448 IEEE TRAXSACTIONS O N AIJTO&lIATIC CONTROL, AUGUST 1974

ALTERS-ITE PltonLnr STATEMEKT [21 11.. 8. Levine and 31. Athans, “On the determination of the optimal con- stant output-feedback gains for linear multivariable systems.” I E E E

Given the plant matrices [A, 13, C], the weighting matrices [Q,R] L31 II . liwakernaak and R. Bivan, L i n e a r Optimal Control &stems. Kew York: Trans. Auiomat. Contr., rol. -\C-l5. pp. 44-48, Feh. 1970.

and the initial state vector XO, find the constant gain matrix (if it (41 A!. . lJllans and 1:. 8ch\\-eppe. -Gradient matrices and caleula- \\‘iley-Interscience. 1972.

exists) F* such that J(Fj in !14j is minimized, when matrix K saris- tlons. M.I .T . Lincoln Labs.. Lexington, Mass, Tech. Kote 1865-53, KoV. fies (1.5j. [5] \\-. P. Levine. T. L. Johnson. and 11. .\tllans. “Optimal limited state vari-

1;. 1965.

alAe ieedback controllers ior linear systems.” I E E E T r a m . Automat. Contr..

gradient matrix operatioos [4] to the Lngr>Ingian s ( F , h - , L ) : \\-herel [61 u. L..Kk?inman. On the h e a r regulator problem and the matrix riccati vol. .\C-16. pp. iqFi93. Dee. 1971.

et(uatlon.” 1l.I.T. Electronic Systems Lal,.. Cambridge. XIass., Tech. Kept. ESL-K-2il. 1966.

171 T. L. Johnson and hl. Athans, “On the design of optimal constrained dy- namic compensators for linear constant systems.” I E E E Trans. A u i o m a f .

The solution to t l l i i problem (,if i t exists) is found by applying

2(F,Z<,Lj = )trKxoxo’ + ; t r ( [(A - HFC)‘G + A-iA4 - RFC)

+ Q + C’F‘KFCIL’} (16) Contr., vol. .XC-l.5, pp. 655-660. Dec. 19TO.

in which I: is an ‘n X n matrix of Lagrange multiplie1x. The first-order necessary conditions are simply

= 0,and- = 0 (1;) On Suboptimal Control via the SimplSed ae I a d t Model of Davison

where ( * is short for “in which F = F*, K = K*, and L = L*.” Making use of the formulas

S. S. LALIBB AKD S. VITTAL RAO

a a a Abstract-Derivation of suboptimal controllers, via simplified

az az az - tr[-VZ] = A\7‘, - tr[SZ‘] = ;\,-, - tr[.\-ZL] = S‘L’ models, implies a knowledge of the relevant aggregation matrices. This note presents a procedure for the evaluation of the aggregation

and matrix for Davison’s model. It is ‘also shown that the resultant feedback system, when the suboptimal control policy is applied, is

- tr[SZLZ‘] = .V‘ZL‘ + .\‘ZL, a stable. az

we establish the following first-order necesary conditions (for no- t.ationa1 convenience, we have omitted the superscript staM from F , K , and L):

F = R-’B’h-LC’(CLC‘)-’ i 1 S )

( A - BFC)L + L ( A - me)’ + X O X ~ J = o (19)

( A - BFC)’K + K(d - BFCj + Q + C’F’KFC = 0. (20j

Bemarks:

1 ) Equations (18)-(20) are equivalent to the results of Levine and Athans [3]. The approach suggested by Levine and .lthnns [2] for removing the dependency of the feedback gains on xu is to as3-ume that x, is random with E{XOXO’} = I and to minimize EIJ(FII =

+trK. If we adopt this approach: (19) reduces to their ( 2 2 ) . A slight generalization of this appears in Levine c,t 01. [ > I , where it is assunled that E{ X O X ~ ’ ) = S O and E{J(F) ) = $trKSo is minin1ized.

2) Our derivation of the necessary conditions? ilsj, (lo), and (20), requires standard formulas for gradient nlatrices of trace fnnr- tions. The derivation of these results in Levine and ;\thana requires an application of Kleinnlnn’s Lenlnla [6] to J ( F j , and also requires a complicated first-order (in e j integral expansion of esp[l,d - B t F j eAFjCjl]. The basic differences between their derivation and OUM stems from the fact that they use the solution for K i t ) in ( i ) , in which matrix F is imbedded in + ( , t j and a, whereas ne use the dynanlic (or algebraic) equation for K!tj, (8). -L to which approach is the most direct, we leave to the reader to decide.

3) All of the results presented in this note can be extended to 1)

ISTRODTJCTIOK

l>etem~nination of the optimal control policy for linear regulator and tracking problenls for higher order plants represents a computa- tionally difficult and cunlbersonle task. Accordingly, there is a need for obtaining suboptinlal solutions for higher order plants through the use of lower order or simplified modeb. -1 large number of results are available [1]-[3] for obtaining lower

order nlodeb of higher order plants. However, it is not clear 8s to how these nmdels can be ut.ilized for developing suboptimal con- trollers which are actually to be inlplenlented on higher order plants. The difiiculty primarily arkes because of the nonavailability of an unambiguotE relationship between the state vector of the plant. and that of the lower order models. Although Aoki [4] men- tions aboltt such a relationship in the form of an aggregation matrix, methods for evaluating the same with reference to the well known lower order models [1]-[3] are not available.

This technical note attempts to bridge this gap by presenting a technique for developing the aggregation matrix for a lon-er order model derived via 1)avison.s method [ 11. An optimal feedback matrix K derived for the model and the aggregation matrix can then be used to implement a subuptima1 feedback controller for the plant. It is also shown in this note that the resultant feedback system, when sttboptinlal contrul policy is applied, is stable.

ET.ILV.\TIOX OF AGGKI-GATIOX X I T R I X

Consider a continuous time, linear dymmic plant represented by

i = -4.r + BU (1)

the case of control of a time-varying linear system w i t h time-varying x is ,l-vector, ~, is ,rl-vector, and the n,atrices -A aIld are of limited state feedback gains, and, 2 ) the case of control of a constant apprclpriate dimel,aiona~ Fnr the follon.ing preaelltation it is Lwumed (time-varying) linear system with constant itirne-varying) output that the matrix d has distinct eigenvalues with negative real parts. measurements and a dynamic compensator (e.g., the dynamic coni- ;\lthtrugh i t is not explicitly shown, yet the results derived in this pensator derived by Johnson and Athans [i]). note are also applicable for the case when d has repeated eigen-

TECHNICAL NOTES AND CORRESPONDENCE 449

i = F z + G u (2)

where z is r-vector and T < n. Let the simplified s t . a t ~ vector z and the original state vector I be related via an aggregat.ion matrix C whereby

z = cx. (3)

Then the matrices F and G are given by

F = CAC’(CC’)-l and G = CB. (4 )

Let

x = My

where M is the modal matrix of A with its columns arranged from left to right in the order of increasing magnitudes of the correspond- ing eigenvalues. From (1) and ( 5 ) we get

zi = AY + r u (6)

where

A = I W ~ A M , r = M-lB. (7)

hsuning that the first T dominant eigenvalues are to be retained in Davison’s [l] simplified model, let

w = Ty (8 )

where

T = [Zr:OlrXn

and where w is an r-vector of the simplified model in modal form. Thus

G = TAT'^ + m u . (9)

In order t.0 convert the modal representation of the simplified model into a general form we utilize a reduced dimensional version of (j), namely

z = l l f o w. (10)

The t.ransformation mat,rix M O is obtained in t,he following manner. Let the first .r columns of the matrix llf be represented by

If on physical grounds certain specific st.ate variables are to be re- tained in the simplified model, then the matrix $10 is written directly from the above matrix. If for example state variables x1,x4, and ~ - 1 ,

are to be ret.ained in the model, then

An entirely different. approach for determining M O has also been discussed in [ 11. Consequent.ly

i = llfoTAl”Uo-’~ + MoTM-~BzL. (11)

The r-dimensional state vector z is given by

z = Mw = MoTllf-lX.

Thus, the aggregation matrix C is given by

C = dfoTM-1.

Having determined t,he appropriate aggregat.ion matrix as given by (13) it is possible to utilize Davison’s model for determining a sub- optinlal cont.roUer [4] for the linear regulator problem.

STABILITY OF TEE FEEDBaCK SYSTEM

Let the optimal control policy for the simplified model (2) with an equivdent performance criterion as developed in [4] be

u = -Kz. (14)

If this control is applied to the higher order plant (I), t.he result,ant feedback system is given by

j : = ( A - BKC)x. (15)

Substitution of (3), (5), and (13) in (6) yields

= (A - rKclkl)y. (16)

The eigenvalues of canonical system (A - rRCM) are identical to the eigenvalues of t.he syst.em ( A - BKC).

Consider the mat.rix (A - I’KCM) in the following form

where

AI = diag (Xl,X2,. . .,X,) AZ = d i w (X,,,, . . .,X,,) Dl = T o ~ ( T X r)submat.rix of (rKMo) = (TrKNo) Dz = Bottom(n - T ) X T submat,rix of (rKMo).

Thus

Hence, the eigenvalues of (A - rKCM) are given by t.he character- istic equation

det{TZ,, - (A - rRCM)) = 0

or

det(rll, - A1 + Dl) .det( TI,+? - Az} = 0. (17)

But n

det{TZi-, - &} = (7 - Xi) = 0. (18) i = r + l

It can be shown that Lhe eigenvalues of ( F - G K ) and t,he eigen- values of (AI - Dl) are identical. Since the propert.ies of controlla- bility and observability ate invariant under the simplification process of Davison, the eigenvalues of the matrix ( F - G K ) have negat.ive real parts.

We have shown that t,he eigenvalues of ( A - BKC) are the sum of the eigenvalues of the matrix ( F - G K ) and the nondominant eigenvalues of the mat,rix A . Hence, for stable A , the resukant feed- back mat.rix ( A - BKC) is asymptotically stable.

NUMERICAL EXAMPLE

Consider the voltage regulator problem of Sannuti and Kokot.ovic [5] for which the plant dynamics is given by

-0.2 0.5 0 0 0 x1

E] = [ 0 -14.28 85.71 :]kj+ [3!]u

-0.5 1 .6 0

0 0 -25 0 0 0 0 -10

and the performance criterion by

-.-

A second-order model of the plant. is derived by retaining t.he eigenvalues at -0.2 and -0.5 and ignoring ones at -10, -14.28,

450 IEEE TR.LVS.X!TIONS ON CONTROL, AGGUST 1974

and -2.5. If it. is assumed that t,he state variables of the model should have the same physical meaning as t.he first. two state vari- ables of the plant then from (12) we get

1 . 0 0 -0.00412 -0.G2244 -0.:33.x .=[ 0 1.0 0.116 0.406 3.3’2 I.

Aoki’s [4] results, which in this c s e involve the solution of a second- order matrix Riccati equation, yield the sltboptimal rontrol law

usob = -0.9M-l~l - 0.19-l~p - 0.0815~3 - 0.0.56-l~r - 0.294S~s.

The feedback system with the suboptinla1 controller has its ejgen- values a t -4.81 f j4.58, -10, -14.28, and -25 and is as>nlp- t.otically stable.

Optimal control law derived by using original fifth-order plant equat.ion gives

u* = -0.92421 - 0 . 1 5 1 ~ - 0.016r3 - 0.C491; - 0.264~:.

It is of interest t o note that a solution of the above problem by Sannuti and Kokotovic [5], in which they ignored the necessity of an aggregation matrix, yielded an unstable feedback system.

CONCLUSIONS

I t has been shown that simplified models due to llavison are meful in deriving suboptimal control policies provided a relationship be- tween t.he state vectors z and z are known. Availability of such a relationship simplifies the control configuration needed for an actual implementation of the suboptimal controller.

[ l ] E. J. Davison. “.\ method for simplifying linear dynamic systems.” I E E E

[ a ] C . F. Chen and L. S. Shieh, -4 nova1 approach t o h e a r model simulifica- T r a n s . Automat. Cohtr . . rol. A:-11. pp. Y3-101, Jan. 1966.

tion.” Ix t . J . C o n f r . . rol. 8. pp. 561-570, Dee. 1Y68. [ 3 ] J. H . .\nderson. “Geometrical approach t o reduction of dynamic systems.”

I’roc. Ins t . L7er. E n g . . vol. 114, pp. 1014-1018, July 196i. [4] 31. Aoki. “Control of large scale dynamic s>-veterns by aggregation.” I E E E

Trona. da tomat . Contr.. vol. AC-13. pp. 2-16-253. June 1968. (51 P. Sannuti and P. V. I<okotovif. “Near-optimum design of linear systems Iry

singular perturbation method.“ I E E E Trans. A u t o m i . Contr., vol. .lC-l4, pp. 15-22, Feb. 1969.

On Boundary Conditions for Adjoint Variables in Problems with State Variable Inequality Constraints

Absfracf-This correspondence discusses boundary conditions for adjoint variables in problems with state variable inequality con- straints. Particular attention is given to inequality terminal con- straints. It is shown that it is important to identify whether the boundary of the state space is ‘‘absorbing” or not. An example is given to show the importance of these considerations in determining optimal trajectories.

INTRODCCTIOS

An aspect of optimal control problems with stale variable in- equality constraints fsee [4]-[6] for comprehensive bibliographies) that does not appear to be adequately discussed in the literature is the boundary conditions for the adjoint variables. Restricting our attention to problems Kith a fiwt-order state variable inequality constraint (see I?]), we mill shorn that the results of both Bryson, Denham, and l)reyfus [2] and also Ppeyer and Bryson [lo] (see also [.i] for summaries) must be modified. Additionall>-, it is pointed

Manuscript received October 23. 1973: revised March 6, 1974. This work 1x-m supported in part 11y the Office of l iaral Research as part of the Foundation RePearch Program at the Saval Posrgraduate School.

The author is xvitli the Department of Operations Research and .ldministra- tire Sciences, Xaval Postgraduate School, AIonterey-, Calil. Y3Y40.

out that in applying such results one must distinguish beheen an “absorbing” boundary of t.he state space and one which is not. (An “absorbing” boundary is defined here to be a boundary such that once a trajectory touches it, t.he trajectory cannot. leave.) For a problem with an “absorbing” boundary of the stat.e space, if a tra- jectory is on the boundary for a finite interval of time, then t.he state constraint essentially arts like a terminal equality constraint as far as the determination of boundary conditions for the adjoint variahles.

Let us consider the problem

subject to

.i = j ( t , x ,u ) (x is an n-vector of state variables); u ( 0 is unrest.ricted scalar control variable; C(t ,x( t ) ) 2 for all t E [O,T] (scalar inequality constraint. on

+(T,x(T)) 5 0 (scalar inequality ternunal constraint,); +(T,x(T)) = 0 (9-vector of t.ermina1 equality constraints);

state variables);

where we assume that all functions are smooth enough to insure the existence of all partial derivatives required in the folloaing analysis. We further asnme that the first time derivative of C(d,x(t)) explicitly contains u and that (d) , , ( t ,x ,u) # 0 along an optimal trajectory. Br;-son et 01. [2] developed necessary conditions of optimality for the above problenl by considering an equivalent problem in which t.he state variable inequality constraint was replaced by the point con- straint C(fentrl-:x) = F and the control inequality constraint on the boundary (C = 0)

d is to he taken as a right-hand limit (i.e., 6(t+)) in ( 1 ) at a point of discontinuity of z c ( t ) . I

I& us consider (for simplicity and without loss of generality) a trajectory with a constrained suharc on which C(t ,x) = 0 for 0 < fl - < t 5 ti 5 T . Several distinct cases must be considered in t.he de- velopment of (3) below. We must consider the terminal inequality constraint C(T,x (T) ) 5 0 in cases in which an optimal trajectory j u t . touches the boundary at the end (i.e., f l = 2’). Then using Valentine’s approach [14], the augmented criterion functional is given by (set ve(t1) = 0 for f l = T )

t l -

J = G ( T , x ( T ) ) - v ~ ( t ~ ) C ( t ~ , x ) + ( H B - P r x ) dl

where b,, bp are control parameters. In those cases of an “absorbing” state boundary and tl < T, we consider the constraint to be C(T, x ( T ) ) = 0. Otherwise, the constraint. C(T,x(T) ) 5 C i s not used as a point constraint in (2). Using standard arguments [ l ] (sign rest.ric- tions on nlultipliers corresponding to inequality constraints being obtained from second-order Conditions [14]),? one obtains the results of Bryson rt al. [2] with the exception that3

the state houndarv at a point iL.e., knrrr = te,it) with t < T. However. this argu- 1 This interpretation allon% t.he treatment oi a trajectory which only touches

ment fails to apply when a trajectory just touches the boundary at t = T (i.e. C C T . ~ ) = o hut C i t . x ) < o for t < T ) . In this latter case no corner occurs at t =’ T. and we lmvep(T) = lim pit).

See (131 for an application of this approach to finitedimensional optimiza- tion.

terminal houndark- segment of finite length. It extends their results to prohlems The result (3) agrees nith t ,at of Brvson el a!. when C ( T . x ( T ) ) < 0 or 8

with an “ahsorl~iog” state boundary or when a trajector>- jrlst touches the boundary ac t = T. \\-e relate these results GO those of Russak [SI. 191 below. (\\-e have introduced a sign change in his results.)

t -r -