The authors propose a novel method for computing state deadbeat feedback gains from systems given in the staircase form. The proposed method uses only manipulations of given matrices, and hence is more direct than the existing method, which requires orthogonal transformations repeatedly. It is shown that the obtained gain is linear quadratic optimal for some weighting matrices </p

Inoue, Akira

Masuda, Shiro

Sugimoto, Kenji

Okayama University Scientific Achievement Repository

EngineeringIndustrial & Management Engineering fieldsOkayama University Year 1990A direct algorithm for state deadbeatcontrolKenji Sugimoto Akira InoueOkayama University Okayama UniversityShiro MasudaOkayama UniversityThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/industrial engineering/107on Dedslon n d  Control rr= 14- I - a;au Honolulu. Hamll Decnnba 1oW A Direct Algorithm for State Deadbeat Control Kenji Sugimot,o, Akira Inoue, and Shiro Masuda Department of Information Technology Faculty of Engineering, Okayama University Okayama 700, JAPAN Abstract This pa.per proposes a new method for coin- puting sta.te deadbeat feedback gains from sys- tems given in the sta.ircase form. The proposed method uses only manipuhtions of given mat,ri- ces and hence is more direct than the existing one which requires orthogonal transformat,ions repea.tedly. The paper also shows that Olie ob- tained gain is LQ opt.ima.1 for soiiie weighting matrices. 1. Introduction As is widely recognized, it is by no means triv- ial to compute stat,e deadbeat, ga.ins in a numer- ically stable way. Emami-Naemi et al. [2] have proposed a method based on t,he linear quadratic (LQ) control theory. Dooren [I] 1ia.s computed, by repertt,ing conversion of a. given sysbem, t8he deadbeat ga.in t,ogether with the result,ing nilpo- tent, state-t,ransit,ion matrix. Both of these meth- ods have excellent numerical reliability. In actual comput,a.tion, however, t,he above met.hods are not stra.ig1itformard in the following sense. In the LQ method [ 2 ] ,  we need to solve the discrete- time R i c a t i  equation for a specia.lly chosen pair of weighting matrices. This is rather indirect and a.rtificia1 heca,use bhose weigltbings have a less physical mea.ning as a. cont,rol per- formance. On the other ha.nd, in the method of [I] we have to comput,e the nilpot.ent, ma.trix a.nd orthogona.1 ma.trices, which are not essentia.1 for the coniputation of the ga.in. In this paper we propose a. new met,liocl for computing the deadbeat, ga,in more directly t1ia.n [l]. As a result, this met8hod reduces bot,li com- putational time and theoret,ica,l complexity. Fur- thermore, it. is shown t.ha,t the obtained feedback ga.in is LQ opt,imal for some weighting ma.trices. Thus the present, approach has both of the ad- vaiihges of the above two methods. 2. Main Resu Consider a reachable system z(t  + 1) = F,s(t) + Got@), where F, E R7’xn,G0 E t t = 1 , 3 , . , .  (1) No assumption is ma.de on nonsiagularit,y of F,. As in [l] and [2], we start by converting (F,, Go) into t,he so-called staircase form: where p is the reachability index, A p  has full row rank r p ,  the diagonal element matrices Fpp are rp  x rp ,  and the rest are of compatible sizes. It is well kno\vn that the form ( 3 )  can easily be obtained via orthogonal transformations. Now we compute the deadbeat gain directly from ( 2 ) .  In order to explain the recursive nature of the algrorithm, we first define a sequence of smalier matrices A - for p = 1, .  . . . p .  Then, F = F I ,  G = GI,  and Namely, &+, is embedded into the preceding F,,. CH2917-3/90/0000-3642$1 .OO Q 1990 IEEE 3642 \ With these notions, the deadbeat gain K is given by the simple formula: A K ,  := A, F,, K p  := A,[I Kp+l]Fp, p = p - 1,.  . . , 1 ,  K := K1 ,. (4) Here, A-  denotes any right inverse of the matrix A .  THEOREM 1. The feedback gain (4) achieves minimal time deadbeat control. Outline of Proof. The closed-loop state- transition matrix is F - G I I  = [ -? ] F (by ( 2 ) ,  (4))  ( 5 )  with - denoting similarity. sively, we finally obtain Proceeding recur- 0 . * e  0 F - G K -  [ e.. A ]  ( 6 )  * i.e., a block triangular form. Hence (4) achieves minimal time deadbeat control (see [l]) .  0 Note that in actual computation, we do not have to perform similarity transformations (5)  and (6).  We need only to compute K,, . . . , K1 according to (4). 3. LQ Optimality Now let us show that the gain (4) is optimal for some weightings. To this end, we give a criterion for optimality in a more general form. THEOREM 2. Consider the system (2), and suppose that G has full column rank. (Then A1 is square and invertible.) If a stabilizing feed- back gain Ii is written as II = AT1[I L]F (7)  for some L,  then Ii is LQ optimal for the index with weightings Q := H T H ,  H := A;’[I L] ,  R := 0 (8) Outline of Proof. Define Z( z )  := ( z l  - F)-’G The return difference matrix for (7) is W ( z )  := I +  I i Z ( Z )  = AT1[I L ] G + A T l [ I  L ] F Z ( z )  = zAT1[I L ] Z ( z )  (9) Hence we have WT(z- l )W(z)  = Z * ( Z - ~ > Q Z ( Z )  (10) On the other hand, let w(z) be the return dif- ference matrix of the LQ optimal gain for Q and R. Then satisfies the Kalman equation -T w ( Z - ~ ) G T G W ( Z )  = z * ( z - ~ ) Q z ( z )  (11) where fl is the solution of the corresponding Ric- cati equation. In view of (10) and (11) we can readily show that K = by slightly modifying the technique in [3]. 0 The deadbeat gain (4) clearly satisfies (7) for L := Kz,  and hence it is LQ optimal. 4. Conclusion We have shown that the gain (4) achieves deadbeat control and is also LQ optimal. Al- though we do not have to solve Riccati equation for any weightings, our feedback gain belongs to the class of optimal control as in [2] References [l]  P. Van Dooren, “Deadbeat Control: A Spe- cial Inverse Eigenvalue Problem,” BIT, Vol. [2] A. Emami-Naeini and G. F. Franklin, “Deadbeat Control and Tracking of Dis- crete-Time Systems,” IEEE Trans. Autom. Control, Vol. AC-27, pp. 176-181, 1982. 24, pp. 681-699, 1984. [3] K.  Sugimoto and Y. Yamamoto, “Solu- tion to the Inverse Regulator Problem for Discrete-Time Systems,” Int. J. Control, Vol. 48, pp. 1285-1300, 1988. 3643 

A direct algorithm for state deadbeat control

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