A Fourier Series Lifting Approach to H∞ Sampled Data Control

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

A FOURIER SERIES LIFTING APPROACH TO Hoo SAMPLED DATA CONTROL C. Zhang The University oJ Melbourne, Departmem oJ Electrical and Electronic Engineering, Paricville, vie 3052, Australia

Abstract: In this paper we present a new approach, called Fourier serie.s liftin.g, to the problem .of H~ sampled data control. The Fourier series lifting is in a generalised form that IS applicable to both contmuous and discrete periodic time-varying systems. By using Fourier series lifting, we show that the H~ sampled data control problem is equivalent to a finite dimensional H~ discrete control problem. The obtamed result allows either exact or approximate numerical computation of the solution. Key Words: Digital control, Fourier series, H ~ control, Sampled data systems, Time-varying systems.

I. INTRODUcrION This paper applies a new lifting technique to the problem of H~ sampled data control. In a sampled data control system the continuous plant is controlled by a discrete feedback controller using the sampled measurements of the plant output. The design is concerned with the system continuous time performance including both the sampled and intersample performance of the system. Due to the periodic nature of the sampling process, sampled data systems are periodic time varying even if the continuous plants are time invariant. One approach to this problem is to apply lifting technique to transform the control of periodic time varying systems to an equivalent time invariant control problem in discrete time, such as (Chen and Francis, 1991 ; Toivonen, 1992; Keller and Anderson, 1992; Bamieh and Pearson, 1992; Dullerud and Francis, 1992).

of the input and output are the generalised Fourier series. For this we call it Fourier series lifting. Effectively, the lifting transforms the system input and output into discrete sequences. The lifted input and output are finite dimensional for discrete systems, and are infinite dimensional for continuous systems. The lifting for the input and output leads to a lifted system with finite dimensional state. The Fourier series lifting is different from the existing lifting techniques for continuous systems in that the Fourier series lifted signals and systems are purely discrete. This is in the sense that the lifted system input and output are constant sequences and the state equation of the lifted system is in terms of constant matrices but not integral operators. We shall show that the Fourier series lifting is applicable to both continuous and discrete systems and it exactly recovers the block lifting.

The lifting technique for discrete periodic time varying systems can be found in some early works (Jury and Mullin, 1959; Sz.Nagy and Fioas, 1970; Davis, 1972). The idea is to group a number of successive inputs and outputs into blocks and process these blocks as new system inputs and outputs. Accordingly, a lifted system can be obtained which maps the input blocks into the output blocks. We shall call such technique block lifting. The block lifting was developed in (Khargonekar et aI., 1985) for analysis and robust control of discrete periodic systems. Thereafter, it has been further developed and applied to a number of control problems. These include design of periodic discrete controllers (Francis and Georgiou , 1988), H2 sampled data control (Chen and Francis, 1991), discretization of continuous controllers (Keller and Anderson, 1992), and LJ sampled data control (Dullerud and Francis, 1992).

For H~ sampled data control, we firstly apply the Fourier series lifting to the continuous plant to obtain a discrete system with infinite dimensional input and output. Then we transform this system to an equivalent one with finite dimen sional input and output. In doing this we follow the idea in (Bamieh and Pearon 1992) and use the loop-shifting technique in (Safonov, 1989). As a result, the H~ sampled data control problem is equivalent to a finite dimensional H~ discrete control problem, to which the standard H~ control methods can be applied , e .g. (Gu, et aI., 1989; Limebeer, et aI. , 1989). Section 2 presents Fourier series lifting for signals and systems and its properties. Section 3 applies the Fourier series lifting to H~ sampled data control.

Technique for lifting continuous systems has been recently developed and applied to sampled data control problems. Yamamoto (1990) studied the continuous tracking problem of sampled data systems by lifting the system to obtain an infinite dimensional state space model. In (Bamieh and Pe arson , 1992; Bamieh, et aI., 1991) , a continuou s system is lifted to a discrete one with finite dimensional state and infinite dimensional input and output. The idea is to uniformly chop the continuous input and output up to pieces of signals and foml these into discrete sequences. Conseque ntly , the lifted system is descrete which operates on the chopped discrete sequence . The system state equation is in terms of infinite dimen sional integrdl operdtors on each sampling interval. This technique has been used to solve Hoo and H2 sampled date control problems in (Bamieh, et aI., 1991; Bamieh and Pearson, 1992). The same technique was also presented independently in (Toivonen, 1992), where an approximate solution is obtained by finite rank approximation of the integral operators.

Notations: y(t) denotes continuous function of t, or discrete function for t = k~ with a sampling period ~ > O. {Yk} = (y(k~)} denotes discrete sequence. <., ·>I •.b) denotes inner product of vectors on [a, b). 11 • IIla.b) denotes norm on L2[a, b) or 12[a, b). 11 • 11 denotes norm on L2 or 12, and also denotes L2 or 12 induced operator norm. 11 • II~ denotes H~ norm of a discrete stable tr.iI1sfer function. 2. FOURIER SERIES LIFTING In an inner product space, an element v is orthogonal to an element Il if
In this paper we extend the block lifting technique to present a generalised lifting for both continuous and discrete systems, and apply it to H ~ sampled data control. The idea of our technique is to represent, in a block sampling interval, ~he system input and output in term s of an onhonormal set whIch spans the input/output space. It turns out that the representations

Consider a vector function yet) on tE [a, b). For a given orthonormal set (U(i)(t)} on tE [a, b) the generalised Fourier series of y(t) can be written as

817

yet) = L U(i)(t) y(i)

properties are preserved by lifting.

(I)

Lemma 2.2 :

i=O

_____

yi)

(i)

The Fourier series coefficients are given by y(i) =
1\

-.-..........

+ G2,

G1G2

,,1\

1\

~

= GIG2,

(G-I)

= G-I

A

(ii) IIGII=IIGII (iii) A controller C stabilises a system G in the sense the operator norm of the closed loop system S(G.C) is bounded if A

A

and only if C stabilises G. Proof: _ (i)

GI+G2

G7Gi =

= F(Gl

+ G2)F-I

= FGIF-l

+ FG2F-1

A

A

= GI

+ G2

FG1G2F-I = FGIF-IFG2F-I = GIG2 (G-l) = (G-I )GG-I = (G-IG)G-I = G-I

If lIy(t)II(a,b) is bounded the Fourier series (1) converges to yet), see for example (Kreyszig, 1978), in the sense that ~

Ily(t) - L U(i)(t) y(i) 11 = 0

"

= GI

GI+G2

(ii)

(2)

It follows from IIFlI = IIF-III = I that

11(;11 = IIFGF-III::; 1IF1II1GIIIIF-III = IIGII

i=O

IIGII = IIF-IGFlI::; IIF-III 11(;11 IIF-III = 11(;1

L lIy(i)1I 2

lIy(t)II(a.b) =

A

Hence IIGII = IIGII. (iii) The closed loop system S(G, C) is a function of G and C involving only algebraic operator additions, multiplications, and inversions_We can use the results in (i) and (ii) to have _ A A

(3)

i=O

Consider a time period ~ > 0 and an orthonormal set (u(i)(t)} on tE [0,

IIS(G, C)II

~) .

This set can be extended to tE [0, (0) by setting = U(i)(t), for k = 1, 2, ... (4) Then we can write the Fourier series for a signal yet) on each time interval tE [~, ~+~) as

A

i=O

D~G

'

[ The Fourier series (5) establishes a correspondence between the signal yet) and a sequence (Yk) . According to this we define the Fourier series lifting for signals as F : Y = F [yet)] = {Yk} (6) The lifting operator F transforms a signal to a discrete sequence.

A

Proof:

A

For a

~-periodic

A

A

known that the operator norm of the discrete system G is the H~ A

norm of the transfer function G(z) (Khargonekar, 1985). That is IIGII

~

A

2

k=O

Izl

=I

}

(8) A

A

A

function of the lifted system is G(z). then IIGII = IIG(z)II~. Proof: The result follows from (8) and Lemma 2.2 (ii). 0 Lemma 2.3 and 2.4 establish a correspondence between periodic time varying systems and the lifted discrete time invariant systems_ This enables the application of the methods for linear time invariant systems to the periodic time varying systems. In the next section we shall apply this result to H "" sampled data control.

2

k=O

A

where Clmax(G(Z» is the maximum singular value of G(z) . We can establish the following result. Lemma 2.4: If G is ~- periodic time varying and the transfer

L lIy(t)II[k6.kM6) = lIy(t)1I10._)

(ii) Since IIFII = sup ( 11 F[y(t)] 11 : lIy(t)1I 10._) follows from (i) and 11 F[y(t»)1I = lIylI that 1IF11 = I.

A

= IIG(z)lI~ = sup ( Clmax(G(Z)) : A

Lemma 2.1: (i) If lIy(t)II(o._) is bounded, then lIylI = lIy(t)1I10._); (ii) 1IF11 = 1. Proof: (i) Using (6) and (3) we have

L IInll =

0 A

time varying system G, the lifted system G is

time invariant and it has a z-transfer function G(z). It has been

A

=

D~G = D~FGF-I = FD~GF- I = FGD~F-I = FGF- ID~ = GD~

Accordingly, the inverse of F is F-l: yet) = F -l[Yk] = ( u(t)T Yk , tE [k~, kM~) ) We show that the norm of the signal yet) is preserved by the lifting in the following lemma.

A

(7)

then the lifted system G is time invariant.

A"":::::~O~:1Tt:~:::~r:;:~:::::::] }

lIyll

= GD~

In panicular, if G is a discrete system with respect to ~ and satisfies (7). it is time invariant. Since the orthonormal set (u(i)(t») defined in (4) is ~-periodic, it is easy to verify that the lifting operator F satisfies D~F = FD~ and D~F-I = F-ID~_ We can further have Lemma 2.3: If a linear system G is ~-periodic time varying,

U(O)(t)] u(1)(t) .

0

D~ : D~[y(t») = y(t-~) where yet) can be continuous or discrete. If a linear system G is ~-periodic time varying, it will satisfy

(5)

where

u(t) =

A

Let D~ be a delay operator defined as

~

yet) = L U(i)(t) Yk(i) = uT(t) Yk

= IIS(G, C)II = IIS(G, C)II

Hence S(G, C) is stable if and only if S(G. C) is stable.

u(i)(t+k~)

= 1 }, then

it

Before we complete this section we show briefly that the block lifting for discrete systems can be recovered from the Fourier series lifting_ For a discrete sequence (y(O), yeT) , y(2T), _.. } with a sampling period T and associated with a block sampling

0

We further define the lifting for systems using F . Consider a finite dimensional causal linear, continuous or discrete, system G. We define the lifted system as

~oo:=NT.:[,y::~: :~:~'{or Si[gO;r~~r]ed

G=FGF-l This transforms the system G to a discrete system with finite dimensional state but it may have infinite dimensional input and output. If u and y are the input and output of the system G. respectively, and G= F[u(t»), the lifting of the output y is

}as

y(~+~-T)

where W is the block lifting operator. The block lifting transforms a p-dimensional sequence to an Np-dimentional sequence. Accordingly, for a discrete system G with a sampling period T the lifted system is

Y= F[y(t») = FGu = FGF-l G= GG This shows that, associated with the system y = Gu , the input A

and output of the lifted system G are the lifted u and y, respectively. Therefore the lifting establishes a correspondence

G=WGW-I

A

On each time period tE

between the system G and the lifted discrete system G. We show in the following that the system algebraic and analytic

818

[k~, k~+~),

if we let

\)(i)(t)

={

~

for t = kt.+iT

x(t) = A x(t) + BI w(t) + B2 u(t)

(9a)

z(t) = Cl x(t) + DII w(t) + D12 u(t)

"

le6 6

(9b)

6

t

6

Cl eA(H) BI \)T(t) dt + DlI \)T(t)l dt

0 \)(t) D12 dt,

"

These matrices are all real and constant. Then G wz is time invariant. Note that the disturbance ~ and the output £ are infinite dimensional after the lifting. However, since the system state is finite dimensional we shall be able to apply the "loop shifting" technique (Safonov, 1989) to transform the system to that with finite dimensional input and output. 1\ ,,1\ 1\ 1\ Assume 11 Dl1l1 < 1. Then (I - D11 T D11) and (I - DlID11 T) have inverse. Define a constant unitary matrix UD as

(I - 011011T)I/2 ] - 011 UD= [ (I - DlITbll)l/2 "" " DlIT

We introduce new variables zand

[l ]

= UD

(9d) y(kt.) = C2 x(kt.) u(t) = u(kt.), for tE [kt., kMt.) (ge) Let G wz denote the continuous system from the disturbance w to the controlled output z. Under the condition that the system stabilisability and detectability are preserved by proper choice of the sampling period t. (Francis and Georgiou, 1988), we state the H~ sampled data control problem as: For the system (9), find a set of possible controllers such that IIGwzll < 1.

wwhich satisfy

~]

[

(12)

£ and ~ can be solved from the above equation as

[~£] [

z]

(l-OllOllT)-I/2 (l-Oll0l1T)-1/201l ] [ = DlI T(I-011011 T)-I/2 (I-Oll TDll)-I/2 w Substituting this solution into (11) yields a new state equation x(kt.+t.) =

z.: =

The system G wz is t.-periodic time varying due to the periodic nature of the sampling process. To deal with this problem we firstly apply the Fourier series lifting to obtain a time invariant discrete system. Since the lifted system will have infinite dimensional input and output, we will further apply the "loop shifting" technique (Safonov, 1989) to obtain a discrete system with finite dimensional input and output. Finally we show that the H~ sampled data control problem is equivalent to the H~ control of the obtained finite dimensional discrete system. On tE [kt., kt.+t.), the solution to the state equation (9) is 1<6+11 x(kt.+t.) = eAII x(kt.) + eA(II.t)BIW(t) dt 1<6 1<6+11 + e A(II-t)B2dt u(kt.) (lOa) k6

A x(kt.) +

8 I WIe + 82 u(kt.)

(13a)

Cl x(kt.) + D12 u(kt.)

(l3b)

y(kt.) = C2 x(kt.) -

where -

1\

(l3c)

1\

1\

1\

1\

1\

1\

1\

-

BI = BI(I - D lI Tb ll )-I/2, -

1\

A= A+BIDlIT(I-DlIDllT)-ICI 1\

1\

1\

1\

1\

1\

1\

Cl = (I - Dl1Dl1T)-I/2 Cl 1\

1\

B2= B2 + BI DIIT(l- DIIDIIT)-I D12 -

1\

1\

1\

-

DI2 = (I - DlIDl1 T)-I/2 D12,

1\

C2 = C2

As a result of the change of variables by (12), the infinite

"

J

dimensional matrix DII has been removed from the new system (13). Further define matrices UB and Uco as

J

UB=8;T1

J

(lOb)

for

I_IT.stn

'\I ti

i7t+n:

for

~ t

_IT I '\I ti cos in: "t1' t

I

- - T- -

1, 3,

A

BI =

-

-

1: -

-

(15)

Substituting this into the system (13) yields x(kt.+t.) =

lie =

Ax(kt.) +

B I Wk + B2 u(kt.)

(I la)

(l6c)

Bz=

A= A,

where

(11 b) (llc)

82,

C2 = C2

- T[~B 0] T[~Rl/2] T[ BI= BIUB=TB ° 0 TBTB 0 =TB

eA(M) BI \)T(t) dt

-

1<6

-

T

-

-

[Cl DI21 = U CD[ Cl D121

819

(l6a) (16b)

Cl x(kt.) + DI2 u(kt.)

y(kt.) = C2 x(kt.)

ktall

J

-T

[~]= [~:~]

equation for the lifted system G wz = FGwzF-I in the following.

A

°O]T

Note that BI B I and [Cl Dd [Cl Dd are finite dimensional. It can be verified that UB and UCD in (14) are unitary matrices. Using these we introduce another change of variables as

A

y(kt.) = C2 x(kt.)

T[~~D

CO

-

= 0

On tE [kt., kt.+t.), let the lifted disturbance be ~ = F[w(t)l and the lifted output be £ = F[z(t)l. We can use (10) to derive a state

1\

[~B 0] TB BOO

I

[Cl D121 [Cl D121 =Tco

f' 2 or t = ,4, ..

Ax(kt.) + BI "A" x(kt.+t.) = A Wk + B2 u(kt.) " " A" Zk = Cl x(kt.) + DII wl< + DI2 u(kt.)

(14)

8 8T =TT

(l0c)

For lifting the continuous periodic time varying system G wz . We consider the well known Fourier trigonometric series. The corresponding orthonormal set is given by

I.y ~

D121T~D[~f2]

UCD=[CI,

matrices, and ~ and ~D are finite dimensional diagonal matrices. These are obtained from the following symmetric factorisations.

1<6 t

e A(t-1:) B2 dt u(kt.) + D12 u(kt.) k6 y(kt.) = C2 x(kt.)

[~i/2J.

where TB and TCD are finite dimensional square unitary

Jt e A(t-1:) BI w(t) dt +Dllw(t)

z(t) = Cl e A(I-k6) x(kt.) + Cl

A= eA6,

l

"

is the control input, ZE IRq is the controlled output, and yE IRP is the measured output. We assume the system is stabilisable and detectable. Also assume there is no direct feed forward from w to y for the system to be stabilisable by using the sampled data control. Consider that the output y is sampled with a sampling period t., and the control input u is piece wise constant in every sampling period tE [kt., kt.+t.). We have

where

J \)(t) [J

DI2 =

(9c) r m where XE IRn is the state, WE lR is the disturbance input, UE IR

\)(i)(t) =

" 6 Cl = f \)(t) Cl e A1: dt

J

o

= C2 x(t)

+ Cl

k6+6 A e (6-t) B2 dt,

DlI =

3. H~ SAMPLED DATA CONTROL This section applies the Fourier series lifting to H~ sampled data control. Consider a linear time invariant system G written in state equation as

y(t)

B2 =

A

otherwise for i = 0, 1, ... , N-1. Then {\)(i)(t)} is an orthonormal set. Associated with this set the Fourier lifting for systems is exactly the block lifting, namely F =W. Therefore we have shown that the proposed Fourier series lifting is indeed a generalised technique and it includes the block lifting as a special case.

[~~ OllCoT~{ L~o ~ ]TCD = [~

=

finite dimensional approximation to the matrix computation in (17) and (18) to certain degree of accuracy. Therefore, the Fourier lifting also provides a good approximation approach to the solution. In fact , the finite dimension approximation can lead to a solution in the same form as that in (Toivonen 1992).

OllCo

This shows that the system (16) is a finite dimensional discrete

z

w

system. Let G wz be the system from to and Gwz(z) be its transfer function . By internal stability, we mean for any initial conditions the system state converges to zero exponentially. Then we establish an equivalence between the H_ control of the sampled data system (9) and that of the discrete system (16) in the following theorem. Theorem 3.1: A discrete controller C stabilises the system (9) and IIGwzll < I if and only if C stabilises the system (16) and

4. CONCLUSION The proposed Fourier series lifting is applicable to both continuous and discrete periodic time varying systems and it covers the block lifting as a special case. For a sampled data system, the lifted system is discrete and time invariant with infinite dimensional input and output and its state equation is written in terms of constant matrices. By using Fourier series lifting, we have shown that the H _ sampled data control problem is equivalent to a finite dimensional H_ discrete control problem. This allows the existing standard H_ control methods to be applied. The result allows either exact or approximate computation of the solution .

IIGwz(z)lI_ < I. Proof: We firstly show the H ~ control problem for the sampled data system G Wl in (9) is equivalent to that for the 11

lifted system G wz in (16). It follows from Lemma 2.2 (ii) and

5. REFERENCES Bamieh, B.A. and J . Boyd Pearson, Jr., (1992), A general framework for linear periodic systems with applications to H- sampled data control,lEEE Trans . on Automatic Control, vol. AC-37, pp.418-435. Bamieh, B .A ., J. Boyd Pearson, Jr., B.A . Francis, and A. Tannenbaum, (1991), A lifting technique for linear periodic systems with applications to sampled data control, Systems & Control Letters, vol. 17, pp.79-88. Chen, T . and B.A. Francis, (1991), Linear time varying H2 optimal control of sampled data systems, Automatica, vol. 27, pp.963-974. Davis, J.H., (1972), Stability conditions derived from spectral theory: Discrete systems with periodic feedback, SIAM 1. Control, vol. 10, pp.I-13. Dullerud, G .E. and B.A. Francis (1992), L I analysis and design of sampled data systems, IEEE Trans . on Automatic Control, vol. AC-37 , pp.436-446. Francis, B.A. and T.T . Georgiou, (1988), Stability theory for linear time invarant plants with periodic digital controllers, IEEE Trans. on Automatic Control , vol. AC-33, pp.820832. Gu, D .W ., M.C. Tsai, S.D . O'Young, and I. Postlethwaite, (1989), State space formulae for discrete time H_ optimization,lnt. 1. Control, vol. 49, pp.1683-1723. Jury, E.!. and F.1 . Mullin , (1959), The analysis of sampled data Control systems with periodically time varying sampling rate, IRE Trans. Automatic Control, vol. AC-24, pp.15-21 . Keller, J.P. and B.D.O. Anderson , (1992), A new approach to the discretization of continuous time controllers, IEEE Trans . on Automatic Control, vol. AC-37, pp.214-221. Khargonekar, P.P., K. Poolla, and A. Tannenbaum, (1985), Robust control of linear time-invariant plants using periodic compensation, IEEE Trans . on Atttomatic Control, vol. AC30, pp. 1088-1 096. Kreyszig, E., (1978), Introductory functi onal analysis, John Wiley & Sons. Inc. Limebeer, D. , M. Green, and D. Walker, (1989 ), Discrete time H_ Control, Proc. of the 28th CDC. Safonov, M.G ., Limebeer,D.1 .N., and R.Y. Chiang, (1989) , Simplifying the H_ theory via loop-shifting, matrix-pencil and descriptor concepts, 1nl. J. Control, vol.50, pp.2467 2488. Sz-Nagy, B. and C . Foias, (1970) , Harm onic Analysis of Operators on Hi/bert Space, New York: American Elseveier. Toivonen, H .T., (1992) , Sampled data control of continuous time systems with an H_ optimality criterion, Automatica, vol. 28, pp.45-54. Yamamoto, Y., (1990), New approach to sampled data control systems - A function space method, Proc. of the 29th CDC, pp.1882-1887 .

11

Lemma 2.4 that IIGwzll = IIGwz
11

Lemma 2 in (Safonov, 1989). That is IIGwz
wto zin (13).

In view of (15) we

-

-

Then we have IIGwAz)lI_ = IIGwz(z)ll_ since UB and UCD are unitary matrices. Also the state of the system (13) and that of the system (16) are identical. Thus the H_ control problems for 0 these two systems are equivalent. To apply Theorem 3.1, we need to transform the system (9) to the form of (16) by computing the following real constant finite dimensional matrices.

A= 81

A+ 81 DIIT(l - DlIDnT)-1 Cl

= T~

(l7a)

[Ltl

(l7b) -

1\

1\

1\

1\

1\

1\

B2 = B2 + BI DlIT (1- DlIDlI1)·1 D12 -

1(2

[Cl D121 = [Leo OllCD' where the matrices TB, LB, T CD, and the following factorisations.

(l7b)

-

C2 = C2

BI(I-DlITf)1I)·181=T~ [~B ~]TB 11

(l8a) T

Dd =Tco 11

11

(l7c)

Leo can be obtained by

[L~D 11

where the inverse of (1- DnDII T) exists since 11 DIIII < 1. We briefly comment the above result in the following. The exact solutions to the matrices in (17) involve computation of the infinite order Fourier series coefficients and the infinite dimentional matrices (18). It is possible that we apply Lemma 2.2 (i) to the matrix computation by firstly carrying out the continuous system integral operations and then doing the lifting. Thus there will be no computation of the infinite dimensional matrices involved, and the solutions will be in the same form as that in (Bamieh and Pearson 1992). This shows that the Fourier lifting provides an alternative approach to the result in (Bamieh and Pearson 1992). On the other hand the well known convergence properties of the Fourier series coefficients allow

820