Available online at www.sciencedirect.com
Automatica 39 (2003) 361 – 366
www.elsevier.com/locate/automatica
Technical Communique
H∞ control of dead-time systems based on a transformation
Qing-Chang Zhong∗
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Technion City, Haifa, 32000, Israel
Received 11 May 2001; received in revised form 20 January 2002; accepted 3 October 2002
Abstract
This paper presents a transformation that makes all the robust control problems of dead-time systems able to be solved similarly as in
the .nite-dimensional situations. With trade-o0 of the performance, some advantages obtained are: (i) the controller has a quite simple
and transparent structure; (ii) there are no any additional hidden modes in the Smith predictor and, hence, there is no additional hidden
possibility to destabilize the system; (iii) it can be applied to systems with long dead-time without any di5culty. Hence, the practical
signi.cance is obvious.
? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Dead-time systems; Dead-time compensator; H∞ control; Smith predictor; Performance index
1. Introduction
Dead-time systems are systems in which the action of control inputs takes a certain time before it a0ects the measured
outputs. The typical dead-time systems are those with input
delays. It is well known that it is very di5cult to control such
systems. Smith (1957) predictor is the .rst e0ective way to
control such systems and many modi.ed Smith predictors
were presented (Palmor, 1996). In recent years, many researchers are interested in the optimal control of dead-time
systems, especially H∞ control, i.e., to .nd a controller to
internally stabilize the system (if so, called an admissible
controller) and to minimize the H∞ -norm of the transfer
matrix from the external input signals (such as noises, disturbances and reference signals) to the output signals (such
as controlled signals and tracking errors).
With the help of the modi.ed Smith predictor, many robust control problems for dead-time systems, such as robust
This paper was presented at The 3rd IFAC Workshop on Time Delay
Systems, Santa Fe, New Mexico, USA, December 2001. This paper was
recommended for publication in revised form by Associate Editor Geir
E. Dullerud under the direction of Editor Paul van den Hof.
∗ Corresponding address. Department of Electrical and Electronic Engineering, Imperial College, Exhibition Road, London SW7 2BT, UK.
Tel.: +44-20-759-46295; fax: +44-20-759-46282.
E-mail address: [email protected] (Q.-C. Zhong).
URL: http://members.fortunecity.com/zhongqc
stability, tracking and model-matching and input sensitivity
minimization, can be solved as in the .nite-dimensional
situations (Meinsma & Zwart, 2000; Nobuyama, 1992).
However, the sensitivity minimization, the mixed sensitivity
minimization and/or the standard H∞ control problems
cannot be solved in this way. Very recently, notable results
have been presented in Mirkin (2000), Meinsma and Zwart
(2000), Tadmor (2000), Nagpal and Ravi (1997), and Zhong
(2003a,b) using di0erent methods. The results in Mirkin
(2000), Meinsma & Zwart, 2000, and Zhong (2003b),
which are formulated in the form of a modi.ed Smith predictor, are quite elegant and the ideas are very tricky, but
the controllers are too involved. Speci.cally, the Smith
predictor is quite complex and, even more, relates to the
performance level . Hence, there exist some problems to
apply the method to systems with long dead-time, as pointed
out by Meinsma and Zwart (2000). Another disadvantage
is that the predictor (see, Fstab in Meinsma & Zwart, 2000,
Theorem 5.3) ; ∞ in Mirkin (2000, Theorem 2), or (s)
in Zhong (2003b, Theorem 2) always includes additional
unstable hidden modes, even with stable plants, because
the hidden modes are the eigenvalues of a Hamiltonian
matrix. It has been pointed out in Van Assche, Dambrine,
and Lafay (1999) and Manitius and Olbrot (1979) that such
hidden modes are not safe and tend to destabilize the system when implemented. Hence, the practical signi.cance
of these results might be limited.
0005-1098/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 2 2 5 - X
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Q.-C. Zhong / Automatica 39 (2003) 361 – 366
This paper presents a transformation on the closed-loop
transfer matrix of dead-time systems. In fact, it is a new
H∞ performance evaluation scheme for dead-time systems.
With this transformation, all robust control problems can be
solved as in the .nite-dimensional situations. The obtained
controller has quite a simple and transparent structure incorporating a modi.ed Smith predictor. The resulting Smith
predictor only depends on the real plant and is independent
of the performance level and of the performance evaluation scheme. There do not exist any additional hidden
modes and, hence, there is no additional hidden possibility
to destabilize the system. This method can be applied to systems with long dead-time without di5culty. The cost to pay
for these advantages is some performance losses from the
conventional H∞ control point of view. When the dead-time
h becomes 0, the transformation becomes null. Hence, it
can be regarded as an extension of the conventional performance evaluation scheme for dead-time systems. This
transformation does not a0ect the original system and is
quite ease to be applied because it just, virtually, subtracts
an .nite impulse response (FIR) block from the original
system.
The rest of the paper is organized as follows: the transformation (or the new performance evaluation scheme) is presented in Section 2; the solution to H∞ control of dead-time
systems with an input/output delay is given in Section 3; an
example is given in Section 4.
2. The transformation
The general control setup for dead-time systems with an
input or output delay is shown in Fig. 1, where
P11 (s) P12 (s)
P(s) =
:
P21 (s) P22 (s)
Tzw (s) = P11 + e−sh P12 K(I − e−sh P22 K)−1 P21 :
is a rational transfer matrix G(s) = D + C(sI − A)−1 B. Truncation operator h {G} and completion operator h {e−sh G}
are de.ned, respectively, as
Ah
A B
−sh A e B
h {G} =
=G(s)
˙
− e−sh G̃(s);
−e
C D
C 0
B
A
h {e−sh G} =
−Ah
Ce
0
Fig. 2. An equivalent structure.
The closed-loop transfer matrix from w to z is
Notation. Assume
A B
G(s) =
C D
Fig. 1. General control setup for dead-time systems.
−sh
−e
A B
C D
=˙ Ĝ(s) − e−sh G(s):
They are slightly di0erent with those de.ned in Mirkin
(2000). Note that these two operators map any rational transfer matrix G into an FIR block. The impulse response of
h {G} is the truncation of the impulse response of G to
[0; h]. The impulse response of h {e−sh G}, which is also
supported on [0; h], is the only continuous function on [0; h]
with the following property: if we add it to the impulse response of e−sh G(s), which is supported on [h; ∞), we obtain
the impulse response of a rational transfer matrix, denoted
above by Ĝ.
This means that there exists an instantaneous response
through the path P11 without any delay. A clearer equivalent
structure is shown in Fig. 2. It is not di5cult to recognize
that, during the period t = 0 ∼ h after w is applied, the output z is not controllable (i.e., not changeable by the control
action) and is only determined by P11 (and, of course, w).
However, the response during this period may dominate the
system performance index. This means the performance index is likely dominated by the response we cannot control.
This is not what we want. It does not make sense to include such an uncontrollable part in the performance index.
Hence, we should eliminate the response during this period
when evaluating the system performance. This is a key idea
in this paper. In general, it is impossible to eliminate the instantaneous response by simply introducing a suitable P11 .
A possible way to implement this idea is proposed below.
There may be other ways to implement this idea.
De.ne an FIR block
1 (s) = h {P11 } = P11 (s) − P̃ 11 (s)e−sh
which is exactly the uncontrollable part in Tzw (s). Subtract
it from the feed-forward path P11 , as shown in Fig. 3, then
Tzw (s) = 1 (s) + Tz w (s);
(1)
Q.-C. Zhong / Automatica 39 (2003) 361 – 366
(A3)
363
A − j!I
B1
C2
D21
has full row rank ∀! ∈ R;
∗
∗
(A4) D12
D12 = I and D21 D21
= I.
Assumption (A4) is made just to simplify the exposition.
∗
In fact, only the non-singularity of the matrices D12
D12 and
∗
D21 D21 is required.
Consider a Smith predictor-type controller
K(s) = K0 (s)(I − 2 (s)K0 (s))−1
as shown in Fig. 3, in which the predictor is designed to be
B2
A
A B2
−sh
−sh
;
2 (s) = h {e P22 } =
−
e
C2 D22
C2 e−Ah 0
Fig. 3. The graphic interpretation of the transformation.
then the system can be re-formulated as
w
z
=
P̃(s)
;
u
y
where
Tz w (s) = e−sh {P̃ 11 + P12 K(I − e−sh P22 K)−1 P21 }:
The fact shown in (1) was recognized in Mirkin and Raskin
(1999) to parameterize all stabilizing dead-time controllers.
There is no instantaneous response in Tz w (s). The only difference between Tz w (s) and Tzw (s) is the FIR block 1 (s),
which is not a part of the real control system but an arti.cial
part introduced into the performance evaluation scheme. It
is this FIR block that makes the control problems so complex and di5cult to be solved. It is shown in Mirkin (2000)
that the achievable minimal performance index Tzw ∞ is
larger than 1 ∞ . Hence,
1 ∞ ¡ Tzw ∞ 6 1 ∞ + Tz w ∞ :
(2)
We optimize Tz w ∞ but not Tzw ∞ in this paper. Once
Tz w ∞ is minimized, the achievable performance Tzw ∞
is not larger than 1 ∞ + Tz w ∞ and so is the optimal
performance. Using inequality (2), one can estimate how
far the proposed sub-optimal controller is away from the
optimal controller.
3. H∞ control of systems with a single delay
Assume that the
generalized process

A B1

P(s) = C1 D11
C2 D21
realization of the rational part of the
in Fig. 1 is taken to be of the form

B2
D12 
D22
and the following standard assumptions hold:
(A1) (A; B2 ) is stabilizable and (C2 ; A) is detectable;
A − j!I
B2
has full column rank ∀! ∈ R;
(A2)
C1
D12
u = K0 (s)y
with e−sh z =z
˙ , where

eAh B1
A
P̃(s) =  C1
0
C2 e−Ah D21

B2
D12 
0
is free of dead-time (but delay-dependent). The closed-loop
transfer function from w to z is
Tz w (s) = e−sh Tz w (s) = e−sh Fl (P̃(s); K0 (s)):
(3)
Hence, the H∞ control problem
Tz w (s)∞ ¡ is converted to
Fl (P̃(s); K0 (s))∞ ¡ :
This is a .nite-dimensional problem which can be solved
using known results. Since this is for the general setup of
systems with an input/output delay, all the H∞ control problems (such as robust stability, tracking and model-matching,
input sensitivity minimization, output sensitivity minimization, mixed sensitivity minimization and the standard H∞
control problem, etc.) can be solved similarly as in the
.nite-dimensional situations.
The solution, which is given in Theorem 1, involves two
Hamiltonian matrices:
∗ B2
A
−2 eAh B1 B1∗ eA h
Hh =
−
−C1∗ D12
−C1∗ C1
−A∗
∗
C1
× [ D12
B2∗ ];
364
Q.-C. Zhong / Automatica 39 (2003) 361 – 366
Jh =
A∗
−2 C1∗ C1
−eAh B1 B1∗ eA
× [ D21 B1∗ eA
∗
∗
h
h
−A
−
∗
e−A h C2∗
∗
−eAh B1 D21
C2 e−Ah ]:
Theorem 1. There exists an admissible main controller
such that Tz w (s)∞ ¡ in Fig. 3 i2 the following three
conditions hold:
(i) Hh ∈ dom(Ric) and X = Ric(Hh ) ¿ 0;
(ii) Jh ∈ dom(Ric) and Y = Ric(Jh ) ¿ 0;
(iii) ((XY ) ¡ 2 .
Moreover, when the conditions hold, one such main controller is
Ah −Lh
;
K0 (s) =
Fh Z h 0
where
Ah = A + Lh C2 e−Ah + −2 YC1∗ C1
+ B2 + −2 YC1∗ D12 Fh Zh ;
∗
Fh = −(B2∗ X + D12
C1 );
Fig. 4. The setup for mixed sensitivity minimization.
Assumptions (A2) and (A4) remain unchanged. Hence,
P̃(s) meets all the standard assumptions. Theorem 5.1 in
Green, Glover, Limbeer, and Doyle (1990) can be directly
used to solve the problem. Substitute P̃(s) into the theorem,
then the above result can be obtained with ease.
Remark 2. It is worth noting that, under this transformation,
either D11 = 0 or D22 = 0 does not make the problem
more complicated. In fact, when h = 0; 2 (s) = −D22 is the
common controller transformation to make D22 = 0 (Zhou
& Doyle, 1997).
4. Example
∗
∗
Lh = −(Y e−A h C2∗ + eAh B1 D21
);
Consider the example studied in Meinsma and Zwart
(2000), see Fig. 4, where
Zh = (I − −2 YX )−1 :
Pr (s) =
Furthermore, the set of all admissible main controllers such
that Tz w (s)∞ ¡ can be parameterized as
1
s−1
W1 (s) =
2(s + 1)
10s + 1
K0 (s) = Fl (M (s); Q(s));
Prd (s) =
and
and
s−1
;
s+1
W2 (s) =
0:2(s + 1:1)
:
s+1
where
M (s) = 




Ah
−Lh
B2 + −2 YC1∗ D12
F h Zh
0
I
I
0
∗
− C2 e−Ah + −2 D21 B1∗ eA h X Zh
and Q(s) ∈ H∞ , Q(s)∞ ¡ .
Proof. First of all, check if P̃(s) meets the standard assumptions (A1–A4).
(A1) (C2 e−Ah ; A) is detectable because A+eAh LC2 e−Ah ∼
A + LC2 and (C2 ; A) is detectable;
(A3)
A − j!I eAh B1
C2 e−Ah
has full row rank ∀! ∈ R.




This problem can be arranged as a standard problem with
2(s + 1)2
(10s + 1) (s − 1)
2(s + 1)
(10s + 1) (s − 1)
0
0.2(s + 1.1)
s+1
P(s)=
D21
has full row rank ∀! ∈ R because
A − j!I
A − j!I eAh B1
∼
−Ah
C2
C2 e
D21

−
B1
D21
Here,
P22 = −
s+1
s−1
−
1
s−1

1
;
s−1

2(s + 1)2
P11 =  (10s + 1)(s − 1)  · W2 (s)
0
is delayed to be W2 e−sh , which does not a0ect the result.
Q.-C. Zhong / Automatica 39 (2003) 361 – 366
Table 1
Performance comparison
h
M–Z’s
Tzw (s)∞
Proposed
Tz w (s)∞
Proposed
Tzw (s)∞
0.2
2
5
0.68
5.22
109.5a
0.58
3.86
78.63
0.88
8.34
183.7
a Note: For long dead-time h, e.g. h ¿ 3:2 s, some matrices are close
to singular or badly scaled; the result of M–Z is likely inaccurate.
The predictor is designed to be
2 (s) = −
e−h − e−sh
:
s−1
There is no additional hidden mode; the only hidden mode
is the same as the mode of the real plant P22 . However, the
predictor obtained in Meinsma and Zwart (2000) is
365
It is much more complicated. There are three hidden modes:
one is the same as the mode of the real plant P22 and the
other two s1; 2 = ±0:2883j are additional.
The achievable performance is listed in Table 1for different delays h. The performance of the proposed method
degrades not too much to pay for the advantages obtained.
The frequency responses of both cases are shown in Fig.
5. Although the H∞ -norm achieved in this paper is slightly
larger than that achieved by the method of Meinsma and
Zwart (noted as M–Z in .gures), the proposed method obtains better performance in a quite broad frequency band
ranging from 1 rad=s to about 10; 000 rad=s. From the engineering point of view, this solution is much better. The singular value plot of the uncontrollable part 1 (s) is shown in
Fig. 6. It is clear that the high gain of Tzw at low frequency
is contributed (or dominated) by the large singular value of
1 (s) at low frequency.
Fstab (s) = −
10:75s2 + 0:245s − 0:3298
(s − 1)(12:03s2 + 1)
5. Conclusions
+
13:16s2 − 0:1316 −sh
e :
(s − 1)(12:03s2 + 1)
This paper introduces a transformation (in fact, a new
performance evaluation scheme) for the H∞ control of
1
0.9
9
8
Proposed
M-Z
6
||T ||
0.7
zw
||Tzw||
Proposed
7
0.8
0.6
M-Z
5
4
0.5
3
0.4
2
0.3
1
0.2
-2
-1
0
1
2
3
4
5
6
10 10
10 10 10 10 10 10 10
0
-2
-1
0
1
2
3
4
5
6
10 10
10 10 10 10 10 10 10
(a)
Frequency (rad/sec)
(b)
Frequency (rad/sec)
Fig. 5. The comparison of Tzw (j!): (a) h = 0:2 s and (b) h = 2 s.
5
4
1
0.3
Singular value of ∆
Singular value of ∆
1
0.35
0.25
0.2
0.15
-1
0
1
2
3
4
5
6
-2
10 10
10 10 10 10 10 10 10
(a)
Frequency (rad/sec)
3
2
1
0
-2
-1
0
1
2
3
4
5
6
10 10
10 10 10 10 10 10 10
(b)
Frequency (rad/sec)
Fig. 6. The singular value plot of 1 (s): (a) h = 0:2 s and (b) h = 2 s.
366
Q.-C. Zhong / Automatica 39 (2003) 361 – 366
dead-time systems. It considerably simpli.es the H∞ control problem of dead-time systems and has many advantages.
In particular, the practical signi.cance is obvious. Since this
is not a solution to the original H∞ control problem of
time delay systems there may exist some performance losses
from the conventional H∞ control or mathematical point of
view. An inequality has been given to estimate how far the
sub-optimal controller is away from the optimal controller;
more accurate analysis of the performance loss is being
undertaken.
Acknowledgements
This research was supported by The Israel Science Foundation (Grant No. 384/99) founded by The Israel Academy
of Sciences and Humanities. The author would like to thank
Dr. Leonid Mirkin, Dr. George Weiss, Prof. Kemin Zhou
and the anonymous reviewers for their helpful suggestions.
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