404 IEEE TRANSACTIONS ON AUTOhIAnC CONTROL, JUNE 1976

--0-8 h

-0.4 I 1

-k Y) 15 20 2s 0 5

Fig. I . Estimates of the feedback controller gain.

asymptotically stable. However, since the feedback gains are selected by minimizing the variance of the error between the actual and a desired (stable) response, a stability problem is not generally expected.

REFERENCES

( I ] H. H. Rosenbrock and P. D. M c M o m , “Good bad and optimal,” IEEE Trans.

[2] W. M. Wonham, “On pole assignment in multi-input controllable linear systems,” Auromr. Conrr., vol AC-16, pp. 552-554. Dec. 1971.

[3] E. J. Davison, “On pole assignment in multivariable linear systems,” IEEE Tram. .IEEE Tram. AUIOIMI. Conrr., vol. AC-12, pp. 6 6 W 5 , Dec. 1967.

(41 0. A. Solheim, -Design of optimal conlrol systems with prescribed eigenvalue” Inr. A U I O ~ I . Conrr., vol. AC.-13, pp. 747-748. Dec. 1968.

[5] M. C. Maki and J. Van De Vegte. ”Optimization of multiple input systems with J. Conrr., vol. 15, no. I , pp. 143-160, 1972.

assigned poles,” IEEE Tram. Auromr. Conrr., vol. AC-19, pp. 130-133, Apr. 1974.

sufficient condition for an optimal period has been obtained in [8] (Proposition 3). However, the approach is limited to a linear system and a certain type of performance index. Altogether, to the best of the author’s knowledge, no sufficient conditions for the optimal time (or the optimal period) has been obtained until now.

THE OPTIMAL CONTROL PROBLEM

In general terms, the problem is one of minimkhg the cost function

J = ;p@[x( t ) ,u ( t ) , r ld t (1) f 0

for the system described by a set of differential equations

i = f [ x ( t ) , u ( t ) , t l x(O)=x, (2)

where the terminal time 9 is unspecified, and the r-vector terminal manifold equation

has to be satisfied at t

Lagrange multipliers, we obtain Combining the equahty constraints with the cost function by means of I:

+ X ’ ( t ) { f [ x ( f ) , ~ ( t ) , t l - ; ) dt. (4) 1 We define a Hamiltonian:

Optimization for Unspecified Terminal Time H [ ~ ( t ) , u ( t ) , N t ) , t I = - ~ I x ( t ) , ~ ( t ) , t I + x ’ ( t ) f [ ~ ( t ) , u ( t ) , t I . (5)

For brevity, dependent functions in parentheses are omitted below wherever possible. Substituting Q from ( 5 ) in (4) and integrating by parts,

1 fr

Y . KOREN

Abstract-This note discusses a continuous optimal control problem in we have which the terminal time is unspecified and where an integral cost func- tional is determined by averaging over the process interval. Sufficient conditions are given for the case where optimization is effected on an J = ~ ‘ N + X ‘ ( O ) . x , - X . ( t ~ ) x ( ~ ) + ~ ~ ( H + j \ . x ) d t . (6 ) instantaneons rather than an integral basis.

A necessary condition for minimization is for the partial derivatives of

INTRODUCTION J with respect to x, u, and 9 to vanish; hence,

In this note we consider an optimization problem where the integral cost functional is determined by averaging over the whole process time interval (9). In other words, equal dimensions of the cost integral ( J ) and its integrand (@) are required. The purpose of this technical note is to obtain sufficient conditions for the optimal time interval fr to be zero. A system which satisfies these conditions is subjected to the simple mode of an instantaneous control. From the industrial point of view, the problem is of practical importance mainly where the cost @ is measured in dollars, as for example, the one that has been presented in [I].

Optimization problems with a similar cost function are related to the periodic control class. The main difference is that in the latter, the process is subjected to control variables which vary periodically with time and, in turn, its state variables vary periodically as well. In many cases the periodic solution is imposed by the nature of the problem and the cycle time have a certain value (see for example [2H5]). A stronger similarity to our problem ar i ses in cases where the period of the control is not fixed a priori and must be determined. A linear system with a quadratic cost function and a free period has been considered in [6] , but an explicit solution of the optimal period was not presented there.

The step-by-step seeking algorithm for the optimal period has been presented in [7], but again an explicit expression was not developed. A

Manuscript received September 12. 1975. The author is with the Depanment of Mechanical Engineering. Technion-Israel

Institute of Technology, Hai fa , Israel.

Equation (7) yields

(8) yields

and (S), with H substituted from (S), yields

Equation (12) is the sought equation for the optimal terminal time.

TECHNICAL NOTES AND CORRESPONDENCE 405

Notice that since

(12) has a trivial solution at $=O. Sufficient conditions for this solution to be unique are stated below. Defining

these conditions can be stated as follows. The optimal terminal time is zero if 1) g(9) a 0 for all r, and the optimal cp increases monotonically with respect to time, or 2) g(t,) G 0 for all 9 and the optimal Q decreases monotonically with respect to time.

EXAMPLE

The cost of a cutting operation may be defined by the following function:

+ = K ~ / u + K ~ u + K ~ x (13)

where u is the cutting speed and x is the wear-width of the cutting tool. K , is a constant proportional to the cost of the cutting machine per unit time, while K2 and K3 represent the cost of regrinding the tool. The wear-rate of the tool, after a short initial period, depends on the wear itself and on the cutting speed. Assuming linearity, the following dif- ferential equation describes the wear process:

. i=ux+bu x(0)=xo>O (14)

where (I and b are positive constants. What concerns the manufacturer is the average cost

with I) unspecified, but ranges between 0 G r, < T < XI. Equation (10) for this particular case yields

- A = a A + K 3 / r , A(t,)=O. (16)

The solution of (16) is

Equation (1 1) yields

By substituting (17) into (18), one obtains

Since (14) is unstable, the term K3x in (13) increases monotonically with time; from (19) it is obvious that the term K , / u decreases monoton- ically with a maximum at t = 0, while K,u increases with time. Therefore, a sufficient condition for + to increase monotonically is

(20)

where uo is obtained from (19) by substituting t =O. Since the maximum r, has been limited to T, (20) might be fulfilled for

specific values of K , , K2, K3, a, 6 , and x,,. If this happens, then t,=O causing A = O as well, and the optimal control law is obtained from (18).

u= v K I / K 2 (21)

which means that the optimal cutting should be performed by a constant cutting speed.

CONCLUSION

Sufficient conditions for the optimal terminal time to be zero in a certain class of optimal control problems have been found. A zero terminal time means that the optimization is effected on an instanta- neous rather than on integral basis, or in other words, -tion of @ rather than J is required. This, in turn, leads to an implementation of simpler control laws and simpler controllers. The problem might be of extreme interest in industrial plants where the cost Q is measured in dollars.

REFERENCES

frolled milling machines” Appendix 11, US. Air Force, Tech. Documentary Rep. “Final report on development of adaptive control techniques for numerically con-

ML-TDR-64-279, Aug. 1964. M. Fjeld, “Optimal control of multivariable periodic processes,” Auromarico, voL 5, pp, 497-506, July 1969. S. Bittanti G. F r o m and G. Guardabassi, “Periodic control: A frequency domain approach,” IEEE Trans. Auromar. Conrr., vol. AC-18, pp. 33-38, Feb. 1973. M. Matsubara, Y. Nisbimura, and N. Takahash, “Optimal periodic confrol of lumped parameter systems,” J . Oprimii. Theory Appl., vol. 13, pp. 13-31, Jan. 1974.

optimization,” J. Oprimiz. Theory Appl., vol. 14, pp. 3149, July 1974. S. Bittanti, A. Locatelli, and C. Maffezmni, “Second-variation methods in periodic

C. Maffezzoni, “Hamilton-Jacobi theory for periodic convol problems,” J . Oprimiz. Theory Appl., vol. 16, pp. 21-29, July 1974. A. Locatelli and S. Rinaldi, “An algorithm for the determinination of the optimal frequency of periodic processes,” Aulomarico, vol. 6, pp. 787-794, Nov. 1970. V. Bertele. G . Guardabassi, and S. Ricci, “Suboptimal periodic control: A describing function approach,” IEEE Trans. Auromal. Conrr., vol. AC-17, pp. 368-370, June 1972.

More A,E+EA,= - D and X = A , X + X A , + D , X ( O ) = C

R. T. LACOSS AND A. F. SHAKAL

Abmuct-The solution of the real matrix equation A , E + EA2 = - D is given in terms of the eigenvalues and eigenvectors of A , E R n I X n 1 , A , E R“2x”2, and the elements of D. The solution is valid if the Ai are symmetric or if they do not have repeated eigenvalaes. The solution can be extended to .hndle nonsymmetric matrices with repeated eigenvalues. The solution of X = A , X + XA2 + D , X(O)= C is also given in closed form for the case of nonrepeated eigenvalues or symmetric Ai.

I. BACKGROUND

A recent paper by Davison [ 11 presented numerical algorithms for the solution of the matrix differential equation

X = A , X + X A , + D , X ( O ) = C (1)

and the related algebraic equation

A I E + E A , = - D (2)

where A I E R ‘1 X n l , A , E R “zx “2, and A , ,A , have eigenvalues in the open left-hand part of the complex plane. He noted that the solution to (1) is

X ( t ) = e A I ’ ( C - E ) e A 2 ‘ + E (3)

where E satisfies (2). He then introduced an approximate expression for

supported by the Advanced Research Projects Ageucy of the Department of Defense.

Lexington. MA 02173.

Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02142.

Manuscript received October 28. 1975; revised January 19, 1976. This work was

R. T. Lacoss is a,ith the Lincoln Laboratory, Massachusetts Institute of Technology,

A. F. Shakal is with the Lincoln Laboratory and the Deparfment of Earfh and