Model validation of multirate systems from time-domain experimental ...
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Model Validation of Multirate Systems From Time-Domain Experimental Data Li Chai and Li Qiu Abstract—The model validation problem using time-domain experimental data is studied for multirate linear fractional uncertain models in this note. As a technical tool, the Carathéodory–Fejér (CF) interpolation problem with a nest operator constraint is first investigated. This problem is itself of interest mathematically and has potential applications in addressing other problems in control, signal processing, and circuit theory. A necessary and sufficient solvability condition for this interpolation problem is given. The validation tests are then presented based on this condition and the lifting technique. Tractable convex optimization methods can be used to solve the validation problems. Index Terms—Carathéodory–Fejér (CF) interpolation, model validation, multirate systems.
I. INTRODUCTION Multirate systems, i.e., digital systems with signals having different sampling rates, have wide applications in control, communication, signal processing, econometrics and numerical mathematics. There are several reasons for this. • In large scale multivariable digital systems, often it is unrealistic, or sometimes impossible, to sample all physical signals uniformly at one single rate. In such situations, one is forced to use multirate sampling. • Multirate systems can often achieve objectives that cannot be achieved by single rate systems [1], [2]. The study of multirate systems goes back to late 1950s [3]. A renaissance of research in multirate systems has occurred since 1980 in control community, signal processing community and communication community. The driving force for studying multirate systems in signal processing comes from the need of sampling rate conversion, subband coding, and their ability to generate wavelets. Multirate signal processing is now one of the most vibrant areas of research in signal processing, see recent book [2] and references therein. In control community, two groups of research stand out: using multirate control to achieve what single rate control cannot as well as the limitation of doing this [1] and the optimal design of multirate controllers [4], [5]. In communication community, multirate sampling is used for blind system identification and equalization [6]. We also notice the cross discipline fertilization between signal processing and control in using H optimization to design filter banks [7], [8]. In this note, we will study the control-oriented model validation problems pertaining to the general multirate systems. There has recently been considerable research devoted to robust or control-oriented model validation [9], [10]. However, the research on model validation of multirate systems is almost nonexisting. Due to the wide applications of multirate systems, their model validation problem should receive comparable attention to those of single rate systems. Model validation is a very important step in the process of control system modeling both in traditional stochastic setting and nonproba-
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Manuscript received February 14, 2000; revised October 16, 2000, March 29, 2001, and September 6, 2001. Recommended by Associate Editor Y. Yamamoto. This work was supported by the Hong Kong Research Grants Council. It was completed when the authors were visiting the National Laboratory of Industrial Control Technology, Zhejiang University, China. The authors are with the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong. Publisher Item Identifier S 0018-9286(02)02078-0.
Fig. 1.
A general multirate LFT uncertain model.
bilistic context. Generally, the model validation problem is to examine if the sets of experimental data are consistent with the model of the plant. A model is said to be invalidated when the validation test fails. Our confidence in the model set is increased if the model is consistent with the data. Since a model is either invalidated or not invalidated, it is actually more accurate to call the validation procedure as model invalidation. More recently, motivated by the considerable research on control-oriented system identification, much attention has been paid on validation of uncertain models consisting of a nominal model and a norm bounded modeling uncertainty [11], [12], [10]. Such uncertainty models are the starting point for robust control. The first study of model validation for linear fractional transformation (LFT) model-sets was carried out by Smith and Doyle [10]. They show that the problem can be solved by a structured singular value type method. Chen [11] considered the general validation problems of linear fractional uncertain models in frequency domain and reduced it to the Nevanlinna-Pick interpolation problem, which can be solved by standard convex optimization methods. Based on the Carathéodory–Fejér (CF) interpolation problem, a purely time-domain formulation for models with an additive uncertainty is presented in [12]. It is shown that the problem can be solved as a convex program involving linear matrix inequalities (LMI). The time domain validation approach in a more general setup which is for LFT uncertain model-sets is studied in [13]. The similar setup is also used to consider the validation problem in a sampled-data framework [14], [15]. In this note, we extend the results in [13] and [12] to multirate systems. The setup is shown in Fig. 1, where Pmr and mr are both multirate systems, and they together form a multirate uncertain system model with Pmr fixed and mr unknown. The model validation problem considered in this note is as follows. Given Pmr , an uncertainty set which mr belongs to, a set of time domain experimental data on ui and yi , and a set E of noise signals, find out if there exists a mr in the uncertainty set such that the experimental data can be reproduced with Pmr and mr together with the noises E . As a technical tool, we first propose and study a CF interpolation problem with a nest operator constraint. This problem is itself of interest mathematically and has potential applications in addressing other problems in control, signal processing, and circuit theory [16]. A necessary and sufficient solvability condition for this constrained CF interpolation problem is given. Then the validation tests are presented based on the above condition. The note is organized as follows. The next section introduces some basic facts about the general multirate systems and shows how to convert a multirate system to an equivalent LTI system with a causality constraint. Section III addresses the tangential CF interpolation problem with nest operator constraint, which are the main tool to obtain the model validation test for general multirate systems.
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if k t < (k + 1) . Extend this definition to spaces 8ip=1 `(mi h) and 8jq=1 `(nj h) in an obvious way. Then the multirate system is said to be causal if Fig. 2.
Pt u = Pt v ) Pt Fmr u = Pt Fmr v
A general multirate system.
Section IV provides the necessary and sufficient validation conditions for the time-domain experimental data. Section V concludes the note. II. GENERAL MULTIRATE SYSTEMS The setup of a general MIMO multirate system is shown in Fig. 2. Here ui ; i = 1; 2; . . . ; p, are input signals whose sampling intervals are mi h, respectively, and yj ; j = 1; 2; . . . ; q are output signals whose sampling intervals are nj h, respectively, where h is a real number called base sampling interval and mi ; nj are natural numbers (positive integers). Such systems can result from discretizing continuous time systems using samplers of different rates or they can be found in their own right. We will assume that all signals in the system are synchronized at time 0, i.e., the time 0 instances of all signals occur at the same time. In this note, we will focus on those multirate systems that satisfy certain causal, linear, shift invariance properties which are to be defined below. Since we need to deal with signals with different rates, it is more convenient and clearer to associate each signal explicitly with its sampling interval. Let `r ( ) denote the space of r valued sequences
`r ( ) = ff. . . ; x(0 ); jx(0); x( ); . . .g : x(k ) 2
r g:
The system in Fig. 2 is a map from 8ip=1 `(mi h) to 8qj=1 `(nj h). It is said to be linear if this map is a linear map. Let l 2 be a multiple of mi and nj ; i = 1; 2; . . . ; p; j = 1; 2; . . . ; q . Let m i = l=mi and nj = l=nj . Denote the sets fmi g and fnj g by M and N , respectively, and the sets fm i g and fnj g by and N respectively. Let S : `r ( ) ! `r ( ) be the forward shift M operator, i.e.,
S f. . . ; x(0 ); j x(0); x( ); . . .g
= f. . . ; x(02 ); j x(0 ); x(0); x( ); . . .g:
for all t 2 . In this note, we will concentrate on causal linear N )-shift invariant systems. Such general multirate system covers (M; many familiar classes of systems as special cases. If mi ; nj ; l are all the same, then this is an LTI single rate system. If mi ; nj are all the same but l is a multiple of them, then it is a single rate l-periodic system [17]. If p = q = 1, this becomes the SISO dual rate system studied in [7]. If mi are the same and nj are the same, then this becomes the MIMO dual rate system studied in [18]. For systems resulted from discretizing LTI continuous time systems using multirate sample and hold schemes in [4], [5], l turns out to be the least common multiple of mi and nj . The study of multirate systems in such a generality as indicated above, however, has never been done before. A standard way for the analysis of such systems is to use lifting or blocking. Define a lifting operator Lr : `( ) ! `r (r ) by the equation shown at the bottom of the page, and let
= diag Lm ; . . . ; Lm = diag Ln ; . . . ; Ln : 1 Then, the lifted system F = LN Fmr L0 system in the sense M is an LTI that F S = SF . Hence, it has transfer function F^ in -transform. HowLM LN
ever, F is not an arbitrary LTI system, instead its direct feedthrough term F^ (0) is subject to a constraint that is resulted from the causality of Fmr . This constraint is best described using the language of nests and nest operators [18], [5]. Let X be a finite-dimensional vector space. A nest in X , denoted fXk g, is a chain of subspaces in X , including f0g and X , with the nonincreasing ordering
X = X0 X1 1 1 1 Xl01 Xl = f0g: Let U ; Y be finite dimensional vector spaces. Denote by L(U ; Y ) the set of linear operators U ! Y . Assume that U and Y are equipped, respectively, with nest fUk g and fYk g which have the same number of subspaces, say, l + 1 as above. A linear map T 2 L(U ; Y ) is said to be a nest operator if
T Uk Yk ; k = 0; 1; . . . ; l:
Define
SM SN
= diag S m ; . . . ; S m = diag S n ; . . . ; S n
Let 5U : U ! U k and 5Y Then, (1) is equivalent to
:
N )-shift invariant Then the multirate system in Fig. 2 is said to be (M; r or lh periodic in real time if Fmr SM = SN Fmr . Now let Pt : ` ( ) ! r ` ( ) be the truncation operator, i.e.,
...
: Y ! Y k be orthogonal projections.
(I 0 5Y )T 5U = 0;
k = 0; . . . ; l 0 1:
(2)
The set of all nest operators (with given nests) is denoted If we decompose the spaces U and Y in the following way:
N (fUk g; fYk g).
Pt f. . . ; x((k 0 1) ); x(k ); x(k + 1) ); . . .g = f. . . ; x((k 0 1) ); x(k ); 0; . . .g
Lr f. . . j x(0); x( ); . . .g !
(1)
U = (U0 9 U1 ) 8 (U1 9 U2 ) 8 1 1 1 8 (Ul01 9 Ul ) Y = (Y0 9 Y1 ) 8 (Y1 9 Y2 ) 8 1 1 1 8 (Yl01 9 Yl )
x(0) .. .
x((r 0 1) )
x(r ) ;
.. .
x((2r 0 1) )
;...
(3) (4)
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then a nest operator T lower triangular form:
2 N (fU g fY g) has the following block k
T21
=
T22
.. .
.. .
Tl1
Denote u =
LM u
, and y =
k
111
0
T11
T
;
.
0 .. .
..
.
0
111
Tl2
LN y
..
Tll
. Then
T (0) 1 1 1 yq (( nq 0 1)nq h)] :
k
Y
k;
k
1
1 1
Here, Uk and Yk represent respectively the input and output signals at or after time kh in the first period. Due to the causality of Fmr , the direct through term of the lifted plant must satisfy
^ (0)Uk
Denote H (U ; Y ) the Hardy class of all uniformly bounded analytic functions on with values in L(U ; Y ), where denotes the ^2 open unit disc. Denote by H (fUk g; fYk g) the set of functions G ^ H (U ; Y ) satisfying G(0) 2 N (fUk g; fYk g). The purpose of this section is to address the CF interpolation problem using functions in H (fUk g; fYk g). Before going into this problem, we need to state a result on matrix positive completion.
1
= fu(0) : ui (rmi h) = 0 if rmi h < khg = fy (0) : yj (rnj h) = 0 if rnj h < khg:
F
A. Matrix-Positive Completion The matrix-positive completion problem is as follows [19]: Given j 0 ij q, satisfying Bij = Bji3 , find the remaining matrices n Bij ; jj 0 ij > q , such that the block matrix B = [Bij ]i;j =1 is positive definite. The matrix-positive problem was first proposed by Dym and Gohberg [19], who gave the following result. Lemma 1: The matrix positive completion problem has a solution iff Bij ; j
= 0; 1; . . . l
or, equivalently
^ (0) 2 N (fUk g; fYk g):
(6)
F
Now we see that each multirate system has an equivalent single rate LTI system satisfying a causality constraint. This causality constraint is characterized by a nest operator constraint as in (6) on its transfer function. We end this section by showing an example. Consider the system shown in Fig. 2. Let p = q = 2; m1 = 2; m2 = 6; n1 = 4; n2 = 3 and l = 12. Then m 1 = 6; m 2 = 2; n 1 = 3 and n 2 = 4. Let u and y be the lifted signals of u and y respectively. Then we have the equation shown at the bottom of the page. Denote the ith column of 8 2 8 identity matrix by ei . Then
U12 = U11 = f0g U10 = U9 = spanf 6 g U8 = U7 = spanf 5 6 g U6 = U5 = spanf 4 5 6 8 g U4 = U3 = spanf 3 4 5 6 8 g U2 = U1 = spanf 2 3 4 5 6 8 g U0 = 8 e
e ;e
e ;e ;e ;e
111
Bii
.. .
111
Bi+q;i
(2h) y1 (4h)
;
i
(4h) y1 (8h)
Let X ; U and Y be finite dimensional Hilbert spaces. The Hilbert space direct sum of n copies of X will be denoted by X n . Assume that U and Y are equipped respectively with nests fUk g and fYk g. Let Ui and Yi ; i = 0; 1; . . . ; n, be linear operators from X to U and from X to Y respectively. Denote
=
.. .
(6h) u1 (8h) u1 (10h) u2 (0) T y2 (0) y2 (3h) y2 (6h) y2 (9h)] : u1
(7)
Bi+q;i+q
Y0
and
Y
Un
u1
= 1; . . . ; n 0 q:
B. CF Interpolation With Nest Operator Constraint
U
:
u1
0
.. .
U0
e ;e ;e ;e ;e ;e
(0) = [u1 (0) y (0) = [y1 (0)
Bi;i+q
Reference [20] gave a detailed discussion of such problem and presented an explicit description of the set of all solutions via a linear fractional map of which the coefficients are directly given in terms of the original data. However, Lemma 1 is enough for us.
e ;e ;e ;e ;e
u
3 0 3 0 3 3 3 0 3 0 3 3 3 3
III. MATHEMATICAL PREPARATIONS
Define for k = 0; 1; . . . ; l
k
0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 3 3 3 0 0 3 3 3 3 0
3 3 0 0 3 3 3 3
where “3” represents an arbitrary number. Note that such matrices are not block lower triangular, but can be turned into block lower triangular matrices by permutations of rows and columns.
(0) = [u1 (0) 1 1 1 u1 ((m 1 0 1)m1 h) 1 1 1 p 0 1)mp h)]T up (0) 1 1 1 up ((m y (0) = [y1 (0) 1 1 1 y1 (( n1 0 1)n1 h) 1 1 1
U Y
3 3 3 3 3 3 3
(5)
:
u
yq
The nests fYk g can be defined in a similar way. Then N (fUk g; fYk g) consists of matrices of the form
=
.. .
Yn
u2
(6h)]T
:
(8)
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The Toeplitz matrix generated by U is defined as
TU
:=
U0
0
U1
U0
.. .
Y
111 0
.. .
..
.
.. .
..
.
0
111
Un Un01
:
^
)
U0
( ^( ) =
1
)
[
(
T
)
for some T 2 N fUk g; fYk g . By the solvability condition of the standard CF interpolation problem [21], the constrained CF interpolation problem has a solution iff
U^ 3 U^ 0 Y^ 3 Y^
0
(11)
U^ =
0
U0
0
0
111 0 0
0
Un01 Un Y0
0 0 0
Un02 Un01
0
111 0 0 1 1 1 0 U0 111 0 0
Yn01 T Yn
0 0
Yn02 Yn01
111 0 0 1 1 1 0 Y0
I Y^
0
=
.. .
.. .
.. .
.. .
0
.. .
.. .
.. .
.. .
.. .
.. .
.. .
:
^ and Y^ do not contribute anything to inequality The zero columns in U (11). Hence, (11) is equivalent to
U 3 U 0 Y 3 Y
0
(12)
where
U
0 =
.. .
0
U0 .. .
0 .. .
Un01 Un02 Ip Un Un01
111 0 ..
.
.. .
111 0 1 1 1 U0
111 0
.. .
..
111
U0
:
111 0 1 1 1 Y0
Yn01 Yn02 T Yn Yn01
.. .
.
U0
0
T3
TU3 TU
TY3
TY
I
0
T
(
)
0
(13)
for some T 2 N fUk g; fYk g . If we decompose the spaces U and Y as in (3)–(4), then a nest operator T 2 N fUk g; fYk g has a block lower triangular form shown in (5). Therefore, the constrained CF interpolation problem has a solution if and only if (13) holds for a block lower triangular matrix T . This is a matrix-positive completion problem. By Lemma 1, such a T exists iff
(5U
I Un )3 .. .
(5U U0 )3
5U
(
)
Un 1 1 1 5U U0 TU3 TU
0 5Y 8Y for k = 0; 1; . . . ; l. Here, 5U
0 (5Y
TY
3 8Y TY )
0
(14)
I
and 5Y 8Y are operators from U to Uk and from Y n+1 to Y n 8Yk? respectively. Using Schur complement
= 0 1 ...
twice, we see that (14) is equivalent to (10) for k ; ; ; l. Finally, notice that (10) when k is implied by (10) when k l. This completes the proof. The solvability condition for the standard CF interpolation problem without constraint is recovered when l .
=0
=
=1
IV. TIME-DOMAIN VALIDATION FOR MULTIRATE LFT UNCERTAIN MODEL
where .. .
0
I Un Un
]
( ) TU3 5U 8U TU 0 TY3 5Y 8Y TY 0 (10) for all k = 1; . . . ; l. Proof: The nest operator constraint on the interpolation function G^ can be considered as an additional interpolation condition 0 0 .. .. TG . = . 0 0 I
.. .
.. .
where TG is the Toeplitz matrix generated by G0 1 1 1 Gn . Note that the dimension of TG depends on an integer n, to simplify the notation, however, we choose to ignore this dependence. In fact, this does not cause any confusion if we always assume that all the matrix operations are compatible. Theorem 1: There exists a solution to the CF interpolation problem with constraint N fUk g; fYk g for the data U; Y if and only if
0
Y0
.. .
Notice that the submatrices of U and Y formed by removing the first block column are block Toplitze matrices and are equal to TU and TY respectively. It follows from Schur complement that (12) is equivalent to:
= TG U
Y
=
(9)
The Toeplitz matrix TY generated by Y is defined in a similar way. The tangential CF interpolation problem with constraint N fUk g; fYk g 1 Gi i for the data U; Y is to find (if possible) a function G i=0 in H1 fUk g; fYk g such that kGk1 < and
(
0
In robust control theory, many problems can be treated in a unified framework using LFT machinery. In fact, additive, multiplicative and coprime factor uncertainty descriptions can all be represented as an LFT on the uncertainty, with a suitable choice of the coefficient matrix [22]. In this section, we will give the validation tests for multirate LFT uncertain models. Suppose we have an uncertain multirate system shown in Fig. 1. Here, ui ; i ; ; p, are input signals whose sampling intervals are m0i h and yj ; j ; ; q; are output signals whose sampling intervals are n0j h. Also, vi ; i ; ; r , and wj ; j ; ; s, are the auxiliary signals whose sampling intervals are mi h and nj h respectively. Assume that both Pmr and mr are lh periodic in real time for some integer l. As discussed in Section II, we can then convert the above multirate LFT uncertain system to an equivalent single rate LTI system with a causality constraint. Let mi0 l=mi0 ; n0j l=nj0 ; mi l=mi ; nj l=nj . Also, let y LN y; u LM u; v LM v; w LN w , and
= 1 ... = 1 ... = 1 ...
= 1 ...
1
=
=
= =
= L0N L0 Pmr L0M N 1 = LN 1mr(LM )01 P
= = 0
LM
= =
01
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for some E 2 E ; where E is a compact convex set representing a bound on the error and
V
Fig. 3.
:=
V0 .. . Vn
; W
W0 .. . Wn
:=
:
The equivalent LTI uncertain model.
Theorem 2 where LN ; LM ; LM ; LN are appropriately defined as in Section II. Then the multirate uncertain system in Fig. 1 is converted to an equivalent LTI uncertain system as shown in Fig. 3. We know that such equivalent LTI system satisfies a causality constraint. Denote
(0)T ; . . . ; v1 ((m 1 0 1)m1h)T T . . . ; vr (0)T ; . . . ; vr ((m r 0 1)mr h)T w(0) = w1 (0)T ; . . . ; w1 (( n1 0 1)n1 h)T T . . . ; ws (0)T ; . . . ; ws ((ns 0 1)nsh)T u(0) = u1 (0)T ; . . . ; u1 (( m01 0 1)m10 h)T T . . . ; up (0)T ; . . . ; up ((mp0 0 1)mp0 h)T y (0) = y1 (0)T ; . . . ; y1 (( n01 0 1)n10 h)T T . . . ; yq (0)T ; . . . ; yq ((n0q 0 1)nq0 h)T : v
(0) =
Define for k
For data U and Y , define
V = fV : TY = TP
TU
+ TP
TV
+ TE ; E 2 Eg:
The uncertain model (15)–(17) is not invalidated if and only if there ; ; l, where exists a V 2 V such that Hk V for k
( ) 0 = 1 ... Hk11 (V ) Hk12 (V ) Hk (V ) = I Hk21 (V ) T Hk11 (V ) = TP TU (5W 8W ) TP TU + TP TU T (5W 8W ) TP TV + TP TV T (5W 8W ) TP TU Hk12 (V ) = TVT Qk Hk21 (V ) = Qk TV 1 Qk = 2 5V 8V 0 TPT (5W 8W )TP
v1
0
:
Proof: First we show Qk so that Qk is well-de2 N fUk 8 Vk g; fYk 8 Wk g , we have fined. From P 2 N fVk g; fWk g . Thus P22 12 H1 fVk g; fWk g since P22 i k P22 k1 . Recall that P22 i=0 P22i . Setting
^ (0) ^
= 0; 1; . . . ; l;
Vk = fv(0) : vi (rmi h) = 0 if rmi h < khg Wk = fw(0) : wj (rnj h) = 0 if rnj h < khg Uk = fu(0) : ui (rmi0 h) = 0 if rmi0 h < khg Yk = fy(0) : yj (rnj0 h) = 0 if rn0j h < khg:
^ (0)
( ^ ) ^ ( )=
( 1
I U
=
0 .. .
0
and Y
=
)
(
P220 P221 .. .
)
:
P22n
= Y , it follows from Theorem 1 that 5V 8V 0 2 TPT 5W 8W TP 0 equivalent LTI system shown in Fig. 3 with such constraints. Assume that an uncertain model of the lifted LTI equivafor all k = 1; . . . ; l. Therefore Qk 0 for all k = 1; . . . ; l, lence of a multirate system is represented by the lower LFT ^ , where the nominal model P^ 2 H1 is given and satisfies and Qk is well-defined. By Theorem 1, there exists a 1^ 2 Fl (P^ ; 1) ^ )k1 in (17) iff ^P22 2 H1 (fVk g; fWk g) and kP^22 k1 1 , and 1^ is the uncertainty H1 (N (fWk g; fVk g)) with k1( ^ k1 . Several time domain experiments are carried satisfying k1 T
2 TW 5W 8W TW TVT 5V 8V TV (18) out so that several input–output pairs of the lifted system are collected for all k = 1; . . . ; l. Substituting (16) into (18) yields U0 Y0 .. .. Hk11 (V ) 0 TVT Qk TV 0: (19) U := ; Y := : . . Un Yn Since Qk 0; it follows by Schur complement that (19) is equivalent to Hk (V ) 0. Hence, the uncertain model is not invalidated if and The model validation problem is to test whether the uncertain model is only if there exists a V 2 such that H (V ) 0 for k = 1; . . . ; l. V k ^ 2 The conditions in Theorem consistent with the experiments data, i.e., whether there exists a 1 2 are the well-known LMI feasibility ^ H1 (NfWk g; fVk g)) with k1k1 such that the following holds: conditions which is numerically feasible. ^
^ (0) 2 N (fUk 8 Vk g; fYk 8 Wk g) and 1^ satisfies ^ 2 N (fWk g; fVk g). From now on, we will only consider the 1(0)
Then P satisfies P
Y W V
= TP U + TP = TP U + TP = T1 W
V V
+E
Note that T P U
(15)
V. CONCLUSION
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The model validation for general multirate systems is studied in this note. Based on the solutions to the constrained CF interpolation
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problem, the time domain validation test is presented for the general multirate LFT uncertain models. These tests can be carried out by solving feasibility problems involving LMIs.
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On the Asymptotically Optimal Tuning of Robust Controllers for Systems in the CD-Algebra Timo Hämäläinen and Seppo Pohjolainen
REFERENCES [1] P. P. Khargonekar, K. Poolla, and A. Tannenbaum, “Robust control of linear time-invariant plants using periodic compensation,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 1088–1096, Nov. 1985. [2] P. Vaidyanathan, Multirate Systems and Filter Banks. Upper Saddle River, NJ: Prentice-Hall, 1993. [3] G. M. Kranc, “Input-output analysis of multirate feedback systems,” IEEE Trans. Automat. Contr., vol. AC-3, pp. 21–28, Nov. 1957. Design of general multirate sampled-data [4] T. Chen and L. Qiu, “ control systems,” Automatica, vol. 30, pp. 1139–1152, 1994. subop[5] L. Qiu and T. Chen, “Multirate sampled-data systems: All timal controllers and the minimum entropy controllers,” IEEE Trans. Automat. Contr., vol. 44, pp. 537–550, Mar. 1999. [6] H. Liu, G. Xu, L. Tong, and T. Kailath, “Recent developments in blind channel equalization: from cyclostationarity to subspaces,” Signal Processing, vol. 50, pp. 83–99, 1996. [7] T. Chen, L. Qiu, and E. Bai, “General mulrirate building blocks and their application in nonuniform filter banks,” IEEE Trans. Circuits Syst., II, vol. 45, pp. 948–958, Aug. 1998. [8] Y. Yamamoto and P. P. Khargonekar, “From sampled-data control to signal processing,” in Learning, Control and Hybrid Systems, Y. Yamamoto and S. Hara, Eds. New York: Springer-Verlag, 1998, pp. 108–126. [9] P. M. Mäkilä, J. R. Partington, and T. K. Gustafsson, “Worst-case control-relevant identification,” Automatica, vol. 31, pp. 1799–1819, 1995. [10] R. S. Smith and J. C. Doyle, “Model validation: A connection between robust control and identification,” IEEE Trans. Automat. Contr., vol. 37, pp. 942–952, July 1992. [11] J. Chen, “Frequency-domain tests for validation of linear fractional uncertain models,” IEEE Trans. Automat. Contr., vol. 42, pp. 748–760, June 1997. [12] K. Poolla, P. P. Khargonekar, A. Tikku, J. Krause, and K. M. Nagpal, “A time-domain approach to model validation,” IEEE Trans. Automat. Contr., vol. 39, pp. 951–959, May. 1994. [13] J. Chen and S. Wang, “Validation of linear fractional uncertain models: Solutions via matrix inequalities,” IEEE Trans. Automat. Contr., vol. 41, pp. 844–849, June 1996. [14] S. Rangan and K. Poolla, “Time domain validation for sampled-data uncertainty models,” IEEE Trans. Automat. Contr., vol. 41, pp. 980–991, July 1996. [15] R. S. Smith and G. Dullerud, “Continuous-time control model validation using finite experimental data,” IEEE Trans. Automat. Contr., vol. 41, pp. 1094–1105, Aug. 1996. [16] T. T. Georgiou, C. I. Byrnes, and A. Lindquist, “A generalized entropy criterion for Nevanlinna–Pick interpolation with degree constraint,” IEEE Trans. Automat. Contr., vol. 46, pp. 822–839, June 2001. [17] L. Qiu and T. Chen, “Contractive completion of block matrices and its application to control of periodic systems,” in Applications of Operator Theory, P. L. I. Gohberg and P. N. Shivakumar, Eds. Boston, MA: Birkhäuser, 1996, pp. 2506–2511. [18] ,“ -optimal design of multirate sampled-data systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 2506–2511, Dec. 1994. [19] H. Dym and I. Gohberg, “Extensions of band matrices with band inverses,” Linear Alg. Appl., vol. 36, pp. 1–24, 1981. [20] H. J. Woerdeman, “Strictly contractive and positive completions for block matrices,” Linear Alg. Appl., vol. 136, pp. 63–105, 1990. [21] C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems. Boston, MA: Birkhäuser, 1990. [22] K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice-Hall, 1998.
H
H
H
H
Abstract—In a previous paper, the authors have shown that a low-gain ( )= ( ) is able to track controller of the form and reject constant and sinusoidal reference and disturbance signal for a stable plant in the Callier–Desoer (CD) algebra. In this note, we investiof the controller ( ) as gate the optimal tuning of the matrix gains the scalar gain 0. The cost function is the maximum error between the reference signal and the measured output signal over all frequencies and bounded reference and disturbance signal amplitudes. Closed forms for asymptotically globally optimal solutions are given. The optimal matrix are expressed in terms of the values of the plant transfer matrix gains at the reference and disturbance signal frequencies. Thus the matrices can be tuned with input-output measurements made from the open loop plant without knowledge of the plant model. Although the analysis is in the CD-algebra, to the authors’ knowledge the main results are new even for finite-dimensional systems. Index Terms—Low-gain control, distributed parameter systems, Callier–Desoer (CD)-algebra, tracking, optimal control.
I. INTRODUCTION In a previous paper [1], the authors solved the following robust regulation problem: Given a stable plant P in the Callier–Desoer (CD)-algebra and reference and disturbance signals of the form
a0 +
n k=1
ak sin(!k t + k ); ak 2
(1)
find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant, see Fig. 1. In [1], it is shown that a low-gain controller C" given by
C" (s) =
n
"Kk s 0 i!k k=0n
(2)
solves the robust regulation problem provided that the positive scalar gain " is small enough and the matrix gains Kk satisfy the stability conditions
(P (i!k )Kk ) + ;
k = 0n; . . . ; n
(3)
where !0 = 0 and !0k = 0!k for k = 1; . . . ; n. Conditions (3) give an easily verifiable condition for the stabilizing matrix gains Kk . Unfortunately there are no analogous conditions for the scalar gain ". The closed-loop system will remain stable if " is below some bound "3 , but there is no easy way to determine "3 . The tuning of " has to be done more or less by trial and error. In the case of constant reference and disturbance signals Logemann and Townley have given an adaptive method of tuning " [2]. However it is a nontrivial problem to generalize their method to signals of the form (1). Therefore Manuscript received November 14, 2000; revised June 4, 2001. Recommended by Associate Editor F. M. Callier. The authors are with Tampere University of Technology, Department of Mathematics, FIN-33101 Tampere, Finland (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0018-9286(02)02079-2.
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Model Validation of Multirate Systems From Time-Domain Experimental Data Li Chai and Li Qiu Abstract—The model validation problem using time-domain experimental data is studied for multirate linear fractional uncertain models in this note. As a technical tool, the Carathéodory–Fejér (CF) interpolation problem with a nest operator constraint is first investigated. This problem is itself of interest mathematically and has potential applications in addressing other problems in control, signal processing, and circuit theory. A necessary and sufficient solvability condition for this interpolation problem is given. The validation tests are then presented based on this condition and the lifting technique. Tractable convex optimization methods can be used to solve the validation problems. Index Terms—Carathéodory–Fejér (CF) interpolation, model validation, multirate systems.
I. INTRODUCTION Multirate systems, i.e., digital systems with signals having different sampling rates, have wide applications in control, communication, signal processing, econometrics and numerical mathematics. There are several reasons for this. • In large scale multivariable digital systems, often it is unrealistic, or sometimes impossible, to sample all physical signals uniformly at one single rate. In such situations, one is forced to use multirate sampling. • Multirate systems can often achieve objectives that cannot be achieved by single rate systems [1], [2]. The study of multirate systems goes back to late 1950s [3]. A renaissance of research in multirate systems has occurred since 1980 in control community, signal processing community and communication community. The driving force for studying multirate systems in signal processing comes from the need of sampling rate conversion, subband coding, and their ability to generate wavelets. Multirate signal processing is now one of the most vibrant areas of research in signal processing, see recent book [2] and references therein. In control community, two groups of research stand out: using multirate control to achieve what single rate control cannot as well as the limitation of doing this [1] and the optimal design of multirate controllers [4], [5]. In communication community, multirate sampling is used for blind system identification and equalization [6]. We also notice the cross discipline fertilization between signal processing and control in using H optimization to design filter banks [7], [8]. In this note, we will study the control-oriented model validation problems pertaining to the general multirate systems. There has recently been considerable research devoted to robust or control-oriented model validation [9], [10]. However, the research on model validation of multirate systems is almost nonexisting. Due to the wide applications of multirate systems, their model validation problem should receive comparable attention to those of single rate systems. Model validation is a very important step in the process of control system modeling both in traditional stochastic setting and nonproba-
1
Manuscript received February 14, 2000; revised October 16, 2000, March 29, 2001, and September 6, 2001. Recommended by Associate Editor Y. Yamamoto. This work was supported by the Hong Kong Research Grants Council. It was completed when the authors were visiting the National Laboratory of Industrial Control Technology, Zhejiang University, China. The authors are with the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong. Publisher Item Identifier S 0018-9286(02)02078-0.
Fig. 1.
A general multirate LFT uncertain model.
bilistic context. Generally, the model validation problem is to examine if the sets of experimental data are consistent with the model of the plant. A model is said to be invalidated when the validation test fails. Our confidence in the model set is increased if the model is consistent with the data. Since a model is either invalidated or not invalidated, it is actually more accurate to call the validation procedure as model invalidation. More recently, motivated by the considerable research on control-oriented system identification, much attention has been paid on validation of uncertain models consisting of a nominal model and a norm bounded modeling uncertainty [11], [12], [10]. Such uncertainty models are the starting point for robust control. The first study of model validation for linear fractional transformation (LFT) model-sets was carried out by Smith and Doyle [10]. They show that the problem can be solved by a structured singular value type method. Chen [11] considered the general validation problems of linear fractional uncertain models in frequency domain and reduced it to the Nevanlinna-Pick interpolation problem, which can be solved by standard convex optimization methods. Based on the Carathéodory–Fejér (CF) interpolation problem, a purely time-domain formulation for models with an additive uncertainty is presented in [12]. It is shown that the problem can be solved as a convex program involving linear matrix inequalities (LMI). The time domain validation approach in a more general setup which is for LFT uncertain model-sets is studied in [13]. The similar setup is also used to consider the validation problem in a sampled-data framework [14], [15]. In this note, we extend the results in [13] and [12] to multirate systems. The setup is shown in Fig. 1, where Pmr and mr are both multirate systems, and they together form a multirate uncertain system model with Pmr fixed and mr unknown. The model validation problem considered in this note is as follows. Given Pmr , an uncertainty set which mr belongs to, a set of time domain experimental data on ui and yi , and a set E of noise signals, find out if there exists a mr in the uncertainty set such that the experimental data can be reproduced with Pmr and mr together with the noises E . As a technical tool, we first propose and study a CF interpolation problem with a nest operator constraint. This problem is itself of interest mathematically and has potential applications in addressing other problems in control, signal processing, and circuit theory [16]. A necessary and sufficient solvability condition for this constrained CF interpolation problem is given. Then the validation tests are presented based on the above condition. The note is organized as follows. The next section introduces some basic facts about the general multirate systems and shows how to convert a multirate system to an equivalent LTI system with a causality constraint. Section III addresses the tangential CF interpolation problem with nest operator constraint, which are the main tool to obtain the model validation test for general multirate systems.
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if k t < (k + 1) . Extend this definition to spaces 8ip=1 `(mi h) and 8jq=1 `(nj h) in an obvious way. Then the multirate system is said to be causal if Fig. 2.
Pt u = Pt v ) Pt Fmr u = Pt Fmr v
A general multirate system.
Section IV provides the necessary and sufficient validation conditions for the time-domain experimental data. Section V concludes the note. II. GENERAL MULTIRATE SYSTEMS The setup of a general MIMO multirate system is shown in Fig. 2. Here ui ; i = 1; 2; . . . ; p, are input signals whose sampling intervals are mi h, respectively, and yj ; j = 1; 2; . . . ; q are output signals whose sampling intervals are nj h, respectively, where h is a real number called base sampling interval and mi ; nj are natural numbers (positive integers). Such systems can result from discretizing continuous time systems using samplers of different rates or they can be found in their own right. We will assume that all signals in the system are synchronized at time 0, i.e., the time 0 instances of all signals occur at the same time. In this note, we will focus on those multirate systems that satisfy certain causal, linear, shift invariance properties which are to be defined below. Since we need to deal with signals with different rates, it is more convenient and clearer to associate each signal explicitly with its sampling interval. Let `r ( ) denote the space of r valued sequences
`r ( ) = ff. . . ; x(0 ); jx(0); x( ); . . .g : x(k ) 2
r g:
The system in Fig. 2 is a map from 8ip=1 `(mi h) to 8qj=1 `(nj h). It is said to be linear if this map is a linear map. Let l 2 be a multiple of mi and nj ; i = 1; 2; . . . ; p; j = 1; 2; . . . ; q . Let m i = l=mi and nj = l=nj . Denote the sets fmi g and fnj g by M and N , respectively, and the sets fm i g and fnj g by and N respectively. Let S : `r ( ) ! `r ( ) be the forward shift M operator, i.e.,
S f. . . ; x(0 ); j x(0); x( ); . . .g
= f. . . ; x(02 ); j x(0 ); x(0); x( ); . . .g:
for all t 2 . In this note, we will concentrate on causal linear N )-shift invariant systems. Such general multirate system covers (M; many familiar classes of systems as special cases. If mi ; nj ; l are all the same, then this is an LTI single rate system. If mi ; nj are all the same but l is a multiple of them, then it is a single rate l-periodic system [17]. If p = q = 1, this becomes the SISO dual rate system studied in [7]. If mi are the same and nj are the same, then this becomes the MIMO dual rate system studied in [18]. For systems resulted from discretizing LTI continuous time systems using multirate sample and hold schemes in [4], [5], l turns out to be the least common multiple of mi and nj . The study of multirate systems in such a generality as indicated above, however, has never been done before. A standard way for the analysis of such systems is to use lifting or blocking. Define a lifting operator Lr : `( ) ! `r (r ) by the equation shown at the bottom of the page, and let
= diag Lm ; . . . ; Lm = diag Ln ; . . . ; Ln : 1 Then, the lifted system F = LN Fmr L0 system in the sense M is an LTI that F S = SF . Hence, it has transfer function F^ in -transform. HowLM LN
ever, F is not an arbitrary LTI system, instead its direct feedthrough term F^ (0) is subject to a constraint that is resulted from the causality of Fmr . This constraint is best described using the language of nests and nest operators [18], [5]. Let X be a finite-dimensional vector space. A nest in X , denoted fXk g, is a chain of subspaces in X , including f0g and X , with the nonincreasing ordering
X = X0 X1 1 1 1 Xl01 Xl = f0g: Let U ; Y be finite dimensional vector spaces. Denote by L(U ; Y ) the set of linear operators U ! Y . Assume that U and Y are equipped, respectively, with nest fUk g and fYk g which have the same number of subspaces, say, l + 1 as above. A linear map T 2 L(U ; Y ) is said to be a nest operator if
T Uk Yk ; k = 0; 1; . . . ; l:
Define
SM SN
= diag S m ; . . . ; S m = diag S n ; . . . ; S n
Let 5U : U ! U k and 5Y Then, (1) is equivalent to
:
N )-shift invariant Then the multirate system in Fig. 2 is said to be (M; r or lh periodic in real time if Fmr SM = SN Fmr . Now let Pt : ` ( ) ! r ` ( ) be the truncation operator, i.e.,
...
: Y ! Y k be orthogonal projections.
(I 0 5Y )T 5U = 0;
k = 0; . . . ; l 0 1:
(2)
The set of all nest operators (with given nests) is denoted If we decompose the spaces U and Y in the following way:
N (fUk g; fYk g).
Pt f. . . ; x((k 0 1) ); x(k ); x(k + 1) ); . . .g = f. . . ; x((k 0 1) ); x(k ); 0; . . .g
Lr f. . . j x(0); x( ); . . .g !
(1)
U = (U0 9 U1 ) 8 (U1 9 U2 ) 8 1 1 1 8 (Ul01 9 Ul ) Y = (Y0 9 Y1 ) 8 (Y1 9 Y2 ) 8 1 1 1 8 (Yl01 9 Yl )
x(0) .. .
x((r 0 1) )
x(r ) ;
.. .
x((2r 0 1) )
;...
(3) (4)
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then a nest operator T lower triangular form:
2 N (fU g fY g) has the following block k
T21
=
T22
.. .
.. .
Tl1
Denote u =
LM u
, and y =
k
111
0
T11
T
;
.
0 .. .
..
.
0
111
Tl2
LN y
..
Tll
. Then
T (0) 1 1 1 yq (( nq 0 1)nq h)] :
k
Y
k;
k
1
1 1
Here, Uk and Yk represent respectively the input and output signals at or after time kh in the first period. Due to the causality of Fmr , the direct through term of the lifted plant must satisfy
^ (0)Uk
Denote H (U ; Y ) the Hardy class of all uniformly bounded analytic functions on with values in L(U ; Y ), where denotes the ^2 open unit disc. Denote by H (fUk g; fYk g) the set of functions G ^ H (U ; Y ) satisfying G(0) 2 N (fUk g; fYk g). The purpose of this section is to address the CF interpolation problem using functions in H (fUk g; fYk g). Before going into this problem, we need to state a result on matrix positive completion.
1
= fu(0) : ui (rmi h) = 0 if rmi h < khg = fy (0) : yj (rnj h) = 0 if rnj h < khg:
F
A. Matrix-Positive Completion The matrix-positive completion problem is as follows [19]: Given j 0 ij q, satisfying Bij = Bji3 , find the remaining matrices n Bij ; jj 0 ij > q , such that the block matrix B = [Bij ]i;j =1 is positive definite. The matrix-positive problem was first proposed by Dym and Gohberg [19], who gave the following result. Lemma 1: The matrix positive completion problem has a solution iff Bij ; j
= 0; 1; . . . l
or, equivalently
^ (0) 2 N (fUk g; fYk g):
(6)
F
Now we see that each multirate system has an equivalent single rate LTI system satisfying a causality constraint. This causality constraint is characterized by a nest operator constraint as in (6) on its transfer function. We end this section by showing an example. Consider the system shown in Fig. 2. Let p = q = 2; m1 = 2; m2 = 6; n1 = 4; n2 = 3 and l = 12. Then m 1 = 6; m 2 = 2; n 1 = 3 and n 2 = 4. Let u and y be the lifted signals of u and y respectively. Then we have the equation shown at the bottom of the page. Denote the ith column of 8 2 8 identity matrix by ei . Then
U12 = U11 = f0g U10 = U9 = spanf 6 g U8 = U7 = spanf 5 6 g U6 = U5 = spanf 4 5 6 8 g U4 = U3 = spanf 3 4 5 6 8 g U2 = U1 = spanf 2 3 4 5 6 8 g U0 = 8 e
e ;e
e ;e ;e ;e
111
Bii
.. .
111
Bi+q;i
(2h) y1 (4h)
;
i
(4h) y1 (8h)
Let X ; U and Y be finite dimensional Hilbert spaces. The Hilbert space direct sum of n copies of X will be denoted by X n . Assume that U and Y are equipped respectively with nests fUk g and fYk g. Let Ui and Yi ; i = 0; 1; . . . ; n, be linear operators from X to U and from X to Y respectively. Denote
=
.. .
(6h) u1 (8h) u1 (10h) u2 (0) T y2 (0) y2 (3h) y2 (6h) y2 (9h)] : u1
(7)
Bi+q;i+q
Y0
and
Y
Un
u1
= 1; . . . ; n 0 q:
B. CF Interpolation With Nest Operator Constraint
U
:
u1
0
.. .
U0
e ;e ;e ;e ;e ;e
(0) = [u1 (0) y (0) = [y1 (0)
Bi;i+q
Reference [20] gave a detailed discussion of such problem and presented an explicit description of the set of all solutions via a linear fractional map of which the coefficients are directly given in terms of the original data. However, Lemma 1 is enough for us.
e ;e ;e ;e ;e
u
3 0 3 0 3 3 3 0 3 0 3 3 3 3
III. MATHEMATICAL PREPARATIONS
Define for k = 0; 1; . . . ; l
k
0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 3 3 3 0 0 3 3 3 3 0
3 3 0 0 3 3 3 3
where “3” represents an arbitrary number. Note that such matrices are not block lower triangular, but can be turned into block lower triangular matrices by permutations of rows and columns.
(0) = [u1 (0) 1 1 1 u1 ((m 1 0 1)m1 h) 1 1 1 p 0 1)mp h)]T up (0) 1 1 1 up ((m y (0) = [y1 (0) 1 1 1 y1 (( n1 0 1)n1 h) 1 1 1
U Y
3 3 3 3 3 3 3
(5)
:
u
yq
The nests fYk g can be defined in a similar way. Then N (fUk g; fYk g) consists of matrices of the form
=
.. .
Yn
u2
(6h)]T
:
(8)
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The Toeplitz matrix generated by U is defined as
TU
:=
U0
0
U1
U0
.. .
Y
111 0
.. .
..
.
.. .
..
.
0
111
Un Un01
:
^
)
U0
( ^( ) =
1
)
[
(
T
)
for some T 2 N fUk g; fYk g . By the solvability condition of the standard CF interpolation problem [21], the constrained CF interpolation problem has a solution iff
U^ 3 U^ 0 Y^ 3 Y^
0
(11)
U^ =
0
U0
0
0
111 0 0
0
Un01 Un Y0
0 0 0
Un02 Un01
0
111 0 0 1 1 1 0 U0 111 0 0
Yn01 T Yn
0 0
Yn02 Yn01
111 0 0 1 1 1 0 Y0
I Y^
0
=
.. .
.. .
.. .
.. .
0
.. .
.. .
.. .
.. .
.. .
.. .
.. .
:
^ and Y^ do not contribute anything to inequality The zero columns in U (11). Hence, (11) is equivalent to
U 3 U 0 Y 3 Y
0
(12)
where
U
0 =
.. .
0
U0 .. .
0 .. .
Un01 Un02 Ip Un Un01
111 0 ..
.
.. .
111 0 1 1 1 U0
111 0
.. .
..
111
U0
:
111 0 1 1 1 Y0
Yn01 Yn02 T Yn Yn01
.. .
.
U0
0
T3
TU3 TU
TY3
TY
I
0
T
(
)
0
(13)
for some T 2 N fUk g; fYk g . If we decompose the spaces U and Y as in (3)–(4), then a nest operator T 2 N fUk g; fYk g has a block lower triangular form shown in (5). Therefore, the constrained CF interpolation problem has a solution if and only if (13) holds for a block lower triangular matrix T . This is a matrix-positive completion problem. By Lemma 1, such a T exists iff
(5U
I Un )3 .. .
(5U U0 )3
5U
(
)
Un 1 1 1 5U U0 TU3 TU
0 5Y 8Y for k = 0; 1; . . . ; l. Here, 5U
0 (5Y
TY
3 8Y TY )
0
(14)
I
and 5Y 8Y are operators from U to Uk and from Y n+1 to Y n 8Yk? respectively. Using Schur complement
= 0 1 ...
twice, we see that (14) is equivalent to (10) for k ; ; ; l. Finally, notice that (10) when k is implied by (10) when k l. This completes the proof. The solvability condition for the standard CF interpolation problem without constraint is recovered when l .
=0
=
=1
IV. TIME-DOMAIN VALIDATION FOR MULTIRATE LFT UNCERTAIN MODEL
where .. .
0
I Un Un
]
( ) TU3 5U 8U TU 0 TY3 5Y 8Y TY 0 (10) for all k = 1; . . . ; l. Proof: The nest operator constraint on the interpolation function G^ can be considered as an additional interpolation condition 0 0 .. .. TG . = . 0 0 I
.. .
.. .
where TG is the Toeplitz matrix generated by G0 1 1 1 Gn . Note that the dimension of TG depends on an integer n, to simplify the notation, however, we choose to ignore this dependence. In fact, this does not cause any confusion if we always assume that all the matrix operations are compatible. Theorem 1: There exists a solution to the CF interpolation problem with constraint N fUk g; fYk g for the data U; Y if and only if
0
Y0
.. .
Notice that the submatrices of U and Y formed by removing the first block column are block Toplitze matrices and are equal to TU and TY respectively. It follows from Schur complement that (12) is equivalent to:
= TG U
Y
=
(9)
The Toeplitz matrix TY generated by Y is defined in a similar way. The tangential CF interpolation problem with constraint N fUk g; fYk g 1 Gi i for the data U; Y is to find (if possible) a function G i=0 in H1 fUk g; fYk g such that kGk1 < and
(
0
In robust control theory, many problems can be treated in a unified framework using LFT machinery. In fact, additive, multiplicative and coprime factor uncertainty descriptions can all be represented as an LFT on the uncertainty, with a suitable choice of the coefficient matrix [22]. In this section, we will give the validation tests for multirate LFT uncertain models. Suppose we have an uncertain multirate system shown in Fig. 1. Here, ui ; i ; ; p, are input signals whose sampling intervals are m0i h and yj ; j ; ; q; are output signals whose sampling intervals are n0j h. Also, vi ; i ; ; r , and wj ; j ; ; s, are the auxiliary signals whose sampling intervals are mi h and nj h respectively. Assume that both Pmr and mr are lh periodic in real time for some integer l. As discussed in Section II, we can then convert the above multirate LFT uncertain system to an equivalent single rate LTI system with a causality constraint. Let mi0 l=mi0 ; n0j l=nj0 ; mi l=mi ; nj l=nj . Also, let y LN y; u LM u; v LM v; w LN w , and
= 1 ... = 1 ... = 1 ...
= 1 ...
1
=
=
= =
= L0N L0 Pmr L0M N 1 = LN 1mr(LM )01 P
= = 0
LM
= =
01
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for some E 2 E ; where E is a compact convex set representing a bound on the error and
V
Fig. 3.
:=
V0 .. . Vn
; W
W0 .. . Wn
:=
:
The equivalent LTI uncertain model.
Theorem 2 where LN ; LM ; LM ; LN are appropriately defined as in Section II. Then the multirate uncertain system in Fig. 1 is converted to an equivalent LTI uncertain system as shown in Fig. 3. We know that such equivalent LTI system satisfies a causality constraint. Denote
(0)T ; . . . ; v1 ((m 1 0 1)m1h)T T . . . ; vr (0)T ; . . . ; vr ((m r 0 1)mr h)T w(0) = w1 (0)T ; . . . ; w1 (( n1 0 1)n1 h)T T . . . ; ws (0)T ; . . . ; ws ((ns 0 1)nsh)T u(0) = u1 (0)T ; . . . ; u1 (( m01 0 1)m10 h)T T . . . ; up (0)T ; . . . ; up ((mp0 0 1)mp0 h)T y (0) = y1 (0)T ; . . . ; y1 (( n01 0 1)n10 h)T T . . . ; yq (0)T ; . . . ; yq ((n0q 0 1)nq0 h)T : v
(0) =
Define for k
For data U and Y , define
V = fV : TY = TP
TU
+ TP
TV
+ TE ; E 2 Eg:
The uncertain model (15)–(17) is not invalidated if and only if there ; ; l, where exists a V 2 V such that Hk V for k
( ) 0 = 1 ... Hk11 (V ) Hk12 (V ) Hk (V ) = I Hk21 (V ) T Hk11 (V ) = TP TU (5W 8W ) TP TU + TP TU T (5W 8W ) TP TV + TP TV T (5W 8W ) TP TU Hk12 (V ) = TVT Qk Hk21 (V ) = Qk TV 1 Qk = 2 5V 8V 0 TPT (5W 8W )TP
v1
0
:
Proof: First we show Qk so that Qk is well-de2 N fUk 8 Vk g; fYk 8 Wk g , we have fined. From P 2 N fVk g; fWk g . Thus P22 12 H1 fVk g; fWk g since P22 i k P22 k1 . Recall that P22 i=0 P22i . Setting
^ (0) ^
= 0; 1; . . . ; l;
Vk = fv(0) : vi (rmi h) = 0 if rmi h < khg Wk = fw(0) : wj (rnj h) = 0 if rnj h < khg Uk = fu(0) : ui (rmi0 h) = 0 if rmi0 h < khg Yk = fy(0) : yj (rnj0 h) = 0 if rn0j h < khg:
^ (0)
( ^ ) ^ ( )=
( 1
I U
=
0 .. .
0
and Y
=
)
(
P220 P221 .. .
)
:
P22n
= Y , it follows from Theorem 1 that 5V 8V 0 2 TPT 5W 8W TP 0 equivalent LTI system shown in Fig. 3 with such constraints. Assume that an uncertain model of the lifted LTI equivafor all k = 1; . . . ; l. Therefore Qk 0 for all k = 1; . . . ; l, lence of a multirate system is represented by the lower LFT ^ , where the nominal model P^ 2 H1 is given and satisfies and Qk is well-defined. By Theorem 1, there exists a 1^ 2 Fl (P^ ; 1) ^ )k1 in (17) iff ^P22 2 H1 (fVk g; fWk g) and kP^22 k1 1 , and 1^ is the uncertainty H1 (N (fWk g; fVk g)) with k1( ^ k1 . Several time domain experiments are carried satisfying k1 T
2 TW 5W 8W TW TVT 5V 8V TV (18) out so that several input–output pairs of the lifted system are collected for all k = 1; . . . ; l. Substituting (16) into (18) yields U0 Y0 .. .. Hk11 (V ) 0 TVT Qk TV 0: (19) U := ; Y := : . . Un Yn Since Qk 0; it follows by Schur complement that (19) is equivalent to Hk (V ) 0. Hence, the uncertain model is not invalidated if and The model validation problem is to test whether the uncertain model is only if there exists a V 2 such that H (V ) 0 for k = 1; . . . ; l. V k ^ 2 The conditions in Theorem consistent with the experiments data, i.e., whether there exists a 1 2 are the well-known LMI feasibility ^ H1 (NfWk g; fVk g)) with k1k1 such that the following holds: conditions which is numerically feasible. ^
^ (0) 2 N (fUk 8 Vk g; fYk 8 Wk g) and 1^ satisfies ^ 2 N (fWk g; fVk g). From now on, we will only consider the 1(0)
Then P satisfies P
Y W V
= TP U + TP = TP U + TP = T1 W
V V
+E
Note that T P U
(15)
V. CONCLUSION
(16)
The model validation for general multirate systems is studied in this note. Based on the solutions to the constrained CF interpolation
(17)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002
problem, the time domain validation test is presented for the general multirate LFT uncertain models. These tests can be carried out by solving feasibility problems involving LMIs.
351
On the Asymptotically Optimal Tuning of Robust Controllers for Systems in the CD-Algebra Timo Hämäläinen and Seppo Pohjolainen
REFERENCES [1] P. P. Khargonekar, K. Poolla, and A. Tannenbaum, “Robust control of linear time-invariant plants using periodic compensation,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 1088–1096, Nov. 1985. [2] P. Vaidyanathan, Multirate Systems and Filter Banks. Upper Saddle River, NJ: Prentice-Hall, 1993. [3] G. M. Kranc, “Input-output analysis of multirate feedback systems,” IEEE Trans. Automat. Contr., vol. AC-3, pp. 21–28, Nov. 1957. Design of general multirate sampled-data [4] T. Chen and L. Qiu, “ control systems,” Automatica, vol. 30, pp. 1139–1152, 1994. subop[5] L. Qiu and T. Chen, “Multirate sampled-data systems: All timal controllers and the minimum entropy controllers,” IEEE Trans. Automat. Contr., vol. 44, pp. 537–550, Mar. 1999. [6] H. Liu, G. Xu, L. Tong, and T. Kailath, “Recent developments in blind channel equalization: from cyclostationarity to subspaces,” Signal Processing, vol. 50, pp. 83–99, 1996. [7] T. Chen, L. Qiu, and E. Bai, “General mulrirate building blocks and their application in nonuniform filter banks,” IEEE Trans. Circuits Syst., II, vol. 45, pp. 948–958, Aug. 1998. [8] Y. Yamamoto and P. P. Khargonekar, “From sampled-data control to signal processing,” in Learning, Control and Hybrid Systems, Y. Yamamoto and S. Hara, Eds. New York: Springer-Verlag, 1998, pp. 108–126. [9] P. M. Mäkilä, J. R. Partington, and T. K. Gustafsson, “Worst-case control-relevant identification,” Automatica, vol. 31, pp. 1799–1819, 1995. [10] R. S. Smith and J. C. Doyle, “Model validation: A connection between robust control and identification,” IEEE Trans. Automat. Contr., vol. 37, pp. 942–952, July 1992. [11] J. Chen, “Frequency-domain tests for validation of linear fractional uncertain models,” IEEE Trans. Automat. Contr., vol. 42, pp. 748–760, June 1997. [12] K. Poolla, P. P. Khargonekar, A. Tikku, J. Krause, and K. M. Nagpal, “A time-domain approach to model validation,” IEEE Trans. Automat. Contr., vol. 39, pp. 951–959, May. 1994. [13] J. Chen and S. Wang, “Validation of linear fractional uncertain models: Solutions via matrix inequalities,” IEEE Trans. Automat. Contr., vol. 41, pp. 844–849, June 1996. [14] S. Rangan and K. Poolla, “Time domain validation for sampled-data uncertainty models,” IEEE Trans. Automat. Contr., vol. 41, pp. 980–991, July 1996. [15] R. S. Smith and G. Dullerud, “Continuous-time control model validation using finite experimental data,” IEEE Trans. Automat. Contr., vol. 41, pp. 1094–1105, Aug. 1996. [16] T. T. Georgiou, C. I. Byrnes, and A. Lindquist, “A generalized entropy criterion for Nevanlinna–Pick interpolation with degree constraint,” IEEE Trans. Automat. Contr., vol. 46, pp. 822–839, June 2001. [17] L. Qiu and T. Chen, “Contractive completion of block matrices and its application to control of periodic systems,” in Applications of Operator Theory, P. L. I. Gohberg and P. N. Shivakumar, Eds. Boston, MA: Birkhäuser, 1996, pp. 2506–2511. [18] ,“ -optimal design of multirate sampled-data systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 2506–2511, Dec. 1994. [19] H. Dym and I. Gohberg, “Extensions of band matrices with band inverses,” Linear Alg. Appl., vol. 36, pp. 1–24, 1981. [20] H. J. Woerdeman, “Strictly contractive and positive completions for block matrices,” Linear Alg. Appl., vol. 136, pp. 63–105, 1990. [21] C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems. Boston, MA: Birkhäuser, 1990. [22] K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice-Hall, 1998.
H
H
H
H
Abstract—In a previous paper, the authors have shown that a low-gain ( )= ( ) is able to track controller of the form and reject constant and sinusoidal reference and disturbance signal for a stable plant in the Callier–Desoer (CD) algebra. In this note, we investiof the controller ( ) as gate the optimal tuning of the matrix gains the scalar gain 0. The cost function is the maximum error between the reference signal and the measured output signal over all frequencies and bounded reference and disturbance signal amplitudes. Closed forms for asymptotically globally optimal solutions are given. The optimal matrix are expressed in terms of the values of the plant transfer matrix gains at the reference and disturbance signal frequencies. Thus the matrices can be tuned with input-output measurements made from the open loop plant without knowledge of the plant model. Although the analysis is in the CD-algebra, to the authors’ knowledge the main results are new even for finite-dimensional systems. Index Terms—Low-gain control, distributed parameter systems, Callier–Desoer (CD)-algebra, tracking, optimal control.
I. INTRODUCTION In a previous paper [1], the authors solved the following robust regulation problem: Given a stable plant P in the Callier–Desoer (CD)-algebra and reference and disturbance signals of the form
a0 +
n k=1
ak sin(!k t + k ); ak 2
(1)
find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant, see Fig. 1. In [1], it is shown that a low-gain controller C" given by
C" (s) =
n
"Kk s 0 i!k k=0n
(2)
solves the robust regulation problem provided that the positive scalar gain " is small enough and the matrix gains Kk satisfy the stability conditions
(P (i!k )Kk ) + ;
k = 0n; . . . ; n
(3)
where !0 = 0 and !0k = 0!k for k = 1; . . . ; n. Conditions (3) give an easily verifiable condition for the stabilizing matrix gains Kk . Unfortunately there are no analogous conditions for the scalar gain ". The closed-loop system will remain stable if " is below some bound "3 , but there is no easy way to determine "3 . The tuning of " has to be done more or less by trial and error. In the case of constant reference and disturbance signals Logemann and Townley have given an adaptive method of tuning " [2]. However it is a nontrivial problem to generalize their method to signals of the form (1). Therefore Manuscript received November 14, 2000; revised June 4, 2001. Recommended by Associate Editor F. M. Callier. The authors are with Tampere University of Technology, Department of Mathematics, FIN-33101 Tampere, Finland (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0018-9286(02)02079-2.
0018–9286/02$17.00 © 2002 IEEE
