2 SUD, S.K., BISWAS, K.K., and SINHA, A.K.: 'Stochastic state-variable model for a paper-machine headbox', ibid., 1977, 124,(12), pp. 1249-1254

3 RUTHERFORD, D.A.: 'Laboratory scale flowbox for a papermachine'. UMIST Control Systems Centre Report 197, 1972

4 ROSENBROCK, H.H.: 'An automatic method for finding thegreatest and least value of a function', Comput. J., 1960, 3, pp.175-84

5 KOPPEL, L.B.: 'Introduction to control theory' (Prentice Hall,1968)

6 SHINSKEY, F.G.: 'Process control systems' (McGraw-Hill, 1979)2ndedn

7 MACFARLANE, A.G.J.: 'Complex variable methods for linearmultivariable systems' (Taylor & Francis, 1978)

8 MUNRO, N.: 'Design of modern control systems' (Peter Peregrinus1982)

9 OWENS, D.: 'Feedback in multivariable systems' (Peter Peregrinus,1978)

CorrespondenceSINGULAR-PERTURBATION METHOD FORDISCRETE MODELS OF CONTINUOUS SYSTEMSIN OPTIMAL CONTROLIn paper 1399 D [IEE Proc. D, Control Theory & Appl.,1981, 128, (4), pp. 142-148], Rajagopalan and Naidu haveproposed a method to derive the optimal control for a classof discrete models of singularly perturbed continuous systems.The method consists of two phases. In the first phase, afirst-order foward difference is utilised to discretise the con-tinuous system and the associated performance index. In thesecond phase, a series solution is employed to obtain theoptimal control (open- and closed-loop). This phase eventuallydepends on the validity of the formula

( / + £ ) - i = I -D (A)

which corresponds to eqn. 11 of the paper, with D = BR'1 B*P(k + 1), where P(k) is the unknown disrete Riccati matrix.It is well known [2] that eqn. A above is valid only if

\\D\\ < 1 (B)

Obviously, the condition of expr. B cannot be verified a priorisince D comprises the Riccati matrix which is yet to be de-termined. Therefore, the results of Rajagopalan and Naidu'sproposed method remain in doubt unless eqn. A with thespecified D is proved to be true. Other formulas to expandeqn. A are available at present [B], however all of themcontain terms of the form (E + F )" 1 where E and F arefactors of the matrix D. This means that the difficultyencountered in eqn. 7 of the paper is still present and thestandard form for the singularly perturbed matrix Riccatiequations (eqns. 10a and b) cannot be obtained throughthe binomial expansion (eqn. 11) as claimed.

MAGDI S. MAHMOUD28th January 1982

Electrical Engineering DepartmentKuwait UniversityPO Box 5969Kuwait

We thank Dr. M.S. Mahmoud for his interest in our paper.We are well aware of the implications of using eqn. A which

corresponds to eqn. 11 of our paper. However, the followingclarifications are given:

(a) Eqn. A is valid only if the norm of D is less than unity.The matrix D contains the sampling interval T, which isassumed to be much less than the small parameter h. Underthis assumption, the norm of D is less than unity, in most ofthe cases. If the norm condition is not satisfied, eqn. A cannotbe used.

(ft) It is commented that 'the condition of expr. B cannotbe verified a priori since D comprises the Riccati matrixP(k + 1) which is yet to be determined'. It should be notedthat the Riccati equation (eqn. 7) in our paper is solvedbackwards in the sense that, at every stage, P(k + 1) is knownand P(k) is determined. Even to start with P(kf = k + 1) isgiven and hence the condition of expr. B can be verified apriori.

(c) A way of resolving this problem is to use the wellknown formula for the inverse of a partitioned matrix [B]:

l {Fl-F2FilF3yl

(C)

Eqn. C again involves the matrix inversions but of reducedorder. Further investigations have been successfully carriedout using eqn. C, and the results will be reported soon [C].

P.K. RAJAGOPALAND.S. NAIDU

4th January 1983

Department of Electrical EngineeringIndian Institute of TechnologyKharagpur-721302India

ReferencesA WILKINSON, J.H.: 'The Algebraic eigenvalue problem' (Clarendon

Press, Oxford, 1965), pp. 60-61B BAR-NESS, Y.: 'Solution of the discrete infinite-time, time-invariant

regulator by the Euler equation', Int. J. Control, 1975, 22, pp. 4 9 -56

C NAIDU, D.S., and RAO, A.K.: 'Singular perturbation analysis ofthe closed-loop discrete optimal control problem', IEE Proceedings(under consideration)

DTC119D

136 IEE PROC, Vol. 130., Pt. D, No. 3. MAY 1983