Singular perturbations and singular arcs--Part I - IEEE Xplore

218

IEEE TRANSACTIONS ON AUTOMATIC CONIXOL,

VOL. AC-20,

NO.

2, APRIL 1975

Singular Perturbations and Singular Arcs-Part I ROBERT E. OMALLEY. JR..

A b s t m c t - S i a r pattubation theory is applied to obtain the asymp totic solution for the nearly singnlar optimal control of a constant linear systemonafinitetimeinterp-alInthelimitastheeontrolcostisredoced to z ero, the initial control is fonnd to have an impulse-like behavior, while the outer solution agrees asymptotically with the familiar solution for a singnlar an. The detailed structure of tbe m ip & is provided by the asymptotic solution.

ONSIDER thelinear,time-invariantstate regulator problem consisting of the differential equation x (Bt )u=(At )x ( t ) +

son, Gershwin, and Lele [SI) have introduced a cheap control cost to advantage for both theoretical and computational study of singular arc problems. To obtain necessary conditions for optimality, we introduce the Hamiltonian

where the n-dimensional costate vector p ( t ) satisfies the terminal value problem

. aH p=--=-Qx-A?,

(1.5)

Along an optimal path, we must have

_ aH -c2Ru+

au

(1-2)

and the scalar cost functional

p(l)=O.

ax

(1.1)

for the n-dimensional state vector x on 0 Q t < 1 (or any other closed, bounded interval); the initial state x ( 0 ) being prescribed:

ANTONY JAMESON

H ( x , p , u , ~ ) = ~ ( x ~ Q X + ~ * u ' R u ) + p ~ ( A x + (1.4) Bu)

I. INTRODUCTION

C

AND

B?=O,

so we can solve for the optimal control

J ( E ) = f ~ ' [ ~ ~ ( t ) Q x ( t ) + ~ ~ ' ( t ) R (1.3) u(~)]~t

u=

-LR-'B%. E2

to be minimized by selection of the r-dimensional control vector u. Here the superscript T denotes transposition, Q and R are symmetric matrices, Q is positive semidefinite, R is positive definite, and E is a small positive parameter. For each fixed E >0, a unique optimal solution can be readily obtained through the classical calculus of variations (cf.,e.g., Anderson and Moore [l]) while singular arcs occur when E = 0 (cf., e.g., Bryson and Ho [2] and Ho [3]) and impulse controls are needed to get on and off the singular arcs. By obtaining the asymptotic solution of the problem as E + O (through the use of singular perturbations theory), we shall find how this comes about. We note that substantial technical difficulties would result if one tried to asymptotically evaluate the usual solution (valid for each fixed c > 0) in the limit E+O. Such problems are important in studying such singular arcs, and also arise in analyzing the limiting possibilities for regulators (cf.. Kwakernaak and Sivan [4] and Friedland [5]), the inverse problem (Anderson and Moore [l]), and "cheap control" (Lions [6]). We observe that Jacobson and others (cf., Jacobson and Speyer [7] and Jacob-

Substituting for u in (1.1) then implies the two-point problem

~'i = E ~ A-XBR - 'B 'p, d = - Q x - A T P>

x ( 0 ) prescribed

p(l)=O

( 1-71

[cf., also (1.2) and (lS)]. Note that this system is singularly perturbed since its differential order reduces from 2n when E > O to n when E = O . The asymptotic theory of singularly perturbed linear boundary value problems is fairly well established (cf., e.g., Wasow [9], O'Malley [lo], [l 11 and Harris [12]). As c+O the limiting solution to such a problem converges (under appropriate hypotheses) to an outer solution within 0 < 1 < 1, while regions of nonuniformconvergence (boundary layers) occur near one or both endpoints. The limiting outer solution will satisfy the system obtained when E = 0.In the closed interval 0 < t Q 1, the asymptotic solution will be expressed as the sum of the outer solution plus boundary layer corrections at t = 0 and t = 1. These boundary layer corrections dependon some stretched variable (like r = r / E a at t = O and a=(l-r)/Ea at t = l , Manuscript received October 26, 1973. Paper recommended by D. L. for some LY >0) and tend to zero as the appropriate Kleinman,Chairman of the IEEE S-CS Optimal Systems Committee. variable tends to infinity. Here, the singular arc will be The work of R. E. O'Malley, Jr. was supported by the Office of Naval ResearchunderContract N00014-67-A-0209-0022. The work of A. provided by the reduced problem (limiting outer solution) Jameson was supported by the U. S . Atomic Energy Commission under of singular perturbations theory, while the impulse at t = O Contract AT(11-1)-3077. R. E. OMalley, Jr. is with the Department of Mathematics, University will be given by the limiting initial boundary layer correcof Arizona. Tucson, Ariz. A. Jameson is with the Courant Institute of Mathematical Sciences, tion. Our detailed singular perturbation results should be New York University, New York. N. Y. 10012. useful in indicating the nature of impulsive control in this

219

O ' M W E Y AND JAMESON: SINGULAR PERTURBATIONS AND SINGULAR ARCS

and other more general situations. Indeed, this (rather than the asymptotic calculation of the singular arc) is our primary objective. In practical applications, however, one should realize that higher order asymptotic .approximations are frequently not too important. The authors give formulas for the general terms of the asymptotic solution because this has been accomplished with .so little additional effort. The asymptotic solution of the two-point problem (1.7) will be found in two cases, namely; Case 0: B of rank n, Q positive definite; and Case I: B of rank r < n, BTQB positive definite. (The problem where r = n is common to Cases 0 and 1.)

The distinction betweenthesetwocaseswaspreviously noted by Ho [3]. Elsewhere we shall discuss the following. Case k:

m 0), the asymptotic solution is

~j

for

x(t,e)-e-l/ex(0)

1

-

u(t,+--

Ee

-I/€

x (0)

(2.2)

for 0 < t Q 1. Thus, the solution features nonuniform convergence at t = O as e+O o n e the function e - t / ' ) unless x(0) = 0. Away from t = 0, the optimal control and the resulting trajectory are asymptotically neghgible, i.e., the outer solution and the boundary layer correction at t= 1 are both trivial. At t =0, the optimal control is, however, . corresponding limiting cost unbounded like l / ~ The

B T ( A T ) ~ - ' Q A ~ >o. -~B

We shall not examine intermediate possibilities. In the cases studied, the limiting solution within (0,l) follows a singular arc while the boundary layer behavior is impulseldce (specifically, the optimal control is like (1 /E)e-'/'m, a > O , at t=O). Before proceeding, we note that many other regulator problems could be solved in a similar manner. In particular, problem (1. I)-( 1.4) could be modified to allow certain terminal costs, time-varying coefficients, and .fixed terminal states. The infinite interval problem and some quasilinear problems are alsosolvable, in addition to some examples with bounded control (cf., O'Malley [13]). Related control problems solved through singular perturbationtheory are discussed in American Society of [14], Collins [ 1 5 ] , Sannutiand MechanicalEngineers Reddy [ 161, and Wilde and KokotoviC { 171. 11. Two EWLES Example I : Consider the scalar problem i(t)=u(t),

OQ tQ 1

however, tends to zero as c+O. We note that the limit of

as c-+O behaves like a del? function 8(t). That is, for any differentiable function f(t).

=f(O) +O(€). Thus, we seem to have u ( t ) - - 8(r)x(O), which is in agreement with Ho [3]. Example 2: Now consider the following problem for control of a harmonic oscillator: jj+y=u

y(O), j (0) prescribed with

x ( 0 ) prescribed with

The linear system corresponding to (1.7) is c 2 i = - px, ( 0 )

p=

prescribed

- xp ,( l ) = O

with u = - p / c 2 . Solving, we obtain the trajectory

is, 1

.I(€)=

(y(t)+€V(t))dt.

Introducing x 1= y and x 2 = y , we put the problem in the form (1.1H1.3) and obtain necessary conditions for an optimum as follows [cf., (1.7)]:

220

IEEE TRANSACTIONS

ON AUTOMATIC CONTROL, APRIL

1975

in the stretched variable o = ( l - t ) / c . Further, the limiting outer solution

The general solution of this linear system is

x,(t.c~=~yk~e'~-~~pk~e-~-Xk,e-~'~~+c~~e'~~'"'~~) satisfies the system (2.4) with E =O and the optimal control

111.

where

CASE

0: B

OF RANK

n,

e POSITIVE DEFINITE

This is the simplest case that can be studied. It is quite unusual in practice, however, since it requires the dimension r of the control vector to be no less than the dimension n of the state vector. We note that it trivially satisfies the controllability-observability assumptions of Ho [3]. Beginning quite naively,letusseek a powerseries solution y ( c 2 ) = 1+c2p' and

00

P(e2)=h2+c2

X ( ~ , E ) -2 X , ( t ) d ,

30

P(t.c)-

q(t)~'(3.1)

;=0

j=O

are all 1 +()(E'). The boundaryconditions provide four linear equations for the unknown ki(E))s. Up to asymptoti- of the Hamiltonian system (1.7). The terms of the series sally negligible terms, then, (3.1) follow by successively equating coefficients of powers E' in (1.7). When E = 0, then y(O)=k,+k2+&3 - E R -'BTPo=O and P o = - Q X o - A 'P,. .i,(O)=~fil-~,&Z--Xk3

O= y e v k ,

+ ye-'nUk2+ pk4.

(2.6)

Gnuing analogously, we find that x,(t)=O=P,(t)

This implies that the ki(e)'s have asymptotic seriesexpansions in E with leading terms

for all j > O .

(3.2)

The trivial outer solution thus constructed will need an initial boundary layer correction, since X(O,E)=O will Y (0) generally be unequal to the prescribed initial vector x(0). k , ( o ) = k,(o) = - 2k4(0)= __ and k3(0)= - j (0). 2 At t = 1, however, there is no need for a boundary layer we note thatthecontrol z4tf,E)= - p 2 ( r - E ) / c 2 and correction. since the trivial outer solution satisfies the the corresponding trajectories xl( t , c ) and x 2 ( t , c ) consist of terminal boundary condition p(l)=O. Thusl we seek an asymptotic solution to (1.7) of the form the outer solution (L'(r,E).X,(t,€),XZ(t.E))

X(t,€)=WZ(T,E)-

oc

oc

j=O

;=0

2 ~Z,(T)€',p(f,E)=Ef(T,E)~E

fi(T)€'

--(y.l.~p)k~e'~+(y,1, -q~El)k~e-'~',

the initial boundary layer correction (L.(T.E).m,(T,E),~7,(T.E))-

inthestretchedvariable boundary layer correction

(f - x i --.E,

(3.3) throughout 0 < t Q 1 where the boundary layer correction ( ~ ( T . E )E,~ ( T , E ) tends ) to zero as the stretched variable

k3ePAT r= t / c

(3.4)

and theterminaltendsto infinity. (Choice of the expansion (3.3) couldbe motivated by ourexample.) first Note that (3.3) will tend asymptotically to thetrivial outer solution as E + O pro( ~ ( ~ , € ) , ~ ~ ( u , € ) , ~ ~ ( ~ , € ) ) ~ ( ~ , € ~ ,vided € ~ ) that ~~e t >- 0. *~ T= [ / E .

22 1

O'hIALLEY Ah?) JAMESON: SINGULAR P E R W A T I O N S AND SINGULAR ARCS

Substituting (3.3) in (1.7)implies

sf,

dm- - d m -

dr

the linear system

and

S = BR - ' B T

All termsin the expansion (3.3) have now been deter-

(3.5) mined as exponentially decaying vectors. By (1.6) and (3.3), the optimal control is

with the initial condition m(O,E)=x(O).

u(t,E)= - LR-lBy(r,E).

(3.6)

(3.15)

E

Equating coefficients of

EJ

for each j 2 0 yields

We observe that it decays exponentially as r i m , and that in the limit as E+O, we have

dmj

-- - - SJ + Amjdr

s-

( t ,E )

-

(3.7)

1

- - Ce - c t / e ~- 1 / 2 (0) ~ E

which corresponds to Ho's result that j>O.

u(0) = - x(O)6(0)

Thus we have when n = r and B = I . The corresponding optimal cost is

lntroducing the positive definite matrices S ' l 2 and +f'(r,E)BR-'BTf(r,~))dr. (3.16)

we have the unique decaying

solution

Since the integrand decays exponentially as r+w, and has an asymptotic series expansion as E+O, the optimal (3.9) cost tends to zero and has an asymptotic series expansion

mo(r)=S'/2e-C'S-'/2 x (0)-

By (3.7), then, to

f o ( r ) = S - ' / 2 C e - c T S - '/2x(0)=Komo(r) (3.10)

.I*(€)-€

2J

~ E ~ .

k=O

with KO- S -1/2CS -]I2. Continuing, we seek the decaying solution of

We note that this conclusion is in agreement with the

for each j 2 1, where

statement of Kwakernaak and Sivan [4] that systems of unlimited accuracy are possible when r 2 n. We further notethat the optimal cost couldbe evaluated as .I*(€) E = T x T ( 0 ) K ( ~ ) x ( Owhere ) E K ( E is ) the usual Riccati gain

is a successively known exponentially decaying vector. Using the appropriate Green's function we obtain the explicit solution

for the infinite interval regulator problem associated with (3.5). In this regard, observe that the boundary layer controllability assumption of Yackel and KokotoviC[18] is implied by the definition of Case 0. When E = 0 the Hamiltonian (1.4) is extremized by taking

d 2mj

-- -SQmj+S1/2hj-1(r), mj(0)=O

(3.11)

dr2

-aH =BBTPO

au

-~ m e - c ( T - s ) ~ j - , ( s ) ~

-

lmeC(T-s)hj-

1

(3.13)

(3.18)

and a singular problem results since a 2H/h 2 = 0. For B of rank ia, (3.18)implies that p = 0, while the costate equation (1.5) implies that x =O since Q is nonsingular. Thus the trivial outer solution for Case 0 corresponds to a singular arc. Example 1 simply illustrates the asymptotic solution of Case 0.

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, APRIL

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