TECHNICAL NOTES AND CORRESPONDENCE 65

Simplification of Some Time-Optimal Switching Fujlctions A. T. FULLER

Absfract-The time-optimal switching functions for certain plants can involve fractional powers of functions of the state coordinates. A technique is given for obtaining equivalent switching functions which do not contain fractional powers, and are thus simpler to generate, especially in digital simulation studies.

INTRODUCTION

Explicit expressions can sometimes be obtained for the time- optimal switching funct,ions of simple plants with scalar saturating control inputs. These expressions oft.en involve square roots or ot.her fractional powers. For example, consider a triple-int,egrat.or plant. wit.h output v

a - = u( t ) ( a = const), d% dt3

with a saturat,ion constraint on the control u

and wit.h state coordinates

dv d2v dt’ dtz

x = a v , y = a - z = a - .

The e.spression for the control which moves t,he state to its origin in minimum time is (see, e.g., Feldbaum [l] )

u = -sgn [z + 423 + yz sgn (y + ++I) + I622 + Y sgn (Y + + Z I Z I ) } ~ / ~ sgn (Y + azlz l )~ (4)

and thus involves a square root. If (4) is generated digitally (e.g., in simulation studies), t.he square-root operation may introduce delay or complexity, and it is preferable to avoid such operat.ions if possible. In this note a technique is given for obt.aining alternative time- optimal switching funct,ions which are free from fractional powers.

A LEXMA ON SIGNUM FCNCTIONS

Lemma: If A, B, m are real and satisfy

A # - B , m>O ( 5 )

then

sgn ( A + B ) = sgn (AIAI” + BIBlm). (6)

Result (6) also holds i f A = -B, provided a value is dejned for sgn (0) . To prove this lemma, suppose h t . that

IA[ > (BI. (7)

Then

sgn ( A + B ) = sgn ( A ) . ( 8 )

Further, ( 5 ) and (7) imply that AIAlm has larger magnitude than BIBl”, so that

s g n (AIAI” + BIB[”) = sgn (AIAI”) = sgn ( A ) . (9)

Result (6) follows from (8 ) and (9), and is thus est.ablished for the c s e I AI > IBI. Similarly, result (6) holds when 1BI > 1 - 4 1 . Finally, if IAl = IBI, t.he result (6) is obvious once sgn ( 0 ) is defined.

The lemma may be applied to the simplification of switching functions, as illustrated in t.he following examples.

FIR~T EX~MPLE A very simple example for which the results are already familiar is

provided by the doubleintegrator plant

Manuscript received June 8, 1973. The author is with the Department of Engineering, Cambridge University.

Cambridge, England.

a -- = u( l ) ( a = const) d2v at2

subject to saturation conshint (2). One time-optimal control law for this plant, with state coordinates z and y as defined in (3), is

u = -sgn [1z[1/2 sgn (x) + 2-1/2y], (11)

this being t,he SERME control law of West. [2]. Applying (6) with

A = 1z)1’2sgn(z), B = 2-1/31, m = 1, (12)

we may change (11) to the well-known form

u = -sgn (z + +g(y(). (13)

SECOND EXAMPLE Consider again the triple integrator plant (I) . From (6) with

m = 1, cont.ro1 law (4) may be replaced by

u = -sgn ( A I A I + B I B ] ) (14)

where

A = X + 923 + yz s g n (y + t z l z l ) (15)

B = (69 + y sgn (y + + ~ l z l ) ] ” ~ sgn (y + +z[zI). (16)

In the derivation of (4), the square root. is defined as positive. Thu.: the square root in (16) is posit.ive, and (16) yields

BIB( = ( $ 9 + g sgn (y + $ z I z ~ ) ] ~ sgn (Y + tzlzl) (17)

= { y + 3 2 2 sgn (Y + )} 3. (18)

Prom (14), ( E ) , and (18) we may write t.he control law as

u = -sgn ( A I A I + C3) (19)

where

A = 2 + 4z3 + YZ SG (y + + z ~ z \ ) (20)

c = y + +zz sgn (y + &I 1, (21 1 which expressions involve only integer powers and are thus simple to synthesize digitally.

REFERENCES (11 A. A. p ldbaum, “On the synthesis of optimal systems aith the aid of phase

space Actomat. Telemekh.. vol. 16. .pp. 129-149, 1955. [2] J. C.’West, discussion in Automatx and &fanual control (Proc. Cranjield

Conf.. 1951), A. Tustin, Ed. London: Buttemorths, 1952, pp. 300-302.

Design of Regulators Using Time-Multiplied Quadratic Performance Indices

N. RAMANI AND D. P. ATHERTON

Absfract-The design of linear regulators, optimal with respect to a time-multiplied quadratic performance index, is considered. Since it is not possible to minimize the normal form of the index with con- stant feedback gains, a suitably modif?ed index is used and a design which is optimal in an average sense is obtained.

I. INTRODUCTION The problem of the design of an optimal feedback regulator for a

linear time-invariant. system

Z = AX + Bu (1)

which minimizes a quadratic performance index

supDorted in part by the National Research Council of Canada Grant ~ 1 1 6 4 6 . ~

New Brnnswick, Fredericton, N. B., Canada.

Manuscript received April 23, 1973: revised August 23. 1973. This paper was

The authors are with the Department of Electrical Engineering, University o f

66 IEEE TIUNSACTIONS ON AUTOMATIC CONTROL, FEBEUARY 1974

J = 1 (2’92 + u’Ru) dt

is well known and t.he optimal controller is given by [ l ]

u = -R-’R’Kr. (3)

IIere ’ denotes transpose, Q (positive semi-definite) and R (positive definite) are aeighting matrices, ( A , R ) is a controllable pair, and K i s the symmetric positive definite solution of t.he algebraic matrix lticcati equat.ion

A‘K + K A + Q = KBR-’B’K. (4)

Extension of t.he above result to the caSe where the quadrat,ic cost is t.imernultiplied for an index of t.he form

J = (tr’Qx + IL’RU) dt ( 5 )

was first attempted by Man and Smith [2] but. later the results were shown to be in error [3]. I t has also been shown [4] that it is not possible to minimize (5) with time invariant gains. This n0t.e de- velops an algorithm to minimize a modified form of ( 5 ) .

11. THE PROBLEM AND ITS SOLLTION

Equation ( 1 ) wit.h u = -Gx; where Gis a mat.rix of constant g ins , becomes

2 = ( A - BG)x = AG

so that ~ ( t ) is given by

r ( t ) = edotz(0) = @(t)Z(O).

Substituting this in ( 5 ) one gets

J = t r [ 1 @’(t ) ( fQ + G‘RG)@(t) dt x(O)x‘(O) . 1 In general, the minimization of J wit.h respect t.0 G, yields a G which depends upon r(O), the initial stat.e. In an attempt t,o remove t.his dependence on r(O), one minimizes the expected value of J :

J l (G) = t.r [ lm +‘(t)(tQ + G’RG)@(t) dt X0 ] (6)

where X 0 = E[roro’] as suggest,& by Levine and At.hans [SI, who give several reasons for using such an index in their work on the det,er- mination of optimd output. feedback gains.

Following Levine and At.hans, it can be shown t,hat,

.- lm R’@’(t - u)(tQ + G’RG)@(t)Xo+’(u) du dt. (7)

Pubstit.ut.ing T = t - u in the second term on the right hand side, and setting aJl/aG = 0 gives

lm RG@(t)Xo@’(t) dt

= lm R’%’( T ) (TQ + u Q + G’RG)@( T ) @ ( u)Xo@‘( u) dT.du

= R’ @’ (T)G’RG%(T)~T @(u)Xo@’(u) d u s,= Lrn

+ H’ Lm @‘(T)7&@(T)dr @(u)Xo@’(u)du

It has been shorn by MacFarlane [6] that for any real symmetric matrix Z and @(t) stable

lm @‘(t)tZ@(t) dt = @‘(t)Zl@(t) dt

where Z1 is the symmetric positive definite solution of

l- (9)

Ao’Z1 + Z1Ao + Z = 0. (10)

Also, it. can easily be shown that for any real symmetric mat.rix Y,

la @‘(t)Y@(t) dt = Y I (11)

=here Yl is the symmetric positive definite solution of

Ao’YI + YIAo + Y = 0. (12)

Using (9)-(12) in (S), it can be shown that. the const.ant gain matrix G which minimizes the modified time-multiplied index J I of (6), is given by the solution of

A0 = A - BG

Ad:, + Ldo’ + X0 = 0

AoLl + LlAo’ + Lo = 0

Ao’Qo + Q d o + Q = 0

Ao‘K + KAo + G’RG + Qo = 0

G = R-’B‘(R + QOLILO-’). (13)

Starting with an assumed G for which AO is stable, and working successively t.hrough the above equations, results in a new value of G. Repeated use of this procedure starting wit.h the new value of G normally provides convergence to t.he required solution.

If one wishes to minimize

Jn(G) = t r [ 1 @ y t ) ( t q + G W ) + ( ~ ) a 0 ] (14)

one can, by proceeding as before, show that G is given by the solution of

A0 = A - BG

AoLo + Ldo’ + Xo = 0

A& + LAO’ + rL-1 = 0, r = 1 , 2 , . . . ,n

Ao’Qo + Q o A o + Q = 0

Ao‘Q, + &,A0 + T&,-I = 0, T = 1,5. . .,n

Ao‘Ro + R d o + G’RG 0

G = R-’B‘(R&, + &,LO + nQn-lL1 + . . . + n.Q1Ln-1 + QaLn)Lo-’- (15)

It should be noted that t.he above equat.ions reduce to (13) for n = 1, with Bo + QI = K .

111. EXAMPLE Consider the linear t.ime-invariant. system

X = AX + Bu, u = -Gx

where

A = [ O B [:I. 1 - 0 . 1

With

TECHNICAL NOTES -4ND CORRESPONDEXCE 67

(15) for n = 0, I, and 2 gives

n = 0: G = [0.76 0.291

n = 1: G = [1.05 0.441

R = 2: G = [ 1 . 2 T 0.571.

Simulat.ion results show, a expect.ed, that the transient responses are less osca1at.ory i.3 n increases.

IT. CONCLGSIONS

h design procedure using tinx-mult.iplied performance indices has been outlined. In any design, the final decision of what constitutes satisfactory performance is left. to the judgment. of the designer; the method presented provides an addit.iona1 range of analytical ap- proaches to those n o r available. It is mathematically straightforward t,o extend t.he procedure to include different time weights on .r and u and for constant. gain output, rather than state variable feedback. Convergence of the algorithm proposed to solve (13) is not. assured but experience shows that normally, successive values of G either converge to the t.rue solution or suggest better starting values.

REFEREKCES

F. T. M a n and €5. W. Smith. ”Design of linear regulat.ors optimal for time- hl. ;ithans and P. L. Falb. Optimal Contro2. S e w Tork: McGran-Hill. 1966.

multiplied performance indices.” I E E E Tram. Automat. Contr. (Short Papers). rol. AC-14. pp. 527-529. Oct. 1969. B. Ramaasami and R. Ramar. ”Comments on ‘Design of linear rexularors optimal for timemultiplied performance indices,”’ I E E E T r a m . Automat.

P. Fortin and G. Parkins, “C0mment.s on ‘The design of linear regulat.ors Contr. fCorresp.1. vol. AC-15. p. 497. . lug. 1970.

optimal for time-multiplied performance indices.”’ I E E E Trans. Automat.

W . 8. Lerine and M. .ithans. “On the determination of the optimal constant Contr. (Tech. Notesand Corresp.), ro l . AC-17, p. 176, Feb. 1972.

output feedback gains for linear multivariable systems.’’ I E E E Trans. Automat. Conttr.. vol. As-15, pp. 44-48, Feb. 1970. A . G . J. MacFarlane. The calculation of functionals of the t ime and fre-

J. Mech. .4ppI. Math. . vol. 16. pt. 2 , pp. 259-271, 1963. quency response of a linear constant coefficient dpnamical system.” Quart.

Weighted Residual Methods in Optimal Control C . P. YEUMAX AKD A . SEN

Abstract-Weighted residual methods ( W R M ) afford a viable approach to the numerical solution of differential equations. Applica- tion of WRM results in the transformation of differential equations into systems of algebraic equations in the modal coefficients. This suggests that WRM can be used as a tool for reducing optimal con- trol problems to mathematical programming problems. Thereby, the optimal control problem is replaced by the minimization of a cost function of static coefficients subject to algebraic constraints. The motivation for this approach lies in the profusion of sophisticated computational algorithms and digital computer codes for the solution of mathematical programming problems. In this note the solution of optimal control problems a s mathematical programming problems via WRM is illustrated. The example presented indicates that reasonable accuracy is obtained for modest computational effort. While the simplest types of modes-polynomials and piecewise con- stants-are employed in this note, the ideas delineated can be applied in conjunction with cubic splines for the generation of com- putational algorithms of enhanced efficiency.

I. INTRODUCTIOX

Within the past fifteen years, enormous effort has been expended on the development of computat.ional schemes for optimal control

supported in part by tbe National Science Foundarion Grant GJ-1075.

>fellon University. Pittsburgh. Pa. 15213.

Manuscript received October 10, 1972; revised June 8, 1973. This work was

C. P. Neuman is Kith the Department of Electrical Engineering. Carnegie-

University. Pittsburgh. Pa. 16213. H e is now rrith the Department of Electrical A. Sen was with the Department of Elect.rical Engineering. Carnegie-Mellon

Engmeering, Kwhington University, St. Louis; 1x0. 63130.

problems [l], [21. The purpose of t.hk technical note is t.o illustrate weighted residual methods (WRM) as a tool for generating sub- optimal control algorithms for lumped parameter systems. Optimal control problems require the minimization of a functional over an admksible set, of cont.rol functions subject to dynamic constraints on t,he state and t.he control. B st,atic analogy k afforded by the mat.he- matical programming problem of minimizing afumtion over a set of coeficimzis subject to algebraic constraints. The underlying motivat,ion for the use of WRM [3] in optimal control problems is to exploit the wealth of knowledge acquired in the area of mat,hematical prc- gramming [4],[5].

The transformation of optimal control problems into mathemat.i- cal programming problems is not a new venture 161, [7]. Heretofore this conversion has been accomplished by discretizing both t.he state and cont,rol variables. Weighted residual methods, e.g., collocation and the Galerkin procedure, employing assumed mode approxima- tions for the state and control afford a viable alternative t.o dis- cretizat.ion methods as is evident from the numerical results pre- sented in thk n0t.e. Furthermore, for standard modal families, e.$., polynonlials and cubic splines, the approach can be systematized and tabulation of commonly used moment and collocation matrices for these modes leads to a marked reduction in the computational requirements for individual optimal control problems.

A simple example is employed to illustrate application of the Galerkin procedure and collocat,ion in opt.ima1 control problems. Excellent. results are obtained for this example with very modest computation. The prsent.ation is strictly introductory and details of general numerical implementation and comparisons wit.h other computational methods are not explored here. These aspects are discussed in [ l o ] .

11. Ax ILLUSTRATIVE EXUXPLE

The following firsborder optimal control problem [SI is considered. Find the control u* (t) which minimizes the cost.

subject t.0 the system dynamics

k ( t ) = --2(z) + u(2) (2 )

and the init.ial condition

r(0) A I O = 1. (3 )

The state x ( z ) and the cont.ro1 ~ ( t ) are scalar time functions. This example is selected because of its simplicity and because a

closed-form solution can be obt,ained by analytical methods. Straight- forward application of Pontryagin’s Minimum Principle [9] yields

u.*(t) = [l + -,&3] Gosh f i t + [ 4 2 + 81 sinh d% (4)

where

cosh @ + 4 sinh fl fi cosh ~ ’ 2 + sinh 4 -

An elementary WRM solut.ion employing two-mode approxima- t.ions for both the st,ate and the control is proposed. The modes select.ed are of the simplest type, polynomials for t.he state and p.iecetuise constant functions for the control. Thus the stat,e approxima- tion is

p = - - 0.98.

2 $( t ) = 2 0 + c i@i ( f ) = 1 f clt + c%,?’. ( 5 )

i = l

The st.ate modes qh(t) = t and &(t) = t2 satisfy zero initial conditions and consequently .the approximation ?( t ) satisfies the specified initial condition (3). The control approximation is

q t ) = a&(t) + a z h ( t ) (6)