PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [National Taiwan University]On: 14 January 2009
Access details: Access Details: [subscription number 905688744] Publisher Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Control
Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713393989
An algorithm for computing the minimum-time regulator of multivariable discrete
linear systems
Wei-Song Lin a
a Department of Electrical Engineering, and Jing Ling Industrial Research Institute, National Taiwan University, Taipei, Taiwan, Republic of China.
Online Publication Date: 01 August 1986
To cite this Article Lin, Wei-Song(1986)'An algorithm for computing the minimum-time regulator of multivariable discrete linear systems',International Journal of Control,44:2,543 — 548
To link to this Article: DOI: 10.1080/00207178608933617 URL: http://dx.doi.org/10.1080/00207178608933617
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf
This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
INT.J. CONTROL, 1986,VOL. 44,No.2, 543-548
An algorithm for computing the minimum-time regulator of
multivariable discrete linear systems
WEI-SONG LINt
This paper constructs an algorithm in general form for computing time-invariant feedback laws which assume deadbeat regulation in minimum time of multivariable discrete-time linear systems. The algorithm simply assumes that the system is irreducible. The key idea is the division of the state space into sub-statespaces according to the time needed to reach the equilibrium state. Based on the spans of all sub-statespaces, iterative formulae for computing all the feedback gains are obtained. The formulae can be programmed easily, and the transient response can be shaped by choosing different sets of feedback gains. An example is worked out to demonstrate the feasibility of the proposed method.
1. Introduction
The deadbeat strategy is unique to discrete-time systems: there is no correspond-ing feature for continuous-time systems. Several works have shown the possibility of obtaining deadbeat performance in discrete-time linear systems and the properties of deadbeat controllers have also been investigated (Cadzow1968,Farison and Fu 1970, Kucera 1971, Leden 1977, Pachter 1977, Sebakhy and Abdel-Moneim 1979). By assuming the state matrix to be non-singular, a direct method of computing the deadbeat controller for multivariable discrete-time linear systems was obtained (Leden 1977). Without the minimum-time limitation, the deadbeat problems were solved by Pachter(1977).On the assumption that the control matrix is of full rank, a method for computing generalized control form of any controllable pair was achieved by Sebakhy(1979).
The purpose of this paper is to present an iterative algorithm for computing the feedback laws which assure deadbeat regulation in minimum finitetime for multi-variable discrete-time linear systems. The only assumption is that the state-space description is irreducible. In § 2 a statement of deadbeat regulation by time-invariant feedback gain matrix is given. The formulae for computing the feedback law is derived in §3. An example is worked out in §4 to illustrate the feasibility of the proposed method.
2. Statement of the problem
Consider a multivariable discrete-time linear system governed by the vector difference equation
X(k
+
1)=
AX(k)+
BU(k) U(k)= HX(k)(I)
(2)
where X(k) is an n-dimensional state vector at the kth iteration, U(k) is an
Received 22 November 1985.
t
Department of Electrical Engineering, and ling Ling Industrial Research Institute, National Taiwan University, Taipei, Taiwan, Republic of China.544
w.-S.
Linr-dimensional control vector at the kth iteration, A is the constant (nxn)system matrix,Bis the constant(nxr)control matrix, and H is the(r xn)constant feedback gain matrix. The only assumption will be that system (I) is an irreducible
n-dimensional system (Swisher 1983). Thus the system is controllable and the following properties hold for the pair(A, B):
rank [B AB ... A"-I B] = n (3)
and
rank [A B] = n
The controllability index is defined as the minimum number, m,such that rank [B AB ... Am-I B]=n
(4)
(5) Obviously, for multi-input systems,mis no larger than the dimension,n,of the system.
The problem of deadbeat regulation in minimum time is the following. For the multivariable system (I) and the feedback law (2) compute the constant feedback gain matrix H such that the system can be driven by unbounded control inputs from any initial state, X(O), to the equilibrium state X. in deadbeat mode, or
(A
+
BH)mx(o)=
X.(6)
with the number of control periods m being the controllability index of the system.Without loss of generality, the equilibrium state will be assumed to be the zero state in the following sections.
3. Derivation of the algorithm
Let R(i), i = 0, 1,2, ... , m represent sub-statespaces such that the minimum time required to bring any state Xi> XiERli) to the equilibrium, R(ol = X" is i control
periods. LetS, be an(nx n)matrix such that the columns ofS,spanR(il' or R(i) = span
(S,). Then any state Xi' XiER(i) can be represented as a linear combination of the columns ofS, or Xi = S,Yi where Yi is an (nx I)matrix.
Since, for completely state-controllable systems, any initial state can be brought to the equilibrium in no more thanmcontrol periods, therefore the whole n-dimensional state space,R",is the union of the sub-statespacesR(il'i= 0, I, ... , m.Consequently, we obtain R"= span (So, SI' ... ,Sm) or
rank [SI S2 ... Sm] = n (7)
Furthermore, for the closed-loop system, any Xi = S,Yi>i= 1,2, ... ,m can be driven to
the zero state in no more than i control periods, therefore we obtain (A
+
BH)iXi = (A+
BH)iSi Yi = 0, for i= 1,2, ... ,m Since (8)can be satisfied by any Yi ,i= I, 2, ... ,m it follows that(A+BH)iSi = 0, for i = I, 2, ... .m
For a specified value of i,the above equation can be written for i and i - I as
(A+ BH)iSi
=
(A+ BH)i-1 {(A+ BH)Si}=
0 and(8)
(9)
(10)
(II)
Algorithm for minimum-time regulator 545 Comparing the second term of (10) to (II) we may form reasonably the relationships
(A
+
BH)S,= S'_I> for i= I, 2, ... .m (12) where So is an (nxn)zero matrix. In accordance with matrix operations, (12) can be rewri tten as[In] [
s, ]
(A
+
BH)S,=
[A B] H S,=
[A B] HS,=5'-1' fori=I,2, ... ,m (13)
(14)
Ithas been assumed in(4)that the nx(n
+
r) matrix [A B] is of full rank n. Therefore [A B] has a right inverse matrix [A B]RIsuch that [A B][A B]RI= In (Cadzow and Martens 1970). In general, the right inverse of[A B] is not unique; one of them is[A B]RI
=
[A B]'{[A B] [A B]'}-IThen a general solution of(13) is found to be
[HS,S, ]
=
[A B]t{[A B] [A B]'} -1S,_I+
{In+r - [A B]t{[A B] [A B]'}- I[A B]}Zi_1 for i = I, 2, ... ,m (15)where Z'-1 is any (n
+
r)xn matrix (Lin etal. 1983). Since So =0, a non-trivial solution of [ S, ], i = 1,2, ... ,mcan be found iteratively by choosing az,
such thatHSi
[ HS ,SI ] is a non-zero matrix. Then form the matrices Sand Q as follows:
S=[SI S2 " , Sm] }
(16)
Q= [HS( HS2 ••• HSm ] = HS
It has been shown in (7) that S is of full rank n. Therefore, a right inverse of S is
SRI
=
SI {SS'} - 1 (17)The feedback gain matrixH for deadbeat regulation can thus be solved from (16) as
H
=
Q{S'{SS'}+
{lnm -
S'{SS'}-IS}
W (18) where W is any (nm x n) matrix. The second term on the right side of (18) is the homogeneous solution. The algorithm for determining an H can thus be summarized as follows.Step I. Compute iteratively the matrices S, and HS, which satisfy (15) until rank
([SI S2 ... S.J) =
n
is obtained and then let the controllability index m be the last value of i.Step 2. Form the matrices Sand Q as shown in (16). Step 3. Compute the feedback gain matrix H from (18).
The solution of the feedback gain matrix H of a multi variable system may not be unique. The designer can shape the transient response of the closed-loop system by
546 W.-5.Lin
choosing a proper H (Klein and Moore 1977). In the algorithm there are two sources of freedom for this choice. They are the selections of2i - " i= I, 2, ... ,min (15) and W in (18). However, in order to obtain minimum-time regulation, 20 in (15) should be
selected such that the columns of [5, H5 ,
J'
span the null row space of[A B].4. Numerical example
Consider the discrete system (I) with system matrices
Note that A is a singular matrix.
5tep 1
Since 20 in (15) can be an arbitrary non-zero (5 x 3) matrix, we can take 0 0
0 I 0
2
0=
0 0 0 0 0 0 0 0 For simplicity of calculations, we can also letZ,= Osd ' forj
>
0Substituting20 and 50
=
0 into (15), we obtain 0·75 -0·25 0·25 -0,25 0·25 0·25[:J
0·25 0·25 0·75 0·25 -0,25 -0,250 0 0
since rank (5,)
=
2 is less than the dimensionn=
3 of the given system. We substitute 5, and 2,=
0 into (15) and obtain-0,0625 -0,0625 -0,0625
[:J
0·3125 0·3125 0·3125 0·0625 0·0625 0·0625 0·4375 0·4375 0·4375 1·25 1·25 ' 1·25 Rank ([5, 52])=
3 shows that the controllability index ism=
2.Algorithm for minimum-time regulator Step 2
Form the matrices Sand Q as follows:
547
[
0"
0·25 0·25 -0,0625 0·0625-_25]
S=
[SI : S2]=
-0,25 0·25 0·25 0·3125 0·3125 0·3125 0·25 0·25 0·75 0·0625 0·0625 0·0625 and [0'25 -0,25 -0·25 0·4375 0-4375 0'4375J Q=
[HS1 :HS 2]=
0 0 0 1·25 1·25 1·25 Step 3Applying (18) and taking W
=
0, we getH=[1'5 2
2·5 5
-1'5J -2,5
As a check, the values ofH are applied to the given system. Assuming the initial state of interest ifx(O)= [I I I ]'.The control vectors and system responses are shown as follows: k U(k) X(k)
o [:]
[:]
[-:]
[:]
2[~]
m
5. ConclusionsA general algorithm has been obtained for constructing time-invariant feedback laws which assure deadbeat response in minimum time of a multivariable discrete-time linear system. The algorithm puts no assumption on the singularity of the system matrix and the rank of the control matrix. The designer can easily convert the algorithm into a computer program. The transient response of the closed-loop system can also be shaped by choosing a different set of generalized eigenvectors.
REFERENCES CADZOW, J. A., 1968,I.£.E.£. Trans. autom. Control,13, 734.
CADZOW,J. A.,andMARTENS, H. R.,1970,Discrete-Time and Computer Control Systems,p. 288.
FARIsciN,J.B., and Fu, F.
c,
1970, I.£.E.£. Trans. autom. Control,15,390.548 Algorithm for minimum-time regulator KLEIN, G., and MOORE, B. C, 1977, I.E.E.E. Trans. auton.. Control, 22,140. KutER, Y., 1971, I.E.E.E. Trans. autam. Control, 16,375.
LilDEN, B., 1977, Automatica, 13,185.
Lrx, W.S., Kuo, T.S. and THALER, G. J.,1983,Int. J.Control, 37,855. PACHTER, M., 1977, I.E.E.E. Trans. autom. Control,22,263.
SEHAKHY, O. A., and ABDEL-MoNEIM, T. M., 1979, I.E.E.E. Trans. aut am. Control, 24,84. SWISHER, G. M., 1983, Introduction to Linear Systems Analysis (Matrix Publishing Co.), p. 563.
