Performance index sensitivity of minimum-fuel systems

Automatica, Vol. 7, pp. 755-759. Pergamon Press, 1971. Printed in Great Britain.

Brief Paper Performance Index Sensitivity of Minimum-Fuel Systems* La sensibilit6 de l'indice de performance des syst mes minimum de carburant Leistungsindex-Empfindlichkeit bei Systemen mit Treibstoffminimum qyBCTBHTeJIt,HOCTb noKa3aTeJI,q pa6oTbI CI,ICTeM C MHHHMyMOM pacxo,aa FOptoqero P. C O U R T I N g " Summary--The purpose of this paper is to investigate the effects of small parameter variations on the performance index of minimum-fuel systems with initial and final manifolds. The pulse-shaped variations produced by the parameter change on the piecewise-constant control are taken into account in order to derive the trajectory sensitivity equation. It is proved that the variation of the performance index is related to those of the trajectory and the parameter by a simple expression. This expression extends, to the class of minimum-fuel problems, a result obtained by KOKOTOVIC et al. [1] for systems with continuous, unconstrained controls.

where x is a real n-dimensional state-vector, q is a real /-dimensional parameter* vector, u is a scalar function bounded in magnitude (lul ~ 1). We minimize the performance index J with

Itl J=q~(x(tt),tl)+

ILl(x,

Olul]dt

q, t)+L2(x, q,

alto

(2)

where (p is a terminal penalty function, Lt and L 2 are scalarvalued functions and L2 is assumed to be strictly different from zero for any x, q, t. The initial and final manifolds are defined by

Introduction THE IMPORTANCEof investigating the problem of performance index sensitivity to small parameter variations of optimal control systems was indicated by DORATO [2] in 1963. The following years several papers were published on this subject. In 1969 KOKOTOVlC, HEELER and SANNOm [l] analyzed the performance index sensitivity of the optimal control problems with initial and final manifolds, fixed final time, and unconstrained control. The result of KOKOTOVlC et al. was recently extended, as described in Ref. [3], to problems with free terminal time and with a terminal penalty appearing in the performance index. In this paper the sensitivity to small parameter variations of minimum-fuel systems is investigated. For these systems the optimal control is piecewise constant. Hence a small parameter variation produces finite control variations during infinitesimal intervals of time. The effect of this type of control variation on the trajectory is considered. Then a relation between the first-order variation of the performance index and the errors on the boundaries is derived. This relation extends the results of Refs [1] and [3] to the case of minimum-fuel systems.

Mo={x: A[x(to), to]=O, an s-dimensional vector function} (s~< n)

M1 ={X:

q'[x(tl), t l ] = 0 , an r-dimensional vector function}.

(r
The final time tl is fixed. We assume that f l , f2, L 1, L2, ~, A, W and all their first partial derivatives are continuous functions of their arguments. We shall use F=fl +f2u and L = L 1+L2[ul when the particular structure with respect to the control is not important. For the nominal value of the parameter qx, the optimal control is defined byt

u*(t, qN) = -- dez[ pr(t)f2(x' q-~ t!] L L2(x,qN,t) J

(3)

Problem formulation Consider a single input* plant 2~= f l ( x ,

q, t ) + f 2 ( x , q, t)ullm-"i .

for te[t0, tl] if the collection of switching times

tis at which

(1)

pr(t)f2(x(t), qs, 0] = 1 L2(x(t), qN, t)

* Received 5 February 1971 ; revised 14 June 1971. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by Associate Editor P. Kokotovi~. This work was partially supported by the National Science Foundation under grant NSF GK-2283 and by the National Aeronautics and Space Administration under grant N G L 33-008-090. t Systems Research Group, Department of Electrical Engineering and Computer Science, Columbia University, N.Y. $, An extension to the multiple input case is straightforward, and does not modify the final result.

forms a set of measure zero. It should be noted that this assumption implies that the system is normal.

* Some initial conditions may be considered as parameters. I dez(a) =

755

+1 a>l 0 lal ~< 1 -1 a<--I

756

Brief papers

The optimal control minimizes lhe Hamiltonian, defined by H=L+pTF. The costate p(t)satisfies the differential equation f

f~T= _ Lx _ prF~

(4)

with the boundary conditions

p( to) = Eflr Ax] rlt= to

(5)

p ( t O - - [ ~ x + ArV.~]rl,= . .

(6)

The Lagrange multipliers (,8, 2) are obtained from the solutions of A [ x ( t o ) , to] = 0

• [x(tl), t,] =o.

The function K(x, q, t) is a continuous function of x, q, : s i n c e f 2, p, L 2 are continuous and LZ is strictly different from zero. Thus an infinitesimal parameter variation generates a new controller u which differs from the optimal It~ by finitc constant values over infinitesimal amounts of time. In order to simplify the notations we assume that the control switches only twice. This assumption may seem restrictive since the definition of the optimal control allows the switching times to form a set of measure zero. We shall see that the number of switching times does not modify the conclusions of Theorems 1 and 2. Since F, L, L~ and Fz are continuous with respect to the parameter q, the control variations indicated in Fig. I, A~u and A2u last only infinitesimal amounts of time eJ and ~2 when q varies. Therelationship between ei a n d f q depends on the control implementation and, therefore, on the designer's choice. In the sequel we shall assume that this relationship is known since the proof of Theorem 2 does not require an explicit definition of U.

u(t)

The sensitivity problem is to consider the first variation of the performance index J about the optimal value J * when there is an infinitesimal change in the parameter, that is,

q=qN+3q. The infinitesimal change in the parameter value determines a new trajectory x * + J x , which is not necessarily optimal, near the optimal trajectory x*. This new trajectory terminates at time tl in x*(tl)+tJx(tl). We now consider how Jq influences the control u, and how we can obtain the variation of the end state 6x(t0.

Time u*

Control variation and trajectory sensitivity equation Two controls u A and u n are said to be nominally optimal if they satisfy the following relation

-I

uA(t, q)--Un(t, q)--u*(t, qN) w h e n e v e r q=qN. FIG. 1. Control variation. Hence there are many nominally optimal implementations o f the optimal control. For instance the open-loop implementation is defined by

UOL= - d e z [ K * ( t ) ]

The control variation 6u(t) is such that~

(7)

6u(t)=

with

K . ( t ) = P r ( t ) f 2 ( x * ( t ) , qN, t) L2(x*(t), qN, t)

with

q, t)]

t~(t 2 - e ' , t.~]

A2u=7+l

t~(t2-e 2, t2]

for

0 elsewhere.

Since there is no feedback in the state or in the parameter the control variation is identically nil (Ju0L = 0). This is not the case for the closed-loop, nominally optimal controller. Since the closed-loop implementation depends on the state, and maybe also on the parameter, the control variation JueL is a function of J x and Jq. In general the closed-loop implementation is defined by

UcL(X, q, t ) = - - d e z [ K ( x ,

A l u = -- 1 t b r

(8)

K(x, q, t)=K (t) when q remains equal to qx.

f A row vector Tv=BT/Oy represents the partial derivative of a scalar function T with respect to an n-dimensional vector y. The Jacobian of an m-dimensional vector funcion Z with respect to y is defined as

The influence of Jq on the trajectory will be analyzed in two steps. First consider the effect of Jq on the control only. It is known, as indicated in Ref. [4] for instance, that the effect of a pulse-shaped variation Atu upon the trajectory is given by the integral of

6,~=Fx[x*(t), qN, u*(t), t]hx for

ts>tis with the initial value

3x(t~)----e'" [ F ( x * ,

qN, u, t ) - F ( x * , qN, U*, t ) ] l t = t q .

(9) Since the system is linear in the control, and since f2 is a continuous function, the pulse-shaped variation Art, changes the terminal state by J~(tl) with

5~(tl)----si"~D(tl,t~)"f2(x*(t), qN, t)Itfti,"A~ Z, = |SZm

l:yl

dZ,,|

y._l

matrix

(8)

(I0)

where O(t, t') is the transition matrix associated with the linear homogeneous differential equation (8).

t [tie, tis+ el) if the variation occurs after tia.

Brief papers Furthermore we know, as shown in Ref. [4], that the joint effect on the trajectory of all the pulse-shaped control variations equals the vector-sum of the individual effect of each pulse-shaped variation. Hence the control variations produce the following change on the terminal state

757

By application of the mean-value theorem and by Taylor series expansions about the nominal values of the arguments, we obtain the first variation tSJ.

6J = [q~xrX]t=,, +

2

~x+(tl) -

~(~(tt).

f

tl

[L~rx + Lqbq]dt to

(ll)

+

i=1

g2(x*(t), ON,

t)'[iu*+&l-lu*l]dt.

(16)

to

Second consider the influence of the parameter variation ~q on the trajectory when the control is not varied, that is, u=u*. The variation of the trajectory is given by the integral oft

6:;c= F x f x + FqSq

In order to eliminate Lx3x, we now repeat the process used by KOKOTOVlCet al. [1]. We add equation (13) premultiplied by prO)and equation (4) post multiplied by 8x to getf

(12)

for t/> to with the initial value tSx(to). Since the variations correspond to infinitesimal tSq, and because the differential equations (8)and (12)are linear, the principle of superposition is applicable.

Theorem 1. When the parameter q changes infinitesimally from its nominal value (i.e. q=qNWt$q), the total trajectory variation of system (I) is given by the integral of

d (pr rx) = - L x f x + pr Fqrq + pTf 2[r#$1t .

(17)

Since (a) Hqrq=pTF~q+ L ~ q (b) p(t) satisfies the boundary conditions (5) and (6). (c) L~Sx is defined by equation (17).

fife = F~Sx + Fqfq +f2(x*(t), qN, t)fu

(13)

for t>to with the initial valale 6X(to).

Proof. The contribution of the last term of equation (13)

(d) tPxrxlt = t, =c~tP, Azc~xlt= to=rA. (e) let lu*+rul-[a*l:ru'( where ~ takes the values (l and ~2 corresponding to Au' and AuL Equation (16) can be written tl

r + 5J = -- ~.Tc~LP+/I,SA,

tl

b+(tl) =

I

~ ( t l , 2)fZ(x*(2), qN, ;t)6ud~,

I

Hqbqd t

.]to

alto

= ~ [""

+

(I)(h, 2 ) f 2 ( x * ( 2 ) , qu, 2) Auid2

which correctly reduces to the expression of tSx+(tl) given in equation (I1) by application of the mean-value theorem. Equation (12) is the trajectory sensitivity equation for an open-loop implementation of the control (tSu0L=0). Equation (13) corresponds to a closed-loop implementation.

Variation of the performance index Theorem 2. For a minimum-fuel system, an infinitesimal

(18)

We now calculate the contribution of the second integral term

if

' [-prf2 + L2~]rudt

to

variation ~q of the parameter produces the following performanee index variation

I

=

• I t'~ i= IJ

tis - e i

" rf2+L2~i]Auidt LP.

(19)

and

tl

6J = - 2 r f t P + f l r r A +

[prf2 + L2(]Sudt. ¢o

i = 1 J t i s -- ei

Hqfqdt

(14)

dto rod and 6A are the variations of the initial and final manifolds.

Proof. The total difference of the performance index is

f

""

[prf2+L2~i]Auidt

t l s - - ~t T 2 2~i [ =d'AW'[pf +L~]

N ,

|t=t's-Oe

i

(20)

given by where 0~[0, 1). Equation (20) is obtained by application of the mean value theorem. We cannot make use of the continuity properties to set 0 equal to zero before we define/~ at time tts. It is proved in the appendix that

A J = q~(x*(q) + 6x(ta), t 1 ) - ~o(x*(q), tl) +

f"

J to

[ L ( x * + 6 x , qN+6q, U*+6U, t)

- L ( x * , qN, u*, t ) ] d t .

(15)

? Here and in the sequel all the partial derivatives are taken about the nominal values of the function arguments,

L\ L }l,=,'J t The symbol [t¢ means that the arguments of the function take their nominal values.

758

Brief papers

Hence the contribution of the last integral term in equation (! 8) becomes

f

" [p'rf2 + L2;.]6udt tO 2

(22)

= ~] d. Au'. [ p V ~ + L~¢']I N i= I

It=tis

The bracket in the right-hand side of equation (22) can be rewritten as follows

[(prf2l_l_L2(,]l~:t, =

L2

+(i

II

N

/_llt=t.~ '

since we assume that L 2 is strictly different from zero for all values of its arguments. By definition of the switching time t*s we have

[pV:-ll --L-T/IN

. = _+ 1.

_JIt=t's

the equality of the performance index sensitivities of ope~ loop and closed-loop implementations (i.e. ~Jox,~JotA. By integrating system (13) and using equation (14), the designer can readily compute the boundary variations and the performance index variation for a chosen control implementation and a given ~q. This process can be repeated to compare several implementations and to make some trade-off between performance index sensitivity and boundary errors. In this regard the research of SINHA and DAI [6] can be readily extended to minimum-fuel problems.

Conclusion The problem of sensitivity to small parameter variations of minimum-fuel systems was investigated in this paper. It is shown that the first variation of the performance index is equal to the sum of terms depending upon the boundary errors with a sensitivity invariant vector derived from the Hamiltonian. Once the designer has selected a nominally optimal control law, the boundary variations corresponding to a parameter variation can be easily calculated. From the relationship between performance index variation and boundary variations, the influence of a parameter change on the perfotmance index is readily determined. Thus this relationship is a useful tool for the designer in his choice of a nominally optimal control law.

Hence

ReJerences

E(")'i ]]1

---~----sgn --~ 2:,,: 2=,,9 0"

(23)

Therefore the second integral term of equation (18) is nil. Thus

M= -2r6tP +flrSA+

.

(14)

Remark 1. The usefulness of theorem 2 may seem to be limited by the fact that 6W depends upon 6x(tO and, therefore, upon the determination of the relationships that exist between the ei's and 6q. It is, in general, impossible to obtain an explicit relationship between an e ~ and d~q. This is due to the non-linear control law and to the way t~x(t), Oq and t~u(t) are related. However such an explicit relationship is not necessary to compute 6x(tl). From the knowledge of 6q we have the initial conditions necessary to start the integration of system (13) with 6u(t) defined by 6u(t) = -- d e z [ K ( x * + 6x, qN + 6q, t)] + d e z [ K * ( t ) ]

[1] P. KOKOXOVlC, J. HELLER and P. SANNUTJ: Sensitivity comparison of optimal controls. Int. J. Control 9, (1969). [2] P. DORATO: On sensivity in optimal control systems. IEEE Trans. Aut. Control AC-8, (1963). [3] P. COURTIN and J. ROOTENBERG: Performance index sensitivity of optimal control systems. Department of Electrical Engineering, Columbia University, Technical Report No. 115, January 1970. [4] M. ATHANSand P. L. FALB: Optimal Control. McGrawHill, New York (1966). [5] P. COURTIN: Performance index sensitivity of optimal control systems. Ph.D. Dissertation, Dept. of Electrical Engineering and Computer Science, Columbia University (1970). [6] N. K. SINHA and S. H. DAI: Reduction of the sensitivity of an optimal control system to plant parameter variations. IEEE Trans. Aut. Control AC-15 (1970).

Appendix: Determination o7~ ~at ti~ Let us study the case of Fig. 2 where the optimal control jumps from minus one to zero at time t~. w(r)

124) where K(x* +6x, qN+6q, t) depends upon the implementation of the closed loop control chosen by the designer. The numerical solution of system (13) gives us Ox(tl) corresponding to a given 6q and a chosen nominally fuel optimal control. The practical problem is, as usual for numerical solutions of non-linear differential systems, the choice of an integrating scheme and that of the step, or steps, of integration.

0

I i

Time u

i -I

Remark 2. The assumption that the final time t~ is fixed can be removed. Theorem 2 has been proven true when tl is free [5]. Remark 3. It can be shown [5] that theorem 2 is still valid if we consider the bang-bang problem and singular minimum-fuel problems. This proof is beyond the scope of the present paper and cannot be included here. Theorem 2 extends to the class of minimum fuel problems the result previously obtained by KOKOTOVl¢ et al. [1] for problems with unconstrained contlols. The conclusions of [1] are still valid for fuel optimal problems. In particular the existence of a terminal manifold precludes in general

H -

u"

FIG. 2. Switching at time tt~.

Depending upon the parameter variation 6q, the varied control u will jump from --1 to 0 before tt~ or after t~s. if the jump occurs before t% we have Aui= +I

and

thus in this case ~ t = _ 1 for t~(tts--e ¢, tis].

Brief papers Similarly if the jump occurs after tl~, we have Au'=-I

and

lu*+~.'[-lu*l=~.'c'--+~

hence ~ = - 1 for t~[t~s, t~,+~). We see that no matter when the jump occurs ~ is equal to minus one when the control component jumps from -- 1 to 0. A careful study of the three other cases allow us to establish Table 1.

TABLE 1. DETERMINATION OF ~t

Jump Jump Jump Jump

--1 to 0 0 to + 1 + 1 to 0 0 to --1

Aut before t~8

Au i after tl,

~

+1 +1 --1 --I

--1 --1 +1 +1

--1 +1 +1 -1

When we relate the values of (i and the definition of the switching times we see that ( ' = - sgn[(Pr-~fz2"~lN L\ L

].

(21)

]it=t,_]

R(nmm6----L'objet du pr6sent article consiste ~ 6tudier les effets de faibles variations de param6tres sur l'indice de performance de syst6mes /t minimum de carburant avec enveloppes initiale et finale. Les variations sous forme d'impulsions produites par la variation des param6tres sur la commande constante par

759

tron~ons sont prises en consideration ann de d6duire l'&luation de sensibilit6 de la trajectoire. II est montr6 que la variation de l'indice de performance est li6e /t celles de la trajectoire et du param~tre par une expression simple. Cette expression g~n~ralise ~t la cat~gorie des probl~mes it minimum de carburant un r6sultat obtenu par Kokotovic et al. (1) pour des syst~mes it commande continue sans contraintes. Zusanunenfassung--Der Zweck dieser Arbeit besteht in der Untersuchung der Wirkung von kleinen Parametervariationen auf den Leistungsindex von Systemen mit Treibstoffminimum. Die impulsf0rmigen Ver~inderungen, die durch Parameteriinderung an der stiickweise konstanten Regelung entstehen, werden in Betracht gezogen, u m den Verlauf der Empfindlichkeitsfunktion abzuleiten. Bewiesen wird, dab die V e ~ n d e r u n g des Gtiteindex mit einem einfachen Ausdruck auf die der Trajektorie und des Parameters bezogen wird. Dieser Ausdruck dehnt ein von Kokotovic und anderen fiir Systeme mit kontinuierlichen, unbeschr[inkten Regelungen erhaltenes Resultat auf die Klasse von Problemen mit minimalem Treibstoff aus. Pe3mMe--I.~eymlo HaCTO~Lt(ei~ CTaTbH IIB.rllleTClt H3ytieHHe BJI~KaH~ He~OYI~I~HXH3MeHeHHI~napaMeTpoB Ha IIOKa3aTeHb pa6oTbI CHCTeM CMHHHMyMOMpacxo~a ropm~er oaMemttmx HaLIa.qbHbIe H KOHetIHI~IeO6OYIOqKH. I/hMeHeHH~, B ~bopMe HMnTm,COB, IIpOH3BO]IHMI)Ie X3MeHeHHeM napaMeTpoB Ha ryco,mo-nocrosrmoe ynpasaetme y'mTbIBatOTCS ~a~ Bbmo~a ypaBHeHHS qyBCTBHTeJIbHOL~FHTpaeKTOpHH. I[oKa3bmaeTc~[ tITO H3MeHeHHe noKa3aTeylll pa6oTbI CBfl3aHO C H3MCHCHHIIMTpaeKTOpHH a riapaMeTpa npoCTblM BbIpa~ermeM. ~TO BHpaxeHHe O606Lt(aeT K K~accy rIpo6YIeM c MHHHMyMOM pacxo~a ropm~ero pe3y,rIbTaT noay~eHa~t KOICOTOBH~eMH Rp. (1) ~ S CHCTeMHenpepbmHoro yripal~ieHmi 6e3 orpam~eHHi~.