A Multiresolutional Controller for Sensitivity Minimization

Copyright© 1996 IFAC 13th Triennial World Congress, San

Franci~co,

2a-034

USA

A MULTIRESOLUTIONAL CONTROLLER FOR SENSITIVITY MINIMIZATION D. J. Clancy* and U. Ozgiiner* 'Th. Ohio St.t. Uni•• nitg, D.p.rtment of Electric.1 Engine.ring, Columb ..., Ohio, USA

Abstract: A multiresolutional signal decomposition allows a signal to have equivalent representations at different scales or levels of detail. This idea can be extended to a control system by having a multiresolutional representation of the plant and of the controller. One important purpose for developing a multiresolutional control system is to have the capability of controlling a plant with a high resolution representation using a low resolution controller. A multireoolutional controller could also be used to control only a particular aspect of a plant. We take" first step towards achieving a multiresolutional control system in this paper. We model the plant using a best, orthonormal wavelet packet basis and then determining the optimal controller using the same wavelet packet basis or another related wavelet packet basis. This paper presents .. theorem for designing a wavelet packet controller which minimizes the sensitivity function for a SISO system. We also present new theoretic..l results on wavelet packets required for making the implementation of this multiresolutional controller very practical. Keywords: Wavelets, Walsh Functions l H2 Control, Disturbance Rejection

1. INTRODUCTION Multiresolution encompasses the ideas of scale and the process of moving from one scale to another. The concept of a muItiresolutional representation refers to a simple hierarchical architecture for interpreting information. The most intuitive environment for such a multiresolutional representation is in the field of image processing. The different resolutions of an image tend to characterize the various physical structures present within the image. A coarse resolution would provide general trends, or larger scale structures present within the image. A fine resolution would accentuate the details, or small scale structures present within the image. The theory of multiresolution analysis was pioneered by (Mallet, 1989) and (Meyer, 1986) and was the impetus for the fast growing field of wavelets. A multiresolutional signal decomposition allows a signal to be represented equivalently at different scales or levels of d...

tail. The theory of multiresolution analysis provides the essential framework for wavelets and wavelet packets. Wavelet packets provide a rich theory with a library of functions and algorithms available for modeling signals. Wavelet packets have the capability to provide a best basis for a signal or class of signals. Wavelet p3Cket coefficients tend to decay quickly and usually require substantially fewer basis functions compared with a Founer basis. Wavelet packet basis functions can be easily modified to account for uncertainties, failures I and changes in the operating environment of a system. This paper involves developing a multiresolutional CODtrol system. First of all, it is necessary to obtain a multiresolutional representation of the plant to be controlled. Then, a controller for the system will be developed which also has a multiresolutional representation. One important purpose for developing a multiresolutionsl control system is to have the capability of con-

1074

trolling a plant with a hillh resolution representation using a low resolutiou controller. Another purpooe for developing a multiresolutional control system is to be able to control a group or claoo of plants with the lI&Dle controller. The controller can have multiple resolutions or only one resolution. A multireoolutional controller could also be used to control only a particular aspect of a plant. We take initial steps towards achieving a multiresolutional control system in tbis paper. Tbe first step te>wards achieving a multiresolutional control system is to use wavelet packet theory iu order to model the plant to be controlled. The main idea is to model the plant using a "best" wavelet packet basis. The word best is in quotes since the best wavelet packet basis may be very dependent on the control objectives, plant characteristics, etc., and therefore could vary greatly from one control system to another. The next step is to determine the optimal controller using the same wavelet packet basis or another related wa.velet packet basis. If the controller has the same wavelet packet basis as the plant,

then it may be possible to easily modify the controller in order to account for uncertainties, changes in operating environment, etc., thus making for a robust/adaptable

controller. This paper presents a theorem for designing a wavelet packet controller which minimizes the sensitivity function for a SISO system. We also present new theoretical results on wavelet packets required for making

the implementation of this multiresolutional controller very practical.

2. MULTlRESOLUTION ANALYSIS The theory of multiresolution analysis (MRA), or multiresolutional signal decomposition, pioneered by (Meyer, 1986) and (Mallet, 1989), provides a natural framework for formulating and understanding wavelet theory. A recapitUlation of these concepts as given in (Mallet, 1989) and (Daubechies, 1992) will now be pr&sented.

Definition 1 (Translation Operator)

T.,,(t) := ",(I - k),

properties: Nesting Property: ... C V_. C V_, C VD C V, C V. C ...

Union Property: (2)

Intersection Property:

n

(3)

z'(I) E Vm+,

(4)

Translation Property:

,,(t) E Vc

==}

".(1) E Vc Vie E 7t

(5)

Basis Property: 3 a scaling function 4> E VD such that {4>~(I)

= 4>(1 - k) : k E 7t} is a Ries. basis for Vc (6)

Equations (4) and (6) imply that {4>r(I): le E 7t} is a Ries. basis for Vm , Vm E 7t with 4>r(l) 2m /24>(2"'1 - /c).

=

When the MRA is orthogonal, the Ries. basis is an orthonormal basis. For an orthogonal MRA, define Pm to be the orthogonal projection operator onto Vm • By (2), lim m _ oo Pmz = z "Ix E 1l. The crux of an orthogonal MRA is that if a set of closed subspaces satisfies (1) (6), then there exists an orthonormal wavelet basis of

1l given by {!pr(t): rn, k E 7t} which for all z E 1l satisfies the relation

Pm+1'" = Pm" + E (z, .pk'(I».pk' (I)

(7)

Oe'" Now, for every m E 7t, define the orthogonal complement of Vm in Vm + 1 as Wm. At every scale Vm+1 = Vm (j) Wm Wm .L Wm, if rn oft rn' Wm C Vml if m < rn' Then, for m

~

(8) (9) (10)

M, m

Vm +1 = VM

In shorthand notation, zm(t) := Dm",(I). entails a. sequence of successive closed approximation

=

"'(I) E Vm

Dm,,(t) := 2m/ 2 ,,(2mt), m E 7t

A multiresolutional signal analysis, in Hilbert space 1l,

= {O}

Dilation Property:

k E 7t

e (Dilation Operator)

V",

mez

In shorthand notation, ",(I) := T.,,(t). Definition

(1)

El}

EB W;

(11)

i=M

From equations (2) and (3), it follows that 1l can b. decomposed into mutually orthogonalsubspaces as fol-

subspaces, Vm with m E .It, satisfying the following

1075

lows

(12) With the orthogonal property for Wm , equation (9), equation (7) equivalently statea that {!IIr(t): I.: E !E} is an orthonormal baais for W m, '1m E !E. The nesting, dilation, and baais properties, equations (1), (4), and (6), imply that a normalized scaling function, defined as f 4o(t)cIt 1 with 40 E Vo c .e'(R), satisfies the following two-scale dilation equation:

=

.p(t) =

J2 2: h(k).p(2t -

k) := Fo.p(t)

(13)

_ell<

J2 2: g(k)4i(2t -

k) := F,40(t)

(14)

.ell<

where (g( k)} is a .quar....ummable sequence of coefficients which defines an operator F1 • A quadrature filter denotes an operator which perform. convolution and then decimation. The operator. defined in equations (13) and (14), namely, Fo and F" are quadrature filters. Typically, Fo consists of a low pass filter (LPF), H, followed by a decimator, ! 2, whereas F, consists of a high pass filter (BPF), G, followed by a decima.tor. The connection between orthogonal wavelets and orthogonal, PR filters banks of finite length was first established in (Daubechies, 1988). The scaling function given by equation (13), 4o(t), and wavelet function given by equation (14), ",(t), which were derived based on a multi resolution analysis, can also be derived by performing an infinite, lowpass iteration on the analysis or synthesis filter bank of particular PR filter banks. 3. WAVELET PACKETS Definition :1 (Binary Representation) For any f E !E+ , where X+ is the positive integers, the infinite binary expansion of f is given by f = Er:o A2-, A E CO, I}. In shorthand notation, CA : k E Z+} := Er:o f. 2l. .....

..

.

Definition .( (Wavelet Packet Frequency Filter Sequence) Let the wavelet packet frequency, f E !E+, 0 ::; f ::; 2N - 1, be represented by {t. : k E Z+}. Define the wavelet packet frequency filter sequence (WPFFS) in the frequency domain, QN-,QN-2·· ·Qo with length N, according to:

Q _ •-

{H'G,

W;,

",~,,,,(t) = 2't'

if fl if ft

=0

=1

(15)

W;/+',

2: h(k)"',(2'+lt -

2p - k)

:=

Fo"'i,,(t) (16)

• ell<

",,/+,(t)

where {h(k)} is a .quar.... ummable sequence of coefficients which definea an operator Fo. Similarly, a normalized wavelet function, '" E Wo C .e' (JR) , .ati.fies the following two-scale dilation equation: ",(t) =

Thil section generalizes the concepts presented in section 2. Wavelet packeta can be viewed 88 a generaliz .... tion of a MRA or as a generalization of an iterated, PR filter bank. Simply put, wavelet packets are partiCUlar linear combinations of waveleta whose translations also comprise a ruesz baais for Vm C 1l, '1m E!E. The twoscale dilation and wavelet equations, (13) and (14), can be generalized for the scale • E Z, frequency , E Z+, and position p E !E, to form the wavelet packet equ.... tions, "'~,,' E and "'~/+'''' E as follows:

= 2't' 2: 9(1.:)",,(2'+'t -

2p- k):= F,"'i,.(t) (17)

lez

Note that (t) of equation (13) is given by 4o(t) = ",g ott) and ,pet) of equation (14) is given by ",(t) = "'~ o(tj. It should be noted that the .caling function spa~es, Vm , and the wavelet function spaces, Wm , discussed in section 2, are represented using the wavelet packet notation of this section according to Won := Vm and Wi := Wm. It can be shown that for each f, f E Z+, and each B, B E Z, the collection p E Z} is a ruesz basis for Wj and that E!E, pE!E, f E Z+} is a rueaz basis for .e'(JR) (Coifman .t al., 1992).

{"'i" :.

{"'i" :

4. ORTBONORMAL WAVELET PACKETS This section presents some new theoretical results for compactly supported, orthonormal wavelet packets (OWPs) at the scale B with dilation factor 2. An OWP refers to a wavelet packet whose basis functions are nOImalized and whose dilation equation coefficients are orthogonal. It is to be assumed throughout this section, and the remaining sections that the equation, ""j,p(t), refers to a compactly supported, OWP basis function at the scale 8 with dilation factor 2. In addition, for the case of Baar wavelet packets (HWPs), it will be shown theoretically that BWPs are closed under multiplication. The following theorem presents an explicit relationship between compactly supported, OWPs with dilation factor 2 at different scales and different frequencies. In other words, any compactly supported, OWP basis function, ·Tii} .0), at the scale 8, can be represented by a summation of wavelet packet basis functions at a finer scale, s + Q', 0: E IN. Theorem 1 (OWP Representation Theorem) Let ,et) be a compactly supported, OWP basis function with dilation factor 2, whose dilation equation coefficients, h and g, correspond to causal, orthogonal, FIR lilters of length r. Represent the frequency of the

"'i

1076

wavelet packet basis function, / E $+, by {fn : n E $+ }. Then, for any '" EN, the wavelet packet basis function, (t), can be represented by the following relationship:

defined by the impulse response coefficients for tbe quadrature filter Fo, h(k), and by the impulse response coefficients for the quadrature filter F" g( k), 88 follows:

"'i ...

"'i,,(t) = 2~Lmo-l(k)"'i"(2'+ol -

2°p_ k)(IS)

,h~O

where:

fa

={

;",-1 2 , :/ /a-I = 0 &:cl ./ /a-I = I 2 '

} with

fo =/;

{ma-I(k) : k = 0, ... , (2a - I)(r - In = Z-I {Qa-I(')}; Qa-I(') = I1:;;;~ Qa-I-n(""); Qn

= {H, ~/

/n

G, ./ /n

=0 } = 1

/or k =0, 1 /or k = 0, 1

~,

h(k) = g(k) =

(2"-I)(r-l)

(\p",

Figure 1 demonstrates the wavelet packet basis functions on a compact interval, I = [a, b), for several wavelet packet subspaces, Wj, based on applying the Haar quadrature filter operators, Fo and F" defined by the coefficients given in equations (5). When moving from left to right in figure 1, it should be noted that the support of the basis functions doubles. This support property holds for the general interval, 1= (-00,00), as well. Also, for a compact interval only, the number of moving from left to right in figure 1. Lastly, the wavelet packet subspaceB,

W8 - W~, shown in figure I, each have

Now it will be important to define explicitly the support interval for a compactly supported, OWP basis function

one

generated using causal, orthogonal, FIR filter sequences

their corresponding basis function.

'# the first

of even length r ;:: 2. The support of "'i,,(t) is given by Supp{"'i,p(l)} = [2-·p,2-'(p+r-I)). The following

eight Paley ordered Walsh functions as

-,

lemma states the orthonormality condition for two unit scale, s = 0, normalized wavelet packet basis functions.

Lemma 1 ((Wickerhauser, 1994))

(,p"",(t), "'12,p,(t»

/2)6(pl - p2)

(19)

-,

The next theorem presents an explicit formula for the inner product of any two compactly supported, OWP basis functions at possibly different scales with dilation factor 2. It should be noted that OWP basis functions at different scales are not necessarily orthogonal!

= ,J

Theorem!! (Wavelet Packet Inner Product Theorem) (t) and "'it.,(t) Given two OWP basis functions at the scales si and s2 respectively, assume WLOG that 81 ;:: s2. Represent the frequency of the second wavelet packet basis function, /2 E $+, by {f2n : n E $+}. The inner product of theae two OWP basis functions is given explicitly by t.he following: If si = s2,

If si

(20)

> s2,

("'i\.PI (t), "'i~.p,(t» = 6(/1 -

j2,,,)

ma.,{m,"_I(k) 6(pl- 2'''p2 - k)}

(21)

where k = 0, ... , (2'" - 1) (r - 1), and sa := si - s2 = '" in Theorem 1.

Definition 5 (Haar WaveJet Packets) The HWPs are

,

..

i

i

,

~

•

~

~

~

'\

.r

'J

Cl

w 0

-,

0 GLJ

C1

D

1/

w 0 LJ

I!.,

.,

Cl

~

• K

w "

•

L-J

l.--.

Cl

"

Cl

0

"'ii,.1

= 6(/1- /2) 6(pl- p2)

"

== /' =K" ...= . = " = ,.j = " == K .. = = " = Wn

("'il.,I(t),"'H",(t»

-,

~I c--1

= 6(/1 -

Wj, is halved when

basis functions, P, in each subspace,

Z-l signifies the inverse z-transform.

(22)

oK

L......I

"

D

0 n

~

~

~

~

~

.. .. ..

~

~

w

Cl

n

~

~

L......I

0

'"

~

~

0

L......I

CL~

L......I

U

W

Fig. 1. The W&vel.et pa.cket b.aaia fu.nctions for sever&l subbased on a.pplying the Haar quadrature spaces, filter operators, Fo and Ft.

W;,

Since each subspace WJ has the fth, Paley ordered Walsh function as its corresponding basis function, it follows that the unit scale HWPs, ",,(t), are the Walsh functions supported on the interval [0, 1). The following theorem is essential to proving that the HWPs are closed under multiplication. Theorem 9 (( Golubov et .1., 1991» Let "'/1 (t) and ",,,(I) be two Walsh functions whose corresponding frequencies, /1 E $+ and /2 E $+ , are represented by (/h : k E $+) and {/2. : k E $+}. The Walsh functions are clooed under multiplication with

1077

their product given by:

(23)

"". (t)""2(t) = "'1'1ll/.(t) where /1 f.Il /2 := Ef=o I/10 - /2. I2> E 2Il'+ •

The main result of this Seetion with respect to the oontroller design methodology discussed in Section 5 will now be presented. This next theorem extends the multiplication property of Walsh functions to include HWP basis functions on the interval [0, 1), and gives explicitly the resultant product. Theorem 4 (HWP Multiplication Theorem) Given two HWP basis functio,:"" (t) and "'j~.(t), at the scales si and .2 respectively, 8B8ume WLOU that .1 ;::: .2. Represent the frequency of the second HWP basis function, /2 E 2Il'+, by {/2n : n E 2Il'+}. For C'[O, 1), the scale of the HWP basis functions must be such that B ;::: 0 and the pooition of the HWP basis function is given by 0 ::; p ::; 2' - 1. The HWP basis functions on C 2 [0, 1) are closed under multiplication, with their product given by: If [f.\-,p~;Ij')n[~,·~;t;·) =0,

"'it••

.pil.•• (tWH,•• (t) If .1

=0

(24)

= .2 and pi = p2, > .2 and (f.\-, p~;t;I) n

.pil.•• (t).pH .•• (t)

to the sensitivity minimization problem involves finding an internally stabilizing controller, C, such that the following performance is achieved:

(27) The seDaitivity function for the feedback control system is given by:

S := (1 + pC)-.

(28)

Th. sensitivity minimization problem has been studied by numerous researchers in terms of minimizing the H' norm and the Hoo norm, see (Doyle et .1., 1989) and the references contained therein.

Assumptions

.pil.., (tWH,•• (t) = 2".p,1,,(2"1 If si

Fig. 2. Feedback Control System

(~, .~;t;')

pi)

(25)

1. W E RH', i.e., W is real, rational, stable and strictly proper. 2. P E HO, i.e., P is stable.

(26)

The first step in our MC development is to perform a Youl .. parameteriz"tion (Youla et 4'., 1976) for the controller C according to:

;e e,

= 2"m,,,,_.(ka) •

.pf1@i.... (2' t - pi) where ka := pI- 2''''p2, ka E {O, ... , 2'''' - I}.

c = Q(1- PQ)-I,

It should be noted that this result, Theorem 4, can be extended to HWP basis functions defined on C'[O, oo) by letting p E 2Il'+ and to HWP basis functions defined on C'(R) by letting p E 2Il'.

jJ

=

Q=

inf

QEH'

IIW -

PQII.

(30)

WQ.

Our goal in developing a multiresolutional controller i.

This section presents a sensitivity minimiza.tion con-

troller based on compactly supported, OWPs at multiple scales with dilation factor 2 which are closed

(1- PQ) # 0 (29)

The performance measure, equation (27), becomes

where

5. MULTIRESOLUTIONAL SENSITIVITY MINIMIZATION

Q E H oo ,

UD-

der multiplication. Presently, only the HWPs have been shown to be clooed under multiplication, see Theorem 4. Throughout this section, any mention of wavelet packets will refer to Haar wavelet packets.

The feedback control system is shown in Figure 2, where P is the plant to be controlled and W is the weight modeling the disturbances. The Wiener-Hopf approach

to represent the plant, P using its "best" orthonOImal, HWP basis given by pet) = E~=o (t). In addition, the disturbance weight W, and tte controller function Q can be represented using their "best" j

<>.,f':,••

orthonormal, HW P bases, which can be either the same orthonormal, Haar wavelet packet basis as the

,fi:'"

plant, (t), or two completely different orthonormal, HW l' In the general c.... , the disturbance weight W, and the controller function Q will be repr ... sented by two different "beat" orthonormal, HWP bases represented as wet) = E~o ,8l:~i:, (t) and q(t) =

tases.

••

1078

E;'=o 'Y.oi>j:, •• (t).

be represented by the following 8 x 8 matrix:

The next theorem presents a practical, suboptimal90lu-

"'M

tion to the minimum sensitivity optimization problem, equation (30), using HWP representations for the sig-

"'f(t) "'~(t)

nals W, P, and Q. Theorem 5 (Multiresolutional Sensitivity Minimizati~) Represent each of the quantities, P, W, and Q in (30) using different orthonormal, HWP bases (t), w(t) .. represented as p(t) '" Ef;o' ", • Ef;o' fJ• (t), and q(t) .. Ef;o' (t). The performance measure is optimized by the gains, 'Y = ['Yo 'Y, ... which Me given by:

"'M "'~(t)

.,Ii:,••

.j;j:,••

'Y.ti>i:,p.

'YN-' f,

"'i(t)

",.(t) "'~(t)

)_ [{ 2di (N+I)_N, if A -- [a'.. J dN(i-I)+; + dNU-I)+,,''f b

i

=j

" J. r J'

}]

= 2 [(fJ®",f [.j;"wW

'" :=

["'0 "'I

...

"'N_I)T

[A, B) is A ® B with each entry an inner product. The resultant minimum sensitivity gains,

r,

mance measure, equation (30) with the upper limit on

the summations in Theorem 5 as iufinity would lead to the optimal, minimum sensitivity gains 'Y'. It should be noted that "Y -+ ,..* as N -+ 00. However, in practice, the upper limits on the snmmatioDs will be some number N - 1. Deriva.tion of the multiresolutional, minimum sensitivity gains presented in Theorem 5 is very straightforward. The inner products are determined us-

ing Theorem 2. The HWP products are determined using Theorem 4. A sampled version of the selected HWP basis functions, "'i,.(t), can be determined using Theorem I as follows:

=

Z-' { 2f

Qr{' Q"_._I_n(Z")}

(32)

where: '" = 10g,(L) L i. the desired length of "'i,,(z) in the z-domain {t. : !: E Z+} is the binary expansion of f E Z+ if f. = 0 } Q• -_ , G, if f. = I .

{H,

As an example, the sampled time domain representation

for the HWP subspaces,

wg - W~, whose corresponding

basis functions are the first eight Walsh functions, can

1

1

I

I I I -1 1 -1

1 1 -1 -1 1 1

I -I -I I -I -I

I I

I

I

-1

-1

I -1 I -I -1 -I -I I -1 -1

I 1

-1

1

1 I -I -1 -I -I

1 1 -1

1

-I 1

I -I -I I -I I I -I

6. CONCLUSIONS

In conclusion, this research is a. first step towards achieving a multiresolutional control system. A sensitivity minimization theorem for a SISO system was presented using HaM wavelet packets to model the plant and the controller. New theoretical results on wavelet packets were also presented which make the implementation of the multiresolutional controller very practical. 7. REFERENCES

presented

in Theorem 5, are suboptimal. Optimizing the perfor-

1

Note that this matrix expresses an equivalent representation of a sampled set of Wa.lsh functions using a HadamMd matrix (Golubov et .1., 1991).

(31)

where: [ d l

=

I I

Coifman, R. R., Y. Meyer and M. V. Wickerhauoer (1992). Size Properties of Wa.velet Packets. In M. B. Rnskai &Dd et a! (Eds.). Wavelela and Their Application•. Jones and Butlett. Boston, MA. pp. 453-470. Daubechies, I. (1988). Orthonormal bases of compaclly supported wavelets. Communication, on Pure and Applied Mathematic, 41, 90~996. Daubechies, I. (1992). Ten Lecture, on Wavelet". Society for Industrial and Applied Mathematics. Philadelphia, PA. Doyl., J. C., K. Glover, P. P. Khargonekar ...d B. A. Funds (1989). State-Space Solutions to Standard H' and H~ Control Problems. IEEE 1hmloctionl on Automatic Control AC34(8), 831-847. Golubov, B" A, Efimov and V, Skvort80v (1991), Walollh S~­ rie6 and 1h2ru/omu, Theory and Application6. Kluwer Ac&demic. Boston, MA. Mallet, S. G. (1989). A theory for multiresolution signa! decomposition: The wavelet representation. IEEE Tron6action6 on Pattern Anal!/6i6 and Machine Intelligence 11, 674-693. Meyer, Y. (1986). Ondelettes et functions splines. In Semi· naire EDP. Ecole Poly technique, Puis, France. Wickerhauser, M. (1994). Adapted Wavelet Anall/.i" From Theory to SofttDa~. A. K. Peters, Ltd. Wellesley, MA. Youl., D. C., J. J. Bongiorno, Jr. and H. A. Jabr (1976). Wiener-Hopf Design of Optimal Controllers - Part I; The Single-Input-Output Case. IEEE Tronsaction6 Automatic Control 21, 3-13.

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