Linear Periodically Time-Varying Discrete-Time

Linear Periodically Time-Varying Discrete-Time Systems: Aliasing and LTI Approximations Tongwen Chen Department of Electrical and Computer Engineering University of Calgary Calgary, Alberta, Canada T2N 1N4 Email: [email protected] Li Qiu Department of Electrical and Electronic Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong Email: [email protected] First version: March 1, 1996; Revised: November 26, 1996

Abstract

Linear periodically time-varying (LPTV) systems are abundant in control and signal processing; examples include multirate sampled-data control systems and multirate lter-bank systems. In this paper, several ways are proposed to quantify aliasing eect in discrete-time LPTV systems; these are associated with optimal time-invariant approximations of LPTV systems using operator norms.

Keywords: periodic systems, multirate systems, optimization, aliasing, discrete-time systems.

1 Introduction Examples of linear periodically time-varying (LPTV) systems are abundant: In control, multirate sampled-data systems are designed to exploit their cost advantage in digital implementation [6, 5]; in signal processing, multirate lter banks, which are typically LPTV, are designed for ecient coding and transmission of digital signals [11]. Dierent from linear time-invariant (LTI) systems, aliasing exists in LPTV systems; this may cause adverse eect for robustness against high frequency uncertainties in periodic control  This

research was supported by the Natural Sciences and Engineering Research Council of Canada and

the Hong Kong Research Grants Council.

1

systems [7] and for perfect reconstruction in multirate lter banks [11]. The rst question in this paper is therefore: How to quantify aliasing eect in LPTV systems? If aliasing is negligible in an LPTV system to be controlled, one can then approximate it by an LTI system with little error. Control design can be then based on the LTI model; this has several advantages: First, robust control design for LTI systems is thoroughly studied and there are now many techniques applicable; second, the controller designed this way normally is LTI too and so is easier to implement than an LPTV controller, resulted from design based on the original LPTV system. Hence the second question in this paper is: How to optimally approximate an LPTV system by an LTI one? Throughout the paper, we will focus on discrete-time MIMO (multi-input multi-output) systems. The two questions are related as follows. Since LTI systems form a subspace within LPTV systems, we consider the following distance problem: Given an LPTV system, compute its distance to the subspace of LTI systems. The distance, to be measured by norms, is a measure of how time-varying the LPTV system is and hence can be used to quantify aliasing; the LTI system achieving the distance is the optimal LTI approximation to the given LPTV system. Two norms will be used for LPTV systems: the Hilbert-Schmidt norm or H2 norm and the `2-induced norm or H1 norm. LPTV systems have no transfer functions in general; however, there are two ways to describe their frequency responses using matrices: The rst one is based on a time-domain technique called lifting in control [8] or blocking in signal processing [11]; the second one is a frequency-domain technique, also independently used in control [7] and signal processing [10]. Though the two techniques are essentially related [9], here we adopt the latter for better insight in the frequency domain. Brie y, the paper is organized as follows. In the next section we discuss a frequencydomain representation for LPTV systems, which is relevant to our studies later. Section 3 studies the distance problem using Hilbert-Schmidt norm and gives complete solutions. Section 4 looks at the distance problem using `2 -induced norm and only partial solutions are obtained. In Section 5 we show the relevance of the work here to an example of LPTV systems in signal processing, namely, the multirate lter-bank system used in, e.g., subband coding. Finally, we conclude in Section 5.

2 Frequency-Response Matrices We begin with the de nition of frequency-response matrices from [7, 10]. (A continuous-time analog of this was introduced in [1].) It is convenient to de ne the exponential signal of frequency f : ef (k) := ej2fk : 2

If H were LTI and stable, the output of H due to this input ef (k) would be H^ (f )ef (k), H^ (f ) being the frequency response. Now let H denote an LPTV system of period m. If the the input to H is again ef (k), the corresponding output is now a linear combination of the complex exponentials of frequencies [10] f; f + m1 ; : : :; f + mm? 1 : De ne the m-dimensional subspace

Sf := span fef ; ef

=m; : : :; ef +(m?1)=mg:

+1

It follows [10] that Sf is an invariant subspace for H . Thus an input to H of the form

u(k) =

mX ?1 n=0

unef +n=m (k);

which is represented by the m-dimensional vector 2 6 6 u^ := 66 4

u0 u1 .. .

um?1

3 7 7 7 7 5

for the obvious choice of basis functions in Sf , will produce an output of the form

y(k) = which is represented by the vector

mX ?1 n=0

2 6 6 y^ := 66 4

yn ef +n=m (k); y0 y1 .. .

ym?1

3 7 7 7 7: 5

This induces an m  m matrix, denoted H^ FR (f ), relating u^ to y^: y^ = H^ FR (f )^u: This matrix is called the alias component matrix in [10] in view of its prior occurrence in the literature on multirate lter banks, and is a generalization of frequency-response function. The matrix H^ FR is called the frequency-response matrix for the LPTV system H from now on. Note that in the de nition of H^ FR (f ), f ranges over the interval ? 21m  f  21m . 3

This frequency-response matrix has an equivalent interpretation as follows. Let the input and output of H be u and y . Denote the Fourier transform of u by u^(f ), a periodic function with period 1 (in f ). Chop one period of u^ into m pieces and form a vector: 2 u^(f )  3  6 u^ f ? m1 777 6 6 ; ? 21m  f  21m : 6 7 .. 6 7 4  .  5 u^ f ? mm?1 Similarly for the Fourier transform y^(f ). It follows that the two vectors are related exactly by the frequency-response matrix: 2 2 y^(f )  3 u^(f )  3   6 6 y^ f ? m1 777 u^ f ? m1 777 6 6 1 f  1 : 6 6 ^ FR 6 (1) = H ; ? 6 7 7 . . . .. 6 7 7 6 2 m 2 m 4  . 5 5 4    y^ f ? mm?1 u^ f ? mm?1 Now we look at how to compute frequency-response matrices. We start with a causal, MIMO, LPTV system H with period m. H is characterized by its impulse response matrix h(k; l) as follows:

y(k) =

k X l=0

h(k; l)u(l):

(2)

Periodicity is equivalent to the condition h(k + m; l + m) = h(k; l); 8k; l; and causality is equivalent to h(k; l) = 0; whenever l > k: De ne  = k ? l to get h(k; l) = h( + l; l). It follows that for any xed  , h( + l; l) is m-periodic in l and hence has a discrete Fourier series:

h( + l; l) =

mX ?1

hn ( )en=m (l) n=0 ?1 1 mX

hn( ) = m

l=0

h( + l; l)e?n=m(l):

(3) (4)

Here hn ( ) = 0 for  < 0 due to causality. Substitute (3) into (2) to get that y can be expressed as a sum of convolutions of hn with en=m u:

y=

mX ?1 n=0

hn  [en=m u]: 4

(5)

Let Hn be the causal, LTI system with impulse response matrix hn (k). Equation (5) represents a time-domain decomposition of the LPTV H as depicted in Figure 1: The input u(k) is channeled into m dierent subsystems numbered 0, 1,


, m ? 1; at the n-th subsystem u(k) is rst modulated by the exponential function en=m (k) and then passed through the LTI system Hn ; the sum of the outputs of Hn forms y . The m LTI systems Hn are called the components of the LPTV system H ; they uniquely characterize H .

u

e0

=m ? - j -

H0

e1

rr r

=m ? - j -

e(

- ?j ?r

H1

? m? =m- Hm? - j 1)

1

rr - ?j

y

-

Figure 1: Decomposition of the LPTV system H . Based on the decomposition in (5), it is easy to get that the LPTV system H becomes LTI i hn = 0 for n = 1; 2;


; m ? 1. To generalize this, let m = m1 m2 with m1 and m2 both positive integers. How to test m1 periodicity of H based on its m LTI component systems Hn ?

Theorem 1 Assume a causal, MIMO system H is LPTV with period m; its associated component systems are denoted Hn(m) : n = 0; 1;


; m ? 1. Let positive integers m1 and m2 satisfy m = m1 m2. Then H is LPTV with period m1 i Hn(m) = 0; n 6= 0; m2; 2m2;


; (m1 ? 1)m2; in this case, the m1 component systems of H associated with periodicity m1 are given by: (m) Hn(m1 ) = Hnm ; n = 0; 1;


; m1 ? 1: 2

The decomposition in (5) or in Figure 1 gives a way to compute the frequency-response matrix for H . Take Fourier transform of both sides of (5) to get

y^(f ) =

  H^ n(f )^u f ? mn n=0

mX ?1

5

=

h

H^ 0(f ) H^ 1(f )


H^ m?1 (f )

2 6 i6 6 6 6 4

3 u^(f )  7 u^ f ? m1 77 : 7 .. 7  .  5 u^ f ? mm?1 1 m?1 m ;


; f ? m , respectively,

Replacing the frequency f in the above equation by f ? and noting the periodicity of u^ [^u(f  1) = u^(f )], one can get a matrix equation in the form of (1), where the frequency-response matrix is: 3 2 H^0 (f )  H^ 1 (f ) 


H^ m?1 (f )  6 6


H^ m?2 f ? m1 777 H^ m?1 f ? m1 H^ 0 f ? m1 6 ; ? 21m  f  21m : H^ FR (f ) = 66 7 .. .. .. 7 4  .  .  .    5 m ? 1 m ? 1 ^ ^ ^ H2 f ? m


H0 f ? mm?1 H1 f ? m (This representation is also given in [14].) The frequency-response matrix is completely characterized by the m transfer functions H^ 0 , H^ 1 ,


, H^ m?1 . Note the row-wise circular structure coupled with the frequency shift. As a special case, if H is LTI, the frequency-response matrix is diagonal: 2 ^ 3 H0 (f ) 0


0   6 7 ^ 0 f ? m1


6 7 0 H 0 1 f  1 : 6 7 ^ HFR (f ) = 66 .. ; ? 7 .. .. 7 2m 2m . 4 .  .  5 0 0


H^ 0 f ? mm?1 (A condition for time-invariance was also obtained in [12] in the time domain using state-space models.) As an example, consider the state-space model with input u, output y , and state vector x: x(k + 1) = A(k)x(k) + B(k)u(k); y(k) = C (k)x(k) + D(k)u(k): Assume that A(k), B (k), C (k), and D(k) are LPTV with period 2; write ( if k is even; 0; A(k) = A A1 ; if k is odd; and similarly for B (k), C (k), and D(k). This system is LPTV with period 2. Its component systems H0 and H1 can be computed from de nitions; they are given by their transfer functions H^ 0 (z ) and H^ 1(z ): First de ne two functions G^ 0(z) = D0 + (C0A1 + zC1)(z ?2 I ? A0 A1 )?1 B0 ; G^ 1(z) = D1 + (C1A0 + zC0)(z ?2 I ? A1 A0)?1 B1 ; 6

then

H^ 0 (z) = 21 [G^ 0(z) + G^ 1 (z)]; H^ 1(z) = 12 [G^ 0(z) ? G^ 1(z)]:

Note that in general the orders of the LTI systems H0 and H1 exceed the dimension in the matrix A(k) but are nite. Of course, these formulas can be generalized to LPTV state-space systems with a general period m. Based on the frequency-response matrices, we shall study two optimal approximation problems involving LPTV and LTI systems; the quantities used to measure degree of closeness of two LPTV systems are the Hilbert-Schmidt norm and the `2-induced norm.

3 Using the Hilbert-Schmidt Norm Any MIMO, causal, LPTV system H is completely determined by its impulse response matrix h(k; l) for 0  k < 1 and 0  l < m ? 1. We say H is stable if mX ?1 X 1 l=0 k=0

trace [h(k; l)0h(k; l)] < 1:

All causal, stable, LPTV systems with period m form a Hilbert space with the Hilbert-Schmidt norm: !1=2 1 mX ?1 X 1 0 kH kHS = m (6) trace [h(k; l) h(k; l)] : l=0 k=0

(H can be regarded as a Hilbert-Schmidt operator if one restricts the input to be de ned on the time set [0; 1;


; m ? 1].) Several ways to evaluate this norm are given below (some of these are also given in [14]): 1. In terms of the component functions hn ( ), n = 0; 1;


; m ? 1, we have:

kH kHS = 2

1 mX ?1 X n=0  =0

trace [hn ( )0hn ( )]:

(7)

Proof: Rearrange the summations in (6) to get 1 mX ?1 X

kH kHS = m1  2

l

trace [h( + l; l)0h( + l; l)]

=0 =0

1 X = m1 trace  =0

2 6 6 6 6 4

h(; 0) h( + 1; 1) .. .

h( + m ? 1; m ? 1) 7

30 2 7 6 7 6 7 76 6 54

h(; 0) h( + 1; 1) .. .

h( + m ? 1; m ? 1)

3 7 7 7 7: 5

(8)

The DFT involved in (3) implies 2 6 6 6 6 4

h(; 0) h( + 1; 1) .. .

h( + m ? 1; m ? 1)

3 2 6 7 6 7 6 7 = 7 6 5 4

I I




I j  m


e ?

I

ej 2=m I .. .

.. .

2 (

.. .

=m I

1)

I ej2(m?1)=m I


ej2(m?1)2 =m I

32 6 7 6 7 6 7 76 54

h0 ( ) h1 ( ) .. .

hm?1 ( )

3 7 7 7 7: 5

Note that the DFT matrix W appears here. Substituting this into (8) and using the property W  W = mI , we get 1 X

kH kHS = 2

 =0

trace

mX ?1 X 1

=

2 6 6 6 6 4

n=0  =0

h0( ) h1( ) .. .

hm?1 ( )

30 2 7 6 7 6 7 76 6 54

h0( ) h1( ) .. .

hm?1 ( )

3 7 7 7 7 5

trace [hn ( )0hn ( )]:

2

2. In terms of the frequency responses H^ n (f ) of the LTI component systems Hn , n = 0; 1;


; m ? 1, we have:

kH kHS = 2

=

mX ?1

kH^ nk

n=0 mX ?1 Z 1=2 n=0

?1=2

2 2

trace [H^ n(f ) H^ n (f )] df:

(9)

Here kH^ n k2 denotes the H2 norm of the LTI system. This result follows from (7) by Parseval's equality. 3. In terms of the frequency-response matrix H^ FR (f ), we have: kH k2HS = kH^ FRk22 :=

Z 1=2m

?1=2m

trace [H^ FR (f )H^ FR (f )] df:

This follows readily from (9) and the de nitions of H^ FR(f ). The distance problem to be studied in this section is: Given a causal, stable, LPTV system H , what is the distance, measured by the Hilbert-Schmidt norm, from H to the subspace of causal, stable, LTI systems? This problem is written:

 := min kH ? GkHS : LTI G

8

The LTI system Gopt achieving this minimum is the optimal LTI approximation to the LPTV H . The quantity  can be used as a measure of aliasing in the LPTV system; or relatively to the size of H , one can use the quantity  := =kH kHS (0    1) to measure aliasing; in this case,  = 0 means H is already LTI and  = 1 means H is anti-LTI, i.e., the optimal LTI approximation Gopt = 0. Theorem 2 The optimal LTI approximation to the LPTV system H is Gopt = H0 and

=

mX ?1 n=1

kH^ nk

!1=2

2 2

:

Proof: Let G be any LTI, causal, stable system with compatible input and output dimensions with H . The component systems Gn for G as an LPTV system with period m are given by ( if n = 0; Gn = G; 0; if 1  n  m ? 1: By property 2 above we have

kH ? GkHS = 2

mX ?1 n=0

kH^ n ? G^nk

2 2

m?1

X = kH^ 0 ? G^ k22 + kH^ n k22 :

n=1

Clearly, this quantity is minimized by taking G^ = H^ 0 and hence the results. 2 This theorem also suggests that one can orthogonally decompose an LPTV system H into H = HLTI + HLTV , where the LTI component is HLTI = Gopt and the anti-LTI (aliasing) component is HLTV . Then kH k2HS = kHLTI k2HS + kHLTV k2HS : To generalize Theorem 2, we can also consider approximating an LPTV system with period m by an LPTV system with period m1 , where m is a multiple of m1. Let H be LPTV with period m as before and write m = m1 m2 with positive integers m1 and m2. The problem is as follows:  := minfkH ? GkHS : G is LPTV with period m1:g: Theorem 3 Given the LPTV system H with period m as above, the optimal LPTV approximation Gopt with period m1 is given by the component systems: Gopt n = Hnm2 ; n = 0; 1;


; m1 ? 1: Moreover, X 1=2 kH^ nk22 ; = where the sum is over all integers n with 0  n  m ? 1 and n 6= 0; m2; 2m2;


; (m1 ? 1)m2. 9

This generalization is relevant if one would like to reduce the number of modeling parameters in LPTV systems by reducing the periodicity number; and the quantity  can be used as an indicator for error incurred in this approximation.

4 Using the `2-Induced Norm The second norm we use for approximation is the `2 -induced norm. For a causal, MIMO, LPTV system H with period m, let u and y be the input and output vectors respectively. The `2 -induced norm of H is de ned as:

kH k := supfkyk : kuk = 1g: If this is nite, we say in this section that H is stable. It is proven in [10, 9] that the ` -induced norm of an LPTV system H equals the 1-norm 2

of the frequency-response matrix:

2

2

kH k = kH^ FRk1 =

max

? 21m f  21m

max[H^ FR (f )];

where max denotes the maximum singular value. This gives a way to evaluate the `2 -induced norm in terms of the frequency-response matrices. The distance problem in this section is as follows: Given a stable, LPTV system H , compute the distance, measured by the `2-induced norm, to the subspace of LTI systems:

1 := inf kH ? Gk:

LTI G Again, the minimizing LTI system, Gopt, can be used as an approximation to H and the quantity 1 as a measure of aliasing in H . These are useful for robust control using unstructured uncertainty models: Using the LTI Gopt as the nominal model, the size of the modeling error is given by 1 .

Theorem 4 Assume H is causal, stable, and LPTV with period 2. The optimal LTI approx-

imation is Gopt = H0 and

1 = kH^ 1k1 := 1max 1 max [H^ 1(f )]: ? 4 f  4

Proof: For any LTI G, the frequency-response matrix for H ? G is: " ^ # ^ (f ) ^ (f )  H ( f ) ? G H     H^ FR (f ) ? G^ FR (f ) = H^ f ? ; ? 14  f  41 : H^ f ? ? G^ f ? 0

1

0 1 2

1

0

10

1 2

0

1 2





In this matrix we note that both H^ 1(f ) and H^ 1 f ? 12 , ? 14  f  41 , appear as submatrices; hence for any LTI G, we have: kH ? Gk = kH^ FR(f ) ? G^ FR(f )k1  kH^ 1k1: So 1  kH^ 1k1 . It can be veri ed that setting G = H0, we get kH ? Gk = kH^ 1k1 and therefore Gopt = H0 and 1 = kH^ 1k1 . 2 Though in this special case the optimal approximation equals that using the HilbertSchmidt norm, the distance 1 is dierent. The proof does not work for general LPTV systems with period m; in this case, only lower and upper bounds for 1 are obtained: i

h

1  k H^ 1(f ) H^ 2 (f )


H^ m?1 (f ) k1 ; 2 0 H^ 1(f )


H^ m?1(f )  3   6 6 H^ m?1 f ? m1 0


H^ m?2 f ? m1 777 6 1  k 66 k : 7 .. .. ... 7 1 . . 5 4     0 H^ 1 f ? mm?1 H^ 2 f ? mm?1


If m = 2, the lower and upper bounds reduce to the same (and hence the results in Theorem 4); however, it is not the case in general. In [3], LTI approximations of periodic systems were also studied; but they are not optimal in the sense of the `2-induced norm.

5 Application to Multirate Signal Processing The results in the preceding sections have applications in quantifying aliasing in multirate digital signal processing systems. Consider the multirate lter bank in Figure 2. In this discrete-time system, G0, G1, F0 ,

x(k)

-

G0

- #2

- "2

-

F0

-

G1

- #2

- "2

-

F1

Figure 2: Multirate lter bank. 11

- ?j -y(k)

and F1 are LTI lters, # 2 denotes the downsampler (decimator) by a factor of two, and " 2 denotes the upsampler (expander) by the same factor. This multirate signal processing system is important in, e.g., subband coding, and has been studied a great deal; see the recent book [11] and the references therein. Typically, the overall system H : x 7! y is required to reconstruct x(k), ideally with at most a time-delay error. So perfect reconstruction is said to be achieved [11] if for some integer m  0, y (k) = x(k ? m). That is, the desired system from x(k) to y(k) is LTI with transfer function H^ d (z) = z ?m : Note that in general the system H is LPTV with period 2 because of the presence of the down- and upsamplers. Compared with the ideal system Hd , the LPTV system H in general suers three types of distortion due to aliasing, magnitude and phase errors. Aliasing distortion can be quanti ed using the norms introduced earlier. It is known that the input and output of the lter bank system H are related in the frequency domain as follows [11]: y^(z) = 21 [G^ 0(z)F^0(z) + G^ 1 (z)F^1(z)]^x(z) + 21 [G^ 0(?z)F^0 (z) + G^ 1(?z)F^1 (z)]^x(?z): Hence the component systems for H are given by H^ 0(f ) = 21 [G^ 0(f )F^0(f ) + G^ 1(f )F^1(f )]; H^ 1(f ) = 21 [G^ 0(f ? 1=2)F^0(f ) + G^ 1(f ? 1=2)F^1(f )]:

In this case, two quantities can be used to measure the degree of aliasing:

1

Z 1=2

jH^ (f )j df; = max jH^ (f )j: jf j =

 =

?1=2

1 4

1

2

1

Their interpretations are as follows:  is the distance from this LPTV system H to the space of LTI systems via the Hilbert-Schmidt norm; and 1 the distance via the `2 -induced norm. (Of course, the optimal LTI approximation to H is H0 using both norms.) In [4], design of the synthesis lters for optimal reconstruction is developed based on H1 optimization. Note that for a given set of analysis lters, the designed optimal synthesis lters do not remove aliasing completely but keep it at a small level for optimal overall performance.

6 Concluding Remarks In Section 4, the distance problem is solved only when LPTV systems are of period 2; in the general case we obtained only lower and upper bounds for the minimum distance. Though no 12

closed-form solutions are obtained in the general case, it is possible to compute the optimal solution within any desired accuracy via numerical optimization, because it is easy to see that the associated optimization problem is convex in nature. In Section 2 we discussed frequency-response matrices for discrete-time LPTV systems. Frequency responses of continuous-time LPTV systems, which include sampled-data control systems as special cases, and their computation are among the recent developments in sampled-data control theory, see, e.g., [1, 2, 13]. How to quantify aliasing and compute optimal LTI approximations in sampled-data systems could be addressed using the ideas of this paper and the ground work in [1, 2, 13].

References [1] M. Araki and Y. Ito, \Frequency-response of sampled-data systems I: open-loop consideration," Proc. IFAC 12th World Congress, vol. 7, pp. 289-292, 1993. [2] M. Araki, T. Hagiwara, and Y. Ito, \Frequency-response of sampled-data systems II: closed-loop consideration," Proc. IFAC 12th World Congress, vol. 7, pp. 293-296, 1993. [3] P. O. Arambel and G. Tadmor, \LTI decomposition and H1 approximation of periodic systems," Proc. ACC, pp. 1598-1602, 1994. [4] T. Chen and B. A. Francis, \Design of multirate lter banks by H1 optimization," IEEE Trans. Signal Processing, vol. 43, pp. 2822-2830, 1995. [5] T. Chen and L. Qiu, \H1 design of general multirate sampled-data control systems," Automatica, vol. 30, pp. 1139-1152, 1994. [6] D. P. Glasson, \Development and applications of multirate digital control," IEEE Control Systems Magazine, vol. 3, pp. 2-8, 1983. [7] G. C. Goodwin and A. Feuer, \Linear periodic control: a frequency domain viewpoint," Systems and Control Letters, vol. 19, pp. 379-390, 1992. [8] P. P. Khargonekar, K. Poolla, and A. Tannenbaum, \Robust control of linear timeinvariant plants using periodic compensation," IEEE Trans. Automat. Control, vol. 30, pp. 1088-1096, 1985. [9] S. Mirabbasi, B. A. Francis, and T. Chen, \Input-output gains of linear periodic discretetime systems with application to multirate signal processing," Proc. ISCAS, pp. II 193196, 1996. [10] R. G. Shenoy, D. Burnside, and T. W. Parks, \Linear periodic systems and multirate lter design," IEEE Trans. Signal Processing, vol. 42, pp. 2242-2256, 1994. [11] P. P. Vaidyanathan, Multirate Systems and Filter banks, Prentice-Hall, Englewood Clis, 1993. 13

[12] P. Van Dooren and J. Sreedhar, \When is a periodic discrete-time system equivalent to a time-invariant one?" Linear Algebra and its Applications, vol. 212/213, pp. 131-151, 1994. [13] Y. Yamamoto and P. P. Khargonekar, \Frequency response of sampled-data systems," IEEE Trans. Auto. Control, vol. 41, no. 2, pp. 166-176, 1996. [14] C. Zhang, J. Zhang, and K. Furuta, \Performance analysis of discrete periodically time varying controllers," preprint.

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