SIAM J. CONTROL OPTIM.
Vol. 45, No. 6, pp. 2207–2223
c 2007 Society for Industrial and Applied Mathematics
PERFORMANCE RECOVERY IN DIGITAL IMPLEMENTATION OF
ANALOGUE SYSTEMS∗
GUOFENG ZHANG† , TONGWEN CHEN‡ , AND XIANG CHEN§
Abstract. In this paper, the generalized bilinear transformation (GBT) is proposed. Compared
with the traditional bilinear, zero-order hold (ZOH) and first-order hold transformations, one advantage of GBT is that it may convert unstable poles (zeros) to stable poles (zeros). It is proved
that controllability and observability are invariant under GBT. After that, it is shown that the
performance of a sampled-data system obtained via GBT approaches that of the analogue system
as the underlying sampling period goes to zero. Performance studied here is characterized in terms
of internal stability and p induced norms for all 1 ≤ p ≤ ∞. This results extends the main results
in [G. Zhang and T. Chen, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, Suppl.,
2003, pp. 28–33] and [G. Zhang and T. Chen, Automatica J. IFAC, 40 (2004), pp. 327–330] from
SISO to MIMO and also removes the limitation on the “A” matrix of the system. Finally, an example
is employed to compare digital implementations via GBT and the ZOH transformation.
Key words. fractional-order hold transformation, generalized bilinear transformation, graph
metric, Hankel singular values, performance recovery, unstable zeros
AMS subject classifications. 34K20, 34K25, 46E15, 47A07, 93B03, 93B07
DOI. 10.1137/050643416
1. Introduction. Sampling a continuous-time system is a fundamental problem
in a variety of scientific areas, such as computer control, system identification, and
signal processing. It is becoming even more conspicuous in light of the huge success of
computer-aided processing and networking [1], [2], [6], [13], [22], [23], [18], [24], [35],
[17]. There are many intriguing problems related to sampling. This paper focuses on
the problem of performance recovery in digital implementation of analogue systems.
Consider the continuous-time closed-loop system Σ1 shown in Figure 1.1, where W is
a finite-dimensional linear time-invariant (FDLTI) stable block, and both G and K
are FDLTI. Suppose that Σ1 is internally stable [6, p. 241] and also satisfies some
input-output performance specifications. Now conduct a digital implementation as in
Figure 1.2 by using Kh obtained via some discretization method, with h being the
underlying sampling period. For convenience, we denote the system in Figure 1.2 by
Σ2 . Note that Σ2 is a sampled-data control system. Within this set-up, the following
question could be raised: Under what condition will Σ2 also be internally stable and
have similar input-output properties as those of Σ1 ?
This problem has been studied extensively in recent years. In Chen and Francis
[5], [6], an upper bound for the sampling period h is derived, for which the sampleddata control system Σ2 is p input-output stable; furthermore, it is shown that performance of Σ2 converges to that of Σ1 as h approaches zero. The digital implementation
∗ Received
by the editors October 24, 2005; accepted for publication (in revised form) August 4,
2006; published electronically February 2, 2007.
http://www.siam.org/journals/sicon/45-6/64341.html
† School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang, China 310038
(gfzhang@hdu.edu.cn).
‡ Department of Electrical and Computer Engineering, University of Alberta Edmonton, Alberta
T6G 2V4, Canada (tchen@ece.ualberta.ca). The research of this author was partially supported by
an NSERC Discovery grant.
§ Corresponding author. Department of Electrical and Computer Engineering, University of Windsor, Windsor, Ontario N9B 3P4, Canada (xchen@uwindsor.ca). The research of this author was
partially supported by an NSERC Discovery grant.
2207
2208
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
d
r
W
r1 e zK
−
6
-?
e u- G
y-
Fig. 1.1. Σ1 : analogue system.
r1
rz
p p p Kh p p p ppp H
W - e - S p p p−
6
d
e u- G
-?
y-
Fig. 1.2. Σ2 : digital implementation of Σ1 .
of an analogue controller will generally degrade the performance of the closed-loop
system. Two examples are given in [21] to illustrate how this degradation is quantified
as a function of the underlying sampling period. Some sufficient conditions that guarantee uniform-in-time input-output performance convergence of sampled-data control
systems are proposed in [28]. A new discretization method is proposed in [16] based on
the principle of controller approximation, where an upper bound is derived to ensure
closed-loop stability. Quite contrary to the above, Oishi [25] illustrates via concrete
examples that, in some circumstances, even as the sampling period tends to zero, the
best sampled-data closed-loop performance may not necessarily converge to the best
analogue closed-loop performance.
All the above results are derived for digital implementation of analogue systems
via the step-invariant transformation (the zero-order hold (ZOH) equivalent). Let
Kd and Kbt denote two different versions of Kh , obtained via the step-invariant and
bilinear transformations respectively. By showing that the p induced norm of Kd −Kbt
approaches zero for all 1 ≤ p ≤ ∞ as h goes to zero, Zhang and Chen [38] and [39]
proved that performance recovery of digital implementation of Σ1 via the bilinear
transformation still holds. However, it is assumed in [38] and [39] that G in Figure 1.1
is a single-input–single-output (SISO) stable block, and furthermore, the “A” matrix
is diagonalizable with all real eigenvalues. Actually, the result holds for p = ∞ no
matter whether G is multi-input–multi-output (MIMO) or G is diagonalizable with
all real eigenvalues. One of the main purposes of this paper is to generalize the results
in [38] and [39] to the cases of 1 ≤ p < ∞. The remainder of this paper is outlined as
follows.
In section 2, we propose the generalized bilinear transformation (GBT) that contains the (traditional) bilinear transformation as a special case. Denoting the resulting
digital controller by Kgbt , we explore the relation between the poles and zeros of K
and Kgbt . One advantage of the GBT is that it may be able to convert unstable
zeros or poles to stable zeros or poles. This is quite desirable since unstable zeros and
poles always impose various limitations on system performance [7], [15], [27], [44],
process identification algorithms [19], [30], network communication bandwidth [33],
[11], and stability of networked control systems [40]. Then we show that controllability, as well as observability, is preserved under the GBT. More important, the
GBT provides a class of discretizations. When control engineers have some practical
digital implementation problem at hand, instead of only the ZOH, first-order hold,
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
2209
or traditional bilinear transformations, now they have infinite many choices at their
disposal. Therefore, for a specified control performance, he may select a particular
discretization method by tuning the parameter in the GBT. Clearly, this new degree
of freedom will enhance the capability of a control engineer.
In section 3, we investigate the limiting behavior of Kgbt . More specifically,
we prove the following result: Given K stable, the p induced norm of Kgbt − Kd
approaches zero for all 1 ≤ p ≤ ∞ as the underlying sampling period h tends to
zero. This result holds no matter whether K is SISO or MIMO, thus generalizing
Theorem 1 in [39]. This result plays a key role in the development in section 4.
Based on the results in section 3, we establish in section 4 that all existing stability
and performance results for the step-invariant transformation (the ZOH equivalent)
can be translated into those for the GBT. Using the methods proposed in this paper,
limiting properties and performance recovery of other types of digital implementations
such as the Al-Alaoui transformation can be dealt with in a similar way.
Section 5 is a case study. An example is used to compare digital implementations
of analogue systems via the ZOH transformation and the GBT. It demonstrates that
GBT converts some unstable system zeros (poles) to stable ones, therefore bringing
in a closed-loop system which has much less performance limitations than that using
ZOH.
Our main purpose in this paper is to show performance recovery in digital implementations of analogue systems via the GBT in fast sampling. However, it is
worthwhile to point out that in some circumstance fast sampling is dangerous. For
example, if a quantizer of fixed sensitivity is inserted into a control loop consisting of
a system and an unstable controller, the simulation in [3] demonstrates that system
performance will grow unbounded as the sampling period goes to zero. Hence, in the
framework of network-based control, where signal quantization is prevalent, very fast
sampling will in general degrade control performance [20]. This understanding warn
us that we should be careful in choosing a suitable sampling period when dealing with
practical problems.
Finally, some words about notation. The norm symbol · represents the Euclidean norm if · is a vector, or its largest singular value if · is a matrix; · p is
the p norm if applied to a vector and is the p induced norm if applied to a system.
Following the convention, for a discrete-time transfer function H(z) with a state-space
realization (A, B, C, D), define
A B
:= D + C(zI − A)−1 B.
C D
2. GBT. The bilinear transformation is motivated by considering the trapezoidal approximation of an integrator. Given an integrator 1/s with input u and
output y, the trapezoidal approximation of
kh+h
y(kh + h) = y(kh) +
(2.1)
u(τ )dτ
kh
is
y (kh + h) = y (kh) + h
u (kh + h) + u (kh)
;
2
i.e., the integral is approximated by the average value of u(kh + h) and u(kh). Now
we approximate the integral in (2.1) using some other combination of u(kh + h) and
2210
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
u(kh). More specifically, consider
y (kh + h) = y (kh) + h (αu (kh + h) + (1 − α)u (kh)) ,
(2.2)
where α ∈ (0, 1). The transfer function of (2.2) is (in z transform)
y (z)
αz + (1 − α)
=h
.
u (z)
z−1
This motivates us to introduce the GBT
1
αz + (1 − α)
=h
,
s
z−1
that is,
(2.3)
s=
z−1
1
.
h αz + (1 − α)
Therefore, under the generalized bilinear transformation, the continuous-time transfer
function K(s) in Figure 1.1 is mapped to Kgbt (z), where
z−1
1
(2.4)
.
Kgbt (z) = K
h αz + (1 − α)
In terms of state-space data, take a minimal realization for K(s), namely, (A, B, C, D),
and assume A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m ; it is straightforward to
derive that Kgbt (z) has a state-space model (Agbt , Bgbt , Cgbt , Dgbt ) with
−1
−1
Agbt = (I − αhA) (I + (1 − α)hA) , Bgbt = (1 − α) (I − αhA)
α
α
Agbt , Dgbt = D +
CBgbt .
Cgbt = C I +
1−α
1−α
hB,
When α = 1/2, we recover the traditional bilinear transformation. In what follows,
without loss of generality, it is assumed that the matrix B is of full column rank, i.e.,
rank(B) = m.
(Conversely, given a discrete-time system Kgbt (z) with state-space data (Agbt ,
Bgbt , Cgbt , Dgbt ), one can also get a continuous-time system K, one of whose statespace realizations is
A = −Ω(I − Agbt ), B = ΩBgbt , C = Cgbt (I + αhΩ(I − Agbt )) ,
D = Dgbt − Cgbt αhΩBgbt ,
−1
where Ω = ((1 − α)hI + αhAgbt ) .)
In what follows we study properties of the generalized bilinear transformation.
Let us first look at poles and invariant zeros. Equation (2.3) yields
(2.5)
z=
1 + (1 − α)hs
.
1 − αhs
Hence, if s is an eigenvalue of the matrix A, then z in (2.5) is that of Agbt . The same
is true for invariant zeros [43]. Actually we have a stronger conclusion.
Theorem 2.1. If s is an invariant zero of K, then z in (2.5) is an invariant zero
of Kgbt . Furthermore, if D = 0, then z = − 1−α
α is also a zero of Kgbt .
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
2211
Proof. We begin with the first part. Let s be an invariant zero of K. Then there
exist 0 = x ∈ Rn and u ∈ Rm such that
(A − sI) x + Bu = 0,
Cx + Du = 0.
(2.6)
It suffices to show that
(Agbt − zI) x̃ + Bgbt ũ = 0,
Cgbt x̃ + Dgbt ũ = 0,
(2.7)
where
(2.8)
ũ = u,
x̃ =
1
1+
x.
α
1−α z
In terms of (2.8), the left-hand side of (2.7) is equivalent to
−1
(Agbt − zI) x̃ + Bgbt ũ = (I − αhA) (I + (1 − α)hA) − zI x̃
−1
+ (1 − α) (I − αhA)
hB ũ
−1
= (I − αhA) {(I + (1 − α)hA − (I − αhA) z) x̃ + (1 − α)hBu} ,
α
α
Agbt x̃ + D +
CBgbt ũ
Cgbt x̃ + Dgbt ũ = C I +
1−α
1−α
= C x̃ + Du
α
−1
+
C (I − αhA) {(I + (1 − α)hA) x̃ + (1 − α)hBu} .
(2.9)
1−α
Rewriting (2.6) as Bu = (sI − A) x, −Cx = Du, and substituting it into the first
equation of (2.9), we have
(Agbt − zI) x̃ + Bgbt ũ
−1
{(I + (1 − α)hA − (I − αhA) z) x̃ + (1 − α)h (sI − A) x}
−1
{(I + (1 − α)hA) x̃ + (1 − α)h (sI − A) x − (I − αhA) z x̃} .
= (I − αhA)
= (I − αhA)
Therefore, we need only to show that
(2.10)
(I + (1 − α)hA) x̃ + (1 − α)h (sI − A) x = (I − αhA) z x̃
and
(2.11)
C x̃ + Du +
= C (x̃ − x) +
α
−1
C (I − αhA) {(I + (1 − α)hA) x̃ + (1 − α)hBu}
1−α
α
−1
C (I − αhA) {(I + (1 − α)hA) x̃ + (1 − α)h (sI − A) x} = 0.
1−α
Clearly, x̃ in (2.8) satisfies (2.10). In light of this, the left-hand side of (2.11) is
converted to
α
−1
C (I − αhA) {(I + (1 − α)hA) x̃ + (1 − α)hBu}
C x̃ + Du +
1−α
α
α
−1
= C (x̃ − x) +
C (I − αhA) (I − αhA) z x̃ = C (x̃ − x) +
Cz x̃
1−α
1−α
α
=C
1+
z x̃ − x = 0.
1−α
2212
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
In summary, (2.7) holds.
Next, assuming D = 0, we prove the second part of the theorem. Since D = 0,
(2.7) becomes
(Agbt − zI) x̃ + Bgbt ũ = 0,
α
α
C I+
Agbt x̃ +
CBgbt ũ = 0,
1−α
1−α
(2.12)
which yields
(2.13)
C 1+
α
z x̃ = 0.
1−α
Clearly, z = − 1−α
α satisfies (2.13). Substitution of it into (2.12) gives rise to
(Agbt − zI)x̃ + Bgbt ũ = 0,
(2.14)
x̃ = zhB ũ.
Because B is of full column rank, the dimension of span(B), namely, the space spanned
by the column vectors of B, is m. Hence, there exist vectors ũ1 , . . . , ũm in Rm such
that B ũ1 , . . . , B ũm constitute a basis of span(B). Define x̃i via the second equation
of (2.14), i = 1, . . . , n − m. Then (x̃i , ũi ), (i = 1, . . . , n − m) are solutions to (2.14)
for z = − 1−α
α . The theorem is proved.
Remark 1. When D = 0, s = ∞ is actually a zero of K. By (2.5), z =
−(1 − α)/α is indeed the corresponding zero of Kgbt . In this regard, Theorem 2.1
tells us that the generalized bilinear transformation may generate new zeros which
have no finite counterparts in the continuous-time domain. Following conventional
notation, we call these zeros discretization zeros. Similarly, given a finite zero in the
continuous-time domain, a finite zero in the discrete-time domain can be obtained via
(2.5). We call such zeros intrinsic zeros. Suppose that K is a SISO system whose
transfer function has relative degree n − m > 0 (note that in this case, D = 0).
If K is discretized via the step-invariant transformation, it is proved in [1] that if
n − m > 2, some discretization zero(s) will become unstable at a sufficiently small
sampling period h. Even when n − m = 2, as h → 0, the unique discretization
zero approaches the point (−1, 0) on the plane from outside the unit disk under a
certain condition (Theorem 5 in [14]). As to the first-order hold transformation, for
n − m ≥ 2, there are unstable discretization limiting zeros for sufficiently small h
(Theorem 9 in [14]). As for the traditional bilinear transformation, namely, α = 1/2
in (2.4), Theorem 2.1 indicates that there are n − m discretization zeros at (−1, 0) on
the plane, so the resulting discrete-time system is at most marginally stable. However,
for α ∈ (1/2, 1), z = − 1−α
α is within the unit circle, implying stable discretization
zeros. It is worth noting that discretization zeros are independent of the sampling
period for the generalized bilinear transformation. Therefore, as far as system zeros
are concerned, the generalized bilinear transformation with α ∈ (1/2, 1) is the best.
In the following, we investigate the stability of poles (resp., zeros) of the discretetime system Kgbt obtained from the continuous-time
system K. Suppose that s =
√
a + jb is a pole (resp., zero) of K, where j = −1. Then by (2.5), we have
1 + (1 − α)hs 2
2
2 2
2
2 2 2
1 − αhs < 1 ⇔ (1 + ah(1 − α)) + h b (1 − α) < (1 − aαh) + b h α ,
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
2213
which is equivalent to
(2.15)
a2
2a
< (2α − 1) h.
+ b2
We have the following observations.
1. When α = 1/2, the GBT reduces to the traditional bilinear transformation.
In this case, inequality (2.15) always holds, provided that a < 0; i.e., the traditional bilinear transformation maps stable poles (resp., zeros) of K to stable
poles (resp., zeros) of Kgbt . If D = 0, Remark 1 shows that discretization
zeros are at (−1, 0) on the plane.
2. When α ∈ (0, 1/2), (2α − 1) < 0. Hence, for such a value of α, the resulting
GBT may map a stable pole (resp., zero) of K to an unstable pole (resp.,
zero) of Kgbt . If D = 0, discretization zeros lie outside the unit circle, thus
leading to a nonminimum phase discrete-time system.
3. When α ∈ (1/2, 1), (2α − 1) > 0. Hence, for such a value of α, the resulting
generalized bilinear transformation may map an unstable pole (resp., zero) of
K to a stable pole (resp., zero) of Kgbt . If D = 0, discretization zeros are all
stable.
Now we see that the traditional bilinear transformation stands in the middle of
the GBT; on the one side lie transformations that may convert a stable zero (resp.,
pole) to an unstable pole (resp., zero) (for 0 < α < 1/2); on the other side stand
transformations that may transform an unstable pole (resp., zero) to a stable zero
(resp., pole) (for 1/2 < α < 1). As is well known, unstable zeros or poles are generally undesirable in a control system because they always impose various kinds of
limitations on system performance, such as frequency-dependent constraints on the
complementary sensitivity function of discrete-time systems [7], the achievable transient performance of tracking and rejection to disturbances applied to the plant output
in some servomechanism problem [27], a discrete-time loop transfer recovery procedure [44], and optimal prediction and estimation algorithms for stochastic models
[19], [30]. Therefore, it is always desirable to avoid unstable poles and zeros. In this
regard, the generalized bilinear transformation with 1/2 < α < 1 has advantages over
the traditional bilinear transformation (α = 1/2) because it may be able to eliminate
unstable zeros or poles of an analogue systems provided that the sampling period is
not too small, which is always the case in practice, as is commented on in section 1.
Moreover, in light of Theorem 2.1, discretization zeros generated via the GBT with
1/2 < α < 1 are all stable, which is also an advantage over the traditional bilinear
transformation. Actually, some other types of bilinear transformations have been used
in the literature. For instance, in [4], a general bilinear relationship is applied to the
problem of deriving discrete analogues of continuous singular perturbation and direct
truncation model reduction. We remark that a similar application can be done for
the GBT presented in this paper.
Because controllability and observability are two fundamental properties of any
control system, in the following we establish that the generalized bilinear transformation preserve these two features.
Theorem 2.2. Assume that α ∈ (0, 1). K is controllable (resp., observable) if
and only if Kgbt is controllable (resp., observable).
Proof. We first prove preservation of observability by contradiction. Suppose that
Kgbt is not observable; then there exist a nonzero vector w ∈ Rn and a scalar z ∈ R
2214
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
such that
(2.16)
Agbt w = zw
and
(2.17)
Cgbt w = 0.
There are two possible cases as follows.
Case 1(z = 0). In this case, (2.16) is equivalent to
(2.18)
Agbt w = 0.
Substituting it into (2.17), we have Cw = 0. According to (2.18), (I + (1 − α)hA) w =
1
1
I)w = 0. Clearly, x = w = 0 and s = (α−1)h
satisfy Ax = sx
0, that is, (A − (α−1)h
and Cx = 0.
α
Agbt )w. Then Cx = 0. However,
Case 2(z = 0). In this case, define x = (I + 1−α
α
we need to verify that x = 0. If x = 0, then w = − 1−α
Agbt w. Combining it with
αz
αz
(2.16), we have (1 + 1−α )Agbt w = 0. Since Agbt w = 0 (otherwise z = 0), 1 + 1−α
= 0,
α−1
α−1
i.e., z = α . According to (2.16), (I + (1 − α)hA) w = z(I − αhA)w = α (I −
αhA)w, which implies w = 0 and contradicts our hypothesis. As a consequence,
z−1
x = 0. Define s = h1 1−α+αh
. Then it is straightforward to verify that Ax = sx and
Cx = 0.
The “sufficient” part can be proved by reverting the preceding procedure. Consequently the case of observability is proved. Preservation of controllability can be
readily proved in a similar way.
Remark 2. We have studied such properties of the GBT as the mapping of zeros
and poles and preservation of controllability and observability. Certainly, there are
more to be investigated. For example, Chen and Weller [9] studied the problem of how
the finite and infinite zero structures, as well as invertibility structures, of a general
continuous-time LTI multivariable system are mapped to those of its discrete-time
counterpart under the bilinear transformation. Similar results can be drawn for the
GBT.
Remark 3. From our point of view, the real significance of GBT is that it induces
a class of controller discretizations. The parameter α provides an extra degree of
freedom in the course of digital implementation of analogue controllers. Therefore,
it is possible for a designer to utilize α to achieve better control performance. For
example, it is now possible to adjust α to assign the poles of the digital controller
to some prespecified place. Take step tracking as another example. For a sampleddata system obtained by approximating the analogue controller via either the stepinvariant transformation or the bilinear transformation, it is well known that it is
step-tracking if and only if the analogue system is step-tracking. However, due to
digital approximation, the transient response in the sampled-data system is worse than
that of the analogue system. Transient response is primarily determined by closedloop poles. Hence, with the aid of GBT, we may expect better transient response
by choosing α carefully. In summary, there are a lot of issues to be investigated
regarding GBT. Due to space limitation, in what follows we discuss only one type
of performance, namely, performance recovery characterized by p induced norms of
systems. In section 3, we show that the p induced norms of Kd − Kgbt approach
zero as the sampling period h goes to zero. Then based on this result, in section
4 we prove performance recovery in digital implementation of analogue systems via
the GBT.
2215
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
3. Convergence in p induced norms. Given an analogue system K with a
state-space realization (A, B, C, D), its step-invariant transformation Kd has a statespace realization Kd (z) = (Ad , Bd , C, D), where
h
Ad = eAh , Bd =
eAτ dτ B.
0
In this section, assuming that K is stable, we investigate the convergence properties
between Kd and Kgbt in terms of p induced norms when the sampling period h goes
to zero. Some preliminaries are necessary for further development.
Lemma 3.1. Assume that the pair (A, B) is stabilizable. Then there exists a
1
constant matrix F such that A+BF and 1−α
A+BF (α ∈ (0, 1)) are all asymptotically
stable.
Proof. Since (A, B) is stabilizable, it is well known that there exists a positive
definite matrix P such that
A P + P A − P BB P < 0.
1
Define F = − 12 1−α
B P . It can be verified that
(A + BF ) P + P (A + BF ) ≤ A P + P A − P BB P < 0
and
1
A + BF
1−α
P +P
1
A + BF
1−α
=
1
(A P + P A − P BB P ) < 0.
1−α
1
By the Lyapunov theorem, A + BF and 1−α
A + BF are both asymptotically stable.
Lemma 3.2. If A + BF is asymptotically stable (in continuous time), Ad + Bd F ,
1
Agbt + Bgbt F , and Agbt + 1−α
Bgbt F are all asymptotically stable (in discrete time)
for sufficiently small h.
Proof. The stability of Ad + Bd F can be proved using the power series. Observe
that
h
Ad + Bd F = eAh +
eAτ dτ BF
0
1
Ah
= e + hI + Ah2 + o(h2 ) BF
2
= I + h (A + BF ) + o(h),
where o(h) satisfies limh→0 o(h)
h = 0. Because A + BF is asymptotically stable, so is
Ad + Bd F for sufficiently small h. To prove the stability of Agbt + Bgbt F , in addition
to the power series, Lemma 3.1 is also required. Observe that
−1
Agbt + Bgbt F = (I − αhA)
−1
= (I − αhA)
−1
(I + (1 − α)hA) + (1 − α)h (I − αhA)
(I + (1 − α)h(A + BF ))
= (I + αhA + o(h)) (I + (1 − α)h(A + BF ))
= I + h (A + (1 − α)BF ) + o(h)
1
A + BF + o(h).
= I + (1 − α)h
1−α
BF
2216
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
1
Coupling this with Lemma 3.1 yields the asymptotic stability of ( 1−α
A + BF ) for
sufficiently small h. Hence, Agbt + Bgbt F is asymptotically stable for sufficiently small
1
h. Finally, we show the asymptotic stability of Agbt + 1−α
Bgbt F . Observe that
Agbt +
1
−1
−1
Bgbt F = (I − αhA) (I + (1 − α)hA) + (I − αhA) hBF
1−α
−1
= (I − αhA) (I − αhA + h(A + BF ))
−1
= I + (I − αhA) h(A + BF )
= I + (I + αhA + o(h)) h(A + BF )
= I + h(A + BF ) + o(h).
1
As a consequence, Agbt + 1−α
Bgbt F is asymptotically stable.
Now we have a set-up for the following lemma that is crucial in further development.
Lemma 3.3. In the graph metric [34], [26], Kd − Kgbt converges to zero as the
sampling period h goes to zero.
This result can be easily derived based on the above two lemmas as well as the
proof of Theorem 2 in [39] by replacing
⎤
⎡
Abt + 2Bbt F Bbt
Mbt (z)
=⎣
2F
I ⎦
Nbt (z)
Cbt + 2Dbt F Dbt
with
Mgbt (z)
Ngbt (z)
⎡
⎢
=⎣
1
1−α Bgbt F
1
1−α F
1
Cgbt + 1−α
Dgbt F
Abt +
⎤
Bgbt
I ⎥
⎦
Dgbt
for a right coprime factorization of Kgbt = Ngbt (Mgbt )−1 .
If K is asymptotically stable and α ∈ (1/2, 1), then it follows from Lemma 3.2
that both Kd and Kgbt are asymptotically stable. In this case, the graph metric
induces the same topology as that induced by the H∞ norm. We have the following.
Corollary 3.4. If K is stable and α ∈ (1/2, 1), limh→0+ Kd (z) − Kgbt (z)2 =
0.
With the aid of Corollary 3.4, we are now in a position to prove the main result
of this section.
Theorem 3.5. Assume that K is stable and α ∈ (1/2, 1). Then
lim Kd − Kgbt p = 0
h→0+
for all 1 ≤ p ≤ ∞.
Proof. Observe that
⎡
Ad
⎣
0
Kd (z) − Kgbt (z) =
C
Let
⎡
Ad
Kcl (z) := ⎣ 0
C
0
Agbt
−Cgbt
0
Agbt
−Cgbt
⎤
Bd
Bgbt ⎦ ,
0
⎤
Bd
⎦.
Bgbt
−1
−αhC (I − αhA) B
−1
Dcl := −αhC (I − αhA)
B.
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
2217
Then
Kd (z) − Kgbt (z) = Kcl (z) + Dcl .
By Corollary 3.4,
lim Kcl (z)2 = 0.
h→0+
h
h
, where σ1h ≥ · · · ≥
= σ1h , · · · , σN
Let the Hankel singular values of Kcl (z) be σH
h
σN ≥ 0. According to the discrete-time counterpart of Theorem 7.8 in [43], we have
Kcl (z)1 ≤ 2
N
σih ≤ 2N σ1h .
k=1
As a result,
lim Kcl (z)1 = 0.
h→0+
Clearly, Dcl → 0 as h → 0. Therefore
lim Kd (z) − Kgbt (z)1 = 0.
h→0+
Furthermore, according to the discrete-time version of Theorem 9.1.2 in [6],
Kd − Kgbt p ≤ Kd − Kgbt 1
for all 1 ≤ p ≤ ∞.
The result is proved.
Remark 4. The above theorem is applicable to MIMO systems. The restriction of
the “A” matrix being diagonalizable with all real eigenvalues in [39] is also removed.
Thus it generalizes Theorem 1 of [39]. In this way, along with Theorem 9.4.1 in [6], if
the step-invariant or GBT is applied to an analogue system, internal stability as well
as other performance specifications of the analogue system can be recovered as the
sampling period tends to zero. This is the topic of the next section.
Remark 5. As commented before, when α = 1/2, the GBT in (2.4) reduces to
the traditional one. For convenience, we denote it by Kbt . It can be shown that K(s)
and Kbt (z) have the same Hankel singular values. Unfortunately, the GBT does not
enjoy this nice property.
4. Performance recovery. In this section, we discuss an application of Theorem 3.5. It is well known that internal stability of an analogue control system can
be recovered if the controller is implemented via the step-invariant transformation.
Similar results can be proved for other types of performance specifications. More
concretely, consider the feedback system Σ1 in Figure 1.1. Now we perform digital
implementation using the step-invariant transformation; then Kh in Figure 1.2 becomes Kd . Assume that both G and Kd are FDLTI and strictly causal. Furthermore,
suppose that W is FDLTI, strictly causal, and stable, introduced as a prefilter before
the sampler for later digital implementation. This closed-loop system is said to be
internally stable [6] if the mapping
I
−K
G
I
−1 r
z
:
−→
d
u
2218
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
r1
rz
p p p Kgbtp p p ppp H
W - e - S p p p−
6
d
-?
e u- G
y-
Fig. 4.1. Digital system Σ3 via the GBT.
pp p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp
pp
p
G1 ppp
pp
ppp pp p Kd p p pp
ppp
pp pp
pp
pp
−
pp pp
p?
pp
ep p p pp
ppp ppp
ppp
p
p
pp pp
pp pp
pp
pp pp p K p p 6
pp pp
pp
p
ppp ppp gbt
p
ppp
p
pp ω pp
pξ
p
pp p p p p p pp p p p p p p p p p p p p p p ppp p p p p p p p p ppp
p
p
p
p
d u
p
p?
r
z
?p pp p p Kd p p p p p pe
p pppp H - e
W - e−- S p p G
6
y
-
Fig. 4.2. The equivalent system of Σ3 .
is bounded L2 (R+ ) → L2 (R+ ).
The following proposition will be used later.
Proposition 4.1 (see [6]). Suppose that Σ1 is internally stable and satisfies
suprp ≤1 zp < , for some > 0. Then
(a) Σ2 is internally stable as h → 0;
(b) suprp ≤1 zp < as h → 0.
If we replace Kd in Figure 1.2 with Kgbt , then we get another sampled-data system, Σ3 , as shown in Figure 4.1. We have the following result that is the counterpart
of Proposition 4.1.
Theorem 4.2. This pertains to the configuration of Figure 4.1. Suppose that Σ1
is internally stable and satisfies suprp ≤1 zp < for some > 0. Furthermore,
assume that K is strictly causal and stable. Then
(a) Σ3 is internally stable as h → 0;
(b) suprp ≤1 zp < as h → 0.
In order to prove this theorem, we observe the following fact: Let G be a stable,
FDLTI, discrete-time system. Then
(4.1)
Gp ≤ G1
for all p ∈ N ∪ ∞.
We can now proceed with the proof of Theorem 4.2.
Proof. First we reconfigure system Σ3 in the way shown in Figure 4.2. The system
G1 can be viewed as a perturbation to the system under it, which is exactly system
Σ2 in Figure 1.2. Therefore, if the norm from ω to ξ tends to zero as h → 0, then
according to the small gain theorem, we can ascertain that system Σ3 is internally
stable. Hence, the problem is reduced to showing property (b) for system Σ3 . For
that, and in view of (4.1), it is sufficient to show limh→0 G1 1 = 0, which is given
by Theorem 3.5.
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
2219
5. Discussion on performance limitations of digital implementations of
analogue systems. In this section, we discuss digital implementation of analogue
systems with examples. If an analogue plant G(s) is preceded by a zero-order holder
and followed by an ideal sampler, then one gets its discrete-time counterpart, Gd (z).
Now suppose there is a certain holder as well as some sampler such that the discretetime version of G(s) is Ggbt (z) obtained via (2.4), i.e., the GBT is physically implementable. Next, we will employ an example to compare these two implementations in
terms of various performance limitations of closed-loop systems, such as limitations of
feedback control given by Bode or Poisson integrals as well as bandwidth limitations
and quantization in network-based control or communication.
Suppose that we have a continuous-time system G(s) whose transfer function is
G(s) =
s3 − 3s + 12s − 10
.
s4 − 10s3 − 7s2 + 64s + 60
√
G(s) has three unstable zeros: z1 = 1 + 3i, z2 = 1 + 3i, and z3 = 1, where i = −1.
The poles of G(s) are p1 = 10, p2 = 3, p3 = −2, and p4 = −1. Note that the first
two are unstable. Let the sampling period be h = 0.5. Assume that Gd (z) is the
discretization of G(z) via the ZOH transformation, and Ggbt (z) is that of G(z) via a
GBT at α = 7/8. Then
Gd (z) =
z3
12.76z 3 − 46.34z 2 + 75.95z − 63.65
− 153.9z 3 + 814.3z 2 − 682.2z + 148.4
and
Ggbt (z) =
0.1765z 4 − 0.3691z 3 + 0.1594z 2 − 0.1595z − 0.02719
.
z 4 + 3.163z 3 − 2.656z 2 − 0.744z + 0.5568
It is easy to see that the zeros of Gd (z) are
z1d = 0.8779 + 1.3742i, z2d = 0.8779 − 1.3742i, z3d = 1.8761,
and its poles are
pd1 = 148.4132, pd2 = 4.4817, pd3 = 0.6065, pd4 = 0.3679.
Hence, Gd (z) has three unstable zeros. Note that the relative degree of G(s) is 1; thus
there are no discretization zeros [1]. Gd (z) has two unstable poles, too. The zeros of
Ggbt (z) are
z1gbt = 1.8889, z2gbt = −0.1429, z3gbt = 0.1724 + 0.7356i, z4gbt = 0.1724 − 0.7356i,
and its poles are
gbt
gbt
gbt
pgbt
1 = −3.8000, p2 = −0.4815, p3 = 0.4667, p4 = 0.6522.
It is worthwhile to observe that z2gbt = −0.1429 is a discretization zero, i.e., − 1−α
α =
1−7/8
− 7/8 = −1/7 ≈ −0.1429 corresponding to the zero at ∞ as indicated by Theorem 2.1. Ggbt (z) has one unstable zero as well as one unstable pole. In the following,
we will compare Gd (z) and Ggbt (z) in terms of performance limitations, such as those
characterized by sensitivity and complementary sensitivity functions as well as bandwidth requirement in network-based control.
2220
GUOFENG ZHANG, TONGWEN CHEN, AND XIANG CHEN
The benefit of feedback is that it compensates for the influence of various sources
of uncertainties and disturbances. For example, a sensitivity function quantifies a
system’s ability to attenuate output disturbance, while a complementary sensitivity
function describes a system’s ability to attenuate noise response as well as maintain
stability robustness. Such performance limitations can be characterized by Bode and
Poisson integrals of sensitivity and complementary sensitivity functions. Due to their
fundamental importance, in what follows we compare Gd and Ggbt in this respect.
First we discuss Ggbt . It is easy to show that Ggbt is stabilizable by a strictly
proper stable controller, F , so to speak (see p. 115 of [6] for one choice of such a
controller). Hence, the resulting open-loop transfer function Ggbt F is strictly proper
too. In light of this, according to Theorem 2 of [36] (resp., Theorem 3.3 of [8]),
the Bode integral of the sensitivity function from −π to π is 2π · (log(pgbt
1 )) ≈ 2π ·
1.3350 ≈ 8.3881. As to Gd , it is not easy to find directly a strictly proper stable
controller M that ensures stability of the closed-loop system. However, by mapping
Gd into the continuous-time domain via the traditional bilinear transformation, one
can use Theorem 1 in [37] to design a strictly proper stable controller, say M . Indeed,
the procedure described by Corollary 3 in [37] can be implemented to find such a
controller. Therefore, the open-loop transfer function Gd M is also strictly proper,
and its unstable poles are given uniquely by Gd . According to Theorem 2 of [36]
(resp., Theorem 3.3 of [8]), the Bode integral of the sensitivity function from −π to π
is around 40.8407, which is much bigger than that of the sensitivity function Ggbt F .
Due to space limitation, we omit the calculation of Bode integrals for complementary
functions of Gd M and Ggbt F . We also omit the Poisson integrals for the sensitivity
function and the complementary function of Gd M as well as those of Ggbt F (interested
readers may follow the discussions on pages 74–83 of [29]). However, our calculation
shows that those performance limitations for Ggbt are much less severe than those of
Gd because of the unstable poles and zeros structures.
Signal quantization is widely adopted in network-based control. Given a plant
G, quantization of state space of G will surely affect the control law to be designed.
Insofar as state feedback is concerned, unstable poles turn out to determine primarily
the quantization of the system state space. For example, it is shown in [10] that the
presence of unstable poles causes nonexistence of a stabilizing state feedback law under
any fixed state quantization. A new network data transmission strategy has recently
been proposed, and its dynamics have been analyzed in depth in [40], [41], and [42],
where it is proved that the presence of unstable poles leads to no stabilizing feedback
laws under this new networked control scheme. In [12], a parameter ρ is employed to
describe the coarseness of quantization. In general, the smaller ρ is, the coarser the
quantization is. Coarser quantization indicates that less control effort is required. Now
we compare the optimal ρ values for both cases of the ZOH and GBT transformations.
According to Theorem 2.2 of [12], the coarsest quantizer ρ associated with Gd (z) is
0.6266, while that of Ggbt (z) is 0.5833. Therefore quantization for Ggbt (z) can be
coarser, signifying less control effort.
Furthermore, a Bode integral formula has been generalized to the case of networked control systems. For example, in a Gaussian network, it is shown in [11] that
the Bode integral is equal to some average directed information [32] that describes the
required transmission rate of a stable communication scheme (Theorem 4.6 of [11]).
It turns out that the Bode integral for the sensitivity function is given by unstable
poles of the plant to be controlled via the network. More concretely, for Gd (z), this
PERFORMANCE RECOVERY OF ANALOGUE SYSTEMS
2221
quantity is
ln pd1 + ln pd2 = 6.5000,
while that of Ggbt (z) is
ln pgbt
= 1.3350.
1
Clearly, a lower transmission rate is required if the original continuous-time plant G is
implemented via the generalized bilinear transformation with α = 7/8. Interestingly,
it is shown [33], [31] that the above value is also the minimal bandwidth required to
guarantee the asymptotic observability and asymptotic stabilizability of a discretetime linear system in network-based control, which are argued to be two fundamentally
important concepts in network-based control.
6. Conclusion. In this paper, we have proposed the generalized bilinear transformation (GBT). We have proved that GBT preserves both controllability and
observability. We have shown that, compared with the traditional bilinear transformation, as well as the zero-order hold (ZOH) transformation, one advantage of GBT
is its ability to convert unstable poles (resp., zeros) to stable poles (resp., zeros). We
have also proved that GBT and ZOH converge to one another as the sampling period goes to zero in the sense of p induced norms for all 1 ≤ p ≤ ∞. Finally, we
have established performance recovery of digital implementations of continuous-time
systems via GBT. We hope that the brief discussion of the GBT in this paper can
lay the groundwork for a subsequent study of its possible applications in the fields of
digital and network-based control, and signal processing.
Acknowledgments. The authors are grateful to the associate editor and the
anonymous reviewers for their constructive comments and suggestions.
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