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Cheap Control of the Time-Invariant Regulator ANTONY JAMESON* Courant Institute of Mathematical Sciences New York University and R. E. O'MALLEY, Jr.t University of Arizona Tucson, Arizona Communicated by J. L. Lions ABSTRACT The asymptotic solution of the linear quadratic state regulator problem is obtained as the cost of the control tends to zero. Matrix Riccati gains are obtained via singular perturba- tions theory and are used to asymptotically calculate the optimal control and the cor- responding trajectories. Several cases are distinguished and applications are discussed. 1. Introduction. Consider the infinite time state regulator problem: 2(0 = Ax(t) + Bu(t), t > O, (1.1) x(0) a prescribed n vector, with the scalar cost functional 1 foo [xrQx+E2urRu] dt J(') = 2 o (1.2) to be minimized. Here, the n x n matrix Q is symmetric and positive semi-definite, while the r x r matrix R is symmetric and positive definite. Further, E is a small positive parameter, and A, B, Q, and R are all time-invariant matrices. Such mathematical problems arise in the control literature as pole-positioning problems (cf. Anderson and Moore (1971) and Kwakernaak and Sivan (1972)) and in inverse-regulator problems (cf. Moylan and Anderson (1973)) among other control applications. Moreover, they are an example of "cheap control" (cf. Lions *Work supported by the U.S. Atomic Energy Commission under Contract No. AT(ll-1)- 3077. ~Worksupported by the Office ofNavalResearch under Contract No. N00014-67-A-0209- 0022. 337 APPLIED MATHEMATICS & OPTIMIZATION, VoL I, NO. 4 © 1975 by Springer-Verlag New York Inc.

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338 A. JAMESON AND R. E. O'MALLEY, JR, (1973)) since the cost of the control u in (1.2) is cheap relative to that of the state x (for nontrivial matrices Q). (Those familiar with boundary layers in singular perturbation theory will find the pictures of Kwakernaak and Sivan (Chapter 3) good motivation for our singular perturbations study. Regrettably, they refer to the opposite situation with E large and control of small magnitude as cheap control.) We note that the introduction of cheap control terms in the cost functional has been used to both computational and theoretical advantage in certain singular arc problems (cf. Jacobson, Gershwin, and Lele (1970) and Jacobson and Speyer (1971)). The corresponding finite interval problem with variable coefficient matrices was discussed by O'Malley and Jameson (1975) who obtained the asymptotic solution as E-+0 by integrating the resulting linear system for the state and costate vectors. Here, we shall instead proceed through a Riccati matrix formulation. For any fixed ~ > 0, it is well known (cf., e.g., Anderson and Moore) that the optimal control (under appropriate stabilizability and detectability hypotheses) is given by the feedback control law u(t) = - ~ R-1Brkx (1.3) where the n x n matrix k is a symmetric and positive semidefinite solution of the quadratic matrix (algebraic Riccati) equation J(kA +Ark+ Q) = kBR- 1BTk. (1.4) Further, the corresponding trajectory is then determined by the linear system ez __dx = (e2A_BR_IBTk)x (1.5) dt where the matrix e2A- BR-1Brk is stable. We shall now obtain similar results for the limiting problem where e-+0 and the usual hypotheses are no longer appropriate. Our principal result will be: Theorem. Consider the state regulator problem (1.1) - (1.2) when B has rank r < n and Br QB is positive definite. Let E = I-B(BrQB)-IBrQ and suppoae Hypothesis (i). There is a symmetric positive semi-definite matrix k o such that koAE+ Er Arko + Er QE = koAB(BT QB) - 1BrArko, BTko = 0 where the linearized equation kAff~+ErArk = O, BTk = 0 for E = E-B(Br QB) - 1Br Arko has only the trivial symmetric solution, and

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Cheap Control of the Time-Invariant Regulator 339 Hypothesis (ii). All solutions EZo of d dt ( Zo) = EA (eZo), t_> 0 tend to zero as t approaches infinity. Then, for every positive integer N, the optimal control u, the corresponding trajectory x, and the optimal cost J* will have the uniform asymptotic approximations: = - v , E + U(t, E) E and N k=O Here, the functions oft or T = t/e tend to zero as t or r, respective@, tends to infinity. It will be shown that these peculiar looking hypotheses are equivalent to the familiar stabilizability-detectability assumptions for a parameter-free state regulator problem of state dimension n-r. As for the analogous finite interval problem (cf. O'Malley and Jameson), several cases should be considered separately. The simpler Case 0 where B and O are of full rank is treated in Section 2, while later cases k, k > 2, where Br(Ar)mQAmB = 0 for m = 0, 1 ..... k-2, and Br(/ir)~-IQAk-IB>O will be briefly discussed in Section 5. In Case 0, the limiting solution will be trivial for t>0 and no special hypotheses are necessary. In all cases, the optimal control features initial impulse- like behavior. [We note that it would also be possible in analogous fashion to obtain a Riccati matrix solution to the corresponding finite interval problem. Then, there would generally be a second boundary layer at the terminal time (cf. O'Malley and Jameson and O'Malley and Kung (1974)).] Also observe that Ho (1972) notes the importance of these separate cases and anticipates our main results for singular arc problems. 2. Case 0: B of Rank n, Q Positive-Definite. Since we expect the singular arc condition BTk = 0 to hold when E = 0 and B is of full rank, we set k -- ~K (2.1) in (1.4). This implies the quadratic equation E(KA + ATK) + Q = KBR- ~B~K (2.2)

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340 A. JAMESON AND R. E. O'MALLEY, JR. for K which will have a unique positive-definite, symmetric solution for E suffi- ciently small since Q is positive definite. Specifically, we can readily obtain an asymptotic solution K of (2.2) with the power series expansion K(,) ~ ~ Kf (2.3) S.=0 where K 0 is the positive definite solution of the reduced quadratic equation KoBR-IBrK o = Q and each Ks., j> 0, is uniquely determined successively from the linear equation ./-1 ~:jBR- ~BTXo + KoBR- ~B~XS. = / 1A + AT/ ~ -- E /<,~R- ~B~KS._ ,. i=l Knowing K, the state equation (1.5) implies the linear initial value problem dx E dt = (cA - BR- IBrK)x, x(0) prescribed (2.4) for the feedback trajectories. The asymptotic solution of this problem follows from well-known singular perturbations theory since BR- 1BrK has positive eigenvalues (cf. O'Malley, (1974)). Its outer solution is trivial, as can be easily observed by seeking a power series solution of (2.4). Thus one might expect its unique solution to have the form x(t, ,) = m(z, e) ~ ~ ms.(,), J (2.5) S.=O where each coefficient ms. tends to zero as the stretched variable -r = t/E (2.6) tends to infinity. The boundary layer correction m must then satisfy dm - (eA-BR-~BrK)m, m(0, ,) = x(0) (2.7) dr for ~-_> O, so dmo = _ BR- 1BTKomo, mo(0) = x(0) dr and /ams. \ dz -BR-1BrK°rns.+Ams.-l' Integrating, too(r) = K o 1/2e-A*Kff2x(O) and ms.('r) K~ 1/2 f~ e-A(,-,)K~/2Ams. - l(s)ds, ms.(0) = 0, j= 1,2,.... j= 1,2,... (2.8)

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Cheap Control of the Time-Invariant Regulator 341 where A = K~/ZBR - ~BrKg/z is positive definite. Thus all the mj's decay exponen- tially as ~---->oo, and the solution (2.5) will be asymptotically negligible as E~0 for each t> 0. The corresponding optimal control (by (1.3), (2.1), and (2.5)) will have an asymptotic expansion of the form U(t, E) "~ ~ 1)(7, ,) '~ ]- ~ Vj(~r), j (2.9) ¢ ej=O where each J vj(z) = -R-IB r ~ Ktm~_t(z) /=0 decays exponentially as -r--->oo. Note, in particular, that the optimal control is generally unbounded at t = 0, but it is asymptotically negligible for each fixed t > 0. Further, observe that it behaves roughly like the delta function lim(!e -t/') c'-* O near t = 0. This impulse-like behavior should be expected from singular arc problems (cf. e.g., Bryson and Ho (1969)). Specifically, the limiting optimal control Vo(r)/e behaves asymptotically like the initial impulse of Ho (1972). The cor- responding optimal cost J*(~) is given by "So J*(e) = ~ mr(r, e)(Q + K(~)BR- IBrK(,))m(r, ,)d, (cf. (1.2), (1.3), (2.1), and (2.5)). It has an asymptotic expansion k=l and tends to zero with ~. Thus, the optimal control in Case 0 is cheap, though initially unbounded. We observe that the limiting trajectory moO-), optimal control 1 - Vo0"), and cost eJ~ are all independent of A. We note that the trivial limiting solution (x(t, ,), u(t, ,), J*(0) ~ (0, 0, 0) for t> 0 corresponds to the trivial singular arc solution obtained for the cor- responding finite interval problem (cf. O'Malley and Jameson). With E = 0 in Case 0, we would expect arbitrarily large controls to drive the controllable- observable system to zero arbitrarily fast (cf. Section 3.3 of Kwakernaak and Sivan). This could be heuristically obtained as follows. Let us seek an asymptotic power series solution k(,) ~ E kj J=Oco Ix(t, ,)~ xj(t),J j=O

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342 A. JAMESON AND R. E. O'MALLEY, JR. for the Riccati gain and the corresponding optimal trajectories. Equating zero order coefficients in (1.4) implies koBR-1BTko = O, so k ° = 0. Next, first order coefficients in (1.4) and (1.5) imply that I Q = klBR-1Brkl and I, BR-1BTklXo = O. Picking the positive definite solution k~, we must then have Xo = O. Likewise, the •2 coefficient implies BR-1BTklX1 = O, so X 1 = 0. Finally, the control law (1.3) implies that the limiting optimal control is given by - ER- 1BTklX2 which tends to zero with E. Since the prescribed initial state x(0) will generally be nonzero, we observe that the power series for X cannot represent the asymptotic solution at t = 0, i.e., nonuniform convergence and a boundary layer correction are generally necessary there. Thus, a singular perturbation analysis is required. 3. Case 1: B of Rank r< n, BTQB Nonsingular; the Direct Approach. (a) The Riccati Equation. Let us first seek an asymptotic series solution k(O ~ ~ ky (3.1) j=O of the quadratic equation J(kA +Ark+ Q) = kBR-1BTk. (1.4) When E = O, we have koBR-1BTko = 0 which implies the singular arc condition BTko = 0. (3.2) Because B is of rank r, this fails to determine k o unless n = r. Further, premulti- plying (3.2) by B T and postmultiplying by B, we have ,2(BrkAB+ Br ArkB+ BT QB) = (BTkB)R-I(BrkB). (3.3) Since BTQB> 0 and BTk = 0(0, we have a determination of BTkB = 0(0 which is positive definite for small values of E > 0. Thus, 1 (BTkB)-IR(BTkB) -1 = -fi [BTQB+BTkAB+BTArkB]-I. (3.4) Alternatively, multiplying (1.4) by B r implies BTk = ,2R(BTkB)- aBT(Q + Ark + kA). (3.5)

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Cheap Control of the Time-Invariant Regulator 343 Substituting (3.5) into (1.4) and using (3.4), we finally obtain kA +Ark+ Q = (Q+kA+Ark)B(BrQB+BrkAB+BrArkB)-IBr(Q+Ark+kA). (3.6) We shall now attempt to solve the problem (3.5)-(3.6) for a symmetric, positive semidefinite solution k(~) having the asymptotic series expansion (3.1). We shall proceed by equating coefficients termwise in (3.5) and (3.6). Setting E = 0 in (3.6) implies that k o will satisfy the parameter-free quadratic equation koAE + Er Arko + QE = koAB(Br QB)- 1Br Ark o (3.7) where E = I-B(BrQB) -1BrQ. Note that the projection E satisfies E z = E, BrQE = 0, and EB = 0. (We recall that E played an important role in the corresponding discussion of Jameson (1973).) Note further that ET QE = QE. Hypothesis (i) of our theorem implies that ko is a uniquely determined matrix of rank no greater than n-r such that koE = Erko = k o. (3.8) In the special case that A is nonsingular, it is easy to further show that the quadratic equation for ko implies Brko = O. Noting that the right-hand side of (3.4) can be expanded as (B r QB + BrkAB + Br A rkB)- ~ = (B r QB) - 1 [1 - EBr(klA + Arkl)B(BrQB)- 1 + O(J)], the coefficient of e in (3.6) yields the linear equation ktAff~+~rArkl = ~1 (3.9) where ~1 =- (Q + koA)B( Br QB)- 1BrktA + Ark 1B(B r QB)- I Br(Q + A rko) - (Q + koA)B(BrQB)- 1Br(klA + Arka)B(BrQB)- 1Br(Q + Arko) is symmetric. Here, ~1 is known because (3.4) implies (Brk~B)- 1R(Brk~B)- ~ = (BrQB)- and this can be solved for (BrklB) - 1 while (3.5) yields Brkl = R(BrklB)- IBr(O + Arko). (3.10) The Fredholm alternative and the second part of Hypothesis (i), then, imply that the symmetric matrix kl is also uniquely determined. In fact, we can verify that k 1 = kxB(BrklB) -1Brkl. Proceeding analogously, higher order coefficients in (3.5) and (3.6) imply the linear equations ~+ L~r ATkj = ~j Brkj = flj (3.11)

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344 A. JAMESON AND R. E. O'MALLEY, JR. where %. and flj are successively determined. Thus, the symmetric solution kj can be found and the expansion (3.1) can be uniquely determined term by term. (b) The State Equation. Recall that the optimal trajectories satisfy the initial value problem = (A- 1 BR_lBrk)x ' x(0) prescribed. (3.12) Since the limiting matrix -1BR-1BTkl is singular, the usual stability hypotheses E for singularly perturbed initial value problems does not apply (cf. Hoppensteadt (1971) or O'Malley (1974)). Further, note that x will be specified by Ex and BTQx since x = Ex + B(B r QB)- IB T Qx. (3.13) Further, multiplying (3.12) by Brk and using (1.4), B T Qx = - Br(Arkx + k2). (3.14) Using (3.13), (3.15), and EB = 0, we have ES¢ = FAx = EA {Ex- B(BrQB)- 1Br(Arkx- kX) }. (3.15) (i) The Outer Solution. We shall seek a formal solution X(t, ~) of the state equation (3.15) and the prescribed boundary value EX(o, ,) = Ex(O) (3.16) such that X(t, ~) ~ ~ Xi(t)d (3.17) j=0 Setting E = 0 in (3.15)-(3.16) and using (3.2) and (3.8), we find that d I T 1 T r -~ (EXo) = EA[ -B(B QB)- B A ko]EXo = EA~(EXo) EXo(0) = Ex(O). (3.18) Hypothesis (ii) then implies that EXo is uniquely determined and tends to zero as t approaches infinity. (Note that since E is not of full rank, the hypothesis does not require EAi~ to be stable.) Now setting E = 0 in (3.14) and using (3.2) and (3.8), we find that Br Q Xo = - Br Arko(EXo). (3.19) Thus the decaying vector EXo yields the decaying vector BrQXo and together these two vectors specify Xo. Higher order terms Xg in the expansion (3.17) can be uniquely obtained in

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Cheap Control of the Time-Invariant Regulator 345 succession as solutions of linear problems of the form EXj(O) = 0 (3.20) B QXj = j where gcj and/~j are decaying functions which are known from previous terms. Thus, we can formally determine an asymptotically stable outer solution X(t, ~) of the state equation (1.5). We hasten to note that X(t, ~) cannot represent the actual solution of the initial value problem near t -- 0 since it generally fails to satisfy the initial condition BrQX(O, ~) = BrQx(O). Thus, a boundary layer correction is needed. (ii) The Boundary Layer Correction. We shall find that the solution of the initial value problem for the trajectories will be uniformly represented throughout t > 0 in the form x(t, ,) = X(t, ,)+m(-r, ,) (3.21) where the boundary layer correction m(-r, ~) has the asymptotic expansion m(% ,) ,,~ ~ m j(l"), J (3.22) j=o whose coefficients mj each tend to zero as r tends to infinity. Note that the outer solution X will be an asymptotically valid solution for t> 0, while the boundary layer correction will be important only near t = 0. By linearity, we have __dm = (cA _ _1 BR- 1Brk)m (3.23) d7 and the split initial condition Em(O, ,) = E(x(O)- X(O, ,)) = 0 and (3.24) Br Qm(O, ~) = Br Q(x(O) - X(O, E)). Since EB = O, (3.23) and (3.24) imply I Emo(r = 0 and (3.25) Emj('r) = EA f~ mj_l(s)ds, j> 1. Further, differentiating (3.23) and using the Riccati equation (1.4), we have d2(BT Qm) dT2 -- [(BrQB)R-1Br(kA+Ark+Q) - BrQ(ABR - 1Brk + BR- 1BrkA) + aZBrQAZ]m. (3.26)

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346 A. JAMESON AND R. E. O'MALLEY, JR. Thus, when E = 0, d2(Br Qm°) = (BT QB)R - lBr(ATko + Q)mo. dT 2 However, since Em o = O, mo = B(Br QB)- 1Br Qmo, and koB = 0 implies that komo = 0. Thus, there remains an initial value problem for BrQmo which has the unique decaying solution Br Qmo('C) = Ra/2e-C~R - 1/2Br Q(x(O)- Xo(0)) (3.27) where C = x/R- 1/2BrQBR- 1/2 (3.28) is positive definite. Higher order terms in (3.26) likewise imply d2(Br Qmj) - (BT QB)R - 1Br(Arko + Q)mj + t~j dr 2 where /L 1 is known and exponentially decaying. Note however that we can, by induction, take m j_ a to be exponentially decaying. Then, using (3.25), the expo- nentially decaying vectors kom j = koEm j can be determined, so we have the initial value problem (Br QmJ) = (Br QB)R- I(BT Qmj) +~J (3.29) I, Br Qmj(O) = _ BT Q Xj(O) where/2 i is known and decaying. Thus, the terms BTQmi can be uniquely obtained successively. Moreover, since the Emj's are also determined, the exponentially decaying boundary layer correction coefficients m s of (3.22) have, in principle, become specified. (c) The Optimal Control and the Corresponding Cost. The control law (1.3) and the representation (3.21) for the optimal trajectories imply that the optimal control takes the form u = U(t, ~)+ _1 v(-c, ~) (3.30) E where the outer solution is R-1 U(t, E) = ~2 BrkX( t, E) and the boundary layer correction, R-1 v(r, ,) - Brkm(r, e), ff tends to zero as r tends to infinity. Note, first, that BTko = 0 implies that 1 U(t, ~) = - - R- IBrkl Xo _ R- 1Br(kl X1 + k2 Xo) + O(E).

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Cheap Control of the Time-Invariant Regulator 347 Further (3.10) implies that - R- ~BTkl Xo = - (BrkaB)- l(BrQXo + BTATkoXo). However, (3.19)and the fact that koX o = koEX o imply that -R-1Brk~Xo = O. Thus, U(t, ~) has the asymptotic series expansion U(t, E)~ ~ Ufl.t)~ i (3.31) j=O where j+l Vj(t) ~ -R-1BT Z kl+xXj+l-l" /=o Likewise, near t = 0 the boundary layer correction v/E is such that v(z, E) ~ ~ vj(r)e ~ (3.32) j=O where each J vj('r) = -R-~B r y' kt+lmj_l('r). l=0 I As for Case 0, then, we generally have the optimal control unbounded like - at E t = 0, but bounded for each fixed t > 0 as E-+0. Knowing the representations (3.30) and (3.21) for the optimal control and the corresponding trajectories, the integrand of the cost functional (1.2) is of the form L (t, when optimal, where L~ and L 2 have asymptotic series expansions as E-~O with the terms of L~ and L 2 decaying to zero as t and % respectively, tend to infinity. Integrating, the optimal cost J* has an asymptotic series expansion J*(,) ,-~ ~ Jt*E t (3.33) l=0 as ~-+0 with leading term 1 f~o X~(t)QXo(t)dt Jg=2o being the limiting cost of the outer solution. We note that here the limiting cost is generally nonzero (unlike Case 0), but that it is unaffected by the large initial impulse of the optimal control (i.e., the optimal control is asymptotically as cheap as the outer solution). In the special case where B -a exists (Case 0), however, note that E = 0, so BTQXo = 0 by (3.19), and X o = 0 -= J~'. We note that the limiting solution (Xo, U0, J~') for t > 0 would be obtained if we sought a power series expansion of the Riccati equation (1.4), the state equa- tion (1.5), and the control law (1.3). Since it satisfies BTko = 0, it follows a singular arc.

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348 A. JAMESON AND R. E. O'MALLEY, JR. 4. Case 1: Solution via a Preliminary Change of Variables. We can obtain somewhat more explicit results and interpret our previous hypotheses in more conventional terms if we first make a change of variables to transform B into Iz_l where I is the r x r identity matrix. Specifically, since the n x r matrix B has rank r < n, there are nonsingular matrices M and N such that (cf., e.g., Wilkinson (1965)). The one-to-one change of variables transforms the original problem (1.1)-(1.2) into a new problem of like form with Furthermore, the symmetry and positive definiteness conditions are preserved. Let us then assume that the change of variables (4.1) has already been made and take (A similar change of variables was also made by Ho (1972).) Also partition the corresponding A and Q matrices as positive definite. (In the special case (Case 0) when r = n, we omit the upper blocks.) Partitioning the Riccati matrix k analogously we have (We note that the e's are inserted in (4.5) based on past experience (cf. O'Malley and Jameson). This simplifies the calculations. They could be omitted, but zero terms would later result.) We then substitute into the quadratic matrix equation (1.4) to obtain the equivalent system

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Cheap Control of the Time-Invariant Regulator 349 We shall seek an asymptotic solution of (4.6) such that Kij's have the expansions Kis(~) ~ ~ K~iz ~t. (4.7) /=0 When E = 0, then, we have I KIloAlx +A~IKllo + Q11 = K12oR-1K12o KlloA12+ Q12 = K12oR-IK22o (4.8) Q22 = K22oR-1K22o The last equation has the unique symmetric, positive definite solution K220 --- R1/2(4 R- 1/2 Q22R- 1/2)R1/2 ' (4,9) while the second equation implies that K12o = (KlaoA12+ Q12)K~R. (4.10) Thus there remains the quadratic equation K11oA51+AT1K11o+Qll (Ka loA12 + -1 -1 r r = Q 12)K220RK220( Q 12 + A 12K15 o) which implies that -1 T T -5 T K550(All-A12Q22 Q12)+(All - 012022 A12)Kllo+(O11- a52Q~sa~2) -1 T = K15oAI2Q2 ~ AlzKllo. (4.11) Let us assume Hypothesis (11)'. The equation (4.11) has a unique symmetric solution K 5 lo > 0 such that the matrix S =-- All-A52Q;25QT2-A12Q~21AT2K150 All -l T = -A12K~2oK120 (4.12) is stable. We note that the results of Martensson (1971) or Ku~era (1972) show that this hypothesis is equivalent to the assumption that the pair (A 11 - A 12 Qfil Qr2, A 52) is stabilizable and that the pair (C, A51-A52Q~ 1 Qrx2) is detectable where cTc = Qll-Q12Q~lQT2 . Further, as Ku6era indicates, these conditions are weaker than controllability and observability. Higher order terms Kis t in (4.7) will satisfy the linear equations obtained by equating coefficients of ~t in (4.6). Thus, when l = I, we have T -1 T -1 T [ KllxAll+A11K511-K12oR K125-K521R K12o _ _ T T _ a x = -K~2oAzl-A21K12o )K A K R-IK K R-1K 115 12-- 120 221-- 121 220 (4.13) J T T = f15 =- -K520A22-AalKI20-A21K220 K220R- 5 K225 + K225R- 5K220 = Yl ~ KlroA12 +AlrK520 +K220A22 +A2~K220.

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350 A. JAMESON AND R. E. O'MALLEY, JR. Solving the last equation, Kz21 = R 1/2 fo exp(-R-1/2K2zoR-1/2-t)R-1/ZylR-:/2 (4.14) exp(- R- 1/2 K22oR- 1/2t )dtRa/2 while the sec0nd equation yields Ki21 -1 -1 -1 = KlllA12K220R-(f11+K120R K2zl)Kzzo R. (4.15) There then remains the equation K, II(Al: -1 T T -I T -- A 12K~2o u 120) -[- (A I 1 - El 2oK22o A 12)K111 -1 ,-1 T = al - al-(fl:+KlzoR K221)K220K120 -1 T -- K120K220(fli Jr" K221 R- 1K 1T20 ) which has the unique solution Kill = -- fo eSt•leSrtdt" (4.16) Higher order Kot's can be analogously obtained in turn. Knowing the Riccati gain k asymptotically, we return to the state equation (1.5). Setting x = (X:~, (4.17) \x2/ (1.5) is equivalent to the singularly perturbed system Y¢1 = AllXl+Aa2x2 (4.18) ~)~2 = E(Az lXl -~ A2 zX2) - R- :(Krlzx: + K22X2). It is natural to seek an asymptotic solution to the initial value problem for (4.18) in the form Xl(t ,) = Xl(t , ~)+,ml(z, e) x2(t, E) X2(t, E)+rn2(~', E) (4.19) X:) has an asymptotic expansion in which for t_> 0 where the outer solution Xz E formally satisfies the system (4.18) for t > 0 and the boundary layer correction (Ern: ] / k \mz ! t has a power series expansion in E whose terms tend to zero as ~- =- tends to E infinity. Since the outer solution satisfies (4.18), when E -- 0 we have i , X10 = A11X1o-[-A12X2o - R- i(K~z0Xlo +/(220)(20 ) SO X2o = -Kz2~Kr12oX1o (4.20)

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Cheap Control of the Time-Invariant Regulator 351 and there remains the linear initial value problem z~10 - 1 T = (Alt-AlzK22oK12o)Xlo , Xlo(0) = Xl(0 ) which has the unique solution Xxo(t ) = eS'xx(O) (4.21) which is exponentially decaying. Likewise, higher order terms of the outer expan- sion satisfy linear systems of the form (Xlj = AllXIj+A12X2j (4.22) R- l(KT2oXlj+ K22oX2j ) = ]/j where ~,j is a successively known, decaying term. Thus each Xij can be uniquely determined recursively as an exponentially decaying function, up to selection of the initial value Xli(0) = -m~,j_ 1(0), j> 1. / x By linearity, the boundary layer correction ~Eml] must satisfy the system \m2 ] din1 = EAllml + A12m2 dr (4.23) dm2 __ --- (~A12-R-1K~2)Emx + (EA22-R-1K22)m2 d~" for r > 0. Thus, when c = 0, dmlo dm2o d~" - A12m2°' dr SO -- R-1K22om2o, m20(~) = R-1/2exp[-R-1/2K220R-l/2"r]gl/2m20(O) " (4.24) where m2o(0) = x2(O)-X2o(0). Further, since mxo-+0 as ~----~, we will have mxo(r) = - ff A12mzo(s)ds (4.25) which implies the initial value Xl1(0 ) = -m~o(0), needed to obtain the second order terms of the outer ex mnsion. Analogously, higher order terms satisfy ( drnl i dr - Alzmzj+Aalml'J-I (4.26) dm2~ -- R-1Kzzom2j+3j dr where ml,j-1 and ~j are known successively as exponentially decaying vectors. Integrating, we'll uniquely find m1~ and m2j as exponentially decaying terms. The control relation (1.3) implies that the optimal control u(t, E) has the form u(t, c) = U(t, ~)+ I v('r, E) (4.27) E

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352 A. JAMESON AND R. E. O'MALLEY, JR. where U and v have asymptotic series expansions as ~-+0 whose terms tend to zero as t or ~-, respectively, tend to infinity. Specifically, for t > 0, the optimal control is asymptotically given by R-1 U(t, ,) = - -- (K~2(E)XI(t, ,) + Kzz(~)Xz(t, ~)) (4.28) E while at t = 0, the boundary layer correction is such that v0", ~) = - R- l(Krz(E)ml(r, ~) + K22(~)m2(% ~)). (4.29) In particular, the optimal control is initially unbounded and impluse-like. The expansions (4.27) and (4.19) for the optimal control and the corresponding states imply that the optimal cost J*(~) has an asymptotic expansion J*(,) ~ ~ J~',' (4.30) 1=0 with leading term J° = (Xro(t)QllXlo(t)+2Xro(t)Q12X20(t)+Xr20(t)Qz2X20(t))dt JZ 2 o being the cost of the outer solution. Like Ho (1972), then, we find that the limiting behavior for t>0 is determined by a dynamic system (for Xlo) of order n-r. We note that the preliminary change of variables and the resulting partitioning have eliminated the critical role played by the E matrix in Section 3. The equi- valence of Hypothesis (H)' of this section and Hypotheses (Hi) and (Hii) of the Theorem are easy to establish. Thus, for B, A, Q, k, and x partitioned as in (4.2), (4.3), (4.5), and (4.17), we have Q~IQT 2 while Brko = 0 implies that The Riccati equation for ko in Hypothesis (Hi), then, reduces to the Riccati equation K0t in Hypothesis (H)'. Further, noting that lQr12Zlo and where S is defined in (H)', Hypothesis (Hii) simply requires the stability of S. This guarantees the uniqueness of the solution k of the linear equation of (Hi). 5. Case k, k > 2. It may be of interest to consider further cases, namely Case k, where Br(Ar) m QA"B = 0 for m = 0, 1 ..... k- 2 and Br(Ar) k- a QA k- 1 B is positive

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Cheap Control of the Time-Invariant Regulator 353 definite. Results are analogous to those for the finite interval problem (cf. Jameson (1973) and O'Malley (1973) for preliminary results). In particular, the appropriate expansions are in powers of ~l/k and the correct boundary layer coordinate is now t/~ 1/k. The limiting control near t = 0 behaves like 1 e_t/,vk. Moreover, this large in- E itial impulse lies in the controllability space spanned by B, AB ..... Ak-IB. That the limiting initial behavior is more complicated than in Case 0 was predicted by Ho (1972). An example of a Case 2 situation is provided by the harmonic oscillator problem y+y = u, y(0), )(0) prescribed, J(~) = ~ f o(Y2(t)+~2u2(t))dt. Its asymptotic solution is readily shown to follow the trajectory y(t, E)= 2Re(C(,/~)expl-to/~t]) where Y(0) + ~/1 _~E oJy'(0) c(`/b = ! 1--~ 1 - i / and o~ = e I"/4. Further, the optimal control is asymptotically given by u(t,O=--Im C(`/~)exp-co t E while the optimal cost is asymptotically given by J,(,) = 2,/; ( Ic(,/;)I2 ] References [1] B. D. O. ANDERSON and J. B. MOORE, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, N.J., 1971. [2] A. E. BRYSON, Jr. and Y. C. Ho, Applied Optimal Control, Blaisdell, Waltham, Mass., 1969. [3] Y. C. Ho, Linear stochastic singular control problems, o r. Optimization Theory AppL 9 (1972), pp. 24-31. [4] V. HOPPENSTEADT, Properties of solutions of ordinary differential equations with a small parameter, Comm. Pure AppL Math. 24 (1971), pp. 807-840. [5] D. H. JACOBSON, S. B. GERSHWlN and M. M. LELE, Computation of optimal singular controls, IEEE Trans. Automatic Control 15 (1970), pp. 67-73.

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354 A. JAMESON AND R. E. O'MALLEY, JR. [6] D. H. JACOBSON and J. L. SPEYER, Necessary and sufficient condition for optimality for singular control problems: A limit approach, J. Math. Anal AppL 34 (1971), pp. 239-266. [7] ANTONY JAMESON, The connection between singular perturbations and singular arcs: Part 2: A theory for the linear regulator, Proceedings, Eleventh Annual Allerton Conference on Circuit and System Theory, October 1973, pp. 686-692. [8] V. KU(~ERA, A contribution to matrix quadratic equations, IEEE Trans. Automatic Control 17 (1972), pp. 344-347. [9] H. KWAKERNAAK and R. SIVEN, Linear Optimal Control Systems, Wiley, New York, 1972. [10] J. L. LIONS, Perturbations Singuh~res dans les Probldmes aux Limites et en ContrDle Optimal, Lecture Notes in Mathematics 323, Springer-Verlag, Berlin, 1973. [11] K. MARTENSSON, On thel"natrix Riccati equation, Information ScL 3 (1971), pp. 17-49. [12] P. J. MOYLAN and B. D. O. ANDERSON, Nonlinear regulator theory on an inverse optimal control problem, IEEE Trans. Automatic Control 18 (1973), pp. 460-465. [13] R. E. O'MALLEY, Jr., Examples illustrating the connection between singular perturbations and singular arcs, Proceedings, Eleventh Annual Allerton Conference on Circuit and System Theory, October 1973, pp. 678-685. [14] R. E. O'MALLEY, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974. [15] R. E. O'MALLEY, Jr. and ANTONY JAMESON, Singular perturbations and singular arcs I, IEEE Trans. Automatic 20 (1975). [16] R. E. O'MALLEY, Jr. and C. F. KUNG, On the matrix Riccati approach to a singularly per- turbed regulator problem, o r. Differential Equations 16 (1974). [17] R. R. WILDE and P. V. KOKOTOVI(~, Optimal open and closed loop control of singularly perturbed linear systems, IEEE Trans. Automatic Control 18 (1973), pp. 616-626. [18] J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. [19] B. FRIEDLAND, Limiting forms of optimal stochastic linear regulators, d. Dynamic Systems, Measurement, and Control, Trans. ASME, Series G, 93 (1970, pp. 134-141. [20] H. G. KWATNY, Minimal order observers and certain singular problems of optimal estimation and control, IEEE Trans. Automatic Control 19 (1974), pp. 274-276. [21] P. J. MOYLAN and J. B. MOORE, Generalizations of singular optimal control theory, Auto- matica 7 (1970, pp. 591-598.

E OMALLEY Jrt University of Arizona Tucson Arizona Communicated by J L Lions ABSTRACT The asymptotic solution of the linear quadratic state regulator problem is obtained as the cost of the control tends to zero Matrix Riccati gains are obtained via ID: 22837

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Cheap Control of the Time-Invariant Regulator ANTONY JAMESON* Courant Institute of Mathematical Sciences New York University and R. E. O'MALLEY, Jr.t University of Arizona Tucson, Arizona Communicated by J. L. Lions ABSTRACT The asymptotic solution of the linear quadratic state regulator problem is obtained as the cost of the control tends to zero. Matrix Riccati gains are obtained via singular perturba- tions theory and are used to asymptotically calculate the optimal control and the cor- responding trajectories. Several cases are distinguished and applications are discussed. 1. Introduction. Consider the infinite time state regulator problem: 2(0 = Ax(t) + Bu(t), t > O, (1.1) x(0) a prescribed n vector, with the scalar cost functional 1 foo [xrQx+E2urRu] dt J(') = 2 o (1.2) to be minimized. Here, the n x n matrix Q is symmetric and positive semi-definite, while the r x r matrix R is symmetric and positive definite. Further, E is a small positive parameter, and A, B, Q, and R are all time-invariant matrices. Such mathematical problems arise in the control literature as pole-positioning problems (cf. Anderson and Moore (1971) and Kwakernaak and Sivan (1972)) and in inverse-regulator problems (cf. Moylan and Anderson (1973)) among other control applications. Moreover, they are an example of "cheap control" (cf. Lions *Work supported by the U.S. Atomic Energy Commission under Contract No. AT(ll-1)- 3077. ~Worksupported by the Office ofNavalResearch under Contract No. N00014-67-A-0209- 0022. 337 APPLIED MATHEMATICS & OPTIMIZATION, VoL I, NO. 4 © 1975 by Springer-Verlag New York Inc.

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338 A. JAMESON AND R. E. O'MALLEY, JR, (1973)) since the cost of the control u in (1.2) is cheap relative to that of the state x (for nontrivial matrices Q). (Those familiar with boundary layers in singular perturbation theory will find the pictures of Kwakernaak and Sivan (Chapter 3) good motivation for our singular perturbations study. Regrettably, they refer to the opposite situation with E large and control of small magnitude as cheap control.) We note that the introduction of cheap control terms in the cost functional has been used to both computational and theoretical advantage in certain singular arc problems (cf. Jacobson, Gershwin, and Lele (1970) and Jacobson and Speyer (1971)). The corresponding finite interval problem with variable coefficient matrices was discussed by O'Malley and Jameson (1975) who obtained the asymptotic solution as E-+0 by integrating the resulting linear system for the state and costate vectors. Here, we shall instead proceed through a Riccati matrix formulation. For any fixed ~ > 0, it is well known (cf., e.g., Anderson and Moore) that the optimal control (under appropriate stabilizability and detectability hypotheses) is given by the feedback control law u(t) = - ~ R-1Brkx (1.3) where the n x n matrix k is a symmetric and positive semidefinite solution of the quadratic matrix (algebraic Riccati) equation J(kA +Ark+ Q) = kBR- 1BTk. (1.4) Further, the corresponding trajectory is then determined by the linear system ez __dx = (e2A_BR_IBTk)x (1.5) dt where the matrix e2A- BR-1Brk is stable. We shall now obtain similar results for the limiting problem where e-+0 and the usual hypotheses are no longer appropriate. Our principal result will be: Theorem. Consider the state regulator problem (1.1) - (1.2) when B has rank r < n and Br QB is positive definite. Let E = I-B(BrQB)-IBrQ and suppoae Hypothesis (i). There is a symmetric positive semi-definite matrix k o such that koAE+ Er Arko + Er QE = koAB(BT QB) - 1BrArko, BTko = 0 where the linearized equation kAff~+ErArk = O, BTk = 0 for E = E-B(Br QB) - 1Br Arko has only the trivial symmetric solution, and

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Cheap Control of the Time-Invariant Regulator 339 Hypothesis (ii). All solutions EZo of d dt ( Zo) = EA (eZo), t_> 0 tend to zero as t approaches infinity. Then, for every positive integer N, the optimal control u, the corresponding trajectory x, and the optimal cost J* will have the uniform asymptotic approximations: = - v , E + U(t, E) E and N k=O Here, the functions oft or T = t/e tend to zero as t or r, respective@, tends to infinity. It will be shown that these peculiar looking hypotheses are equivalent to the familiar stabilizability-detectability assumptions for a parameter-free state regulator problem of state dimension n-r. As for the analogous finite interval problem (cf. O'Malley and Jameson), several cases should be considered separately. The simpler Case 0 where B and O are of full rank is treated in Section 2, while later cases k, k > 2, where Br(Ar)mQAmB = 0 for m = 0, 1 ..... k-2, and Br(/ir)~-IQAk-IB>O will be briefly discussed in Section 5. In Case 0, the limiting solution will be trivial for t>0 and no special hypotheses are necessary. In all cases, the optimal control features initial impulse- like behavior. [We note that it would also be possible in analogous fashion to obtain a Riccati matrix solution to the corresponding finite interval problem. Then, there would generally be a second boundary layer at the terminal time (cf. O'Malley and Jameson and O'Malley and Kung (1974)).] Also observe that Ho (1972) notes the importance of these separate cases and anticipates our main results for singular arc problems. 2. Case 0: B of Rank n, Q Positive-Definite. Since we expect the singular arc condition BTk = 0 to hold when E = 0 and B is of full rank, we set k -- ~K (2.1) in (1.4). This implies the quadratic equation E(KA + ATK) + Q = KBR- ~B~K (2.2)

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340 A. JAMESON AND R. E. O'MALLEY, JR. for K which will have a unique positive-definite, symmetric solution for E suffi- ciently small since Q is positive definite. Specifically, we can readily obtain an asymptotic solution K of (2.2) with the power series expansion K(,) ~ ~ Kf (2.3) S.=0 where K 0 is the positive definite solution of the reduced quadratic equation KoBR-IBrK o = Q and each Ks., j> 0, is uniquely determined successively from the linear equation ./-1 ~:jBR- ~BTXo + KoBR- ~B~XS. = / 1A + AT/ ~ -- E /<,~R- ~B~KS._ ,. i=l Knowing K, the state equation (1.5) implies the linear initial value problem dx E dt = (cA - BR- IBrK)x, x(0) prescribed (2.4) for the feedback trajectories. The asymptotic solution of this problem follows from well-known singular perturbations theory since BR- 1BrK has positive eigenvalues (cf. O'Malley, (1974)). Its outer solution is trivial, as can be easily observed by seeking a power series solution of (2.4). Thus one might expect its unique solution to have the form x(t, ,) = m(z, e) ~ ~ ms.(,), J (2.5) S.=O where each coefficient ms. tends to zero as the stretched variable -r = t/E (2.6) tends to infinity. The boundary layer correction m must then satisfy dm - (eA-BR-~BrK)m, m(0, ,) = x(0) (2.7) dr for ~-_> O, so dmo = _ BR- 1BTKomo, mo(0) = x(0) dr and /ams. \ dz -BR-1BrK°rns.+Ams.-l' Integrating, too(r) = K o 1/2e-A*Kff2x(O) and ms.('r) K~ 1/2 f~ e-A(,-,)K~/2Ams. - l(s)ds, ms.(0) = 0, j= 1,2,.... j= 1,2,... (2.8)

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Cheap Control of the Time-Invariant Regulator 341 where A = K~/ZBR - ~BrKg/z is positive definite. Thus all the mj's decay exponen- tially as ~---->oo, and the solution (2.5) will be asymptotically negligible as E~0 for each t> 0. The corresponding optimal control (by (1.3), (2.1), and (2.5)) will have an asymptotic expansion of the form U(t, E) "~ ~ 1)(7, ,) '~ ]- ~ Vj(~r), j (2.9) ¢ ej=O where each J vj(z) = -R-IB r ~ Ktm~_t(z) /=0 decays exponentially as -r--->oo. Note, in particular, that the optimal control is generally unbounded at t = 0, but it is asymptotically negligible for each fixed t > 0. Further, observe that it behaves roughly like the delta function lim(!e -t/') c'-* O near t = 0. This impulse-like behavior should be expected from singular arc problems (cf. e.g., Bryson and Ho (1969)). Specifically, the limiting optimal control Vo(r)/e behaves asymptotically like the initial impulse of Ho (1972). The cor- responding optimal cost J*(~) is given by "So J*(e) = ~ mr(r, e)(Q + K(~)BR- IBrK(,))m(r, ,)d, (cf. (1.2), (1.3), (2.1), and (2.5)). It has an asymptotic expansion k=l and tends to zero with ~. Thus, the optimal control in Case 0 is cheap, though initially unbounded. We observe that the limiting trajectory moO-), optimal control 1 - Vo0"), and cost eJ~ are all independent of A. We note that the trivial limiting solution (x(t, ,), u(t, ,), J*(0) ~ (0, 0, 0) for t> 0 corresponds to the trivial singular arc solution obtained for the cor- responding finite interval problem (cf. O'Malley and Jameson). With E = 0 in Case 0, we would expect arbitrarily large controls to drive the controllable- observable system to zero arbitrarily fast (cf. Section 3.3 of Kwakernaak and Sivan). This could be heuristically obtained as follows. Let us seek an asymptotic power series solution k(,) ~ E kj J=Oco Ix(t, ,)~ xj(t),J j=O

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342 A. JAMESON AND R. E. O'MALLEY, JR. for the Riccati gain and the corresponding optimal trajectories. Equating zero order coefficients in (1.4) implies koBR-1BTko = O, so k ° = 0. Next, first order coefficients in (1.4) and (1.5) imply that I Q = klBR-1Brkl and I, BR-1BTklXo = O. Picking the positive definite solution k~, we must then have Xo = O. Likewise, the •2 coefficient implies BR-1BTklX1 = O, so X 1 = 0. Finally, the control law (1.3) implies that the limiting optimal control is given by - ER- 1BTklX2 which tends to zero with E. Since the prescribed initial state x(0) will generally be nonzero, we observe that the power series for X cannot represent the asymptotic solution at t = 0, i.e., nonuniform convergence and a boundary layer correction are generally necessary there. Thus, a singular perturbation analysis is required. 3. Case 1: B of Rank r< n, BTQB Nonsingular; the Direct Approach. (a) The Riccati Equation. Let us first seek an asymptotic series solution k(O ~ ~ ky (3.1) j=O of the quadratic equation J(kA +Ark+ Q) = kBR-1BTk. (1.4) When E = O, we have koBR-1BTko = 0 which implies the singular arc condition BTko = 0. (3.2) Because B is of rank r, this fails to determine k o unless n = r. Further, premulti- plying (3.2) by B T and postmultiplying by B, we have ,2(BrkAB+ Br ArkB+ BT QB) = (BTkB)R-I(BrkB). (3.3) Since BTQB> 0 and BTk = 0(0, we have a determination of BTkB = 0(0 which is positive definite for small values of E > 0. Thus, 1 (BTkB)-IR(BTkB) -1 = -fi [BTQB+BTkAB+BTArkB]-I. (3.4) Alternatively, multiplying (1.4) by B r implies BTk = ,2R(BTkB)- aBT(Q + Ark + kA). (3.5)

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Cheap Control of the Time-Invariant Regulator 343 Substituting (3.5) into (1.4) and using (3.4), we finally obtain kA +Ark+ Q = (Q+kA+Ark)B(BrQB+BrkAB+BrArkB)-IBr(Q+Ark+kA). (3.6) We shall now attempt to solve the problem (3.5)-(3.6) for a symmetric, positive semidefinite solution k(~) having the asymptotic series expansion (3.1). We shall proceed by equating coefficients termwise in (3.5) and (3.6). Setting E = 0 in (3.6) implies that k o will satisfy the parameter-free quadratic equation koAE + Er Arko + QE = koAB(Br QB)- 1Br Ark o (3.7) where E = I-B(BrQB) -1BrQ. Note that the projection E satisfies E z = E, BrQE = 0, and EB = 0. (We recall that E played an important role in the corresponding discussion of Jameson (1973).) Note further that ET QE = QE. Hypothesis (i) of our theorem implies that ko is a uniquely determined matrix of rank no greater than n-r such that koE = Erko = k o. (3.8) In the special case that A is nonsingular, it is easy to further show that the quadratic equation for ko implies Brko = O. Noting that the right-hand side of (3.4) can be expanded as (B r QB + BrkAB + Br A rkB)- ~ = (B r QB) - 1 [1 - EBr(klA + Arkl)B(BrQB)- 1 + O(J)], the coefficient of e in (3.6) yields the linear equation ktAff~+~rArkl = ~1 (3.9) where ~1 =- (Q + koA)B( Br QB)- 1BrktA + Ark 1B(B r QB)- I Br(Q + A rko) - (Q + koA)B(BrQB)- 1Br(klA + Arka)B(BrQB)- 1Br(Q + Arko) is symmetric. Here, ~1 is known because (3.4) implies (Brk~B)- 1R(Brk~B)- ~ = (BrQB)- and this can be solved for (BrklB) - 1 while (3.5) yields Brkl = R(BrklB)- IBr(O + Arko). (3.10) The Fredholm alternative and the second part of Hypothesis (i), then, imply that the symmetric matrix kl is also uniquely determined. In fact, we can verify that k 1 = kxB(BrklB) -1Brkl. Proceeding analogously, higher order coefficients in (3.5) and (3.6) imply the linear equations ~+ L~r ATkj = ~j Brkj = flj (3.11)

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344 A. JAMESON AND R. E. O'MALLEY, JR. where %. and flj are successively determined. Thus, the symmetric solution kj can be found and the expansion (3.1) can be uniquely determined term by term. (b) The State Equation. Recall that the optimal trajectories satisfy the initial value problem = (A- 1 BR_lBrk)x ' x(0) prescribed. (3.12) Since the limiting matrix -1BR-1BTkl is singular, the usual stability hypotheses E for singularly perturbed initial value problems does not apply (cf. Hoppensteadt (1971) or O'Malley (1974)). Further, note that x will be specified by Ex and BTQx since x = Ex + B(B r QB)- IB T Qx. (3.13) Further, multiplying (3.12) by Brk and using (1.4), B T Qx = - Br(Arkx + k2). (3.14) Using (3.13), (3.15), and EB = 0, we have ES¢ = FAx = EA {Ex- B(BrQB)- 1Br(Arkx- kX) }. (3.15) (i) The Outer Solution. We shall seek a formal solution X(t, ~) of the state equation (3.15) and the prescribed boundary value EX(o, ,) = Ex(O) (3.16) such that X(t, ~) ~ ~ Xi(t)d (3.17) j=0 Setting E = 0 in (3.15)-(3.16) and using (3.2) and (3.8), we find that d I T 1 T r -~ (EXo) = EA[ -B(B QB)- B A ko]EXo = EA~(EXo) EXo(0) = Ex(O). (3.18) Hypothesis (ii) then implies that EXo is uniquely determined and tends to zero as t approaches infinity. (Note that since E is not of full rank, the hypothesis does not require EAi~ to be stable.) Now setting E = 0 in (3.14) and using (3.2) and (3.8), we find that Br Q Xo = - Br Arko(EXo). (3.19) Thus the decaying vector EXo yields the decaying vector BrQXo and together these two vectors specify Xo. Higher order terms Xg in the expansion (3.17) can be uniquely obtained in

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Cheap Control of the Time-Invariant Regulator 345 succession as solutions of linear problems of the form EXj(O) = 0 (3.20) B QXj = j where gcj and/~j are decaying functions which are known from previous terms. Thus, we can formally determine an asymptotically stable outer solution X(t, ~) of the state equation (1.5). We hasten to note that X(t, ~) cannot represent the actual solution of the initial value problem near t -- 0 since it generally fails to satisfy the initial condition BrQX(O, ~) = BrQx(O). Thus, a boundary layer correction is needed. (ii) The Boundary Layer Correction. We shall find that the solution of the initial value problem for the trajectories will be uniformly represented throughout t > 0 in the form x(t, ,) = X(t, ,)+m(-r, ,) (3.21) where the boundary layer correction m(-r, ~) has the asymptotic expansion m(% ,) ,,~ ~ m j(l"), J (3.22) j=o whose coefficients mj each tend to zero as r tends to infinity. Note that the outer solution X will be an asymptotically valid solution for t> 0, while the boundary layer correction will be important only near t = 0. By linearity, we have __dm = (cA _ _1 BR- 1Brk)m (3.23) d7 and the split initial condition Em(O, ,) = E(x(O)- X(O, ,)) = 0 and (3.24) Br Qm(O, ~) = Br Q(x(O) - X(O, E)). Since EB = O, (3.23) and (3.24) imply I Emo(r = 0 and (3.25) Emj('r) = EA f~ mj_l(s)ds, j> 1. Further, differentiating (3.23) and using the Riccati equation (1.4), we have d2(BT Qm) dT2 -- [(BrQB)R-1Br(kA+Ark+Q) - BrQ(ABR - 1Brk + BR- 1BrkA) + aZBrQAZ]m. (3.26)

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346 A. JAMESON AND R. E. O'MALLEY, JR. Thus, when E = 0, d2(Br Qm°) = (BT QB)R - lBr(ATko + Q)mo. dT 2 However, since Em o = O, mo = B(Br QB)- 1Br Qmo, and koB = 0 implies that komo = 0. Thus, there remains an initial value problem for BrQmo which has the unique decaying solution Br Qmo('C) = Ra/2e-C~R - 1/2Br Q(x(O)- Xo(0)) (3.27) where C = x/R- 1/2BrQBR- 1/2 (3.28) is positive definite. Higher order terms in (3.26) likewise imply d2(Br Qmj) - (BT QB)R - 1Br(Arko + Q)mj + t~j dr 2 where /L 1 is known and exponentially decaying. Note however that we can, by induction, take m j_ a to be exponentially decaying. Then, using (3.25), the expo- nentially decaying vectors kom j = koEm j can be determined, so we have the initial value problem (Br QmJ) = (Br QB)R- I(BT Qmj) +~J (3.29) I, Br Qmj(O) = _ BT Q Xj(O) where/2 i is known and decaying. Thus, the terms BTQmi can be uniquely obtained successively. Moreover, since the Emj's are also determined, the exponentially decaying boundary layer correction coefficients m s of (3.22) have, in principle, become specified. (c) The Optimal Control and the Corresponding Cost. The control law (1.3) and the representation (3.21) for the optimal trajectories imply that the optimal control takes the form u = U(t, ~)+ _1 v(-c, ~) (3.30) E where the outer solution is R-1 U(t, E) = ~2 BrkX( t, E) and the boundary layer correction, R-1 v(r, ,) - Brkm(r, e), ff tends to zero as r tends to infinity. Note, first, that BTko = 0 implies that 1 U(t, ~) = - - R- IBrkl Xo _ R- 1Br(kl X1 + k2 Xo) + O(E).

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Cheap Control of the Time-Invariant Regulator 347 Further (3.10) implies that - R- ~BTkl Xo = - (BrkaB)- l(BrQXo + BTATkoXo). However, (3.19)and the fact that koX o = koEX o imply that -R-1Brk~Xo = O. Thus, U(t, ~) has the asymptotic series expansion U(t, E)~ ~ Ufl.t)~ i (3.31) j=O where j+l Vj(t) ~ -R-1BT Z kl+xXj+l-l" /=o Likewise, near t = 0 the boundary layer correction v/E is such that v(z, E) ~ ~ vj(r)e ~ (3.32) j=O where each J vj('r) = -R-~B r y' kt+lmj_l('r). l=0 I As for Case 0, then, we generally have the optimal control unbounded like - at E t = 0, but bounded for each fixed t > 0 as E-+0. Knowing the representations (3.30) and (3.21) for the optimal control and the corresponding trajectories, the integrand of the cost functional (1.2) is of the form L (t, when optimal, where L~ and L 2 have asymptotic series expansions as E-~O with the terms of L~ and L 2 decaying to zero as t and % respectively, tend to infinity. Integrating, the optimal cost J* has an asymptotic series expansion J*(,) ,-~ ~ Jt*E t (3.33) l=0 as ~-+0 with leading term 1 f~o X~(t)QXo(t)dt Jg=2o being the limiting cost of the outer solution. We note that here the limiting cost is generally nonzero (unlike Case 0), but that it is unaffected by the large initial impulse of the optimal control (i.e., the optimal control is asymptotically as cheap as the outer solution). In the special case where B -a exists (Case 0), however, note that E = 0, so BTQXo = 0 by (3.19), and X o = 0 -= J~'. We note that the limiting solution (Xo, U0, J~') for t > 0 would be obtained if we sought a power series expansion of the Riccati equation (1.4), the state equa- tion (1.5), and the control law (1.3). Since it satisfies BTko = 0, it follows a singular arc.

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348 A. JAMESON AND R. E. O'MALLEY, JR. 4. Case 1: Solution via a Preliminary Change of Variables. We can obtain somewhat more explicit results and interpret our previous hypotheses in more conventional terms if we first make a change of variables to transform B into Iz_l where I is the r x r identity matrix. Specifically, since the n x r matrix B has rank r < n, there are nonsingular matrices M and N such that (cf., e.g., Wilkinson (1965)). The one-to-one change of variables transforms the original problem (1.1)-(1.2) into a new problem of like form with Furthermore, the symmetry and positive definiteness conditions are preserved. Let us then assume that the change of variables (4.1) has already been made and take (A similar change of variables was also made by Ho (1972).) Also partition the corresponding A and Q matrices as positive definite. (In the special case (Case 0) when r = n, we omit the upper blocks.) Partitioning the Riccati matrix k analogously we have (We note that the e's are inserted in (4.5) based on past experience (cf. O'Malley and Jameson). This simplifies the calculations. They could be omitted, but zero terms would later result.) We then substitute into the quadratic matrix equation (1.4) to obtain the equivalent system

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Cheap Control of the Time-Invariant Regulator 349 We shall seek an asymptotic solution of (4.6) such that Kij's have the expansions Kis(~) ~ ~ K~iz ~t. (4.7) /=0 When E = 0, then, we have I KIloAlx +A~IKllo + Q11 = K12oR-1K12o KlloA12+ Q12 = K12oR-IK22o (4.8) Q22 = K22oR-1K22o The last equation has the unique symmetric, positive definite solution K220 --- R1/2(4 R- 1/2 Q22R- 1/2)R1/2 ' (4,9) while the second equation implies that K12o = (KlaoA12+ Q12)K~R. (4.10) Thus there remains the quadratic equation K11oA51+AT1K11o+Qll (Ka loA12 + -1 -1 r r = Q 12)K220RK220( Q 12 + A 12K15 o) which implies that -1 T T -5 T K550(All-A12Q22 Q12)+(All - 012022 A12)Kllo+(O11- a52Q~sa~2) -1 T = K15oAI2Q2 ~ AlzKllo. (4.11) Let us assume Hypothesis (11)'. The equation (4.11) has a unique symmetric solution K 5 lo > 0 such that the matrix S =-- All-A52Q;25QT2-A12Q~21AT2K150 All -l T = -A12K~2oK120 (4.12) is stable. We note that the results of Martensson (1971) or Ku~era (1972) show that this hypothesis is equivalent to the assumption that the pair (A 11 - A 12 Qfil Qr2, A 52) is stabilizable and that the pair (C, A51-A52Q~ 1 Qrx2) is detectable where cTc = Qll-Q12Q~lQT2 . Further, as Ku6era indicates, these conditions are weaker than controllability and observability. Higher order terms Kis t in (4.7) will satisfy the linear equations obtained by equating coefficients of ~t in (4.6). Thus, when l = I, we have T -1 T -1 T [ KllxAll+A11K511-K12oR K125-K521R K12o _ _ T T _ a x = -K~2oAzl-A21K12o )K A K R-IK K R-1K 115 12-- 120 221-- 121 220 (4.13) J T T = f15 =- -K520A22-AalKI20-A21K220 K220R- 5 K225 + K225R- 5K220 = Yl ~ KlroA12 +AlrK520 +K220A22 +A2~K220.

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350 A. JAMESON AND R. E. O'MALLEY, JR. Solving the last equation, Kz21 = R 1/2 fo exp(-R-1/2K2zoR-1/2-t)R-1/ZylR-:/2 (4.14) exp(- R- 1/2 K22oR- 1/2t )dtRa/2 while the sec0nd equation yields Ki21 -1 -1 -1 = KlllA12K220R-(f11+K120R K2zl)Kzzo R. (4.15) There then remains the equation K, II(Al: -1 T T -I T -- A 12K~2o u 120) -[- (A I 1 - El 2oK22o A 12)K111 -1 ,-1 T = al - al-(fl:+KlzoR K221)K220K120 -1 T -- K120K220(fli Jr" K221 R- 1K 1T20 ) which has the unique solution Kill = -- fo eSt•leSrtdt" (4.16) Higher order Kot's can be analogously obtained in turn. Knowing the Riccati gain k asymptotically, we return to the state equation (1.5). Setting x = (X:~, (4.17) \x2/ (1.5) is equivalent to the singularly perturbed system Y¢1 = AllXl+Aa2x2 (4.18) ~)~2 = E(Az lXl -~ A2 zX2) - R- :(Krlzx: + K22X2). It is natural to seek an asymptotic solution to the initial value problem for (4.18) in the form Xl(t ,) = Xl(t , ~)+,ml(z, e) x2(t, E) X2(t, E)+rn2(~', E) (4.19) X:) has an asymptotic expansion in which for t_> 0 where the outer solution Xz E formally satisfies the system (4.18) for t > 0 and the boundary layer correction (Ern: ] / k \mz ! t has a power series expansion in E whose terms tend to zero as ~- =- tends to E infinity. Since the outer solution satisfies (4.18), when E -- 0 we have i , X10 = A11X1o-[-A12X2o - R- i(K~z0Xlo +/(220)(20 ) SO X2o = -Kz2~Kr12oX1o (4.20)

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Cheap Control of the Time-Invariant Regulator 351 and there remains the linear initial value problem z~10 - 1 T = (Alt-AlzK22oK12o)Xlo , Xlo(0) = Xl(0 ) which has the unique solution Xxo(t ) = eS'xx(O) (4.21) which is exponentially decaying. Likewise, higher order terms of the outer expan- sion satisfy linear systems of the form (Xlj = AllXIj+A12X2j (4.22) R- l(KT2oXlj+ K22oX2j ) = ]/j where ~,j is a successively known, decaying term. Thus each Xij can be uniquely determined recursively as an exponentially decaying function, up to selection of the initial value Xli(0) = -m~,j_ 1(0), j> 1. / x By linearity, the boundary layer correction ~Eml] must satisfy the system \m2 ] din1 = EAllml + A12m2 dr (4.23) dm2 __ --- (~A12-R-1K~2)Emx + (EA22-R-1K22)m2 d~" for r > 0. Thus, when c = 0, dmlo dm2o d~" - A12m2°' dr SO -- R-1K22om2o, m20(~) = R-1/2exp[-R-1/2K220R-l/2"r]gl/2m20(O) " (4.24) where m2o(0) = x2(O)-X2o(0). Further, since mxo-+0 as ~----~, we will have mxo(r) = - ff A12mzo(s)ds (4.25) which implies the initial value Xl1(0 ) = -m~o(0), needed to obtain the second order terms of the outer ex mnsion. Analogously, higher order terms satisfy ( drnl i dr - Alzmzj+Aalml'J-I (4.26) dm2~ -- R-1Kzzom2j+3j dr where ml,j-1 and ~j are known successively as exponentially decaying vectors. Integrating, we'll uniquely find m1~ and m2j as exponentially decaying terms. The control relation (1.3) implies that the optimal control u(t, E) has the form u(t, c) = U(t, ~)+ I v('r, E) (4.27) E

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352 A. JAMESON AND R. E. O'MALLEY, JR. where U and v have asymptotic series expansions as ~-+0 whose terms tend to zero as t or ~-, respectively, tend to infinity. Specifically, for t > 0, the optimal control is asymptotically given by R-1 U(t, ,) = - -- (K~2(E)XI(t, ,) + Kzz(~)Xz(t, ~)) (4.28) E while at t = 0, the boundary layer correction is such that v0", ~) = - R- l(Krz(E)ml(r, ~) + K22(~)m2(% ~)). (4.29) In particular, the optimal control is initially unbounded and impluse-like. The expansions (4.27) and (4.19) for the optimal control and the corresponding states imply that the optimal cost J*(~) has an asymptotic expansion J*(,) ~ ~ J~',' (4.30) 1=0 with leading term J° = (Xro(t)QllXlo(t)+2Xro(t)Q12X20(t)+Xr20(t)Qz2X20(t))dt JZ 2 o being the cost of the outer solution. Like Ho (1972), then, we find that the limiting behavior for t>0 is determined by a dynamic system (for Xlo) of order n-r. We note that the preliminary change of variables and the resulting partitioning have eliminated the critical role played by the E matrix in Section 3. The equi- valence of Hypothesis (H)' of this section and Hypotheses (Hi) and (Hii) of the Theorem are easy to establish. Thus, for B, A, Q, k, and x partitioned as in (4.2), (4.3), (4.5), and (4.17), we have Q~IQT 2 while Brko = 0 implies that The Riccati equation for ko in Hypothesis (Hi), then, reduces to the Riccati equation K0t in Hypothesis (H)'. Further, noting that lQr12Zlo and where S is defined in (H)', Hypothesis (Hii) simply requires the stability of S. This guarantees the uniqueness of the solution k of the linear equation of (Hi). 5. Case k, k > 2. It may be of interest to consider further cases, namely Case k, where Br(Ar) m QA"B = 0 for m = 0, 1 ..... k- 2 and Br(Ar) k- a QA k- 1 B is positive

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Cheap Control of the Time-Invariant Regulator 353 definite. Results are analogous to those for the finite interval problem (cf. Jameson (1973) and O'Malley (1973) for preliminary results). In particular, the appropriate expansions are in powers of ~l/k and the correct boundary layer coordinate is now t/~ 1/k. The limiting control near t = 0 behaves like 1 e_t/,vk. Moreover, this large in- E itial impulse lies in the controllability space spanned by B, AB ..... Ak-IB. That the limiting initial behavior is more complicated than in Case 0 was predicted by Ho (1972). An example of a Case 2 situation is provided by the harmonic oscillator problem y+y = u, y(0), )(0) prescribed, J(~) = ~ f o(Y2(t)+~2u2(t))dt. Its asymptotic solution is readily shown to follow the trajectory y(t, E)= 2Re(C(,/~)expl-to/~t]) where Y(0) + ~/1 _~E oJy'(0) c(`/b = ! 1--~ 1 - i / and o~ = e I"/4. Further, the optimal control is asymptotically given by u(t,O=--Im C(`/~)exp-co t E while the optimal cost is asymptotically given by J,(,) = 2,/; ( Ic(,/;)I2 ] References [1] B. D. O. ANDERSON and J. B. MOORE, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, N.J., 1971. [2] A. E. BRYSON, Jr. and Y. C. Ho, Applied Optimal Control, Blaisdell, Waltham, Mass., 1969. [3] Y. C. Ho, Linear stochastic singular control problems, o r. Optimization Theory AppL 9 (1972), pp. 24-31. [4] V. HOPPENSTEADT, Properties of solutions of ordinary differential equations with a small parameter, Comm. Pure AppL Math. 24 (1971), pp. 807-840. [5] D. H. JACOBSON, S. B. GERSHWlN and M. M. LELE, Computation of optimal singular controls, IEEE Trans. Automatic Control 15 (1970), pp. 67-73.

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354 A. JAMESON AND R. E. O'MALLEY, JR. [6] D. H. JACOBSON and J. L. SPEYER, Necessary and sufficient condition for optimality for singular control problems: A limit approach, J. Math. Anal AppL 34 (1971), pp. 239-266. [7] ANTONY JAMESON, The connection between singular perturbations and singular arcs: Part 2: A theory for the linear regulator, Proceedings, Eleventh Annual Allerton Conference on Circuit and System Theory, October 1973, pp. 686-692. [8] V. KU(~ERA, A contribution to matrix quadratic equations, IEEE Trans. Automatic Control 17 (1972), pp. 344-347. [9] H. KWAKERNAAK and R. SIVEN, Linear Optimal Control Systems, Wiley, New York, 1972. [10] J. L. LIONS, Perturbations Singuh~res dans les Probldmes aux Limites et en ContrDle Optimal, Lecture Notes in Mathematics 323, Springer-Verlag, Berlin, 1973. [11] K. MARTENSSON, On thel"natrix Riccati equation, Information ScL 3 (1971), pp. 17-49. [12] P. J. MOYLAN and B. D. O. ANDERSON, Nonlinear regulator theory on an inverse optimal control problem, IEEE Trans. Automatic Control 18 (1973), pp. 460-465. [13] R. E. O'MALLEY, Jr., Examples illustrating the connection between singular perturbations and singular arcs, Proceedings, Eleventh Annual Allerton Conference on Circuit and System Theory, October 1973, pp. 678-685. [14] R. E. O'MALLEY, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974. [15] R. E. O'MALLEY, Jr. and ANTONY JAMESON, Singular perturbations and singular arcs I, IEEE Trans. Automatic 20 (1975). [16] R. E. O'MALLEY, Jr. and C. F. KUNG, On the matrix Riccati approach to a singularly per- turbed regulator problem, o r. Differential Equations 16 (1974). [17] R. R. WILDE and P. V. KOKOTOVI(~, Optimal open and closed loop control of singularly perturbed linear systems, IEEE Trans. Automatic Control 18 (1973), pp. 616-626. [18] J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. [19] B. FRIEDLAND, Limiting forms of optimal stochastic linear regulators, d. Dynamic Systems, Measurement, and Control, Trans. ASME, Series G, 93 (1970, pp. 134-141. [20] H. G. KWATNY, Minimal order observers and certain singular problems of optimal estimation and control, IEEE Trans. Automatic Control 19 (1974), pp. 274-276. [21] P. J. MOYLAN and J. B. MOORE, Generalizations of singular optimal control theory, Auto- matica 7 (1970, pp. 591-598.

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