Overlapping Decomposition for Multirate Decentralized Control

Copyright © IFAC Large Scale Systems, Rio Patras, Greece, 1998

OVERLAPPING DECOMPOSITION FOR MULTIRATE DECENTRALIZED CONTROL Hiroshi Ito •

*

Dept. of Control Engineering and Science, Kyushu Institute of Technology, 680-4 kawazu, Iiz1i.ka, Fukuoka 820-8502, Japan

Abstract: This paper proposes an extension and contraction approach to multirate decentralized design of large-scale control systems. The extension which is a generalized notion of overlapping decomposition is defined comprehensively from a viewpoint of hybrid signals. Stabilization and robust control via the overlapping decomposition technique are investigated. Copyright@1998 IFAC Keywords: Overlapping, Decomposition methods, Multirate, Sampled-data control, Decentralized control, Robustness

ping decompositions, expansions, extensions and contractions in feedback control design have been demonstrated by examples in Siljak (1991) and Bakule and Rodellar (1995). The first objective of this paper is to give a novel and comprehensive definition of the extension and contraction for multirate control of uncertain systems. The second objective is to show how robust multirate stabilizing controllers can be designed using the extension and contraction. To the best of the author's knowledge, multirate robust control via overlapping decomposition was initiated by Ito (1996) in the literature and a primary results was obtained for the first time. For large-scale systems, it is natural and practically useful to consider multirate control law since: Each subsystem has its own bandwidth and characteristic frequencies; Each subsystem has physically allowable sampling rate; Multirate control has more potential than single-rate control; There is no need to use the same sampling period for control agents in different locations. The notation is quite standard. 'Rn denotes the ndenotes Banach dimensional real vector space . space of all Lebesgue measurable functions (ndimensional vector-valued) on the time set [0,00) which are p-integrable, 1 ~ p ~ 00. Let denote the extended space of I!' denotes the space of real valued one-sided sequences of vectors whose dimension is n. The dimension n is omitted if it is clear from context.

1. INTRODUCTION Especially for design of complex and large-scale systems, decomposition and decentralization are very attractive and important methodologies. Although most of approaches to decentralized control has been developed using disjoint decompositions, there is a large class of systems which inherently require sharing of information among individual control agents. The advantage in using overlapping decomposition instead of disjoint one is obvious. For large-scale systems whose subsyst~ are strongly connected each other, disjoint decomposition {completely disjoint modeling of subsystems for decentralized design or completely decentralized structure of control} may easily fail to produce satisfactory results. In such a situation, overlapping decomposition (overlapping modeling of subsystems for decentralized design or overlapping decentralized control) is superior and becomes very useful. Since the pioneering work of Ikeda and Siljak (1980), overlapping decompositions, expansions and contractions for designing partially decentralized controllers have been discussed in some papers (Ikeda et al., 1984; Ikeda and Siljak, 1986; Ohta et al., 1986; mar, 1993) and the framework of the inclusion and extension principle has been proposed. The problem of LQ optimal controller design was considered in Ikeda et al. {1981} and iftar and Ozgiiner (1990) . Stankovic et al. {1996} dealt with a stochastic inclusion for optimal LQG design. Recently, the extension concept has been specialized to mechanical systems described by matrix second-order equations (Bakule and Rodellar, 1995). These papers only dealt with nominal stability and nominal performance. Advantages of using overlap-

.c;

.c;.

.c;,1:

2. MULTIRATE CONTROL SYSTEMS Consider the hybrid system Eh shown in Fig.1. Here, G is a finite-dimensional linear time-invariant 259

~h

el

I

e~h d

d1

:- - --------------------.

1

I

:d

e/.

C

~mc

md

It :

G

.. i I

I

fll

Fig. 2. Multirate control systems .

: .. I

I

~----- - ----------------~

Fig. 1. Hybrid system.

C={

where

(FDLTI) continuous-time system defined by

Cm.[O

G: x=Ax+Bug, Yg =Cx+Dug, x(t) E'Rn

(1)

li[k]

[~~I;, l'

s

=

[:.~s;, 1

E

h j =lljCj, h=

[ ~~:. 1' c= [ ~~:. 1,

hm • hj(k(KjT) + t)=cj(k), MH : C E

T.

(2)

P

,

The

e

{C : (Mj, Pi)-causal, C :~;~;}

j = 1, 2, (3)

... ,mc

(8)

which are called multirate controllers. Note that if the operator C is a finite-dimensional linear shift-invariant(FDLSI) system whose directfeedthrough matrix of the state-space realization satisfies (Mj, Pi)-causality conditions, Wi/CWp always belongs to .csv(Mj , Pi ) and it is (Mj, Pi )causal (Ito et al., 1994; Meyer, I990b). The controller ~mc is described on the lifted spaces by

Cm.

O
= [~,cTf ~ y = [eT,IT]T .

(~sr)H=H(~SrJ ~7)

where Sj denote the right shift operator on the jth component of the signal vector. For details of lifting and (Pi, Mj)-shift-varying operators, we refer the reader to Ito et al. (1994), Meyer (1990a), Meyer (1990b) and references therein. Now, consider the feedback system shown in Fig.2(a). It is assumed that the digital controller ~mc belongs to the set

en· ~ hE .c;,: .

~

. z[k + I} = FZfk] + Gq[k] v[k} = Hz[k + Kq[k] ,

(9)

mc'

(4)

where z(t) E 'RP, q = Wpl and v = Cq. In Fig.2(a), the output v is applied to the input C of ~h' i.e., C = Wiiv. Next, define another FDLTI continuous-time system described by

Let N be the least common multiple among Li (i 1,2, ... , l/) and Kj (j 1,2, . . . , mc). The schedule 0 Ms and MH is NT-periodic. Then, define Pi := N / Li and M j := N / K j for i = 1,2, . .. ,l, and j = I,2, .. . ,mc. Let M = 2:;;1 M j and a discrete-time lifting operator W M : f-.... -+ lM is defined as

=

:

:= W pi

{H: linear F-ti

Both C and 1 in Fig.1 are discrete-time signals which is supposed to be used by a digital feedback controller. Li and K j are positive integers. In this setting, the hybrid system ~h is defined as a mapping between hybrid signals U and y: ~h: u

j

.csv(Pi , Mj ) :=

.c~e ~ 1 E f! .

The basic sampling period is denoted by multirate hold (zero-order) MH is

Cm. [1]

il, -t iP in the same manner as is the lifted sequence of the output signal I. The map from C to 1 of ~h is a (Pi, M j )shift-varying operator. The set of (Pi,Mj)-shiftvarying operators is denoted by

operator W p

= si(kL(r), i = I,2, ... ,l,

Ms : s

.(6)

c is called the discrete-time lifted signal. Let = L!~1 Pi and we define a discrete-time lifting

WM.

Both d and e are continuous-time signals with which performance cost functions of the hybrid system ~h are defined. We assume that the transfer function from U g to s is strictly proper. In Fig.I, the multirate sampler Ms is defined by

1=

,. .}

Here,

P

ug=[dT,hTf, ug(t)E'R m, d(t)E'Rmd, h(t)E'Rm. Yg=[eT, sTf, Yg(t)E'RI , e(t)E'RI ., S(t)E'Rlf .

li =SiSi,

[~:f~IJ ,[~:I:ll

=

(; : i = Ai + BUg, Yg = ei + DUg.

(10)

The dimensions are i(t) E'Rn, ug(t) E'Rm. and yg(t) E'Ri, where n ~ ft, m ~ rn, 1 ~ i and

c= {c[k]}~ = WMC

~~f~l

ug=[F,hT ]TE'Rm. cl[2M1

-

1]

4

+m.., Yg=[eT,sTfE'Ri.+i , .

The multirate sampler Ms and the multirate hold MH connect continuous-time signals and discretetime signals as j Mss and h = MHe. The basic sampling period of Ms and MH is supposed to

,... (5)

c2[M2]

=

Cm.[2Mm. -1 260

be T. It is assumed that N T is a common sampling period. In other words, N is a common multiple among Li (i = 1, 2, ... , i,) and K j (j = 1,2, ... , me) which satisfy Pi := N/Li and M j := N/K j . The hybrid system Eh is defined as Eh : U =

[JT,cTf

f-t

ii

=

[eT,p']T .

lFi:={jE{I , 2, ... ,n} : Kj=Ki }, i=I,2, ... ,n.

Moreover, the constant matrix F defined by

(11)

F=

Here, the subsystem mapping c into j is (Pi, Mj)shift-varying. Since G has larger dimensions of the state, input and output signals than G, the hybrid system Eh also is larger than I:h in size. Let W M and W.(> denote the lifting operators for c and j, respectIvely. For the multirate control system in Fig.2(b) , the digital controller Eme belongs to

{

l~ e-wew- } e- : (M- j , Pi)-causal , ~ MP

e is FDLSI

[~11 ::~ F~nl' where

Fij=O't/j

tI. JF,{19)

Fnl . . . Fnn

is called the matrix representation of J=. The linear operator J= is defined on multirate discrete signals. However, it is observed that the operator is well-defined in the "real world" which goes in the continuous time-variable t. In fact, the operator defined is "memoryless" or "static" in such a sense. Due to the above definition, one can define transformations of the control input and the measurement output as follows:

. (12)

On the lifted spaces, the controller is described by

E

. z[k + 1] =: Fi[k]-+: Gq[k] v[k] = Hi[k] + Kq[k] ,

(20)

(13) (21)

me '

where i(t) E 'Ri, q = wpj and c = W~lV . The controller Erne has larger dimension than I: me .

These two linear operators are assumed to be multirate memoryless. The matrices Re and S, are matrix representations of Re and S" respectively. It is also assumed that me ~ me and i, ~ If. Define the following sets of integers.

3. EXTENSION AND CONTRACTION

JRc .i :={jE{I, 2, ... , me}: K j =Ki} ' i=I, ... ,m e(22)

In this section, we will define extension and con-

Js t •i :={jE{I,2, ... ,l,}: Lj=Li }, i=I, ... ,i, (23)

traction for the hybrid systems and the multirate controllers. The extension and contraction are generalized notions of overlapping decomposition and composition of decentralized control laws. Let us define the following transformation matrices for signals of the hybrid systems:

Obviously, Re is in .cs v (Mi , M j ) and it is (Mi , Mj)causal. Also, Sf is an operator in .csv(Pi,Pj ), which is (Pi,Pj)-causal. Here, we should aware that (20) and (21) not only define the transforma-tion Re and Sf themselves, but also they implicitly require MH (Ms) and MH (Ms, respectively) to have a certain relationship. In order to fulfill rank(Rc) = me, the pair of MH and MH should be chosen appropriately. For example, all the elements of the linear diagonal operator MH must appears in MH. H it is not, rank(Re) < me holds and there is not any transformation 'Re which is admissible for such a pair (M H, MH) . The pair of MH and MH is eligible if MH is an overlapping augmentation of M H : for instance, the pair

T : x(t) E'Rn f-tx(t)E'Rn, rank(T)=n (14) P P P : i[k] E'R f-t z[k] E'R , rank(P) = p (15) m rn Rd: d(t) E'R " f-td(t) E'R ", rank(Rd) =m~16)

Se: e(t)E'RI. f-te(t)E'R i .,

rank(Se)=le . (17)

It is assumed that n ~ n, p ~ P, md ~ md and le ~ le. By contrast with the above signals, we must be careful in defining linear transformations of the control input c and the measurement output f since these signals ace discrete-time signals consisting of components defined at different instants. Operations among signals which occur at different instants do not make sense. In order to avoid operations which ace not well-defined, we need the notion of multirate memoryless operators. Definition 1. Let J= : C f-t C, t!' -+ I!' be a linear operator, where c = [Cl, C2, ... , en] T and C = [Cl! C2," . , en]T. Suppose that the sequence Ci is a discrete signal defined on continuous-time [0, (0) with the interval Ki and the sequence ~ is a discrete signal defined on [0, (0) with the interval Ki . Then, the operator J= is said to be a multirate memoryless operator if the operator can be described by

MH=

1

11.10 0 01£20 , [ o 01£3

MH=

~

1 0 0 00 0] 01£20 0 0 01£20 0 o 0 01£3 0 o 0 0 01£1

is the case. In a similar manner, the definition of Ms must depend on Ms. Every element of the linear diagonal operator Ms appears in Ms if

rank(S,) = If holds. Now, we are ready to define the notion of extension for the hybrid system Eh'

Definition 2. The system Eh is an extension of the system Eh if there exist transformations as 261

in (14), (16), (20), (17) and (21) such that for any initial state x(O) E 'Rn of G and any input d(t) E nni", 0 ~ t < 00 and e[k] E 'R.ni., k = 0,1,2, ... of I;h, the choice

Moving back to the example (27), it will be demonstrated that the overlapped system I;h can be defined appropriately as an extension of Eh. Let (A, E, C, D) be the state-space description of G which defines I:h. Matrices A, B, C and D are

x(O) =Tx(O), d(t) =~d(t), \1't ~ 0, c='R.ca:24) implies that x(tjx~),il)=Tx(tjx(O),u),

e(tj x,il)=See(tj x,u),

J[.; x, ill

= Sd[·; x, u]

\1't~O \1't~O

.

(25) (26)

It is emphasized that the extension for the hybrid

system is defined with discrete-time measurement and discrete-time control signals since these signals are parts of the controller and one cannot Define a large-dimensional hybrid system I;h as use continuous-time signals for digital control. Extensions for control should not be introduced to (35) Eh = 0 Ms G 0 MH ' continuous-time signals. On the other hand, exogenous signals and controlled signals are continuoustime signals and those are not signals used by where the state-space realization of G are the controller. In order to evaluate control pera12 0 a13] b12 b13 b14} formance and robustness correctly, those signals should remain continuous-time in the definition of A _ a21 a22 0 a23 B _ b21 ~2 b23 b23 b24 ~6) the extension. - a21 0 a22 a23 ' - b21 b22 b23 b23 b24 . .. . a31 0 a32 a33 b31 b32 b33 b33 b34 The overlappmg decomposltlOn of hybnd systems can be discussed within the framework of the exC12 0 d12 d 13 d13 tension defined in the above. For example, consider _ C21 C22 0 C23 _ 0 0 0 0 0 a system Eh in which the state, the input and C = C31 C32 0 C33 ,D = 0 0 0 0 0 .(37) output are partitioned as C3l 0 C32 C33 0 0 0 0 0 C41 0 C42 C43 0 0 0 0 0 X=[Xl,X2,X3]T, U=[d,Cl,C2,C3f, (27) - - - 9 = [e il h h]T It is verified that these matrices (A,B,C,D) , " , together with transformations (29) satisfy (31). where the sampler and the hold are Hence, the overlapped hybrid system I;h is an [11. 1 0 0] extension of Eh· It is straightforward to extend s1 0 0] Ms = 0 S2 0 ,MH = 0 11.2 0 .(28) these state space formulas to more complex cases [ o 0 Sa 0 0 11.3 of overlapping structure. Actually, Theorem 1 suggests that the technique shown in mar and Let X2, C2 and h overlap to obtain a new hybrid Ozgiiner (1990) developed for continuous-time syssystem I;h of larger dimension. The overlapping tems can be used and we can obtain (A,B,C,D) structure required can be described by of G by choosing complementary transformation matrices appropriately. [1 0 0J 1 0 0] Rd=l, [1 0 0 0] T= ~ Re= 0 1 1 0 ,S,= ~ 29) Now, we shall move onto controller contraction. [ o 0 1 Se=l, 0 0 0 1 00 1 Definition 3. The controller I;me is contractible to the controller Eme if there exist transformations The sampler and hold required for the overlapped as in (14), (16), (20)and (15) such that for any system I;h are initial state x(O) E 'Rn of G and any input d(t) E 1 'Rni",O~t
Theorem 1. The system I;h is an extension of I:h if and only if there exist transformation matrices T, R and S such that

H the hybrid system of larger dimension is

an extension of the original system, any largedimensional multirate controller is contractible to a smaller one. Theorem~. H I;h is an extension of I: h, then any controller I;mc is contractible to I:mc by choosing

T A=AT, TBR=B, SC=CT, SDR=D(31) R=

[~

(40)

lJ, S= [~e 3/]

hold and Rc and S / satisfy 262

~--r-----..~_

and 2: me is well-posed. Furthermore, for any initial states x(O) E 'Rn and Z[O] E 'RP, and for any input d(t) E 'Rmd , 0 :::; t < 00, the choice i(O) = Tx(O), z[O] = Pz[O] d(t) = Rctd(t), 'fit ~ 0

2: me (a)

implies that the closed loops

(c)

= fr,

G

= as"

H

= keH,

K

rh and rh satisfy

i{t; ~o, d) =Tx(t; {a, d), e(t;~, d) =See(t; { , d)(43)

Fig. 3. Overlapping decentralized control design: (a) Hybrid plant; (b) Extension and design; (c) Contraction and implementation. F

(42)

z[k;{o,d]=Pz[k;~o,d'], k=0,1,2, ... , 'fIt~O(44)

where ~o = [iT{O),zT[OllT. H time exponentially stable, so is

= keKS, (;i1)

rh

rh'

is continuous-

where these matrices define 2: me and f: rne on the lifted spaces. Here, Rc and S, are constant 1 matrices defined by Rc = WM'ReW~ and S, = WpS,W;l.

The following shows that discrete-time stability of the large-dimensional system implies continuoustime stability of the small-dimensional system.

Theorem 3. H Erne in (13) satisfies (Mj, Fi)causality conditions, 2: me defined by (41) satisfies (Mj, Pi)-causality conditions.

rh in Fig.2{b) with d =0 satisfies that there exist positive real numbers a, f3 such that rh with

This theorem guarantees the causality of the contracted controller if the large-dimensional system satisfies causality conditions. In other words, the contraction 2: me of an admissible controller Eme is admissible as well in terms of (8) and (12).

lIi{kNT)II:::; lI~ollf3e-o~ Ilz[k]ll:::; lI~ollf3e-ok, 'fIk

A

A

Theorem 5. Suppose that f:h is an extension of I:h and that 2:rne is a contraction of Eme and it is

defined by (41). Hthe extended closed-loop system

for any {o = [i{O)T, z[O]T]T E 'RiHp , then, the system rh in Fig.2{a) with d 0 satisfies

=

Ilx(t)lI:::; II{ollf3ee-ot/N~ Ilz[klll:::; lI{ollf3e- ok,

4. STABILIZATION

for any initial state {o = [x{O)T, z[OlT]T E 'R n+p and some positive real number f3e .

In the previous section, extension and contraction for open-loop systems are defined and investigated. This section considers closed-loop systems and discuss behavior and stability of the closedloop. The multirate control feedback system rh in Fig.2{a) consisting of 2:h and 2:rne is said to be continuous-time exponentially stable if there exist positive real constants a g , a me , f3g , f3rne such that the associated unforced system satisfies

IIx{t)11 :::; lI{ollf3g e- o ,t

;

This theorem suggests that the pair of extension and contraction preserves the relative stability. The relative stability of rh in the continuous-time domain is determined by discrete-time pole placeLet the quadruple (A, Bc, C" D'e) ment of denote the state-space realization of the discretetime lifted subsystem of Eh mapping from is to j. The discrete-time modes of are eigenvaIues of

rh.

rh

'fit ~ 0

0]

[Bc [A o fr + 0

Ilz[k]1I :::; lI{ollf3rnee-o",ck ; 'fIk = 0, 1,2", "

for any initial state {o = [x{Of,z[01T1T. On the other hand, the system rh is said to be discrete-time exponentially stable if there exist positive real constants ad, a me , f3d, f3me such that the associated unforced system satisfies

I _k]-l [0C, H]0 a0] [-D'e I

rh

IIx{kNT) 11 :::; II{ollf3de-odk, IIz[klll:::; lI{ollf3mee- o".c k

for any initial state {o = [x{O)T, z[0]T1T . The diagram in Fig.2{b) is a feedback system defined on the extended spaces of larger dimensions. Exponential stability for the large-dimensional system f h is defined in the same way as the small-dimensional feedback system rh' The following theorem clarifies the explicit relationship between the large-dimensional system and the smalldimensional system.

.

rh

If is discrete-time exponentially stable, is continuous-time exponentially stable as well. Figure 3 illustrates overlapping decentralized control design within the framework of extension and contraction. Suppose that Eh in Fig.3{b) is an extension of 2:h in Fig.3{a). Decentralized control with Erne for the extended hybrid system Eh is shown in Fig.3{b). Then, Fig.3{c) illustrates the implementation of the contraction as an overlapping decentralized controller 2: me • 5. ROBUST CONTROL Although Theorem 4 does not take account of uncertainty, the framework developed in the previous section is also useful for robust control design via the extended spaces. In this section, we assume that Eh is an extension of Eh and that Erne is a contraction of Erne.

Theorem~. Suppose that Eh is an extension of 2:h and that 2: mc is a contraction of Erne and it is

defined by (41). If the extended closed-loop system

rh in Fig.2(b) consisting of th and t me is wellposed, the system

'fit

rh in Fig.2(a) consisting of 2:h

263

z

achieves robust Lp-gain performance if the maphas Lp-induced norm :5 1. ping from J to e of

rh

J

... _------------.

These corollaries suggest that in order to design a robust mulritate controller, Lp-induced norm {sub)optimal control can be performed in the ~­ tended signal spaces. Indeed, a norm-preservmg transformation technique (Ito et al., 1996) is available for finding such controllers. Then, robust decentralized multirate control design with overlapping decompositions is reduced to a popular problem of discrete-time decentralized l£OO-type control on the extended spaces (e.g. Veillette et al. (1992), paz (1993), Morari and Zafiriou (1989), Ito et al. (1995a) and Ito et al. (1995b». The controller f: me obtained in the extended spaces is always contractible to Eme in the original signal spaces by using Theorem 2. See Ito (1996) for details of the p = 2 case.

... _----------_ ....

(a)

(b)

Fig. 4. Robustness of stability.

Fig. 5. Robustness of performance. Consider the uncertain hybrid system shown in Fig.4(a). The system II denotes uncertainty arising from various sources in the system. We suppose that the map II : e 1-+ d is any causal linear N Tperiodic operator having Lp-induced norm< 1 for a real number p E [1,00]. Let ~ denote this set of admissible uncertainties ll. The uncertain system shown in Fig.4{a) is said to be robustly Lp-stable if the nominal part rh is continuoustime exponentially stable and if the map between (w, z) to (d, e) is Lp-induced norm bounded for any II E &. It is easily shown that an Lp-stable uncertain hybrid system is continuous-time exponentially stable if II is FDLTI. Figure 4(b) depicts the large-dimensional hybrid control system defined on extended signal spaces. For simplicity, we assume Rd = Imd and Se = 11 •. This assumption is reasonable since uncertainty lies in the actual control system. However, in some cases it may be easier to represent the uncertainty in the extended spaces and estimate the size of the uncertainty there (e.g. uncertainty modeling on the extended spaces). In such cases, Eh can absorb the priori knowledge of transformation matrices ~ and Se. The following shows that robust stability can be obtained from the extended system.

rh

Corollary 1. Suppose that is continuous-time exponentially stable. The system shown in Fig.4{ a) is robustly Lp-stable if and only if the mapping has Lp-induced norm :5 1. from J to e of

rh

Another uncertain system is shown in Fig.5{a). The uncertainty II : e2 1-+ d2 is any causal linear operator having L2-induced norm< 1. Let ~TV denote this set of ll. The uncertain system shown in Fig.5{a) is said to achieve robust Cp-gain performance if the system is robustly Cp-stable and if Lp-induced norm of the map between d1 to el is less than or equal to 1 for any II E &TV. Let d = [di,Jf]T and e = [er,er]T. Figure 4{b) depicts the large-dimensional hybrid closed-loop control system. We assume ~ = Imd and Se = 11. again.

rh

Corollary f. Suppose that is continuous-time exponentially stable. The system shown in Fig.5{a) 264

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