proof One may verify by direct computation that
			(R R2	+ S S2)(R1R		+ S1S )
		−	1	1	2	2
U3 = R1R1	+ S1S1
				R2R2 + S2S2

(this follows from (15.4)). Now work backwards through the proof of Lemma 15.7 to see that

U3 = R1

− θR2 R1

− θ R2

+ S1

− θS2 S1

− θ S2

− U1 − θU2

U1 − θ U2

for any admissible θ RH∞. Therefore, if kU3k∞ ≥

2 then

|R1 − θR2|2

+ |S1 − θS2|2 ∞ ≥

through of the deﬁnitions of R

, R

, S

, and S

shows that this

However, a simple working

implies that for any stabilising controller RC we must have

|WuT¯L|2

+ |WpS¯L|2 ∞ ≥

as desired.

Next we show that with T1 and T2 as deﬁned in Step 10, the modiﬁed robust performance problem is indeed equivalent to the model matching problem.

15.9 Lemma With T1 and T2

as deﬁned in Step 10 we have

|WuT¯L|2

+ |WpS¯L|2 ∞ <

kT1 − θT2k ,

where

ρ2 + θP1

RC =

ρ1 − θP2

Proof As outlined at the end Section 15.1.2, the condition

|WuT¯L|2 + |WpS¯L|2 ∞ <

is equivalent to

|U1 − θU2

|2 + U3

∞ < 2.

[P1P2 P2P2 ]−

P1 P2

22/10/2004

15.1 Reduction of robust performance problem to model matching problem 565

566

15 An introduction to H∞ control theory

22/10/2004

Also note that by deﬁnition, U

(iω)

≥

0 for all ω

, and that U

= U , the latter fact

−

W have poles on i

. Thus, our claim will follow if we can show that

implying that U3

is even. Therefore, provided that

k∞

2 −

admits a spectral

has no poles on i

. This is true since the imaginary axis zeros of P P

and

factorisation. Now we compute

[P1P2 P2P2 ]−

1 2

[P1P2 P2P2 ]− agree in location and multiplicity. Thus we have shown that U1 is analytic in

finish

+. That U1 RH∞+

will now follow if we can show that U1 is proper.

|U1 − θU2|

+ U3 ∞ <2

U1(iω)

θ(iω)U2(iω)

+ U3(iω)

15.2 Optimal model matching I. Nevanlinna-Pick theory

−

θ(iω)U2(iω)

The ﬁrst solution we shall give to the model matching problem comes from a seemingly

U1(iω)

+ U3(iω) <

−

unrelated interpolation problem. To state the problem, we need a little notation. We let

RH+,C be the collection of proper functions in C(s) with no poles in

+. Thus RH+,C is just

|U1(iω) −

θ(iω)U2(iω)|

− U3(iω), ω R

∞

|U1(iω) −

θ(iω)U2(iω)|

− U3(iω)]

− U3(−iω)],

ω R

like RH∞, except that now we allow the functions to have complex coe cients. Note that

< [2

[

k·k∞ still makes sense for functions in RH∞+,C. Now the interpolation problem is as follows.

|U1(iω) − θ(iω)U2(iω)|2 < |U4(iω)|2 , ω R

15.11 Nevanlinna-Pick interpolation problem Let {a1, . . . , ak} C+ and let {b1, . . . , bk} C

U−1(iω)U (iω)

−

θ(iω)U−1(iω)U (iω)

< 1,

that if (ak, bk) is a Pick pair, then so is (¯aj, ¯bj), j = 1, . . . , k. j j

{

}

collections of distinct points. A Pick pair is then a pair (a , b ), i

1, . . . , k

. Suppose

By deﬁnition of T1 and T2, the lemma follows.

Find R RH∞ so that

It is necessary that T1, T2 RH∞ in order to ﬁt them into the model matching problem.

(i)

k∞

≤

1 and

The next lemma ensures that this follows from the constructions we have made.

(ii)

R(aj) = bj, j = 1, . . . , n.

15.10 Lemma T1, T2 RH∞+ .

This problem was originally solved by Pick [1916], and was solved independently by

Proof 1First

note that

< 1

by the time we have gotten to deﬁning T

and T

. Therefore

Nevanlinna [1919]. Nevanlinna also gave an algorithm for ﬁnding a solution to the interpola-

tion problem [Nevanlinna 1929]. In this section, we will state and prove Pick’s necessary and

U4 = [2 − U3]

is strictly

proper, and so is invertible in RH

∞

. The lemma will then follow

su cient condition for the solution of the interpolation problem, and also give an algorithm

if we can show that U1, U2 RH∞.+

for determining a solution.

First let us show that U2 RH∞. By deﬁnition, U2 is the left half-plane spectral factor

15.12 Remark We should say that the problem solved by Pick is somewhat di erent than

P1P1 P2P2 (WpWp + WuWu ).

As such, it is the product of the two quantities

the one we state here. The di erence occurs in three ways.

1. Pick actually allowed kRk∞ = 1. However, our purposes will require that the H∞-norm

[P1P1 P2P2 ]+,

[WpWp + WuWu ]+.

of R be strictly less than 1.

Since each of P

RH+ , [P

P P

P ]+

RH+

. Since

2. Pick was not interested in making the restriction that if a every Pick pair have the

property that its complex conjugate also be a Pick pair.

∞

+ N

N D

Pick was interested in the case where the points a1, . . . , an lie in the open disk D(0, 1)

WpW + WuW =

7→1+s

DpD DuD

of radius 1 centred at 0. This is not a genuine distinction, however, as the map s

1−s

we have

bijectively maps C+ onto D(0, 1), and so translates the domain of concern for Pick to

N D

+ N

D ]+

our domain.

+ W

]+ =

Finally,

Pick allowed interpolating functions to be general meromorphic functions,

DpDu

bounded and analytic in C+. For obvious reasons, our interest is in the subset of such

Since W

, W

RH+

, it follows that [W

W + W

RH+

. Thus U

RH+ .

∞

functions that are in R(s), i.e., functions in RH∞.

Now let us show that U1 RH∞. A computation shows that

ρ2P2WpWp − ρ1P1WuWu U1 = [WpWp + WuWu ]− V.

The inner function V is designed so that this function has no poles in C+. We also claim that U1 has no poles on iR. Since Wp, Wu RH+∞ and since they have no common imaginary axis zeros, it follows that [WpWp + WuWu ]− has no zeros on iR. Clearly, neither P1 P2 nor

15.2.1 Pick’s theorem Pick’s conditions for the existence of a solution to Problem 15.11 are quite simple. The proof that these conditions are necessary and su cient is not entirely straightforward. In this section we only prove necessity, as su ciency will follow from our algorithm for solving the Nevanlinna-Pick interpolation problem in the ensuing section. Our necessity proof follows [Doyle, Francis, and Tannenbaum 1990]. For the statement of Pick’s theorem, recall that M Cn×n is Hermitian if M = M = Mt.

22/10/2004

15.2 Optimal model matching I. Nevanlinna-Pick theory

567

A Hermitian matrix is readily veriﬁed to have real eigenvalues. Therefore, the notions of deﬁniteness presented in Section 5.4.1 may be applied to Hermitian matrices.

15.13 Theorem (Pick’s Theorem) Problem 15.11 has a solution if and only if the Pick matrix, the complex k × k symmetric matrix M with components

	¯
Mj` =	1 − bjb`	, j, ` = 1, . . . , k,
	aj + a¯`

is positive-semideﬁnite.

Proof of necessity Suppose that Problem 15.11 has a solution R. For c1, . . . , ck C, not all zero, consider the complex input u: (−∞, 0] → C given by

	k
	Xj
u(t) =	cjeaj t.
	=1
This can be considered as an input to the transfer function R, with the output computed
by separately computing the real and imaginary parts. Moreover, if hR denotes the inverse
Laplace transform for R, the complex output will be		Check

Z ∞

y(t) = hR(τ)u(t − τ) dτ

kZ ∞

= cj hR(τ)eaj (t−τ) dτ

j=1 0

kZ ∞

	Xj	hR(τ)e−aj τ dt
=	cjeaj t	hR(τ)e−aj τ dt
	=1	0

=cjeaj tR(aj)

j=1

=cjbjeaj t,

j=1

where we have used the deﬁnition of the Laplace transform. By part (i) of Theorem 5.21, and since kRk∞ ≤ 1, it follows that

	0		0
	Z−∞ \|y(t)\|2 dt ≤ Z−∞ \|u(t)\|2 dt.
Substituting the current deﬁnitions of u and y gives
Z 0	k cjbjeaj t	2 dt ≤ Z	0	k	cjeaj t	2 dt
	X			X

−∞

Z 0

−∞

Z 0

−∞

j=1

−∞ j=1

kZ 0 k

X			a¯`t		−∞	X	¯	a¯`t
cje	aj t	c¯`e	a¯`t	dt ≥		aj t	¯	a¯`t	dt
cje		c¯`e		dt ≥		cjbje	c¯`b`e		dt
j,`=1						j,`=1
k
X				¯	(aj +¯a`)t	dt ≥ 0.
cjc¯`(1 − bjb`)e						dt ≥ 0.

j,`=1

568	15 An introduction to H∞ control theory								22/10/2004
We now compute	0
	0						1
	Z−∞ e(aj +¯a`)t dt =						1	,
	Z−∞ e(aj +¯a`)t dt =				aj + a¯`			,
thus giving
	k		¯
	c		1 − bjb`	c¯		≥	0,
	j,`=1	j	aj + a¯` `			≥
	X
which we recognise as being equivalent to the expression
	x Mx ≥ 0,
where	x = c¯...1 .
	x = c¯...1 .
			c¯n

Thus we have shown that the Pick matrix is positive-deﬁnite.
15.2.2 An inductive algorithm for solving the interpolation problem									In this section

we provide a simple algorithm for solving the Nevanlinna-Pick interpolation problem in the situation when the Pick matrix is positive-deﬁnite. In doing so, we also complete the proof of Theorem 15.13. The algorithm we present in this section follows [Marshall 1975].

Before we state the algorithm, we need to introduce some notation. First note that by the Maximum Modulus Principle, it is necessary that for the Nevanlinna-Pick interpolation problem to have a solution, each of the numbers b1, . . . , bk satisfy |bj| ≤ 1. Thus we may assume this to be the case when we seek a solution to the problem. For b C satisfying |b| < 1, deﬁne the Blaschke function Bb R(s) associated with b by

B (s) = 1s2−¯bs,			2			b R
	b
b	s−+	2Re(b)s + b
	\|		\|		,	otherwise.
	2				,	otherwise.
	2		s	2
	1 + 2Re(b)s + \|b\|		s

Some easily veriﬁed relevant properties of Blaschke functions are the subject of Exercise E15.4. Also, for a C deﬁne a function Aa R(s) by

s − a

(s) =

s2+ a¯

2Re(a)s + a

−

otherwise.

+ 2Re(a)s + |a|

Again, we refer to Exercise E15.4 for some of the easily proven properties of such functions. Let us begin by solving the Nevanlinna-Pick interpolation problem when k = 1.

15.14 Lemma Let a1 C+ and let b1 C have the property that |b1| < 1. The associated Nevanlinna-Pick interpolation problem has an inﬁnite number of solutions if it has one solution, and the set of all solutions is given by

Re(R)		R(s) = B−b1	R1(s)Aa1 (s) , R1 C(s) has no poles in	C	+, and kR1k∞ < 1 .

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15.2 Optimal model matching I. Nevanlinna-Pick theory

569

Proof First note that the Nevanlinna-Pick interpolation problem does indeed have a solution, namely the trivial solution R0(s) = b1.

Now let R1 C(s) have no poles in C+ and suppose that kR1k∞ < 1. If

R(s) = B−b1 R1(s)Aa1 (s)

then R is the composition of the functions

s7→R1(s)Aa1 (s)

s7→M−b1 (s).

The ﬁrst of these functions in analytic in C+ since both R1 and Aa1 are. Also, by Exer-

k k ¯

cise E15.4 and since R1 < 1, the ﬁrst of these functions maps C+ onto the disk D(0, 1).

The second of these functions, by Exercise E15.4, is analytic in D(0, 1) and maps it onto itself. Thus we can conclude that R C(s) as deﬁned has no poles in C+ and that kRk∞ < 1. What’s more, we claim that R(a1) = b1 if R1(a1) = b1. Indeed

R(a1) = B−b1 R1(a1)Aa1 (a1) = B−b1 (0) = b1,

using the deﬁnitions of Aa1 and Bb1 . It only remains to show that Re(R) solves Problem 15.11. Clearly, since b1 must be real, it follows that Re(R(a1)) = b1. Furthermore, since

kRe(R)k∞ < kRk∞, it follows that kRe(R)k∞ < 1. Finally, Re(R) can have no poles in C+

since R has no poles in C+.

Now suppose that R R(s) solves Problem 15.11. Deﬁne R1 C(s) by

R1(s) = Bb1 (R(s)).

Aa1 (s)

The function in the numerator is analytic in C+ and has a zero at s = a1. Therefore, since

the only zero of Aa1 is at zero, R1 is analytic in C+. Furthermore, the H∞-norm of the numerator is strictly bounded by 1, and since the H∞-norm of the denominator equals 1, we conclude that kR1k∞ < 1. This concludes the proof of the lemma.

As stated in the proof of the lemma, if |b1| < 1, then the one point interpolation problem always has the trivial solution R0(s) = b1. Let us also do this in the case when k = 2 and

¯
we have a2 = a¯1 6= a1 and b2 = b2 6= b2.
¯	} C have the property that \|b1	\| <
15.15 Lemma Let {a1, a2 = a¯1} C+ and let {b1, b2 = b1	} C have the property that \|b1	\| <
1. Also suppose that a1 6= a2 and b1 6= b2. The associated Nevanlinna-Pick interpolation

problem has an inﬁnite number of solutions if it has one solution, and the set of all solutions is given by


Re(R)	R(s) = B−b1		R1(s)Aa1 (s) , R1 C(s) has no poles in	C	+, and kR1k∞ < 1 .
Proof Problem		15.11 has the solution

Rs(s) =

570

15 An introduction to H∞ control theory

22/10/2004

The lemmas gives the form of all solutions to the Nevanlinna-Pick interpolation problem in the cases when k = 1 and k = 2 with the points being complex conjugates of one another. It turns out that with this case, one can construct solutions to the general problem. To do this, one makes the clever observation (this was Nevanlinna’s contribution) that one can reduce a k point interpolation problem to a k − 1 or k − 2 point interpolation problem by properly deﬁning the new k − 1 or k − 2 points. We say how this is done in a deﬁnition.

15.16 Deﬁnition For k > 1, let

{a1, . . . , ak}, {b1, . . . , bk} C

be as in Problem 15.11. Deﬁne
k˜ =	k	−	1,	bk R
	(k	−	1,	otherwise.
		−

The Nevanlinna reduction of the numbers {a1, . . . , ak} and {b1, . . . , bk} is the collection

of numbers
˜	˜
{a˜1, . . . , a˜k˜}, {b1	, . . . , bk˜} C
deﬁned by
With this in mind, we state the algorithm that forms the main result of this section.

15.17 Algorithm for solving the Nevanlinna-Pick interpolation problem Given points

{a1, . . . , ak}, {b1, . . . , bk} C

as in Problem 15.11, additionally assume that |bj| < 1, j = 1, . . . , k, and that the Pick

matrix is positive-deﬁnite.
1.
15.2.3 Relationship to the model matching problem	The above discussion of the

Nevanlinna-Pick interpolation problem is not obviously related to the model matching problem, Problem 15.3.

15.18 Model matching by Nevanlinna-Pick interpolation Given T1, T2 RH+∞.

15.3 Optimal model matching II. Nehari’s Theorem

In this section we present another method for obtaining a solution, or an approximate solution, to the model matching problem, Problem 15.3. The strategy in this section involves signiﬁcantly more development than does the Nevanlinna-Pick procedure from Section 15.2. However, the algorithm produced in actually easier to apply than is that using NevanlinnaPick theory. Unfortunately, the methods in this section su er from on occasion producing a controller that is improper, and one must devise hacks to get around this, just as with Nevanlinna-Pick theory. Our presentation in this section follows that of Francis [1987].

22/10/2004	15.3 Optimal model matching II. Nehari’s	Theorem	571
15.3.1	Hankel operators in the frequency-domain The	key tool for the methods

of this section is something new for us: a Hankel operator of a certain type. To initiate this discussion, let us note that as in Proposition 14.9, but restriction to functions in RL2, we have a decomposition RL2 = RH−2 RH+2 . That is to say, any strictly proper rational function with no poles on iR has a unique expression as a sum of a strictly proper rational function with no poles in C+ and a strictly proper rational function with no poles in C−. This is no surprise as this decomposition is simply obtained by partial fraction expansion. The essential idea of this section puts this mundane idea to good use. Let us denote by Π+ : RL2 → RH+2 and Π− : RL2 → RH−2 the projections.

Let us list some operators that are readily veriﬁed to have the stated properties.

1.The Laurent operator with symbol R: Given R RL∞ and Q RL2, one readily sees that RQ RL2. Thus, given R RL∞ we have a map ΛR : RL2 → RL2 deﬁned by ΛR(Q) = RQ. This is the Laurent operator with symbol R.

2.The Toeplitz operator with symbol R: Clearly, if R RL∞ and if Q RH+2 , then

RQ RL2. Therefore, Π+(RQ) RH+2 . Thus, for R RL∞, the map ΘR : RH+2 → RH+2 deﬁned by ΘR(Q) = Π+(RQ) is well-deﬁned, and is called the Toeplitz operator with symbol R.

3.The Hankel operator with symbol R: Here again, if R RL∞ and if Q RH−2 , then RQ RL2. Now we map this rational function into RH+2 using the projection Π−. Thus, for R RL∞, we deﬁne a map R : RH+2 → RH−2 by R(Q) = Π−(RQ). This is the Hankel operator with symbol R.

15.19 Remarks 1. Note that the Toeplitz and Hankel operators together specify the value of the Laurent operator when applied to functions in RH+2 . That is to say, if R RL2 then

ΛR(Q) = ΘR(Q) + R(Q).

2.It is more common to see the Laurent, Toeplitz, and Hankel operators deﬁned for general analytic functions rather than just rational functions. However, since our interest is

entirely in the rational case, it is to this is that we restrict our interest.

The Laurent, Toeplitz, and Hankel operators are linear. Thus it makes sense to ask questions about the nature of their spectrum. However, the spaces RL2, RH−2 , and RH+2 are inﬁnite-dimensional, so these issues are not immediately approachable as they are in ﬁnite-dimensions. The good news, however, is that these operators are “essentially” ﬁnitedimensional. The easiest way to make sense of this is with state-space techniques, and this is done in the next section.

It also turns out that the Laurent, Toeplitz, and Hankel operators are deﬁned on spaces with an inner product. Indeed, on RL2 we may deﬁne an inner product by

	1	∞
hR1, R2i2 =		Z−∞ R1(iω)R2(iω) dω.	(15.5)
hR1, R2i2 =	2π	Z−∞ R1(iω)R2(iω) dω.	(15.5)

Note that this is an inner product on a real vector space. This inner product may clearly be applied to any functions in RL2, including those in the subspaces RH−2 and RH+2 . Indeed, RH−2 and RH+2 are orthogonal with respect to this inner product (see Exercise E15.5). One may deﬁne the adjoint of any of our operators with respect to this inner product. The adjoint of the Laurent operator with symbol R is the map ΛR : RL2 → RL2 deﬁned by the

relation

hΛR(R1), R2i2 = hR1, ΛR(R2)i2

572 15 An introduction to H∞ control theory 22/10/2004

for R1, R2 RL2. In like fashion, the Toeplitz operator has an adjoint ΘR : RH+2 → RH+2

deﬁned by

hΘR(R1), R2i2 = hR1, ΘR(R2)i2 , R1, R2 RH+2 , and the Hankel operator has an adjoint R : RH−2 → RH+2 deﬁned by

h R(R1), R2i2 = hR1, R(R2)i2 , R1 RH+2 , R2 RH−2 .

The following result gives explicit formulae for the adjoints.

15.20 Proposition For R RL2 the following statements hold:

(i)ΛR = ΛR ;

(ii)ΘR = ΘR ;

(iii)R(Q) = Π+(ΛR (Q)).

Proof (i) We compute
	1	∞
hΛR(R1), R2i2 =		Z−∞ ΛR(R1)(iω)R2(iω) dω
hΛR(R1), R2i2 =	2π	Z−∞ ΛR(R1)(iω)R2(iω) dω
	1	∞
=		Z−∞ R(iω)R1(iω)R2(iω) dω
=	2π	Z−∞ R(iω)R1(iω)R2(iω) dω
	1	∞
=		Z−∞ R1(iω)R(−iω)R2(iω) dω
=	2π	Z−∞ R1(iω)R(−iω)R2(iω) dω
	1	∞
	1	∞
=		Z−∞ R1(iω)R (iω)R2(iω) dω
=	2π	Z−∞ R1(iω)R (iω)R2(iω) dω

= hR1, ΛR (R2)i2 .

This then gives ΛR = ΛR as desired. (ii) For R1, R2 RH+2 we compute

hΘR(R1), R2i2 = hΛR(R1), R2i2

=hR1, ΛR(R2)i2

=hR1, ΛR (R2)i2

=hR1, ΘR (R2)i2 ,

and this part of the proposition follows.

(iii) For R1 RH+2 and R2 RH−2 we compute

h R(R1), R2i2 = hΛR(R1), R2i2

=hR1, ΛR(R2)i2

=hR1, ΛR (R2)i2 ,

and from this the result follows.

In the next section, we will come up with concrete realisations of the Hankel operator and its adjoint using time-domain methods.

22/10/2004	15.3 Optimal model matching II. Nehari’s Theorem	573
15.3.2	Hankel operators in the time-domain The above operators deﬁned in the ra-

tional function domain are simple enough, but they have interesting and nontrivial counter-

¯	(−∞, ∞) those functions
parts in the time-domain. To simplify matters, let us denote by L2

of time that, when Laplace transformed, give functions in RL2. As we saw in Section E.3, this consists exactly of sums of products of polynomial functions, trigonometric functions,

		¯	¯	(−∞, ∞)
and exponential functions of time. Let us denote by L2(−∞, 0] the subset of L2				(−∞, ∞)
		¯	¯	(−∞, ∞)
consisting of functions that are bounded for t < 0, and by L2			[0, ∞) the subset of L2	(−∞, ∞)
consisting of functions that are bounded for t > 0. Note that
¯	¯	¯
L2	(−∞, ∞) = L2(−∞, 0]	L2[0, ∞).

¯	(−∞, ∞) can be uniquely decomposed into a sum of two func-
That is, every function in L2

tions, one that is bounded for t < 0 and one that is bounded for t > 0. It is clear that this decomposition corresponds exactly to the decomposition RL2 = RH−2 RH+2 that uses the partial fraction expansion. Let us also deﬁne projections

	¯ +	¯	¯	[0, ∞)
	Π	: L2	(−∞, ∞) → L2	[0, ∞)
	¯ −	¯	¯	(−∞, 0].
	Π	: L2	(−∞, ∞) → L2	(−∞, 0].
If we employ the inner product
			∞
	hf1, f2i2 = Z−∞ f1(t)f2(t) dt				(15.6)
¯	¯		¯
on L2	(−∞, ∞), then obviously L2	(−∞, 0] and L2[0, ∞) are orthogonal. We hope that it will

be clear from context what we mean when we use the symbol h·, ·i2 in two di erent ways, one for the frequency-domain, and the other for the time-domain.

The following result summarises the previous discussion.

15.21 Proposition The Laplace transform is a bijection

(−∞, ∞) to RL2,

(i) from L2

−

, and

(ii) from L2

(−∞, 0] to RH2

(iii) from L2

[0, ∞) to RH2 .

Furthermore, the diagrams

¯ +

¯ −

/¯

(−∞,

∞)

/¯

(−∞, 0]

L2(−∞, ∞)

L2[0, ∞)

RL2

/ +

RL2

/ −

Π+

RL2

Π+

RL2

commute.

We now turn our attention to describing how the operators of Section 15.3.1 appear in the time-domain, given the correspondence of Proposition 15.21. Our interest is particularly in the Hankel operator. Given Proposition 15.21 we expect the analogue of the frequency

¯	∞	¯	−∞
domain Hankel operator to map L2[0,				RL∞. To do this, given
		) to L2( , 0], given R

R RL∞, let us write R = R1 + R2 for R1 RH−2 and R2 RH+∞ as in Proposition 14.9. We then let Σ1 = (A, b, ct, 01) be the complete SISO linear system in controller canonical form with the property that TΣ1 = R1. Therefore, the inverse Laplace transform for R1 is

574	15 An introduction to H∞ control theory									22/10/2004
									¯
the impulse response for Σ1. Note that σ(A) C+. Thus if r1 L2(−∞, 0] is the inverse
Laplace transform of R1 we have					(0,
		1			(0,		t > 0,
		r		(t) = −cteAtb, t ≤ 0
								¯	[0, ∞) and for t ≤ 0, let us
which is the anticausal impulse response for R1. Now, for u L2									[0, ∞) and for t ≤ 0, let us
deﬁne	¯R(u)(t) = Z0					∞ r1(t − τ)u(τ) dτ.
	¯R(u)(t) = Z0					∞ r1(t − τ)u(τ) dτ.				(15.7)
¯							¯
We take R(u)(t) = 0 for t > 0. We claim that R is the time-domain version of the Hankel
operator R. Let us ﬁrst prove that its takes its values in the right space.
¯	¯
15.22 Lemma R(u) L2(−∞, 0].
Proof For t ≥ 0 we have
	¯R(u)(t) = −cteAt Z0 ∞ e−Aτ bu(τ) dτ.
	in	C+	,	−	A has all eigenvalues in			C−	, so the integral converges.
Since A has all eigenvalues At		C+		−				C−	¯	¯
Also, for the same reason, e		is bounded for t ≤ 0. This shows that R(u) L2(−∞, 0], as
claimed.

Now let us show that this is indeed “the same” as the frequency domain Hankel operator.

15.23 Proposition

Let R RL∞ and write R = R1 + R2

with R1 RH2− and R2 RH∞+ .

Let r1 L2(−∞, 0] be the inverse Laplace transform of R1 and deﬁne R as in (15.7). Then

the diagram

/¯

[0, ∞)

L2(−∞, 0]

−

RL2

commutes.

Proof Let u L2

[0, ∞) and denote y = R(u) L(−∞, 0]. Denote as usual the Laplace

transforms of u and y by uˆ and yˆ. We then have

R(ˆu) = Π−((R1 + R2)ˆu).

Note that R

uˆ

RH+. Therefore, Π−((R

+ R

)ˆu) = Π−(R

uˆ). Now let us compute the

˜ ˜

inverse Laplace transform of R1uˆ. Let Σ = (A, b, c˜ , 01) be a complete SISO linear system

deﬁned so that T

. Now we compute

= uˆ. Thus A has all eigenvalues in C

−

∞

L −1(R1uˆ)(t) =

Z−∞ r1(t − τ)u(τ) dτ

Z ∞

= − cteAt 1(τ − t)e−Aτ bu(τ) dτ

−∞

Z ∞

= − cteAt 1(τ − t)e−Aτ bu(τ) dτ,

22/10/2004 15.3 Optimal model matching II. Nehari’s Theorem 575

¯− −1

since for τ < 0, u(τ) = 0. Now note that Π (L (R1uˆ)) is nonzero only for t < 0 so that we can write

(Π¯−(L −1(R1uˆ)))(t) = −cteAt Z0	∞ eAτ bu(τ) dτ.
Thus Π¯−(L −1(R1uˆ)) = ¯R(ˆu). By Proposition 15.21 this means that
L −1(Π−(R1uˆ)) = L −1( R(ˆu)) = ¯R(u),
¯
or, equivalently, that R ◦ L = L ◦ R, as claimed.

Up to this point, the value of the time-domain formulation of a Hankel operator is not at all clear. The simple act of multiplication and partial fraction expansion in the frequency-domain becomes a little more abstract in the time-domain. However, the value of the time-domain formulation is in its presenting a concrete representation of the Hankel operator and its adjoint. To come up with this representation, we introduce some machinery harking back to our observability and controllability discussion in Sections 2.3.1 and 2.3.2. In particular, we begin to dig into the proof of Theorem 2.21. We resume with the situation

where Rt = R1 + R2	RL∞ with R1 RH2− and R2 RH∞+ . As above, we let Σ1 =
(A, b, c , 01) be the canonical minimal realisation of R1. With this notation, we deﬁne a
¯		n	by
map CR : L2[0, ∞) → R			by
			CR(u) = − Z0 ∞ e−Aτ bu(τ) dτ,
and we call this the controllability operator. Similarly, we deﬁne OR : R							n	¯
								→ L2(−∞, 0]
by				OR	(0,	t < 0,
			(		(x))(t) = cteAtx,	t ≥ 0

and we call this the observability operator. Note that the adjoint of the controllability

n → ¯ ∞

operator will be the map R : R L2[0, ) satisfying

	,	¯	n	,
hCR(u), xi = hu, CR(x)i2	,	u L2	[0, ∞), x R	,

¯ −∞ → n

and the adjoint of the observability operator will be the map OR : L2( , 0] R satisfying

	¯		n	.
hOR(x), yi2 = hx, OR(y)i ,	y L2	(−∞, 0], x R		.
		¯		¯	[0, ∞).
In each case, the inner product of equation (15.6) is being used on L2			(−∞, 0] and L2		[0, ∞).

The following result summarises the value of introducing this notation.

15.24 Proposition With the above notation, the following statements hold:

(i)

the diagrams

CR vv

JJ OR

HH CR

/ ¯

(−∞, 0]

/ ¯

[0, ∞)

L2[0, ∞)

(−∞, 0]

commute;

(0,

t > 0;

(ii)

(

(x))(τ) =

−bte−Atτ x,

t ≤ 0

576		15 An introduction to H∞ control theory		22/10/2004
	0
(iii)	OR(y) = Z−∞ eAttcy(t) dt;		τ > 0;
	R	(0, R	τ > 0;
(iv)	(¯ (y))(τ) =	− 0∞ bteAt(t−τ)cy(t) dt,	τ ≤ 0
(v)	CR is injective;

Proof (i) It su ces to show that the left diagram commutes, since if it does, the right diagram will also commute by the deﬁnition of the adjoint. However, the left diagram may

be easily seen to commute by virtue of the very deﬁnitions of CR, OR, and R.

(ii) This follows from the deﬁnition of CR and the inner product in equation (15.6). (iii) This follows from the deﬁnition of OR and the inner product in equation (15.6).

(iv) This follows from the right diagram in part (i), along with parts (ii) and (ii).

(v) It su ces to show that CR(x) = 0 if and only if x = 0. If (CR(x))(τ) = −bte−Atτ x = 0 for all τ then successive di erentiation with respect to τ and evaluation at τ = 0 gives

−btx = 0, btAtx = 0, . . . , (−1)nbt(At)n−1x = 0.

Since (A, b) is controllable, this implies that x = 0.

Thus the above result gives a simple way of relating a Hankel operator and its adjoint to operators with either a domain or a range that is ﬁnite-dimensional. In the next section, we shall put this to good use.

15.3.3 Hankel singular values and Schmidt pairs Recall that a singular value for a linear map A between inner product spaces is, by deﬁnition, an eigenvalue of A A. One readily veriﬁes that singular values are real and nonnegative. Our objective in this section

is to ﬁnd the singular values of a Hankel operator R using our representation R in the time-domain.

As in the previous section, for R RL∞ we write R = R1 + R2 with R1 RH+2 and R2 RH−∞. We let Σ1 = (A, b, ct, 01) be the complete SISO linear system in controller canonical form so that TΣ1 = R1. We next introduce the controllability Gramian

Z ∞

CR = e−Atbbte−Att dt,

and the observability Gramian

Z ∞

OR = e−Attccte−At dt.

These are each elements of Rn×n. We have previously encountered the controllability Gramian in the proof of Theorem 2.21, and the observability Gramian may be used in a similar manner. However, here we are interested in their relationship with the Hankel operator R. The following result gives this relationship, as well as providing a characterisation of CR and OR in terms of the Liapunov ideas of Section 5.4.

15.25 Proposition With the above notation, the following statements hold:

(i)(At, CR, −bbt) is a Liapunov triple;

(ii)(A, OR, −cct) is a Liapunov triple;

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