Frequency response of sampled-data systems

Pergamon

0005-1098(95)00162-X

Frequency MITUHIKO

Response of Sampled-data

ARAKI,?

YOSHIMICHI

ITOS and TOMOMICHI

Auromarica, Vol. 32, No. 4, pp. 483-497, 1996 Copyright 0 19% Elsevier Science Ltd Printed m Great Britain. All rights reserved ocm-1098/% $15.00 +o.cKl

Systems* HAGIWARAt

A frequency-domain paradigm for the analysis and design of sampled-data control systems that is exactly parallel to the continuous-time case is established. Key Words-Sampled-data systems; intersample behavior; frequency-domain analysis; frequency response; &-induced norm; sensitivity and complementary sensitivity; performance of digital control.

Abstract-This paper develops a frequency-domain theory that provides a method to analyze and design sampled-data control systems, including their intersample behaviors. The key idea is to consider the signal space %* A {x(t) 1x(t) = x;= __x, exp (j@ + jnw,r), G= ax [Ix, 11z< m}, where w, is the sampling angular frequency. It is shown that a stable sampled-data system equipped with a strictly-proper pre-filter before the sampler maps %* into ZV (=I,) in the steady state. That mapping is denoted by Q(jq) and is referred to as an ‘FR operator’. It is proved that the norm of the sampled-data system as an operator from Lz to L2 is given by maxp ll@(jp)\llz,, where -$w, (3) is obtained (the convergence of the right-hand side is assured, as we show later). Now select one frequency (pn= cp+ nw, (n is an integer) out of the frequency set of (2) and add the sinusoid of this angular frequency, x(t) = x, exp (jqJ)(x,

is a constant vector),

to the system of Fig. 1. In parallel derivation of (3), we can derive heady@)

=

i m=-cc

C(.k%)~(j%)

X 4_i4xn

exp

(4)

to the

~s(@?

(ku).

(5)

Frequency

response of sampled-data

Fig. 1. Open-loop sampled-data

Here we find that the set of frequency components contained in the steady-state output is the same as the set from which the input is selected. This implies that the frequency mapping made by a sampled-data system in the steady state is closed with respect to that set (i.e. the set given by (2)). In other words, the frequency-separation property holds in a generalized sense, if we treat the frequency set (2) as a whole. Our intention is to make use of this fact to establish a ‘frequency-domain theory for sampled-data systems’. 2.2. FR matrices of sampled-data systems To develop a theory in the direction explained above, the range of Q is restricted to Q E &I = (-bs,

h],

x$ x(t)

x(t) =

g n=__X” exp (jQJ), .ZW Il&ll12< m}*

system.

A,&) = A& + jn%), P(jQ) =

A member of Z,, will be called an SD sinusoid of angular frequency Q, where SD stands for ‘sampled-data’. An SD sinusoid is uniquely determined by Q and by the bidirectional series consisting of the coefficient vectors x,: x= (. . . , XT_*,XT,, x;f, XT, XT, . ..)=d2.

K(b)

ym exp(hd

(9)

is obtained, where Ym = n=

2

= WQ + jmw,),

C&Q> = C(jQ + jm4, provided that the series of (10) are convergent. In addition, if the power of ysteady(t) is proved to be finite, it can be claimed that the sampled-data system maps an SD sinusoid of angular frequency Q to an SD sinusoid of the same angular frequency in the steady state. Conceming the convergence of the series (10) and the finiteness of the power of ysteady(t), we have the following result.

Theorem 1. Assume that A(s) is strictly proper, C(s) is proper, A(s) and C(s) do not have poles at s = j(Q + mu,), m = 0, *l, . . . , B(z) is proper and does not have a pole at z = exp (jQr), and H(s) is a finite-response C1 hold. Then, if x E 12, the series of (10) converges absolutely for all m, and the inequality IIyIIi,‘K holds for independent y

=

llxll~~

some nonnegative of x, where (.

* f , yT_*,

y~,,y;f,yT,yT,

(12) constant

*. 3’.

K

(13)

(8)

x will be referred to as the generalized amplitude of the SD sinusoid and treated as an infinite-dimensional column vector. x, will be called the nth. component vector of x and denoted by [xl,,. Now, add an SD sinusoid to the system of Fig. 1. From (5) and the linearity of the system,

2

~~(@‘% (11)

(7)

ysteadyW =

485

(6)

and the signal set ZV is defined as the set of all signals that consist of sinusoidal components with the angular frequencies of (2) and that have finite power, i.e.

I I

systems

Here a finite-response C, hold is defined as a hold circuit whose impulse response is piecewise C1 and terminates in a finite time (see Ap endii A). Note that stability of A(s), C(s) and B (z) is not assumed explicitly in Theorem 1. This is because the mapping (10) can be defined without referring to the steady state, and because we need such a treatment later. The proof is given in Appendix B, together with the proof of Theorem 2 given below. Theorem 1 implies that a linear bounded operator a( j Q) from l2 to l2 (= %J

Q,,(jQh

-m

Q,,(ja> = c,(jQ> *f&&Q)

* P(jQ>

(m, n =O, ll, X2,. . .),

’

A&Q,>

(10)

NQO):

is associated

X-Y

with the sampled-data

(14) system of

486

M. Araki

Fig. 1. This operator can be expressed of an infinite-dimensional matrix Q (jp) Y=

in terms as

et al. Then, for any positive integer N such that

(13

Q(j~)x,

where

IIQW

IIQh)

(16)

The operator o(jp) will be called the FR operator (from x(t) to y(t)) of the sampled-data system in Fig. 1, and Q(jq) the FR matrix of the system or the matrix expression of the FR operator g(jp), where FR stands for ‘frequency response’. Qmn(jq) will be referred to as the m. n component block of Q(jq). The FR matrix Q(jq) of the system in Fig. 1 can be expressed as

Q(k) = c(j~)P(jvMj~)‘. a(jq)

(17)

and c( jq) are

a(jcp)‘r = [. . .

A

I(jp)

Aoh)

C ,(jiF)H

A,t.k)

.I3

t(h)

In view of the above expression, it is evident that Q(jq) becomes a bounded operator if the two series IIA,(jcp)ll, II =O, kl,. , and liC,,,(jcp) H,,,(jcp)ll, m = 0, f 1, , belong to 1:. The proof of Theorem 1 exactly follows this idea. The induced norm IIQ(jq)l/,+ can be calculated as

llQ~kll~+

exists an

- QI.&v)lI121,2< E.

(21)

Qm.v dig)

Q ~,4i~)

QNmdh)

QNdb)

5 K(jqW”’

(22)

IIQ(jp) - Q~NI(j~)III,jI, 5 Kn,axNp”2 holds. This by QrNl(jp) case.

(23)

implies that Q(jq) is approximated uniformly with respect to cp in this

2.3. FR matrices of continuous-time systems and composite systems Consider a stable, finite-dimensional, linear, time-invariant continuous-time system represented by the transfer-function matrix D(s). If an SD sinusoid ,x(t) is added to this system, the steady-state output I _~+d~(t) =

C o(j9 ,,L 7

+ jnw,)x,, exp (j9 + jfiw,)

(24) is obtained. This implies that an FR operator @(jp) can be associated with the continuous-time system D(s) too, and its matrix expression

Q(k)

= [QAj~)>I is given by Q,,,h) = &iv + jnw,> Cm = n), Q,,,Li~) = 0 (m f n),

(25)

or equivalently

Q(b)

= d(jv),

. . ,D-,(j~), &(j~), DI(j9), . .I9 ~A9) = Wjv + h-4) (26) d(jp) = diag [.

= (20)

The above formula is a direct consequence of the next theorem, which asserts that the FR matrix can be approximated by Qt,vl(jp), where QINl(jq) is an infinite-dimensional matrix obtained from the matrix Ql,Vl(jp) of (20) by supplementing zero for i, ; component blocks with ji] > N or ]il > N. Theorem 2. Assume that A(s), H(s) satisfy the assumptions

- Q&j~)IIlzil~

where K( j,) = 2V%K, = 2V%K, K2 /Ip I/. (Note that K, and Kz satisfying (B.4) in Appendix B generally depend on cp. ]]p ]I evidently depends on cp, too.) If A(s) and C(s) do not have poles on the imaginary axis and B(z) does not have poles on the unit circle then K(jq) is continuous on cp E [-hw,, $w,] and so is bounded. Therefore K,,,, = max, K( jq) exists and

h llQ~~r~(.i~)ll~ CFE I,,, (19) = ,‘L,~,r

where Qt,,,,(jq) is the (2N + 1) X (2N + I) matrix at the center of the FR matrix Q(jq):

Qdjd

E, there

Remurk 1. Concerning the dependence of the result of Theorem 2 upon the angular frequency cp, the following is important. By the proof, (21) can be made more exact as

Q(b) =

where

number

b(z), C(s) and of Theorem 1.

We need this FR-operator representation of a continuous-time system when we study general sampled-data systems in the following. Next consider the parallel connection of p stable systems S,, . , S,,. Here S, are sampleddata systems of Fig. 1 with A(s) being strictly proper or continuous-time systems studied above. Then, since each Sj maps EV into EV in the steady state, the parallel connection does the same. It is evident that the FR matrix of the parallel connection is the sum of the FR matrices of s, ( ) s,,.

Frequency

response of sampled-data

Consider also the serial connection of the systems S,, . . . , S,. It is straightforward from the preceding results that the serial connection maps .%‘,into 2’ in the steady state and its FR matrix is the product of the FR matrices of S,, . . . , S,. Feedback connections of systems will be the topic of Section 4. 3. &-INDUCED

NORM OF SAMPLED-DATA SYSTEMS

In this section, the FR operators studied in the preceding section will be related to the L2 operators associated with stable sampled-data systems. The result can be extended to general feedback sampled-data systems, the details of which will be described at the end of the next section. Chen and Francis (1991a) studied L2 stability of sampled-data systems with the structure of Fig. 1. They treated the case when A(s) and B(z) are stable, C(s) = Z, H(s) is the standard zero-order hold, and A(s) is strictly proper, and proved that the output y(t) belongs to L2 if the input x(t) belongs to Lz. Their result can be extended to the case of general C(s) and H(s) as follows (refer to the remark at the end of Section 2 of Chen and Francis 1991a). Theorem 3. Consider the sampled-data system of Fig. 1. Assume that A(s), B(z) and C(s) are stable, A(s) is strictly proper, and H(s) is a finite-response C, hold. Then, if the input x(t) belongs to Lz, the output y(t) also belongs to L2 and satisfies (27) IIY(f) IIL25 K II-Q) IIL2’

where K is a nonnegative of x(t).

constant independent

Based on Theorem 3, a linear operator .Zfrom L2 to Lz,

bounded

2 x(t) -y(t),

X,(jq)

. .lT, [. . . , X-,(j~)T,X,(j~)T,Xl(jrp)T,. = X(jq

+ jnq)

(n = 0, *l,

+2,. . .). (29)

In other words, Xs,(jq) is the vector obtained by vertically lining up the segments of the ordinary Fourier transform in the intervals L

=

((n

-

tbJs,

(n

+

1 Z)%l, II =o,

*l, *2,. . .

.

(30)

487

Xs,(jq) will be called the SD Fourier transform of x(t) E Lz, and Xn(jq) the nth component uector of Xs,(jq). The L norm of the (transient) output y(t) of the sampled-data system in Fig. 1 is given in terms of the SD Fourier transform of the input and the FR matrix of the system as follows. Theorem 4. Consider the sampled-data system of Fig. 1. Assume that A(s), B(z) and C(s) are stable, A(s) is strictly proper and H(s) is a finite-response C1 hold. Then, if the input x(t) belongs to Lz, the output y(t) also belongs to L2, and its norm is given by

IIYwll2,,

(31)

where Q(jp) is the FR matrix from x(t) to y(t), and Xsr,( j,) is the SD Fourier transform of x(t). Proof: The first conclusion is included in Theorem 3. The second conclusion is proved in three steps. Step 1. The set of x(t)s that have bounded supports and that are piecewise-constant is dense in L,. So, it suffices to show (31) under the assumption that x(t) belongs to this set. By this assumption, there is a nonnegative constant K, such that

(This can be proved in the same way as Lemma A of Appendix A.) Also, by the above assumption and by the assumption that A(s) is strictly proper, w(t) of Fig. 1 becomes continuous in (0, ~0) and satisfies

(28)

can be associated with the sampled-data system of Fig. 1. 2 will be called the L2 operator (from x(t) to y(t)) of the sampled-data system. Let X(jq) be the Fourier transform of be the infinitex(t) E L*, and let Xs,(jq) dimensional column vector given by Xsb(jp) =

systems

lim w(t) = 0.

(33)

r-+0

Therefore, by the impulse modulation formula (Jury, 1958), the Fourier transform @(jo) of the impulse series G(t) is given by W(jo) = i _i: W(jw + jnw,) n co =k 2 n

=

A(jw + jnoJX(ju

+ jnwJ,

(34)

where W(jw) is the Fourier transform of w(t). Step 2. Since y(t) E L,, Parseval’s equality can be used: Y*( jo)Y( jw) dw.

(35)

M. Araki et al.

488 By decomposing the interval of integration I,,,, (35) can be rewritten as

into

where Ym(jq) is the mth component vector of the SD Fourier transform of y(t). Step 3. From Fig. 1, the Fourier transform Y(jw) of y(t) is related to that of *(t) by Y(jw) = C(jw)H(jw)B(ejwr)&(jw).

Theorem 5. Consider the sampled-data system of Fig. 1. Assume that A(s), b(z) and C(s) are stable, A(s) is strictly proper and H(s) is a finite-response C, hold. Then the induced norm of the L2 operator Zof the system is given by

II211 L21L2 =

(37)

From (34) and (37).

max

V

Il&(j~)ll~,i~,

(42)

where @(jq) is the FR operator of the system.

Y,(jq) = C(jq + jmw,)H(jq

+ jmw,)B(eJ’P”“““~‘)

1

1 2 A(jq t ,,=~ 7

x

The next result, which asserts that the L,-induced norm of the sampled-data system of Fig. 1 is equal to the H, norm of the associated is obtained from the above FR operator, theorem.

+ jn4X,(jv).

(38)

Proof

In the same way as the derivation of (31) the following equality can be derived:

By the relation eJ”‘W,r= 1 and by (10) and (11). x

Y,,,(jv) =

2

Q,,,,,(j~)X,(jv).

m = 0, *l

I..’

,,:-x

(39) By (32), the series {X,,(jp), II = 0, *l, .} belongs to 12. Therefore, by Theorem 1, the series on the right-hand side of (39) are series and the convergent. absolutely {Ym(j,), m = 0, +l, . .} also belongs to I?. Therefore, the termwise integration is valid:

= t

,,,c I,,

YXjq)y,,(jq)

dq.

(40)

r

By (39) and by (29) and (16), the integrand on the left-hand side can be expressed as

= XX,(j~)Q*(j~)Q(j~)Xs~(j~).

Therefore, the following from (31):

inequality

is evident

I/~11L2iL2 5 max II Q(jv)ll,21,2.

P

(44)

Note that IIQ(jcp)llr,,,, is continuous in &,= k&J,, t4, so that the right-hand side of (44) is well defined; the continuity of ]IQ( jq) [lIZilL follows from (19) because A(s) and C(s) have no poles on the imaginary axis, B(z) has no poles on the unit circle, A(s) is strictly proper and H(s) is a finite-response Ci hold. Next, let us establish the opposite inequality. Suppose that max, IIQ(jcp)llIzilZ is attained at cp= q,,,,, E $,. It follows from (19) and the structure of QtNl that for any E > 0, there exists N and xN E i2 with [x~],, = 0 (InI > N) such that

(41)

Finally, (31) follows from (36), (40) and (41). Cl

IIYNilr,> lIxNlII,

IIQ[N](j~max)l112~12 - E = IIQ~~(j~,,,)ll

(45)

Remark 2. In the above proof, (39) was derived

under the assumption that the input x(t) has a bounded support and is piecewise-constant. But, by using the same argument employed in Step 1. this equality can be validated for the case when the input x(t) is a general member of L,, too. In that case, the infinite sum on the right-hand side may not converge at a set of specific values of cp of measure 0, and the equality should be interpreted as the limit-in-the-mean of the right-hand side is equal to the left-hand side’.

- E.

where yN = Q(jpmax)xN. Since the system is stable and the SD sinusoid x,(t) of angular to xN actually frequency qrnax corresponding consists of finite sinusoid components, x;(t) E L2 can be chosen so that llYNll/*

IIYM)IIL2

IIXN II/,

llxiJ(t) II L2

T satisfies the above inequality for sufficiently large T because of the orthogonality of two sinusoids with different frequencies.) The above two inequalities imply

IIQ~~~(jcpmax)II- 2~ < Noting

IIYmllL,

Ilx%t) IIL2 .

(47)

that IIY~O) IILJ IlxX4 IIL, c IWII L21Lzand

y(t) = Y*(t) + Y*(f),

4. FREQUENCY-DOMAIN

+ k(f),

(48)

Y(f) + 5(t) = 77(f)*

the 0

From this, the following relations among the SD Fourier transforms UsD( jq), Ysn( j,), Rsr,( jq), Dsn(jq) and Zs,(jq) of the signals u(t), y(t), r(t), d(t) and t(t) are obtained:

E+ 0. This completes

FEEDBACK

u(t) = h(l)

(49)

IlQ(j~maxN112~12 5 IWII LzlL2 by letting

489

be expressed in terms of the SD Fourier transforms of signals and FR operators of systems. In the block diagram of Fig. 2, we move the hold circuit H(s) to the left over the summation circle. We then find that the feedback control system is composed of the four parts shown in Fig. 3, where the signals of Fig. 2 and those of Fig. 3 are related by

using (19)

follows proof.

systems

EQUATIONS

Next, let us consider the sampled-data feedback control system of Fig. 2, where r(t) is the reference signal, u(t) and y(t) are the manipulating input and the controlled output of the plant, d(t) is the disturbance entering into the plant, and t(t) is the detection noise. 6(t), t?(t) and ti((t) are digital signals, and are treated as impulse series. The plant and the pre-filters are finite-dimensional time-invariant linear continuous-time systems, so that the transfer matrices P(s), P,(s), F(s) and F,(s) are rational and proper. The digital compensator is a finite-dimensional time-invariant linear discretetime system so that the pulse transfer matrices t’(z) and er(z) are rational and proper, too. It is assumed that the pre-filters F(s) and F,(s) are stable and strictly proper, that the closed-loop system is stabilized, and that the hold circuit H(s) is a finite-response C1 hold. Under the above assumptions, it can be proved, as in the preceding section, that if r(t), d(t) and t(t) belong to L2, u(t) and y(t) also belong to L2 (Chen and Francis, 1991a). This guarantees the existence of Fourier transforms of u(t) and y(t) and allows us to talk about the frequency-domain characteristics of the system. In the following, the frequency-domain relation of the outputs (u(t), y(t)) to (r(t), d(t), t(t)) will

Usn(jq) = C(jq)Rsb(jq) - C(j~)[Ysn(j~)

ydbo) = p(jv)“SD(j~)

+

Wjv))l,

Pd(jv)DSD(jv)p

(W t51)

where C(jq) and C,(jq) are the FR matrices of the sampled-data parts in Figs 3(a, b) given by C(jq) =

Ln(bd17

G444 = fL(jd ‘,&+VXjv) (m, n =o, *l, *2,.

(52)

. ‘),

Wso) = [C,Aj9)1, G&f)

= fL(k) i ~r(@9EAjv)

(53)

(m, n =o, *l, *2,. . .), K&o>

= Wjv

+ bw),

F,(b)

= F(j,

+ h.4,

FAjq) r---------1

L--_-_-_--J

sampler

digital-compensator Fig. 2. Sampled-data

+

feedback

control

prefilter system.

= K(jq + jn&

(54)

M. Araki

490

Fig.

3. Decomposition of the feedback control system. (a) Sampled-data feedback controller. (b) Sampled-data ence controller. (c) Plant model for the manipulating input. (d) Plant model for the disturbance.

and P(jp) and P,(jq) are the FR matrices of the continuous-time parts in Figs 3(c, d), given by

P(jq) = block diag [P,,(jq)], Pm(jp) = P(j,

+ jmw,)

(m = 0, *l.

3~2,.

.), (55)

Pdjv) = block diag [~f.t,l(jv)J3 (m =O, *I, 32..

p,l.,,,(jq)=P,l(jp+jmw,)

= W~PGdj~) - C(j~)P,dj~P%djO (57)

+ PA~Pdjv)

- P(jv)C(j~)&D(j~>7 where

I

is

the

FR

.L,,,tjcp)= 1 - P,,,(.i~W,dj~)fi.(eJ‘V,,(i~). (59)

s,,,,,(jcp) = -P,,,(j~)K(jcp)

matrix

representing

j~(eV%j~)~ W1# II,

where c(z) discrete-time i-(z)

is the pulse transfer function feedback system given by = Qz)[l+

&z)Qz)]

(5X) the

(60) of the

‘,

(61)

i’(z) = Z[F(s)P(s)H(s)]

F(s)P(s)H(s)

(Br is the Bromwich

P(j~F%v)lYdv) = VjvYXvPdjv)

Theorem 6. Let P(jp) and C(jp) be the FR matrices of (55) and (52). Then the FR matrix I + P( jq)C( j,) is invertible almost everywhere in I,,, and the inverse [I + P(jp)C(jp)]’=

---___-

C@Mv>lUdjv)

- C(j~Bdi~).

the equations.

Wv) 2 L,,(.k>l is givenby

It should be noted that the right-hand sides of (50) and (51) constitute infinite series and that these series may not converge at certain values of cp. However, as stated in Remark 2, they have limits-in-the-mean if the system satisfies the assumptions stated at the beginning of this section and the inputs belong to L1. The equalities (50) and (51) should be understood as meaning that the limits-in-the-mean of the right-hand sides are equal to the respective left-hand sides. Equations (50) and (51) will be referred to as the frequency-domain feedback equations of the sampled-data control systems. By substitution, we obtain

[I +

identity mapping on 17. To solve we need the following theorems.

rcfcr-

.). (56)

[I +

et ul.

ds

path)

(Note that p(z) is the pulse transfer matrix the plant seen from the digital compensator.)

(62) of

Thrown 7. Let P(jp) and C(jq) be the FR matrices of (55) and (52). Then the FR matrix I + C( jp)P( jq) is invertible almost everywhere in I,,, and the inverse [I + C(jq)P(jq)] ’=

Frequency S’(jq) 4 [Sk,(jq)]

response of sampled-data

is given by

systems

491

modulation

formula, are more useful (Qy,,mn, are the component blocks of %;;mQY:; : . . respectively, and the arguments jq and da’ are omitted):

m #n.

(64)

Qyd.mn

=

Pd,m- P,,,H,,, ; (I + @)-‘tF,P,, (m = n),

The proof of Theorem 6 is given in Appendix C. Theorem 7 can be proved similarly. Based on the above theorems, the feedback equations can be solved as follows. First, we apply Theorem 6 to (58), to obtain Ysb(jq) =

Q,Av)&djv)

+ Q,djPD)DdjP)

+ Q,.&vPdjvo),

Qd,mn= -

P,,,H,,, b (I+ ep)-‘eF,P,,,

(m#n)

Qys%wl=-P,H,s(Z+

@-‘cF, (70)

(65)

where

Qy&+‘) = Q,&>

s(j~)pd(jvo)F

Q ud,mn=

-H,,, i (I + @-‘eF,,P,,,,

Q uf,mn=

-H,,,;

(71)

(66)

= -Stj4oP(h4C(j~o).

(I + @-‘CF,,.

Second, we apply Theorem 7 to (57), to obtain

+ Q,&v)&djv),

(67)

where

QAv)

= S’(jvo)C(jv),

Qudh’) = Qdjd

-s’(jv)c(jv)pd(jv)y

(68)

= -S’(kNjv).

The FR matrices QJj,), QJjq), . . . will be called the closed-loop FR matrices from r(r) to y(t), from d(r) to y(l), . . . of the sampled-data feedback control system. By substituting (65) into (50) or (67) into (51), we can obtain other expressions for the closed-loop FR matrices. For instance, QJjq) can be expressed as

Q,4~)

= Wv)S’(jrpYXv).

(69)

The expressions (66) and (69) for the FR matrices are useful for studying the performances of the sampled-data control systems, as will be found in the next section. However, the right-hand sides of these equations result in infinite sums, and so they are not necessarily convenient for calcaluting the component blocks of the closed-loop FR matrices. For such purposes, the following equations, which are obtained from (66) by applying the impulse

Remark 3. We can easily see from (70) and (71) that the component blocks of the closed-loop FR operators Q,,, QydpQyt, Q,,, Qud and Quz have the same form as that in (10) given for the open-loop case, except that Qd has an additional term corresponding to (25) because there is a ‘continuous-time path’ D(s) = Z from d to y. It is generally true that any closed-loop FR-matrix of the system in Fig. 2 can be represented as

QW

= c(b) . P(k).

a(MT + d&o) (72)

by suitably defining A(s), B(z), C(s) and D(s) (note that the corresponding b(z) is stable, by the assumption of closed-loop stability). If we replace A(s), b(z) and C(s) with these corresponding (pulse) transfer-function matrices introduced as appropriate, Theorems l-5 are true also for the closed-loop system in Fig. 2 mutatis mutandis.

Theorem 1 is true if the following additional condition is satisfied: (Al) 0, *l,.

D(s)

has no poles

at s =jpm

(m =

. .).

Theorem 2 is true if Al and the following additional condition are satisfied: (A2) D(s) is strictly proper. Similarly, the arguments in Remark 1 are ture

M. Araki et al.

492

if A2 and the following additional (which includes Al) are satisfied:

condition

(A3) D(s) has no poles on the imaginary axis. Although Theorem 2 is not true when A2 is not satisfied (because Q(jq) is not compact in that case), (19) is still true if Al is satisfied. (This follows basically from (B.17) in Appendix B). Theorems 3 and 4 are true for the sampleddata system of Fig. 2 even without the stability assumptions of A(s) and C(s) (note that closed-loop stability is assumed throughout this section). Theorem 5 is true for the sampled-data system of Fig. 2 if the stability assumptions of A(s) and C(s) are replaced by l

(A4) A(s) and C(s) have no poles on the imaginary axis and if A3 is also satisfied. All of the above can be proved by suitably modifying the arguments in the open-loop seeting, and the details are omitted. 5. SENSITIVITY COMPLEMENTARY

FR OPERATOR y(jcp) AND SENSITIVITY FR OPERATOR .%jv)

For the sampled-data feedback control system of Fig. 2, defined the sensitivity FR operator P’(jq) and the complementary sensitivity FR operator F(jq) by their matrix expressions as follows:

PhNjv))l II (73) T(b) = [I+ P(j~)C(j~)>l-‘P(jcP)C(j~). S(jp) = ]I +

By Theorem 6, the component blocks of S(jp) are given by (59) and (60). From the above definition, S(jq) + T(jq) = I

(74)

is evident. From this relation, the following expression for the component blocks of T(jp) is obtained: T,, = P,,&)K,(j~)

i e(e?F,(jq).

(75)

In the following, we show that Y(j,) and F(jq) have parallel meanings to the sensitivity function and the complementary sensitivity function of continuous-time control systems. The absolute and relative modeling errors of the plant will be denoted by AP(s) and A(s) respectively; i.e. the transfer matrix of the real plant is assumed to be as follows. where we assume that A(s) is stable: P,,,,(s) = P(s) + AP(s) = [I + iz(s)

(76)

First, let us consider the sensitivity of the transfer characteristics. As the representation of the transfer characteristics, we use the closedloop FR operator Q,,(j,) from the reference to the controlled output. Cruz and Perkins (1964) considered the open-loop control system of Fig. 4. In the same way as in the derivation of (69), it can be shown that, when P,(s) = P(s), the FR operator from r(t) to yO(t) of this system is equal to Q,,(jq) of (69). Therefore we can use this sytem as the open-loop equivalence to the closed-loop control system of Fig. 2 when d(t) = 0 and t(t) = 0. Let us compare the variations of the controlled outputs of the open-loop system (Fig. 4) and the closed-loop system (Fig. 2) when the plant varies as given in (76). The variation AYs&jq) of the output of the open-loop system is given by AYs&jq)

= AP(j~)Usb,O(j~)

(77)

where AP(jq) is the FR operator to the absolute error AP(s):

corresponding

AP(jp) = block diag [. . . ,AP_,(jp),

AP,(jq),

Apdjv)

= Af’h

+ jm4,

(78)

and U,&jq) is the SD Fourier transform of the manipulating input of the open-loop system of Fig. 4. On the other hand, from (50) and (51), the following equations are obtained for the variations AUs, and AYs,,(jq) of the manipulating input and the controlled output of the feedback control system of Fig. 2: AUs,

= -C(jq)AYsp(jq),

AYsrJjq) = P&j~)AUsp(j~)

+ AP(jq)Usp(jq), (79)

Therefore AYsb(jp) = ]I +

~,,,~~jcp)~(j~o)l~‘~~~jcp~~~~~jcp~. (80)

Here Us,(j~) is the SD Fourier transform of the manipulating input of the feedback control system when the transfer function of the plant takes its nominal value P(s). Since the system of Fig. 2 and that of Fig. 4 are equivalent when P,(s) = P(s), we have Usb,O(j~) = Us&q). So, by comparing (77) and (SO), we can conclude that the improvement (or deterioration) of the sensitivity due to the feedback control is given by [I + Prea,(j~)C(j~))]-‘. This means that we can use S(jp) of (73) as the evaluation of the sensitivity improvement at the nominal point. Second, let us consider the ability of rejecting

Frequency

response of sampled-data

L__-_______-

systems

493

-____

open-roop cf&r&r- - - - - - - - ’ (P(s) ia 5xed to ita nominal value) Fig. 4. Open-loop equivalence to the control system of Fig. 2.

the disturbance. From Fig. 2, we find that the disturbance d(t) effects the controlled output ~0) as Ysb(jq) = PAjp)Dsb(jq) (81) if the feedback control is removed. By comparing (81) with (65) and (66), we conclude that the disturbance-rejection ability of feedback control is represented by S(jq). Third, let us consider robust stability. When the modeling error of (76) exists, the control system becomes as depicted in Fig. 5, where only the essential parts are shown explicitly. In order to study the stability of Fig. 5, let us consider the system I: obtained by breaking up the dotted line between y(t) and yl(t) in Fig. 5. The system Z has the input y,(t) and the output y(t). By identifying t,(t) of Fig. 5 with t(t) of Fig. 2, the system Z can be regarded as composed of the system of Fig. 2 (with nominal P(s)) and the block A(s) added at the tail of the t(t) line. Since we assume that the system of Fig. 2 is stable for the nominal P(s) and that A(s) is also stable, the system Z is stable. The FR matrix from yl(t) to y(t) of the system C is

Q,&W&)9

where

WQP) =

block diag [. . . , A-,&),

&(jv), h(k), . . . I, b&P) = A(jQf jm%). (82)

By Theorem 5, the norm of the L2 operator from yl(t) to y(t) of Z is given by II%IIL.*IL_* = FEY 11Q,&Qo)A(jQ)

11/2//2.

_‘&

(83)

Now, the system of Fig. 5 (with the dotted line

H(8)

P(s)

v(t)

+’ +

e(s) T

FM

Fig. 5. Sampled-data feedback control system with modeling error.

closed) is obtained by applying the unity feedback around the system Z, and, therefore, its L2 stability is guaranteed by the small-gain condition:

lI~IILtIL* < 1.

W)

If we compare (66) with (73), Q,,(jQ) = -T(jQ). From this and from (83), we conclude that the system of robustly L,-stable with respect to the error (76) if IIT(jQ)A(jQ)llrjr,<

Based

on

COrT@XUXItary

(85),

we

1

can

SenSitiVity

we find (84) and Fig, 2 is modeling (85)

VQ E 10.

conclude OpeITitOr

r(

that j Q)

the rep-

resents the degree of robust stability. Fourth, we must show that T(jQ) gives the effect of the detection noise, but this is evident from the equality &(jQ) = -T(jQ). 6. REMARKS

6.1. On disturbance rejection and robust stability In the previous section, it was shown that Y( jQ) and Y(jQ) possess parallel properties to the sensitivity and complementary sensitivity functions of continuous-time control systems. Here we should note the following points, which are reflections of the essential nature of sampled-data control and which give important indication about the formulation of sampled-data design problems. First, concerning the disturbance-rejection ability represented by Y( jQ), IIs(jQ)ll/,,,,2

1

VQ E lo

(86)

follows from the fact that the diagonal component block S,,,,(jQ) converges to the identity matrix. This means that if we measure the disturbance-rejection ability without a frequency-dependent weight (which takes the characteristics of the disturbance itself and also those of the anti-aliasing filter into consideration) then we always reach the conclusion that no improvement is made by sampled-data feedback at any angular frequency Q E IO. This is a natural consequence of the fact that sampleddata control can only deal with slow phenomena and cannot suppress a high-frequency distur-

M. Araki

494

bance, which is converted into the low-frequency range by the frequency-folding nature of the sampler if no anti-aliasing filter is equipped. Second, concerning the robust stability condition (85) given in terms of Y(jq), the requirement becomes inhibitingly strict if we separate T(jq) and A(jq) as lIT(j~)ll~lil,<

[llNj~)II~~I~J

’ vq E 4,. (87)

This is caused by the fact that the modeling error /IA(j becomes very large for large w, and hence IIMb> IIW? becomes very large for every cp E lo. 6.2. Relation to the Goodwin-Salgado method Goodwin and Salgado (1994) treated the case e,(z) = C(Z) and t(t) = 0 in Fig. 2, and derived the following frequency-domain relation between y(t) and (d(t), r(t)): Y(jw)

= &.4j~)~cis(eJw) - &s(jw) X D(jw

+ jn4

+ jnq),

(88)

where &s@J) = H(jw)P(jw)t(ej”‘)[Z

+ B(eiw’)t(e’“‘)]

‘, (89)

&djW) = 1 - &djw)F(jw) &s(z)

=

Z[F,@Y+)l.

(90)

The above relations correspond exactly to (the expanded form of) (65) with Z( j,) = 0. Such a treatment is powerful enough to analyze the influences of the reference and the disturbance on the plant output in the frequency domain, and has the advantage of simplicity. On the other hand, our treatment has the advantages that the robust stability can be dealt with in the same framework, that sP(jp) and .Y(j,) can be defined so as to possess exactly parallel meanings to the continuous-time case, and that the H, and HZ control problems can be formulated in the frequency domain (suggested below: see also Section 7). 6.3. H, and Hz design problems In our framework, the H, and Hz problems can be formulated as follows. As an example, consider the problem of minimizing the influence of the disturbance. Let the L, operator from the disturbance d(t) to the controlled output y(t) be denoted by ZYd. then, by Theorem 5,

II=Yycr IIL~u.~= y;

IIQ_d_i~) II,+.

Thus the equivalent

problem to the

(91)

of minimizing Il.&~l,.~,,,, is mini-max optimization of

IIQ,&~>Il~Z~~2= IIS(j~,>P~~l(j~)ll~~,,,over cpE lo, or, in other words, the minimization of the H, norm of the weighted sensitivity operator S(jp)P,(jq). Thus the &-type problem formulated previously using the induced norm of Lz operators (Bamieh 1992; Hayakawa et al., 1994: and Pearson, Kabamba and Hara, 1993) recovers its original meaning as the ‘mini-max’ optimization in the frequency domain. If we view the problem in this way, the advantages of multirate digital con1986) and trollers (Araki and Hagiwara, generalized sampled-data hold functions (Chammas and Leondes, 1978; Kabamba, 1987) can be understood in a transparent way. Namely, by using such controllers, we obtain the freedom of adjusting the factor H,,(jq) in (59) and (60) and thus gain the possibility of attaining a smaller value of min [max,

+ &.(ju)D(jm)

c F(jw II- ~,~r*o

et al.

II~~jcp>~~~j~)Il~~~~,l.

6.4. Extension of the FR operator to the Laplacr domain The idea of SD-Fourier transformation employed in Section 4 can be extended to the Laplace transformation. Namely, define the SD Laplace transform of a signal x(t) t

x

= {x(t) I x(t)e

‘I’ E L2 for some real number

as the infinite-dimensional given by

x,+)

vector Xsn( U)

= 1. , x_,(~j’, Xo(ajr, X,(Q)1, . . .I’,

X,l(o) = X(a where

column

c,}

(n = 0, *1, &2,

+ j,,*)

u belongs

.), (92)

to

J,, = {CT1- 4 W, < lm (T c iw,}. Then, in parallel to the derivation Remark 2), the following equality for the system in Fig. 1: Y,,(a)

=

Q(~Fsd~),

(93)

of (39) (note can be proved

(94)

where Q(a) is the analytic (in the elementwise sense) extension of the matrix Q(j,) from the segment IO of the imaginary axis to the strip J,, of the complex plane. This type of equation can be used for analyzing the transient behavior of sampled-data control systems such as the one given by Fig. 2.

7. (‘ONCLUSIONS

We have defined a signal space 2, by (7) and have shown that a stable sampled-data system with a strictly proper pre-filter before the sampler maps ,%q into %+ (Theorem 1). This

Frequency

response of sampled-data

mapping was termed the FR operator a(jp) of the sampled-data system, and the infinitedimensional matrix Q(jq) representing @(jq) the FR matrix. The FR operator completely characterizes the frequency-domain property of the sampled-data system by (15). It has been shown that the induced norm of the sampleddata system as an operator from L2 to L2 is given by max, I[@(jq) Ilr,,l, (Theorem 5). This tells us that the FR operator satisfies the basic relation between the frequency-domain and the timedomain characterizations that ‘the induced norm as a mapping in the time domain is equal to the supremum of the norm of the frequency transfer matrix’, and can be regarded as the exact sampled-data counterpart of the frequency transfer function of a continuous-time system. The above results have been used to define sampled-data versions of the sensitivity function and the complementary sensitivity function, which have exactly parallel properties to the continuous time case, such as y( j,) + y( jq) = S. They are obtained by solving the frequencydomain feedback equations (50) and (51) for the sampled-data system of Fig. 2 in an explicit form. How to design sampled-data control systems using the above results has also been explained briefly. After the first versions of this paper were presented, substantial advances were made on various problems of sampled-data systems, using the theory developed in this paper. Some of these are listed here. For the purpose of computing the norm of the FR operator, the direct computation of the right-hand side of (19) does not seem efficient. A more refined method has been obtained in Hagiwara el al. (1995) based on the relation (19) and an ZZ, problem of sampled-data systems is also solved there. Its importance lies in that it enables us to give the frequency response gains at each frequency cp in a closed form, rather than only the L,-induced norm (as in e.g. Chen and Francis, 1990; Bamieh and Pearson, 1992) given as the maximum of the frequency response gains over cp E I0 (see (42)), whereas a general treatment of this problem was already given by Yamamoto and Khargonekar (1993). ZZ2 problems of sampled-data systems with both continuous-time and discrete-time inputs and outputs are formulated and solved in Hagiwara and Araki (1995a), where the extended notion called the hybrid FR operator plays an important role. Robust stability problems under possibly unstable linear timeinvariant/ r-periodic perturbations are solved in Hagiwara and Araki (1995b).

systems

495

Acknowledgement-The authors are grateful to Professor Y. Yamamoto for valuable discussions and comments. REFERENCES Araki, M. and T. Hagiwara (1986). Pole assignment by multirate sampled-data outout feedback. ht. J. control, 44. 1661-1673. . Araki, M. and Y. Ito (1992). On frequency-response of samoled-data control svstems. In Proc. 21st SICE Symp. . . on Control Theory, Kahya, pp. 19-24. Araki, M. and Y. Ito (1993). Frequency-response of sampled-data systems I: open-loop consideration. In Preprints 12th IFAC World Congress, Sydney, Vol. 7, pp. 289-292. Araki, M., T. Hagiwara and Y. Ito (1993). Frequencyresponse of sampled-data systems II: closed-loop consideration. In Preprints 12th IFAC World Congress, Sydney, Vol. 7, pp. 293-296. Bamieh, B. and J. B. Pearson (1992). A general framework for linear periodic systems with applications to H, sampled-data control. IEEE Trans. Autom. Control, AC-37,418-435.

Chammas, A. B. and C. T. Leondes (1978). On the design of linear time invariant systems by periodic output feedback. Part I. Int. J. Control, 27. 885-894. Chen, T. and B. Francis (1990). On the &-induced norm of a sampled-data system. iyst. Control Leti, 15,211-219. Chen. T. and B. A. Francis (1991a). Irmut-outout stabilitv of . sampled-data control systems. IEEE Trans. Autom. Control, AC-36,50-58. Chen, T. and B. A. Francis (1991b). H, optimal sampled data control. IEEE Trans. Autom. Control, AC-36,387-398. Cruz, J. B. and W. R. Perkins (1964). A new approach to the sensitivity problem in multivariable feedback system design. IEEE Trans. Autom. Control, AC-9, 216-223. Dorato, P. (ed.) (1987). Robust Control. IEEE Press, New York. Goodwin, G. C. and M. Salgado (1994). Frequency domain sensitivity functions for continuous time systems under sampled data control. Automatics, 30. Hayakawa, Y., S. Hara and Y. Yamamoto (1994). H, type problem for sampled-data control systems-a solution via minimum energy characterization. IEEE Trans. Autom. Control, AC-39,2278-2284. Hagiwara, T. and M. Araki (1995a). FR-operator approach to the Hz analysis and synthesis of sampled-data systems. IEEE Trans Autom. Control, AC-40,1411-1421. Hagiwara, T. and M. Araki (1995b). Robust stability of sampled-data systems under possibly unstable additive/ multiplicative perturbations. In Proc. American Control ConjI, Seattle, WA, pp. 3893-3897. Hagiwara, T., Y. Ito and M. Araki (1995). Computation of the frequency response gains and HE-norm of a sampled-data system. Syst. Control Lett., to appear. Jury, E. I. (1958). Sampled-Data Control Systems. Wiley, New York. Kabamba, P. T. (1987). Control of linear systems using generalized sampled-data hold functions. IEEE Trans. Autom. Control, AC-32, 772-783. Kabamba, P. T. and S. Hara (1993). Worst-case analysis and design of sampled-data control systems. IEEE Trans. Autom. Control, AC-38,1337-1357. Khargonekar, P. P. and N. Sivashankar (1991). Hz optimal control for sampled-data systems. Syst. Control Lett., 17, 425-436. Sun, W., K. M. Nagpal and P. P. Khargonekar (1993). H, control and filtering for sampled-data systems. IEEE Trans. Autom. Control, AC-38,1162-1175. Tadmor, G. (1992). H, optimal sampled-data control in continuous time system. Znt. J. Control, 56,99-141. Toivonen, H. T. (1992). Sampled-data control of continuoustime systems with an H, optimality criterion. Automatica, 28,45-54. Yamamoto, Y. (1994). A function space approach to sampled-data control systems and tracking problems. IEEE Trans. Autom. Control, AC-39, 703-713. P.

496

M. Araki

Yamamoto, response

Y. and P. P. Khargonekar (lYY3). Frequency of sampled-data systems. In Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, TX. pp. 799804 (to appear in IEEE Trans. Autom. Control). Yamamoto, Y. and M. Araki (1994). Frequency responses for sampled-data systems-their equivalence and relationships. Linear Algebra Applic., 205/m, 1319-1339.

APPENDIX

A-FINITE-RESPONSE

c’i HOLD

A multidimensional hold circuit is a system series of impulsive vectors o(r) = c

k ,I

u,6(r

that converts

a

=

2

Schwarz’s used, where

inequality for real numbers I is a set of integers:

.- kr)

Proof

I. By assumption on g(z), ilp /I is finite. on A(s) and C(S), and by Lemma A on the C, hold.

of Theorem

1

(A.1

kr)u,,

-

h,, is also

Since the value of cp is fixed in Theorems I and 2, the argument j, is generally omitted. The capital KS (e.g. K, K,, Kz.. .) are nonnegative numbers, which may depend on cp-

IV,,

h(t

a,, and

(R.3)

By assumptions finite-response

into a vector-valued piecewise-continuous function u([). I1 linearity and time invariance are assumed, the relation of 14(f) to the uk can be generally expressed as 14(r)

et ul.

(A.21

are satisfied

II 5 K,f(n).

Il~dLlI

5 K&m)

llQ,,,,ll 5 Kd(mIf(n), where

(B.3)

for some K, and K,. Therefore

lipll. By (B.5),

K1 = KIKz

(B.5)

(B.3) and (B.2),

,4-0

where h(t) = [hJt)] is a matrix-valued finite-response C, hold if h,,(r) = 0

for

function.

it is called a

I @(0, 7;,]

h(t)c

” dt

(A.4)

I’0

the following

lemma

I/Q,&,~ II 5 Kd(m)

5

A.

nonnegative

For a constant

c‘,

finite-response K,, such that

hold.

there

is

llylI,,~fi&

(A.5)

1).

B-PROOFS

OF THEOREMS

the theorems,

Assume

the function

l ’ i lnl~

N’ 2 N +

Qtk)x.

f(n)

constant

K,,.

I AND

(B.‘))

, _ x,, (In N),

(B-10)

0 (InI W

1

y = Q’x’

f‘(n)= is used. that

2. Let

Then y of (B.9) can be expressed

kt”‘dt

(A.5) holds for some nonnegative

To prove defined by

llxll,~ 5 SK3 IIxII,~. tB.8)

L-et N be an integer greater than 1, and decompose x into two parts as x = x’ + XI’, where the component vectors of x’ and x” are given by

p

APPENDIX

(B.7)

a

(A.6) Therefore

and the

By (B.7) and (B.2)

Proof: Note that the integral of (A.4) is actually a finite-interval integral. since h(r) = 0 outside (0. 7;,]. Decompose [0, 7;,] into subintervals [I,, I(+ ,I, I - I, . I,, so that all h,-(t) are continuously differentiable in each subinterval. We then have

(“” h(r)e

I/x//I~.

(iY,~,II5 fi&./‘(m)

Y=

(lwl 5 1)

r-1

(B.6) convergent

holds.

(lwl?

H(jw) = i

*KJ(m) IlxllI,.

Therefore the series of (LO) is absolutely norm of .v,, satisfies

Pro@ of Theorem Lemma

i f’(n1 II& II I,- 1

5- K3f(m)

(A.3)

and the h,-(t) are piecewise-C,. The finite-response C, hoLd includes most of the hold circuits, such as the standard zero-order hold. the first-order hold and the generalized sampled-data hold (Chammas and Leondes. 1978: Kabamba. 1987). Concerning the transfer-function matrix H(s) of the finite-response C, hold given by H(s) =

i

,! = w

as

+ Q’x” -t Q”x,

where Q’ and Q” are FR matrices are given by

whose

(B.1 I) component

blocks

0

2

of an integer

II

In (B.11). the first term Q’x’ actually entries, and turns out to be

contains

finite nonzero

(n = 0). (i-f # 0)

(B.1)

1S 2. Then we can easily verify

Q’x’ = Q,,v,x. On the other

hand.

to prove

llQ(jv)x

(21), it is enough

(B.13) to show that

- Q[,~I(Jv)xtIIz < E 11xlL2

(B.14)

holds for any x E I,. Therefore. in view of (B.11) and (8.13). the proof is complete if llQ’x”II,, and (lQ”~ll,~ are majorized numbers by ejyN) 1Ixll/,r where 0(N) expresses nonnegative that converge to zero as N tends to x.

Frequency

response of sampled-data

First, consider IIQ’fIl,,. By definition of x” and Q’, [Q’x”],

=

Qmn~

,&

i 0

(lml s N),

APPENDIX (B.15)

(Iml ‘W

By (B.15), (BS), (B.3) and (B.2),

lIIQ’~lm II2 5 [ ,&

Kdm)f(n)

IL II]’

497

C-PROOF

OF THEOREM

6

The argument j, is omitted unless needed explicitly. Since multiplication of FR matrices is associative, it is enough to show that (I + PC)S = I and S(I + PC) = I for S given by (59) and (60). We only prove the first equality, because the second can be proved in parallel. By the assumptions stated at the beginning of Section 4, P(s), P&) and H(s) do not have poles at s = j, + jnrus, and c(z) and e,(z) do not have poles at z = exp (jq) for almost all values of (p in I,. Therefore P,(jp), P&jq), H,(jp), t?(ejq’) and e(ej-3 have finite values almost everywhere in 1,). Let p and fT be the infinite-dimensional column and row matrices respectively, given by

Therefore

IlQ’~“ll,~ = (,$,

systems

II~QWmi,2)“2

I

P_,H_, P=

i

PoHo

, f’=[..

.)

F-,,

‘F”,

&,

.].

(C.1)

1 IJ 4 HI

(B.17)