432 IEEE TRANSACTIONS ON ALTONATIC CONTROL JUNE 1976

to the author's knowledge has not been noted before. If B is of rank m( < m), it can be shown that the above result applies with m in (3) replaced by Also, because of duahty, a similar result is valid for the observability of multioutput systems.

REFERENCES

111 W. A. Wolovich, Linear Mulriwriable Sy5rem. New York: Springer, 1974, pp. 8 U l .

Continuous-Time Implementation of Optimal-Aim Controls R. D. BARNARD

Abstract-A modified optimal-aim control strategy for special classes of nonlinear systems is presented, and a theorem leading to an explicit, continuous-time implementation of the strategy is given.

I. IhTRODUCTION

In the present correspondence we consider a special modification of the optimal-aim control strategy discussed in [1]-[3]. This modified strategy applies to nonlinear regulation systems characterized by state equations of the form

X ( r + ) = f ( X ( r ) , r ) + B u ( r ) (1)

and conditions of the form

r E T = h p r l ) , x ( r ) E R"

f(u,,r)=O V t E T

B ' B > 0

where T is a time interval; R is a p-dimensional Euclidean space with norm 1 1 . 1 1 and inner product (., .) ;a is a bounded set of admissible control values containing the zero element; x is a state function of T into R"; u is a control function of T into R'"; j is a nonlinear function of R" X T into R " ; B is an n X rn matrix with transpose B ' ; a, is a stable equilibrium state into which the control is required to drive the system; and j is a component index for elements v = ( v , ) E R p . The original and modified strategies are described geometrically in the state-space dia- grams of Fig. ](a) and (b), respectively. In the former case an optimal- aim control is defined as a piecewise continuous function u such that at each state x ( t ) on the system trajectory r the state-space angle 6, between vectors p(x ( t ) )=o , -x ( r ) and x ( r + ) = f ( x ( r ) , t ) + Bu(r), namely, the angle between the direction of equilibrium state a, and the direction of system movement, is a minimum among the angles 0 between p ( x ( r ) ) and the admissible state-derivative vectors 6 =f(x(t), t ) + Bo determined by admissible control values wee. the system being therefore driven, or aimed. toward a, as directly as control constraints allow. In the latter case an optimal-aim control is defined as a piecewise continuous func- tion u such that the distance do between p(x ( t ) ) and i ( r + ) is a minimum among the distances d between p(x ( t ) ) and 6, the system being therefore aimed perhaps not as directly but almost as effectively. Two important advantages of this modification are 1) that each minimum do along r exists uniquely and. hence, no supplementary optimization criteria are required (cf. [I, sec. 1111). and 2) that the related control law has the explicit, continuous-time implementation shown in the system diagram of Fig. 2 and, hence, no on-line, discrete-time calculations are required (cf. [ I , sec. IV]).

Science Foundanon under Grant Eng 75-13781.

State Unwersity. Detroit. MI 48202.

Manuscnpt received Februar). 9, 1976. This work was supported by the National

The author is with the Department of Electncal and Computer Engineering, Wayne

Fig. I. (a) Optimal-aim angle criterion. (b) Optimal-am distance criterion.

In the sections below we formulate the modified strategy more pre- cisely, establish a theorem justifying the implementation of Fig. 2, and outline a useful and straightforward extension of these notions and results to a more general class of systems.

11. FOFMULATTON AND RESULTS

To formalize the geometric discussion above, we give the following analytic definition.

Definition: A continuous state function x : T+R" and a piecewise continuous control function u : T-R" are termed an optimal-aim pair, or solution. relative to state equation (1) and conditions (2) if for all t E T

where

p ( x ( t ) ) = u , - x ( r )

~ ( ~ ( 1 ) . 1 ) = { 6 € R " ~ 6 = f ( . ~ ( t ) , t ) + B o . w € ~ } .

These conditions are termed collectively the optimal-aim strategy. Regarding the generation and implementation of optimal-aim pairs,

we have the following basic result, the proof of which appears in the Appenmx.

Theorem: A continuous state function x : T+R" and a piecewise continuous control function u : T+R " are an optimal-aim pair iff for all t E T

u(r)=r(DB'(u,-x(rj-f(x(r).rj)-(DB'~-~)u(r)) (4)

TECHKiCAL NOTES AND CORRESPONDENCE 433

Fig. 2. Optimal-aim implementation.

IV. CONCLUSIONS

The implementation established by the theorem of Section I1 repre- sents a major advantage of the modified optimal-aim strategy over the original for the class of systems considered. As in the original case, several important problems remain relatively unexplored. Among these are the determination of existence and controllability criteria and the analytic comparison of optimal-aim and classical optimal-control solu- tions. There is in preparation a follow-up paper giving detailed qualita- tive comparisons of the original and modified strateges for several special examples.

where r is a mapping of R”’ into R m with component values APPEND=

THEOREM PROOF

It is noted first that for any time t and corresponding state x ( t ) , set Cl and, hence, sets B(O) and A ( x ( t ) , t ) = f ( x ( t ) , t ) + B (0) are closed and convex; consequently, there exists a unique element 6 , ~ A ( x ( t ) , t ) for which

Z is the m X m identity matrix, and D is the m X m diagonal matrix given llP(x(t))-6oll G IIP(x(t))- 611 V6 W x ( z ) , t ) . by either

In addition, the fixed-point equation N (to) = w constitutes a stationary- D=diag(2(A,+AI)-’) point condition that follows drectly from applying the Lagrange-

multiplier technique to the minimization of IIp[x(r))-f(x( t ) ,r)- Bull subject to w En. Since matrix B‘B is positive definite and mapping N

in terms of the minimum and maximum eigenvalues h, and X, of B ’ B , satisfies the contraction condition respectively, or

D = ( B ’ B ) - l I I~(w‘ ) -N(w‘ ‘ ) I I=” r (z(~ ’ ) ) - r (z(~ ‘ ‘ ) ) I l

IIr(z’)--(z‘’)ll <[ sup according as B ’ B is nondiagonal or diagonal. Furthermore, for each r ’ , t ” E R m ~ ~ z ’ - z “ ~ \ time t and corresponding state x ( ? ) the mapping N : R m 4 1 with values

= IIz(to‘)-z(w’’)11

is a contraction the unique fixed point of which is u(t)€P, and for any where arbitrary R m the sequence generated by

z ( w ) = D B ‘ ( o , - x ( t ) - f ( x ( t ) , t ) ) - ( D B ’ B - Z ) o

IIDB‘B-Zjl=P<I, ~ r ( w ( ” ) = w ( ~ + l ) , k=o,1,2,. . . (5)

converges to ~ ( t ) with error then there exist unique elements ooEQ and &E Q for which

where P=(AL -&,)(Xl +A,)-’ or /3=0 according as B‘B is nondiagonal or diagonal.

Clearly, since the leedback control loop in the system of Fig. 2 represents an explicit, contiuuous-time implementation of (4), any con- tinuous state function and piecewise continuous control function gener- ated by this system constitute an optimal-aim pair. Also, it is worth noting that iteration (5) can be used for generating approximate optimal- aim pairs by means of on-line, digital computers, an approach which is treated at length in [ I ] and [4].

6 , = f ( x ( t ) , f ) + B ~ o

N ( G ) = & .

Therefore, we observe that optimal-aim conditions hold iff i ( t + ) = 6 , and u(t)=w,, and to show that they hold iff N(u(r ) )= u ( r ) , we need prove only that &=too. For an indirect proof, assume &#a,,, set

? ? = p ( x ( r ) ) - f ( x ( r ) , z ) y = DB’q - ( D B ‘ B - 1);

- w = ( s , ) . a=(.,),

and consider the vectors &e, and 5” E R given by the relations

Inasmuch as the proof of the theorem above involves only conditions of the system at an arbitrary time t and corresponding state x ( t ) , the notions and results of Section I1 extend immediately to a more general class of systems. In particular, control matrix B, constraint set Q with associated limits gi and a,, and mappings r and N in ( 1 x 5 ) can be replaced by time- and state-dependent matrix B ( x ( f ) , t), set Cl(x( t ) , t ) with associated limitsg,(x(t), t ) and sj;(x(t), t). and mappings r ( x ( t ) , f ; .) and N ( x ( f ) , t ; . ) , respectively, characterizing certain forms of adaptive systems; equilibrium state u, can be replaced by an aim state a ( x ( t ) , t ) in R”, i.e., a state toward which the system is to be driven momentarily, characterizing certain forms of tracking systems; and reference vector p(x ( t ) ) can be replaced by vector p ( x ( f ) , r ) = a ( x ( r ) , t ) - x ( t ) . Similar extensions are discussed more fully in [4].

which lead directly to the following conditions: 1) [=5‘-E“ 2) g>o,g.>o 3) g>O*a.=‘i(y)=N.(j)=, J-

4) q>o*,,.=r,(y)=N.(+ J wj - *(w-a, &o

=N&-Lj, 5”)=0.

’ .

434 IEEE TRANSACTIONS ON AbTOMATTC C O ~ O L , mi 1976

a contradction proving that G and wo are equal.

I11

I21

P I

141

REFERENCES

R. D. Barnard “An optimal-aim control strateg for nonlinear regulation systems,” IEEE Trans. RU~OIMI. Conrr., vol. AC-20, pp. 200-208. Apr. 1975.

systems by on-line optimal-aim control,- presented at the 6th Annu. Pittsburgh R. D. Barnard and J. Meisel, “Transient stability augmentation of electric power

Conf. Modeling and Simulation. Pittsburgh, PA, Apr. 1975.

aiming-strateg?; algorithm,” presented at the 5th Power-System Conf., Cambridge, J. Meisel and R. D. Barnard “Transient stabiliry augmentation using a l o c a l i z e d

England, Sept. 1975.

Nerworlcr, vol. 5, pp. 307-329, Oct. 1975. R. D. Barnard, ”Optimal-aim regulation and tracking in large-order networks,” J.

. .