CORRESPOKDENCE 137

ooan 11c.21 0022

21.co3 22.0-c 23.000 I1.0CO 12.022 13.ouo

31.303 32.03.1 33.000

21.000 22.0”C 1l.QOO 12.000

51-000 32.0CO 41.000 42.020

22.003 42.01iO

13.C03 23.OJO 1 2 . C O O 3.3.C:lO

1 4 . 0 3 3 24.090

0.3 C.9 2‘I.O’JO

13.000 0.0 14.000

c.u 0.0 0.0

0.0 0.0 0.0

0 ST?? I C

23.C30 13.COO

33.030 43.000

62. 0CO 32.0GO 44. coo 34.000

12.COO 0.C

20.coo 0.0

14.000 0.0

24.000 l4.0C8O

34.000 44.ccu

Case A = 3

Case N = 4

Fig. 3.

REFEREXES A Recursive Method for Determining [I] S. Barnett and C. Storey, “The Liapunov matrix equation and the Laplace TrmsfOm of exp (At)

~ 0 1 . 4C-12, pp. lli-118, February 196i. Schwarz’s form,” I E E E Trans. Aufamatic Control (Correspondence),

[ 2 ] C. F. Chen and L. S. Shieh, ‘ ‘ A n0t.e on expanding P A + ATP = -Q.” I E E E T r a n s . Automatic Control (Correspondence), 5-01, Abstract-A method for recursively determining the Laplace

[31 s, Bar;lett, c, F, cben, and L, s. Shieh, on .A on transform of the state transition matrix for a linear time-invariant AC-13 pp. 122-123 February 196s.

espanding P A + ATP = -Q’,” I E E E Trans. Automatic Control system is presented. The method does not require matrix inversion, (Correspondence). vol. AC-13 p. i56 December 196s.

[4] D. L. Kleinman. “On an i&at,ix-e technique for Riceat,i equation nor does the ComPled“ of the operations involved increase with comPutat,ions.” I E E E Trans. Automatic ConlroI (Correspondence). the dimensionality of the system. It is equally applicable to systems

[5] S. P. Bingulac ?,nd M. Stoji6 “An algorithm for smt.hesis of feed- with distinct or multiple roots. vol. AC-13, pp. 114-115, February 1968.

slax<a, 1969. back controller, Publ. of Mithematical Institute, Belgrade, Yugo-

[6] A . A . Krasovskii, “New method of control system analyt,ical design,” Pror. 1969 I F A G C0n.g. (Tarsaw). Manuscript received July 21, 1969.

-

138

I. INTRODUCTION It is well known that the general linear nth-order constant.

coefficient system

CD" + an-1Dn--l + - - * + a 1 D + aaly ( t )

= [b,Dm + bm-1D'"-' + * - + ba}u ( t )

where m 2 n and D = djdt , may be put into the canonic form

i ( t ) = AZ( t ) + hu(t) (1) where [I]

0 1 0 0 0 e . .

(2)

--a0 -01 -a2 - - - -a.n-1

The development rhich follows will assume that the system has been put into t.he form of (1) and (2). The Laplace t.ransform of e*t,L{e4tJ = CIS - A ] - 1 , will be det.ermined recursively, wit,hout employing matrix inversion.

11. DEVELOPNEW Let. the Zth power of the system matrix be represented by

0111' CYlPZ f f l " ':

Az = [CY? CY: a n 1 l (la2 * - - f f n n

where, from ( 2 )

a i j 1 = 1, j = i + I = 0, j # i + 1, i = 1,2,-..,n - 1.

ff .1 = -a . "I ,-I.

The elements of A'+' are then n

ff . J + l = z ) u i m k j l

k=1

= f f ( i + l ) j l , i = 1,2,. - .,n - 1. For i = n

" ff . Z i l = n1 ffnk'ffkj

k=l

= f f . n(l-1)' - aj-Iann'

where the fact that A and A' commute has been used.

1 4

The elements in the nth row of the t,ransit.ion mat.ri.. are given by

IEEE TEUNSACTIONS OX AUTONATIC CONTILOL, FEBRUARY 1970

Using (4),

It can easily be seen from (7) that the elements of the nth row of t,he state transition nlatris must sat.isfy (8 ) ,

d+,s(t)/dt = &(~--l)(t) - Q M + ~ ~ ( ~ ) (8)

where & ~ ( t ) 0, subject to the initial conditions

+"k(O) = 0, n # k

+,m(O) = 1.

Using (3) in (5) and an argument entirely analogous to the preceding, it may easily be est,ablished that the remaining of the transition matrix elements must satisfy

d+i j ( t ) /d t = @(<+I)j(L), i = 1,2,...,n - 1, j = 1,2,...,n (9)

subject to the initial condit.ions

+ i j ( t ) 1t-o = 6 i j

where 6 i j is the Kronecker delta.

(8) , nhich gives The development proceeds by taking the Laplace transform O F

S & k ( S ) - @ n e ( O ) = &(.k-n (S) - Q . L I & ~ ( S ) .

Now take the Laplace t.ransform of (9) and obtain

&j(S) = B ( i + l ) j ( s ) / s + 6 i j / s . (11)

The essential results are summarized in Prcperty 1.

Property I If the system matrix A i s in the canonic form of (2), then the

Laplace transform of t,he transhion matrix may be determined recursively by init.ia1ly finding

n-1

&n (s) = Sn-I/ (sn + ais<) i=o

and t.hen generating

&k(S) = - [ ( C Q i s i ) / S k ] ~ t , n ( S ) , k = 1,2,-.*,7Z - 1. A-1

i=O

One then uses this last r o ~ of the transition matrix to recursively generate the rows above it. by applying

&(s) = I $ ( j + l , j ( S ) / S + G i j / S , i = 1,2 ,..., n - 1.

CORRESPONDENCE 139

The t,ransition matrix in the time domain may be easily determined by inverting the Laplace transform using part.ia1 fraction exipansion techniques. Therefore, by knowing the roots of the syst.em char- acteristic equation, one may directly determine the transition mat.rix in the time domain.

It should be noted that the relationships given in (3) and (4) will be useful in numerical determinations of the transition matrix. For instance, if a method such as that proposed by Liou [2] were to be employed, then only the last row of the matrix need be checked for the desired degree of accuracy as the other rows may easily be seen to converge faster than the last. row.

C. P. HATSELL Dept. of Elec. Engrg.

Duke University Durham, X. C.

REFEREKCES

[l] hi. Atham and P. L. Falb, Optimal C a l r o l . S e w Pork: McGram- 121 M L. Liou “A nox-el method of evaluating transient response,”

Hill, 1966, pp. 143-147. P r k . IEEE: vol. 54, pp. 20-23, January 1966.

On the Lyapunov Design of Systems with Zeros in the Right-Half Plane

Abstract-The effect of a zero in the right-half plane on the controller derived through the stability theorem of Lyapunov is considered through a simple illustrative example. It is shown that the controller should have the form of a linear saturating function instead of the bang-bang form for the minimum-phase systems.

Many industrial processes are known to exhibit a dead-time delay in their txansfer characterist.ics. It is also known t,hat such delay effects can often be approximated by placing a zero of t,he transfer function in t.he right-half plane (RHP). This usually causes an unsatisfactory design of linear controllers for these plants since one of t,he root loci branches off towards t.he RKP. -4 design technique which has been successfully employed to derive cont.rollers that ensures system stability is based on the stability theorem of Lyapunov. Several interesting applicat.ions of this approach have been reported in the literature (see, for example, Grayson [l]). All of the systems considered, however, are of t.he minimum phase t.ype. It has been established that the desired cont.roller for such system is of the bang-bang t,ype. The aim of this correspondence is to point out that a RHP zero causes t.he optimal controller to have a saturating form.

To illust.rate the point consider a simple feedback regulator system as shown in Fig. 1. The control signal u.(e) is yet unknown and should be so chosen as to ensure that the controlled system remains asymptotically stable with the error of regulation suitably bounded. (As discwed in [l] this choice of u ako minimizes a quadratic cost function.) The plant transfer function G(s) is taken to be

G ( s ) = (s - b ) / ( s 2 + als + ~ a ) . (1)

This corresponds to the system differential equation

+ U I ~ + ~ o y zi - bu. (2)

Defining t.he state variables x1 = y (3)

x2 = y (4)

I I 1 I

Fig. 1.

the state equation can be written as

X + A X = C z i - B u (5) where

x = (21 zdT, A = [ -1.1, B = (0 bIT, C = (0

Since for the regulat.or the desired value of the state is the origin of the state space, the error vector e ( t ) is simply the negative of X ( t ) . Using t.his t.he sysstem equation c m be rewritten as

i + Ae = Bv, - Czi. (6)

Nom it is assumed that t,he autonomous part of (5) is st,able. Ac- cordingly, 6he system (7) is ako stable

6 = -Ae. (’7)

Let t,he V function for (7) be

V ( e ) = eTe. (8)

Taking the t.ime derivative of both sides and using (7) we can write

V ( e ) = -eTDe (9)

where D = AT + A . Since from Lyapunov’s stability theorem P ( e ) is to be a negative definite, it follows that D is a positive defmite matrix.

Consider now the controlled system ( 6 ) . Choosing the same V function as before its time derivative can be mitten as

p ( e ) = -eTDe + (BTu - CTzi)e + eT(Bu - Czi). (10)

Since the first term on t.he right-hand side has been seen to be negative definite, p ( e ) can be const.rained to be negative definite if

eT(Bu - Czi) < 0. (11)

This inequality can be expressed in the scalar form as

&(h - zi) < 0 (12) where i1 = -y.

are met The inequality (12) can be satisfied if the following conditions

a) sgn u = sgn e’, b) ba > ti.

A third condition is usually present in the form of power con- straint which requires that

c) u 5 L.

These t h e e relations are sufficient to find out the form of the desired controller. Apparently, i t is not possible to satisfy b) by choosing u to be a bang-bang function. This follows from the fact that this condition requires zi to be bounded which in turn requires that the slope (du/del) be bounded. This restriction on the slope is obvious if it noted that

i = (dU/del)C.

The quantity on the left being bounded the slope can not be in- finite unless & is zero. This latter condition is of course not met at the instant when the error itself is passing through zero. Because Manuscript received June 19, 1969.