Proceedings of the 41st IEEE
Conference on Decision and Control
Las Vegas, Nevada USA, December 2002

FrA05-2

Approximating Sampled-Data Systems
with Applications to Digital Redesign
Yutaka Yamamoto1

Brian D. O. Anderson2

Abstract

continuous-time performance, rather than the mere samplepoint behavior. This requires approximation of the generalized plant in the sampled-data setting, and desirably
a convergence analysis. Keller and Anderson [1, 8] haveproposed to use a fast-sampling/fast-hold approximation
with its approximating performance taken to be the closedloop behavior. Related design problems have been studied
by Madievski and Anderson [10] for H ∞ control and by
Bamieh et al. [4] for L1 control problems. Later it wasproven by Yamamoto et al. [14] that such fast-sampling approximations converge uniformly in the frequency domain
to the limiting sampled-data system. The following important issues still remain open, however:
1. How do we guarantee that a designed controller K(based on plant approximation/discretization) converges to the “right” controller in the limit? Since
we do not know in advance which controller will bepart of the closed-loop before the design, a convergence theorem with a fixed K would not be enough.
We need some kind of uniformity in the controllers K
in the convergence process to guarantee this. (While
Bamieh et al. [4] successfully gave an explicit convergence bound for the L1 control design problem, there
is an important difference here. Although sampling iscontinuous with respect to the L∞ norm where it iswell defined, it is not continuous with respect to theL2 norm. This makes the estimate of the convergence
rate here in the H ∞ context more delicate, and quite
different from that studied in [4].)Despite the existence of methods for a direct optimal design of sampled-data control systems, it is often desired toapproximate sampled-data systems with discrete-time ones.
This occurs frequently in the context of digital redesign,in which one retains intuition in the continuous-time context. This paper investigates and gives conditions underwhich such a method works. Under some conditions, weguarantee that fast-sampling approximation works well forH ∞ sampled-data design, and give an estimate for such an
upsampling factor. As an application, we propose a new
method for obtaining an FIR controller (possibly with firstorder approximation). A comparison is made with an existing method.
1 Introduction
Modern sampled-data control theory enables us to design
directly a digital controller that makes an analog performance optimal. It is also known that for finite-dimensional
plants this design problem, for example, H ∞ control problem having the mixed nature of both discrete-time andcontinuous-time, can be reduced to an equivalent discretetime, finite-dimensional problem [3, 6, 7].
In spite of these results, however, there are still several reasons that lead us to approximate the design problem in avariety of ways by a more conventional approximate design problem, and then obtain a digital controller. One reason is that we may be able to use a design software of our
choice, and base our design intuition on such softwares weare familiar with. Once we have obtained an approximant,
it is not difficult to invoke such a more conventional design package. Yet another advantage is that we may wish to
rely on some design intuition we have developed, and thisis in many cases easier to maintain by resorting to approximations, rather than using the direct optimization methods
mentioned above. So-called digital redesign is regarded assuch a case: we first obtain a continuous-time controller,
and then attempt to discretize it, while attempting to maintain the desirable continuous-time performance.

2. How fast should fast-sampling periods be? While
there exist some empirical studies on estimates andindications in some special cases, there does not exist a precise estimate that guarantees quality of approximation for a particular rate, to the best of theauthors’ knowledge (again, except [4] for the L1
control problem, but not for the frequency responsecomputation/H ∞ design).
This paper studies these problems. We first invoke the socalled FR-operator T by Araki and others [2]. Its principal
submatrix T (M, M ) of size M tells us how many aliased
components we need to obtain a desired degree of precision. We then obtain an estimate on the fast-sampling rateusing T (M, M ), based on the approximation estimate ofsinusoids via piecewise step functions arising from sample
and hold actions.

In such attempts, we of course wish to capture the
1 yy@i.kyoto-u.ac.jp, Department of Applied Analysis andComplex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, JAPAN. Author to whom all correspondence
should be addressed.
2 Brian.Anderson@anu.edu.au, Research School of Information Sciences and Engineering, Australian National University, Canbera
ACT 0200, AUSTRALIA.
3 nagahara@acs.i.kyoto-u.ac.jp, the same address as Yutaka Yamamoto.

0-7803-7516-5/02/$17.00 ©2002 IEEE

Masaaki Nagahara3

We also apply the results to the digital redesign problem,and show that the FIR (finite-impulse response, i.e., those3724

having finitely many Markov parameters) approximation
problem of a stable controller for sampled-data systems canbe reduced (with first-order approximation when the original problem is not one-block) to an LMI. In the field ofdigital filters, there is always a strong need for replacing an
IIR (infinite-impulse response) filter by an FIR filter. The
latter has the advantages that it is intrinsically stable andfree from various problems such as limit cycles arising fromfinite-wordlength precision. We show that the obtained LMI
can be solved via a convex optimization method such as the
cutting plane method. We illustrate the result by comparing
it with the example studied in [10].

Sh/N

K(z)

u

Hh/N

Hh

Figure 2: Fast sampling approximation
vergence result of [14] requires that the controller be fixed,
and this assumption is (in effect) not satisfied for synthesis
problems. To this end, we must guarantee the uniform convergence, with respect to controller K, of the norms of the
fast-sampling approximants if we are going to use such approximants in designing the controller. This motivates thefollowing theorem.

Consider the sampled-data control system in Fig. 1, consistingof a continuous-time generalized plant P =
P11 P12and a discrete-time controller K with samP21 P22
pler Sh and zero-order hold Hh with sampling period h.
The closed-loop transfer operator from w to z will be denoted by Tzw (K)(z).
z
w
P (s)
y
u

K(z)

w
P (s)

Sh

2 Uniform Approximation via Fast-Sampling

Sh

z
y

Theorem 2.1 Let S be a set of stable controllers K suchthat i) every K ∈ S is stabilizing, and ii) S is compact withrespect to the H ∞ -norm. Then the frequency response gainn
Tzw
(K)(ejωh ) of the n-step fast-sampling approximantnTzw (K) converges to Tzw (K)(ejωh ) uniformly in K ∈S. This convergence is also uniform in ω ∈ [0, 2π/h).
Proof Fix  > 0, and take any K ∈ S. By the convergence result of [14], there exists N such that


 n

Tzw (K)(ejωh ) − Tzw (K)(ejωh ) <  for all n ≥ N,

Hh

Figure 1: Sampled-data control system
We assume that P21 is strictly proper, which is necessary toassure that sampling is well-defined. Likewise, we assume
that P11 and P12 are also strictly proper.

and this is uniform in ω. We will thus omit the dependenceon ω below. Take the least such N and name it N (K, ).
nSince Tzw
(K) − Tzw (K) is continuous with respect to
K as Lemma 2.3 below shows, there exists a neighborhood
B(K, δ) := {K  : K  − K < δ} of K such that


 n

Tzw (K  ) − Tzw (K  ) < 

It is well known that for a finite-dimensional plant P , thesampled-data H ∞ controller design problem of Fig. 1 canbe reduced to a norm-equivalent discrete-time H ∞ design
problem [3, 6, 7]. Nonetheless, discretization via fastsampling and fast-hold as studied in [10, 14] is often effective as explained in the introduction.

for all n ≥ N (K, ) and K  ∈ B(K, δ). This yields acovering of the controller set:

Let us thus consider fast-sampling approximants for thedesign problem (The N -step closed-loop transfer operator
N(K)(z).) Fig. 2; for defrom w to z will be denoted by Tzw
tailed formulae, see, e.g., [14]. The idea is to approximate
the inputs by step functions of step size h/N for sufficiently
large N and also approximate outputs by taking their samples at every h/N seconds rather than the original sampling
period h. It may appear trivial that such an approximation
gives the right answer, for example, in computing the frequency response. The situation is however much more delicate because we do not know in advance what kind of input
gives rise to the gain of the frequency response which is actually the operator norm at each frequency. It is proven in[14] that the frequency response gain of such approximants
converges to that of the limit uniformly for 0 ≤ ω ≤ 2π/h,
for a fixed controller K.

S = ∪K∈S B(K, δ).
By the compactness of S, there exists a subcovering
S = B(K1 , δ1 ) ∪ · · · ∪ B(Km , δm ).
Taking Nmax := {N (K1 , ), . . . , N (Km , )}, we readily
have that n ≥ Nmax implies



 n
(1)
Tzw (K) − Tzw (K) < 
for all K ∈ S.
Remark 2.2 A typical example of S may be given by theset of FIR controllers of order less than or equal to a prefixed
integer q and with a prespecified norm bound as
S := {K : ord K ≤ q, K ≤ M, K is stabilizing}.

While this is sufficient for analysis purposes, we still need to
go one step further. In synthesis, we do not know in advancewhich controller will appear in the closed-loop. The con-

Since the number of coefficients of K is at most q + 1, thiscan be regarded as a subset of Rq+1 . The norm K may

3725

be interpreted in the sense of any norm in Rq+1 . More generally, one can also consider IIR controllers, having both
numerator and denominator, with fixed order. To avoid unstable pole-zero cancellations, we may have to exclude asmall open neighborhood around unstable poles in considering S.
Theorem 2.1 states that if we agree to fix a compact set S
and search our controller K in this set, there exists N for aN(K) < γ then Tzw (K) < γ +.
given  such that if Tzw
That is, a controller designed by the N -step approximationindeed approximates the performance of the solution to theoriginal problem without approximation. Furthermore, if
Tzw (K) < γ and if K is known to be in S, then it can befound with error tolerance  by taking N large enough.

3.1 A Special CaseIn what follows we first make the following assumptions:(A) P11 = 0.
(B) Each Pij is an SISO system.
(C) |P21 (jω)| is rational and monotone decreasing forω > 0.
The first assumption is somewhat specialized, but it greatly
simplifies the subsequent arguments. We will discuss a generalization in the next subsection. The second assumption isjust for convenience of notation, and it does not lead to any
real loss of generality. The third assumption is also placed
to simplify the subsequent arguments. If this is not satisfied by a possible peak in the high frequency region, thenwe should simply replace it by a suitable SISO anti-aliasing
filter to estimate the norm of T .

It remains to show the following lemma on the continuity of
the closed-loop operator with respect to K.
Lemma 2.3 Consider the closed-loop operator Tzw (K)where K is assumed to be stable and stabilizing. Then
Tzw (K) is continuous in K with respect to the H ∞ norm.
Sketch of Proof Let K  − K∞ < δ with K, K  stabilizing. Put ∆ := K  − K, and we can easily rewrite Tzw (K  )involving ∆ and the nominal closed-loop operator Tzw (K).
Then for sufficiently small δ, ∆∞ <  enables us to invoke the small-gain theorem, and the result follows.


Denote by T (L, M ) the principal submatrix{T ij }0≤i≤L,0≤j≤M of T . We allow L, M to be ∞as a special case, in which case the respective ≤ in theindices should be replaced by <.
Let us first note the following:

3 Estimating the Upsampling Factor N

Lemma 3.2 Assume
∞ (A)–(C) above, and let C0 :=2
sup0≤ω≤2π/h
n=1 |P21 (jωk )/P21 (jω)| , where ωk :=ω + 2kπ/h. By assumption (C), this quantity is finite. Then

Theorem 2.1 above guarantees that we can rely on fastsampling approximations for designing a sampled-data H ∞stable controller, by taking the upsampling factor N sufficiently large. Our next question is naturally how large suchan N should be.

T  ≤ C0 T (∞, 0).

In other words, only the base-band inputs are enough toestimate the norm of T .

To the best of our knowledge this question has not been
investigated in depth in the literature even for the analysis
problems (except [4] for L1 control problem). We give ananswer to this by invoking certain a priori bounds and estimates for plants and controllers. Consider Fig. 1 again.

Proof Let ejωk t be the k-th aliased input. When it is applied to the system, it first goes through P21 (s) with steadystate output P21 (jωk )ejωk t . After sampling, this becomesa discrete-time signal {P21 (jωk )ejωh }∞Hence the=0 . steady-state output corresponding
to
the
input
αk ejωk tjωt
is also realized by an input ( k αk P (jω.
k )/P (jω))e
2
Since | k αk P (jωk )/P (jω)| ≤ C0by
k |αk |
Schwarz’s inequality, the norm of T is easily seen to bebounded by the right-hand side of (3).


Let Tzw (ejωh ) denote the frequency response operator associated with the closed-loop transfer operator Tzw . We
will omit the dependence on K hereafter. This operator is also equivalent [13] to the notion of the so-called
FR operator [2]. We denote this operator by T (ω). In
essence, T (ω) is nothing but the (infinite) matrix representation of Tzw (ejωh ) with respect to an orthonormal basis
2{exp(j(ω + 2nπ/h)t}∞n=−∞ in L [0, h] [13].

Remark 3.3 Since |P21 (jωk )/P21 (jω)| decays √by arithmetic progression, C0 is roughly estimated by π/ 6.

Let us first note the following formula for T .
Proposition 3.1 Let Pij , Tzw and T (ω) be as above, and∗desuppose that the closed-loop system is stable. Let P22
note the pulse-transfer function associated with P22 , and let∗K)−1 . Then the (i, k) entry T ik (ω) of T is
V := (I − P22
given by
T ik (ω)(3)

Let us next investigate how large M for T (M, 0) should betaken to estimate T (∞, 0).
Proposition 3.4 Given  > 0, there exists M such that= P11 (jωi ) +(2)
1
P12 (jωi )Ĥ(jωi )K(ejωh )V (ejωh )P21 (jωk )h|T i0 | ≤
√ ,C0 π/ 6|i| > M.

This in turn yields the estimate
T − T (M, 0) < .

Outline of Proof Follow the proof in Yamamoto andAraki [13] given for the unity feedback case.


3726

(4)

(5)

Proof Note first that Ĥ(jω) = (1 − e−jωh )/jω is strictly
proper and monotone decreasing. Hence (2) and assumption(A) imply the existence of M .

frequency range. Furthermore, when close to the Nyquist
frequency,
both |C(ejωh )| and |P21 (jω)| are bounded by√
1/ 1 + ω 2 T 2 , so that they contribute to the decay of
T i0 (ω) at least up to the Nyquist frequency. On the other
hand, ψ(ω) governs the error in the low frequency range,
and this is determined by the upsampling factor N . It iseasy to compute the order of ψ(ω) and see that it is of orderωh/N for small ω (by Taylor expansion), and bounded by
2 for large ω (note |ejωh/N | = 1). For N = 5, this is about
0.62 at the Nyquist frequency, and increases monotonicallyup to N π/hrad/sec. But then the high-frequency roll-offof Ĥ(jω)/h becomes prevalent, and the overall product decays approximately by −40dB/dec. A numerical computation exhibits that the overall maximum is attained at around1rad/sec with value of 0.022 in gain (i.e., about −33dB).
Remark 3.8 The above estimate is overly conservative.
This conservativeness arises from using ψ(ω) for the sampling error. While it is certainly correct for a single sinusoid, we should note that the gain of T (ω) is obtained as
the response against the worst-case input. Thus, even foreach frequency, it is possible to consider a piecewise constant input, different from the one obtained by sampling thesinusoid, but giving rise to less error. In fact, just by shifting the sampled input by half the upsampling step size, one
achieves a substantially small error in L2 -norm. For an input ejωt , this is just a phase shift (advance) by h/2N , andis achieved by multiplying ejh/2N . The resulting sampled
output has phase advance by h/2N against the output resulting from ejωt . In the case of the proposition above with
N = 5, the overall L2 error in one upsampled step h/N isat the level of 0.007—approximately 1/10 of the error estimated by ψ(ω). This explains the true reason why the fastsampling approximation works so well in computation of
the frequency response and H ∞ designs for sampled-datasystems. (In fact, even N = 3 works in the case above.)

To compute the induced norm T − T (M, 0), note first thatas in Lemma 3.2
T − T (M, 0) ≤ C0 T (∞, 0) − T (M, 0)

(6)

holds. Since T (∞, 0) − T (M, 0) is a single column vector,
we have, by the Schwarz inequality,T (∞, 0) − T (M, 0)2 ≤|T i0 |2
i>M

≤



i>M

This, along with (6), implies (5)2= π 2 2 /6.
|i − M |2relative degree
Remark 3.5 If P12 is strictly proper with r, then the last estimate can be improved to i>M 2 /|i −M |2(r+1) .
Remark 3.6 Suppose P12 = 1/(1 + T s) with 1/T ∼π/2h, i.e., the bandwidth is about half the Nyuqist rate. Thisimplies that M = 3 means the bandwidth 6π/h > 5π/h =10 × π/2h = 10/T , i.e., 1 decade. Hence it rougly decays
at least by −20dB. More generally, if P12 is strictly proper,
then it decays by order 2, and hence M = 3 guarantees  to
be of order 10−2 .
Proposition 3.4 tells us how many blocks we should taketo compute T . This in turn means that we need sinusoids
ejωt up to the frequency ω = 2M π/h to attain the desired
accuracy as specified in this proposition. The rest of thework is then to estimate the upsampling factor N to obtain
the accuracy of (4). We have the following estimate for atypical case.
Proposition 3.7 Assume sup0≤ω≤2π/h |P21 (jω)| = 1 forsimplicity, and put C(z) := K(z)V (z), ψ(ω) := |1 −ejωh/N√ |. Suppose also that |P12 (jω)| is bounded by
1/ 1 + ω 2 T 2 for some first order plant 1/(1 + T s) with
1/T ≈ π/h. Likewise, we also √assume |C(ejωh )| and|P21 (jω)| are also bounded by 1/ 1 + ω 2 T 2 at least up
to the Nyquist frequency π/h. Then |T i0 (ω)| decays by−40dB/dec, and attains the maximum error level of 0.022
when N = 5.

We now turn our attention to estimating the approximationlevel in the fast-hold in the input term. In view of Lemma
3.2, it is enough to estimate T (∞, 0). This means we need
only consider the single sinusoid ejωt for each ω. If weapproximate this by the fast-hold of step N , there will again
be an error proportional to ψ(ω) above. However, what isdifferent here is that we can arbitrarily change the fast-holdinput function uN (t) to attain the norm of the operator. In
particular, we can change it so that the sampled values after
going through P21 (s) at the original sampling period match
those derived from input ejωt .

Outline of Proof By the assumptions,
|T i0 (ω)| ≤ P12 (jωi )(Ĥ(jω)/h)C(ejωh )P21 (jω).
at each frequency. Sampling ejωt at t = kh/N , k =0, 1, 2, . . . induces the error ψ(ω)ejωt which is bounded by
ψ(ω). Then the total error in |T i0 (ω)| induced in fast sampling is bounded by
ψ(ω)|P12 (jω)(Ĥ(jω)/h)C(ejωh )P21 (jω)|.

We start with the following:
Lemma 3.9 Suppose there exists T > 0 such that




1


|P21 (jω)| ≤ 
1 + jT ω 

(7)

(8)

Let V denote the operator obtained by replacing P21 (s) by
1/(1 + T s) in T (∞, 0). Then the error in the norm ofT (∞, 0) induced by the fast-hold Hh/N is less than thatof V induced by Hh/N .

Here |Ĥ(jω)/h| = |(1 − ejωh )/jωh| is bounded bymin{1, 2/ω}, which decays at −20dB/dec in the high frequency range. The same can be said of |P21 (jω)| in the high3727

Proof Since |P21 | is assumed to be monotone decreasing,(8) is not restrictive.

decay for P11 . But aside from this, the argument in Remark
3.8 applies, so the error in the low frequency range can be
expected to be very low, even for relatively small N .

Note that in view of (2) T (∞, 0) can be decomposed as
T (∞, 0) = ΓP21 for some Gamma. Then we haveT (∞, 0)(ω)

=≤|P21 (jω)|Γ(jω)
|1/(1 + jT ω)|Γ(jω)

=1/(1 + jT ω)Γ(jω) = V (ω).

On the other hand, since there is no sampling in the input term, we cannot invoke Lemma 3.10 for estimating theerror arising from the fast-hold inputs. This requires further study, but at least, the worst-input analysis as given in
Remark 3.8 will help ensure that the convergence is much
faster than might be normally expected.

This means that the error arising from fast-hold Hh/N canbe bounded by that of V .


4 Digital Redesign via Fast-SamplingThis lemma reduces the problem to the simple caseP21 (s) = 1/(1 + T s).

In this section, we treat a digital redesign problem by using
the fast-sampling method. We will show that the problem
reduces to a convex optimization problem.

Lemma 3.10 Let P21 (s) = 1/(1 + T s), and the sampling
period h not pathological, i.e., the zero-order hold equivalent of P21 be minimal. Then for any N ≥ 1, and for any
input u = ejωt , there exists a piecewise constant input v(with step length h/N ) with the same norm as u such thatthe sampled output value of P21 at kh, k = 0, 1, 2, . . . are
identical to those produced by u.

Consider the continuous-time feedback system shown in
Fig. 3. In the figure, P is a continuous-time plant and K
is a continuous-time controller.
w +
z
K
P
−

Proof We prove that starting from the zero initial state, it
is possible to find v with the same norm as u such that at
t = h the attained state is equal to that produced by input
u. Observe first that without the norm constraint there isalways such a v because the zero-order hold equivalent system remains reachable by hypothesis, and N ≥ 1. Thus weneed only to guarantee that the minimum norm among suchv does not exceed that of u. This is governed by the normof the reachability (controllability) Grammian h
T
eAt BB T eA t dt
(9)

Figure 3: Continuous-time feedback system
Many methods are known for designing such a controller,
while the implementation often needs a discrete-time controller. Therefore the feedback system may be realized asa sampled-data system shown in Figure 4. In the figure,
w+
−

0

or by

N


(e

K
F

Sh

KF

Hh

P

z

Figure 4: Sampled-data feedback system

Ah/N kT) B̃ B̃ (eAT h/N k)

F is an anti-aliasing filter for sampling which is stable andstrictly proper.

(10)

k=0

Then our problem is stated as follows:

for the hold-equivalent system, where (A, B, C) is a mini h/N Ate Bdt. It is routine
mal realization of P21 and B̃ = 0
to check that (10) coincides with (9) for a first-order system.

Problem 1 Given a stable continuous-time controller K(s)which stabilizes the feedback system in Fig. 3, find an FIR
controller KF which approximates the performance of thecontinuous-time system.
To achieve a good approximation, we optimize the H ∞ error between the systems in Fig. 3 and Fig. 4. Let K  :=Hh KF Sh F and Tzw (·) denote the transfer operator from w
to z depending on the situation, e.g. Tzw (K) := P K(I +P K)−1 . Then we express the problem above as an H ∞optimization problem:

Then the least norm of such v does not exceed that of theleast norm of a continuous-time input attaining the same final state at t = h. They by lifting this piecewise constant
input in the period h, we obtain an input which gives rise to
the same sampled values at t = kh, k = 0, 1, 2, . . . .
This lemma guarantees that the fast-hold device does notdeteriorate the norm estimate of T . This is a consequenceof the fact that we look for the worst-case inputs (hence theintermediate behavior of the input in fast-hold need not be
the same as the original sinusoid ejωt ).

Problem 2 Given γ > 0, find an FIR controller KF which
satisfies
Tzw (K) − Tzw (K  )∞ < γ

3.2 Outline of the general case P11 = 0
When P11 = 0, the argument above applies to the estimate
on the effect of sampling of outputs. The difference is that
we cannot count exclusively on the effect of the hold Ĥ(s)and the discrete-time controller for the high-frequency rolloff, so that it is reasonable to assume −20 or −40dB/dec(11)

In order to make the problem a convex optimization, we first
apply a first-order approximation in K − K  to the error
systemTzw (K) − Tzw (K  ) ≈ W (K − K  )V3728

where W := (I + P K)−1 P and V := (I + P K)−1. Therefore the sampled-data error constraint (11) can be converted
to the error constraint

Frequency

response

of

error

20
0
-20

W (K − K )V ∞ < γ.

Gain [dB]

-40



(12)

-60
-80
-100

Then we apply the fast-sampling approximation to thesampled-data H ∞ optimization problem (12), which can bea discrete-time H ∞ optimization:

-120
-140
-160
-2
10

10

-1

0

Frequency

G1 (z) − G2 (z)KF (z)G3 (z)∞ < γ

(13)

10
[rad/sec]

10

1

Figure 5: Gain of Error Tzw (K)−Tzw (K  ): optimal FIR (solid),
optimal IIR (dot) and balanced truncation (dash)where G1 , G2 and G3 are fast-sampling approximations ofW (s)K(s)V (s), W (s)Hh and Sh F (s)V (s) respectively.
N −1
αk z −k . Since G1 +Let the FIR controller KF := k=0
G2 KF G3 is affine in KF (or the parameters αk ), the problem of minimizing the left side of (13) can be expressed asa convex optimization, which can be easily solved by somemethods, e.g. the LMI or the cutting-plane method [5].

Frequency

response

of

T

zw

20
0
-20

Gain [dB]

-40
-60
-80
-100

5 Design Example

-120
-140

In this section, we study the example given in [10]. The
upsampling factor N for fast-sampling is 5, and the orderof the FIR controller is 63. For comparison, the following
controllers are also given:

-160
-2
10

10

-1

Frequency

010[rad/sec]

10

1

Figure 6: Gain of Tzw (K  ): optimal FIR (solid) and balanced
truncation (dash), and original Tzw (K)(dash-dot)∞• H -optimal sampled-data IIR controller (designedby the fast-sampling approximation).

[4]
B. Bamieh, M. Dahleh and J. B. Pearson, “Minimizationof the L∞ -induced norm for sampled-data systems,” IEEE Trans.
Autom. Control, AC-38: 717–732, 1993.
[5]
S. P. Boyd and C. H. Barratt, Linear Controller Design,
Limits of Performance, Prentice Hall, 1991.
[6]
T. Chen and B. A. Francis, Optimal Sampled-Data Control
Systems, Springer, 1995.
[7]
P. T. Kabamba and S. Hara, Worst case analysis and design of sampled data control systems, IEEE Trans. Autom. Control, AC-38: 1337–1357, 1993.
[8]
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control systems and tracking problems,” IEEE Trans. Autom. Control, AC-39: 703–712, 1994.
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• reduced order IIR controller of the continuoustime H ∞ -optimal controller obtained by frequencyweighted balanced truncation [10].
Fig. 5 shows the gain comparison of the error Tzw (K) −Tzw (K  ) (see (11)) for the optimal FIR, the optimal IIR andthe balanced truncation. It can be seen that the maximum
gain of the error (i.e. H ∞ norm) with FIR is better than thatwith the balanced controller.
To illustrate the difference in the performance with thesecontrollers, we show the gain of the sampled-data feedback
system Tzw (K  ) and the original continuous-time Tzw (K)in Fig. 6. In the figure, we can see that the system with thebalanced controller has a peak around ω = 0.8 [rad/sec].
On the other hand, the FIR controller does not show such apeak although it shows larger errors in the high frequency
range.
Acknowledgment The authors wish to thank Yoko Koyanagi forher help in computing the numerical example.
References[1]
B. D. O. Anderson and J. P. Keller, “Discretization techniques in control systems,” in Control and Dynamic Systems, Academic Press, 66: 47–112, 1994.
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M. Araki, Y. Ito and T. Hagiwara, “Frequency-response ofsampled-data systems,” Automatica, 32: 483–497, 1996
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B. Bamieh and J. B. Pearson, “A general framework for linear periodic systems with applications to H∞ sampled-data control,” IEEE Trans. Autom. Control, AC-37: 418–435, 1992.

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