A Case Study for the Delay-type Nehari Problem

Qing-Chang Zhong

Abstract— In this paper, a case study is given for the delay-type Nehari problem so that it can be better understood.

Index Terms— Delay-type Nehari problem, algebraic Riccatiequation, H∞ control, Smith predictor

I. INTRODUCTION

The H∞ control of processes with delay(s) has been anactive research area since the mid 80’s. It is well knownthat the Nehari problem [1] played an important role inthe development of H∞ control theory. This is still truein the case for systems with delay(s) [2]. Some papers,e.g. [3], [4], were devoted to calculate the infimum of thedelay-type Nehari problem in the stable case. It was shownin [3] that this problem in the stable case is equivalentto calculating an L2[0, h]-induced norm. For the unstablecase, Tadmor [5] presented a state-space solution using thedifferential/algebraic matrix Riccati equation-based methodand Zhong [6] proposed a frequency-domain solution usingthe J-spectral factorization. The solvability condition of thedelay-type Nehari problem is formulated in terms of the non-singularity of a delay-dependent matrix in [6]. The optimalvalue γopt is the maximal γ ∈ [0, ∞) such that this matrixbecomes singular when γ decreases from ∞. All sub-optimalcompensators are parameterized in a transparent structureincorporating a modified Smith predictor. The solution ismathematically elegant. However, it seems not trivial to findthe solution for a given system, even for a simple system.In order to better understand this problem, a case study fora first-order system is given here.

II. SUMMARY OF THE THEORETICAL RESULTS

The Delay-type Nehari Problem(NPh) is described asfollows. Given a minimal state-space realization Gβ(s)

.=[

A B−C 0

]= −C(sI − A)−1B, which is not necessarily

stable and h ≥ 0, characterize the optimal value

γopt = inf{∥∥Gβ(s) + e−shK(s)

∥∥L∞

: K(s) ∈ H∞} (1)

This work was supported by the EPSRC under Grant No. EP/C005953/1.Q.-C. Zhong is with the Department of Electrical Engineering and

Electronics, The University of Liverpool, Brownlow Hill, Liverpool L693GJ, UK. Email: zhongqc@ieee.org, URL: http://come.to/zhongqc.

and for a given γ > γopt, parameterize the suboptimal set ofproper K(s) ∈ H∞ such that∥∥Gβ(s) + e−shK(s)

∥∥L∞

< γ. (2)

The result about this problem is summarized as follows. See[6], [2] for more details.

Assume that Gβ has neither jω-axis zeros nor jω-axispoles. Define the following two Hamiltonian matrices:

Hc =

[A γ−2BB∗

0 −A∗

], Ho =

[A 0

−C∗C −A∗

]and

H =

[A γ−2BB∗

−C∗C −A∗

],

Σ =

[Σ11 Σ12

Σ21 Σ22

].= Σ(h) = eHh, (3)

The algebraic Riccati equations (ARE)

[−Lc I

]Hc

[ILc

]= 0 (4)

and [I −Lo

]Ho

[Lo

I

]= 0 (5)

always have unique solutions Lc ≤ 0 and Lo ≤ 0 such that

A+γ−2BB∗Lc =[

I 0]Hc

[ILc

]and A+LoC

∗C =

[I −Lo

]Ho

[I0

]are stable, respectively.

Then, the optimal value γopt is given by

γopt = max{γ : det Σ̂22 = 0},

where

Σ̂22 =[−Lc I

]Σ

[Lo

I

]. (6)

Furthermore, for a given γ > γopt, the suboptimal set ofproper K ∈ H∞ such that (2) holds is given by

K = Hr(

[I 0Z I

]W−1, Q) (7)

where ‖Q(s)‖H∞ < γ is a free parameter and

Z = −πh{Fu(

[Gβ II 0

], γ−2G∼

β )}, (8)

Proceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005

MoA12.1

0-7803-9568-9/05/$20.00 ©2005 IEEE 386

W−1

=

[A + γ−2BB∗Lc Σ̂

−∗

22(Σ∗

12+ LoΣ∗

11)C∗

−Σ̂−∗

22B

−C I 0

γ−2B∗(Σ∗

21− Σ∗

11Lc) 0 I

].

(9)

Here, πh, called the completion operator, is defined as

πh(G).=

[A B

Ce−Ah 0

]− e−sh

[A BC D

](10)

for G =

[A BC D

]= D + C(sI − A)−1B.

It is easy to see that γopt satisfies∥∥ΓGβ

∥∥ ≤ γopt ≤ ‖Gβ(s)‖L∞

(11)

because, on one hand, the delay-type Nehari problem issolvable [7] iff

γ > γopt.=

∥∥ΓeshGβ

∥∥ , (12)

where Γ denotes the Hankel operator and, on the other hand,it can be seen from (1) that

γopt ≤ ‖Gβ(s)‖L∞

(at least K can be chosen as 0). The symbol eshGβ in (12)is non-causal and, possibly, unstable. Hence, γopt ≥

∥∥ΓGβ

∥∥.

III. A CASE STUDY

ConsiderGβ(s) = −

1

s − a

with a minimal realization

Gβ =

[a 1−1 0

],

i.e., A = a, B = 1 and C = 1. According to the assumptions,a �= 0. This gives

Hc =

[a γ−2

0 −a

], Ho =

[a 0−1 −a

]

and

H =

[a γ−2

−1 −a

].

The ARE (4) and (5) can be re-written as

L2

cγ−2 + 2aLc = 0 and L2

o + 2aLo = 0.

When a < 0, i.e., Gβ is stable, the stabilizing solutions are

Lc = 0 and Lo = 0;

When a > 0, i.e., Gβ is unstable, these are1

Lc = −2aγ2 and Lo = −2a.

1In this case, A + γ−2BB∗Lc = a − 2a = −a and A + LoC∗C =

a − 2a = −a are stable.

It is easy to find that the eigenvalues of H are λ1,2 = ±λ

with λ =√

a2 − γ−2. Note that λ may be an imaginarynumber. Assume that γ �= 1/ |a| temporarily. With twosimilarity transformations with

S1 =

[1 −a + λ0 1

]and S2 =

[1 0

− 1

2λ1

],

H can be transformed into

[λ 00 −λ

], i.e.,

S−1

2S−1

1HS1S2 =

[λ 00 −λ

].

Hence, the Σ-matrix defined in (3) is

Σ = eHh

= S1S2

[eλh 00 e−λh

]S−1

2S−1

1

= S1

[eλh 0

− 1

2λ(eλh − e−λh) e−λh

]S−1

1

=[

eλh +a−λ2λ

(eλh− e−λh)

a2−λ2

2λ(eλh

− e−λh)

−12λ

(eλh− e−λh) e−λh

−a−λ2λ

(eλh− e−λh)

].

When γ = 1/ |a|, i.e., λ = 0, the above Σ still holds if thelimit for λ → 0 is taken on the right-hand side, which gives

Σ|λ=0=

[1 + ah a2h−h 1 − ah

].

In the sequel, Σ is assumed to be defined as above for λ = 0.

A. The stable case (a < 0)

In this case, Gβ is stable and Lc = Lo = 0. Hence, Σ̂22

defined in (6) is

Σ̂22 = Σ22 = e−λh −a − λ

2λ(eλh − e−λh).

When γ ≥ 1/ |a|, the number λ is positive and hence theeigenvalues of H are real. It is easy to see that Σ̂22 is alwayspositive (nonsingular)2. According to Section II, there is

γopt < 1/ |a| .

When 0 < γ < 1/ |a|, the number λ = ωi, where ω =√γ−2 − a2, is an imaginary and hence the eigenvalues of H

are imaginaries. However, Σ̂22 is still a real number because

Σ̂22 = e−ωhi −a − ωi

2ωi(eωhi − e−ωhi)

= cos(ωh) −a

ωsin(ωh).

Substitute ω =√

γ−2 − a2 into it, then

2Σ̂22 = 1 − ah when γ = 1/ |a| = −1/a.

387

Σ̂22

ah

aγ

Fig. 1. The surface of Σ̂22 with respect to ah and aγ (a < 0)

−10−9−8−7−6−5−4−3−2−10

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

ah

aγ

aγopt

Fig. 2. The contour Σ̂22 = 0 on the ah-aγ plane (a < 0)

Σ̂22 = cos(ah

aγ

√1 − a2γ2)−

aγ√1 − a2γ2

sin(ah

aγ

√1 − a2γ2).

This can be shown as the surface in Fig. 1, with respect to thenormalized delay ah and the normalized performance indexaγ. This surface crosses the plane Σ̂22 = 0 many times, ascan be seen from the contours of Σ̂22 = 0 shown in Fig. 2.The top curve in Fig. 2 characterizes the normalized optimalperformance index aγopt with respect to the normalizeddelay ah. On this curve, Σ̂22 becomes singular the first timewhen γ decreases from +∞ (or actually, ‖Gβ‖L∞

) to 0.Since

∥∥ΓGβ

∥∥ = 0 and ‖Gβ‖L∞

= 1/ |a|, the optimal valueγopt satisfies 0 ≤ γopt ≤ 1/ |a|, i.e., −1 ≤ aγopt ≤ 0. Thiscoincides with the curve aγopt shown in Fig. 2.

Now discuss the (sub-optimal) compensator for γ > γopt.

Re|a|−|a|

Im

0

γ=+∞γ=1/|a|

(γ 0)

Fig. 3. The locus of the hidden poles of Z v.s. γ

According to (8) and (10), there is

Z(s) = −πh(

⎡⎣ a γ−2 0

−1 −a 10 γ−2 0

⎤⎦)

= γ−2γ−2Σ∗

21+ (e−sh − Σ∗

11)(s − a)

s2 + γ−2 − a2. (13)

The locus of the hidden poles of Z(s), i.e. the eigenvalues ofH , is shown in Fig. 3. When γ ≥ 1/ |a|, Z(s) has two hiddenreal poles symmetric to the jω-axis. When 0 < γ < 1/ |a|,Z(s) has a pair of hidden imaginary poles. In either case,the implementation of Z needs to be careful; see [8].

The W−1 given in (9) is well defined for γ > γopt as

W−1 =

⎡⎣ a Σ−∗

22Σ∗

12−Σ−∗

22

−1 1 0γ−2Σ∗

210 1

⎤⎦

and Π22 =[

Z I]W−1

[0I

]is

Π22 =[

Z 1] ⎡⎣ a −Σ−∗

22

−1 0γ−2Σ∗

211

⎤⎦

= 1 +Σ−∗

22

s − a(Z − γ−2Σ∗

21)

= 1 + γ−2Σ−∗

22

e−sh − Σ∗

11− Σ∗

21(s + a)

s2 + γ−2 − a2.

It is easy to see that Π22 is stable. As a matter of fact, asrequired, Π22 is bistable for γ > γopt. The Nyquist plotsof Π22 for different values of aγ are shown in Fig. 6 forah = −1. The optimal value γopt is between −0.44/a and−0.45/a. This corresponds to the transition from Fig. 6(b)to Fig. 6(c), in which the number of encirclements changesaccordingly to the change of the bi-stability of Π22: the

388

Nyquist plot encircles the origin when γ < γopt and henceΠ22 is not bistable but the Nyquist plot does not encircle theorigin when γ > γopt and hence Π22 is bistable.

B. The unstable case (a > 0)

In this case, Gβ is unstable and Lc = −2aγ2, Lo = −2a.Hence, Σ̂22 defined in (6) is

Σ̂22 = e−λh − 4a2γ2eλh + (eλh − e−λh) ·

−2aγ2λ2 − 2a3γ2 + 4λa2γ2 + a + λ

2λ

=−2aγ2λ2 − 2a3γ2 + a + λ − 4λa2γ2

2λeλh −

−2aγ2λ2 − 2a3γ2 + a − λ + 4λa2γ2

2λe−λh

=−2aγ2λ2 − 2a3γ2 + a

2λ(eλh − e−λh) +

eλh + e−λh

2(1 − 4a2γ2)

=−4a3γ2 + 3a

2λ(eλh − e−λh) +

eλh + e−λh

2(1 − 4a2γ2). (14)

Σ̂22 can be re-arranged as

Σ̂22 = −4a(a2γ2 − 1) + a

2λ(eλh − e−λh) −

4(a2γ2 − 1) + 3

2(eλh + e−λh).

Hence, Σ̂22 is always negative (nonsingular)3 when γ ≥1/ |a|, noting that a and λ are positive. According to SectionII, the optimal performance index is less than 1/ |a|, i.e.,

γopt < 1/ |a| .

When 0 < γ < 1/ |a|, the eigenvalues ±λ of H are onthe jω-axis. Substitute λ = ωi with ω =

√γ−2 − a2 into

(14), then

Σ̂22 = (1 − 4a2γ2) cos(ωh) + (3 − 4a2γ2)a

ωsin(ωh).

Similarly, with the substitution of ω =√

γ−2 − a2 =γ−1

√1 − a2γ2 , then

Σ̂22 = (1 − 4a2γ2) cos(ah

aγ

√1 − a2γ2) +

aγ(3 − 4a2γ2)√1 − a2γ2

sin(ah

aγ

√1 − a2γ2).

3Σ̂22 = −ah − 3 when γ = 1/ |a| = 1/a.

Σ̂22

ah

aγ

Fig. 4. The surface of Σ̂22 with respect to ah and aγ (a > 0)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ah

aγaγopt

Fig. 5. The contour Σ̂22 = 0 on the ah-aγ plane (a > 0)

This surface is shown in Fig. 4 and the contours of Σ̂22 = 0on the ah-aγ plane are shown in Fig. 5. The top curve in Fig.5 characterizes the normalized optimal performance indexaγopt with respect to the normalized delay ah. On this curve,Σ̂22 becomes singular the first time when γ decreases from+∞ to 0.

Since I − LcLo = 1 − 4a2γ2, there is∥∥ΓGβ

∥∥ = 1

2a. As

a result, the optimal value γopt satisfies 1

2a≤ γopt ≤

1

a, i.e.,

0.5 ≤ aγopt ≤ 1. This coincides with the curve aγopt shownin Fig. 5.

Now discuss the (sub-optimal) compensator for γ > γopt.

In this case, the FIR block Z remains the same as in(13) and the form is not affected because of the sign of

389

a. However, W−1 is changed to

W−1 =

[−a Σ̂−∗

22(Σ∗

12 − 2aΣ∗

11) −Σ̂−∗

22

−1 1 0γ−2Σ∗

21 + 2aΣ∗

11 0 1

]

and Π22 =[

Z I]W−1

[0I

]is

Π22 =[

Z 1] ⎡⎣ −a −Σ̂−∗

22

−1 0γ−2Σ∗

21+ 2aΣ∗

111

⎤⎦

= 1 +Σ̂−∗

22

s + a(Z − γ−2Σ∗

21− 2aΣ∗

11)

= 1 + γ−2Σ̂−∗

22·

s−as+a

e−sh−(s−a)Σ∗

21+(2a2γ2

−1−2aγ2s)Σ∗

11

s2+γ−2−a2

= 1 + γ−2Σ̂−∗

22·

s−as+a

e−sh−(s−a)(Σ∗

21+2aγ2Σ∗

11)−Σ∗

11

s2+γ−2−a2 .

It is easy to see that Π22 is stable and invertible. As a matterof fact, as required, Π22 is bistable for γ > γopt. The Nyquistplots of Π22 for different values of aγ are shown in Fig. 7for ah = 1. The optimal value γopt is between 0.73/a and0.74/a. This corresponds to the transition from Fig. 7(b)to Fig. 7(c), in which the number of encirclements changesaccordingly to the change of the bi-stability of Π22.

IV. CONCLUSIONS

In this paper, a case study is given to show the delay-type Nehari problem using a first-order system. The stablecase and the unstable case are all discussed. The systemis normalized so that the (normalized) optimal value canbe shown as a function of the delay. It has been shownthat solving the problem involving only a first-order systemis not easy at all. When the system order gets higher, thecomputation of the optimal value becomes very complicated.One might have to use (11) for a compromise.

REFERENCES

[1] B.A. Francis, A Course in H∞ Control Theory, vol. 88 of LNCIS,Springer-Verlag, NY, 1987.

[2] Q.-C. Zhong, Robust Control of Time-Delay Systems, Springer-VerlagLimited, London, UK, 2006, to appear.

[3] K. Zhou and P.P. Khargonekar, “On the weighted sensitivity mini-mization problem for delay systems,” Syst. Control Lett., vol. 8, pp.307–312, 1987.

[4] D.S. Flamm and S.K. Mitter, “H∞ sensitivity minimization for delaysystems,” Syst. Control Lett., vol. 9, pp. 17–24, 1987.

[5] G. Tadmor, “Weighted sensitivity minimization in systems with a singleinput delay: A state space solution,” SIAM J. Control Optim., vol. 35,no. 5, pp. 1445–1469, 1997.

[6] Q.-C. Zhong, “Frequency domain solution to delay-type Nehari prob-lem,” Automatica, vol. 39, no. 3, pp. 499–508, 2003, See Automaticavol. 40, no. 7, 2004, p.1283 for minor corrections.

[7] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of LinearOperators, vol. II, Birkhäuser, Basel, 1993.

[8] Q.-C. Zhong, “On distributed delay in linear control laws. Part I:Discrete-delay implementations,” IEEE Trans. Automat. Control, vol.49, no. 11, pp. 2074–2080, 2004.

−1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Re

Im

Π22−1

Π22

(a) aγ = −0.3

−120 −100 −80 −60 −40 −20 0 20−80

−60

−40

−20

0

20

40

60

80

ReIm

Π22−1

Π22

(b) aγ = −0.44

−10 0 10 20 30 40−30

−20

−10

0

10

20

30

Re

Im

Π22−1

Π22

(c) aγ = −0.45

0 0.5 1 1.5 2−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re

Im

Π22−1

Π22

(d) aγ = −0.7

Fig. 6. The Nyquist plot of Π22 (a < 0 and ah = −1)

390

−4 −2 0 2 4 6 8−6

−4

−2

0

2

4

6

Re

Im

Π22−1

Π22

(a) aγ = 0.2

−50 −40 −30 −20 −10 0 10−30

−20

−10

0

10

20

30

Re

Im

Π22−1

Π22

(b) aγ = 0.73

−50 0 50 100 150−80

−60

−40

−20

0

20

40

60

80

ReIm

Π22−1

Π22

(c) aγ = 0.74

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Re

Im

Π22−1

Π22

(d) aγ = 1.5

Fig. 7. The Nyquist plot of Π22 (a > 0 and ah = 1)

391