A study of weighting factors of the quadratic performance index

Journal of The Franklin Institute DEVOTED

Volume

287,

Number

TO

SCIENCEAND THEMECHANICARTS

February

2

1969

A Study of Weighting Factors of the (&adratic Performance by

W.

R.

Index”

WAKELAND

Engineering Science Department Trinity University, San Antonio, Weighting

ABSTRACT: to a large weighting

extent factors

factors

used in performance

the

optimal

system

for

the ISE

performance

Graphical

relationships

measures

are presented.

derived

TexaJ

expressions.

between

which

The usefulness

factors

weighting

optimal

Analytical

design

expressions

are derived for

several

and some time response

factors

of negative

sets of curves. A sample design problem design aids presented

index

weighting

Negative

indices for

results.

are produced

weighting

is presented,

factors

in some

the

basic systems. performance cases

is illustrated

the solution

determine

relating

by the

by several

of which utilizes

the

herein.

Introduction

When a designer chooses to use an optimization process in the design of a control system, he establishes a “cost functional”. This cost functional is an analytical expression, related to the system, which the designer desires to maximize or minimize depending on whether the expression is composed of desirable or undesirable terms. In any event, the designer optimizes by choosing parameters or a control law which will cause an extremal of the cost functional. If the cost functional relates to an undesirable quality such as cost, or error, and the extremal is minimal then the designer has optimized the system with respect to the cost functional. In most systems it is analytically impractical, or impossible, to include in the cost functional all the design requirements and considerations. Consequently, it is not unusual to find that the system which results from the optimal selection of parameters or control law has some undesirable characteristics. Obviously, some trade-off in design must take place, but what is it to be 1 The establishment of the cost functional * This paper is a portion of a Ph.D. dissertation submitted in April 1968. The research was performed under the supervision of Professor C. F. Chen, Electrical Engineering Department, University of Houston.

101

W. R. Walceland may have been somewhat arbitrary and therefore the designer with clear conscience could modify it. But how should he modify the cost functional to obtain an improvement in the system characteristics 1 Weighting factors used in performance indices determine to a large extent the system which results from an optimization process. This paper determines, for several basic systems, the relationship between the weighting factors of the integra,l squared error (ISE) cost functional and the coefficients of the characteristic equation and also some time-response performance measures. With such guidance, the designer can determine from the beginning the correct tradeoff between the cost functional and other system characteristics. If the system is represented by its error state variables, li: = AE,

(1)

then

I=

“E* QE dt (2) s0 In Eq. 2, E is the error state vector, E* is E

is the ISE cost functional. transposed, and Q is the weighting factor matrix. If the symmetric matrix Q is a diagonal matrix, the cost functional becomes the generalized quadratic performance index,

I=

s

omLhW) + q2G(t) + . . .4%W)l

4%

Early work by Hall (l), N ims (2), and Graham and Lathrop (3) developed cost functionals which were one-dimensional. For instance, Hall proposed I = J-Te”(t) dt, Nims offered I = STte(t) dt and Graham and Lathrop examined many. Graham and Lathrop developed optimal forms for control systems by the selection of the coefficients to minimize the cost functional I = STt / e(t) 1dt. The resulting choice of parameters is dependent upon the nature of the variation of the cost functional. Designer intuitive feel for what is a good response is necessary in cases where the cost functional does not minimize within a practical region. When the cost functional is not limited to one dimension the weighting factor matrix Q gives the designer great potential for controlling or guiding the resulting selection of parameters or control law. Hsia (4) showed that it was sometimes possible to choose a weighting factor matrix Q which will yield a special case linear controller for a plant with a saturation element. Tuel (5) developed a canonical form for the weighting factor matrix Q which has the minimum number of parameters required in the performance index for the computation of the optimal control law. The selection of this matrix Q is a very important part of optimal design using the generalized quadratic performance criterion. In spite of this, there is very little, if any, guidance in the literature in the selection of the weighting factors. Neither are there any prevailing principles or procedures for the selection and use of weighting factors. This paper treats analytically the optimization of three basic systems using the ISE cost functional. The method of optimization follows generally

102

Journal of The Franklin

Institute

Weighting Factors of the Quadratic Per$nmunce Index that of Aizerman (6). All systems are normalized and Lathrop (3).

Analytical

in the manner of Graham

Development

Consider a system whose open loop transfer function is G(s) =

4 S(S2+ a, WgS + aa w;,

(4)

and the closed loop is then (5)

The state equations can be chosen as:

i C2(t) cl(t)

)i

C3M

=

-W$

-a2wg

[Yl= [d By translation

!! )ij

01

01

0

C2@) cl(t)

0 0

+

(6)

1

C3V)

--%wO

[r(t)12

0 01

(7)

of the reference axis:

E=R-C

(8)

and dropping the function of (t) notation for convenience, equations result :

i:)i e2 e1

e3

=

01

0 0

-fiJi

0 1

--CJ2CiJ;

-a1wo

i)

the following state

e2 e1

!i

e3

*

(9)

1

e1

bl = [-l,O,Ol

e2 +r,

(10)

e3

or

&=AE. The cost functional

to be minimized

I=

Vol.287,Xo. 2,February1969

(11)

is

cOE*QE, s0

(12)

103

W. R. Wakelund where the weighting factor matrix Q is defined as

(13)

Since the extremal is concerned with a maximum or minimum and not the value itself, the matrix Q can be normalized with respect to ql. In a manner similar to Aizerman (6) if it is assumed that I = E*PE, where P is a symmetrical

2 if [PA+A*P]

(14)

constant matrix, then = E*[PA+A*P]E,

is forced to -[Q]

(15)

then d1 - -E*QE dt-

(16)

and I = -E*PEr evaluated then

(17)

at 0 and co. Since for systems under consideration 1=

[ei(co)] = null,

[e(o)1* PI [WI.

(18)

Equation 16 determines P as a function of qi and the parameters of the system, which are present in A. Optimization with respect to a parameter ai requires the partial derivative allaa,. The vector differentiation operator Va is defined in Tou (7) as

a aa, Va=

&

,

(1%

2 a

(11aa, where m = number of parameters. ai, in general terms, V,I =

Then, for an extremal for all parameters

[Ml[al- PI = 0,

(20)

where, [M] is an m x (n - 1) matrix since q1 = 1 and I is a function of only (n - 1) pi; [qi] is a column vector of n - 1 rows ; [K] is an m x 1 column matrix ; and mij and k, are functions of the system parameters. If minimum cost functionals are obtainable in all parameter planes, then solving these m equations simultaneously leads to the optimization of m

104

Journal

of The Franklin

Institute

Weighting Factors of the Quadratic Performance

Index

parameters. However, for the purpose of this paper it is of interest to know what Q matrix is required for given sets of parameters ai. Therefore, for the synthesis of a Q matrix which gives these desired results, solving n- 1 equations in qi, there results

(21)

[sil = VW1 WI.

The simultaneous solution of the n- 1 equations determines a set of weighting factors which gives a cost functional extremal at the particular values of the parameters chosen. This solution must be tested to ascertain if the extremal is a minimum. Second-order

System

Consider the classical second-order, a unit step input, i.e.

G(s) =

4

type-one

s + 25w, s2+25wos+4

E(s) =

s(s + 254

system which is subjected to

(22)

the state equations g = AE are

i:)=i

0

el

1

e2

cl(O)

= 1,

Ii 1 e1

-2250~~

-0JE

e,(O)

e2 ’

= 0,

the cost functional I = JF(e! + q2 ei) dt, assume I = E* PE. Forcing [PA + A* P] to

determines the elements of P as Pll =

w;(ag” + 1) + 45w;

1 +w;q;

PI,=&

p22= 0

Evaluating

q2 OJ; ’

45w;

*

at t = 0 and t = co gives I=

c2+ 1 +qzw; 4

-*

(23)

Then as in Eq. 20

VI=

Vol.287.No. 2,February

1969

(~),,- (;;y) =o.

(24)

105

W. R. Wakeland For a fixed w0 the relationship between qz and 5 is given by the first equation of (24):

Figure 1 illustrates how the damping ratio of a second-order system is determined by the weighting factor qz in the optimization based upon the ISE cost functional. Note that the well-recognized characteristic of the onedimensional ISE criterion to give a rather lightly damped solution shows in Fig. 1 by the intersection of the curves at 5 = 0.5 for q2 = 0.

3.0-

2.092 I.0-

FIG.

1. Second-order variation of 5 with qz for a fixed wO.

The relationship which results in the extremal value of I with respect to w0 is the hyperbola, q2wg=

452fl

(26)

which is illustrated in Fig. 2. By inspection of the functional I in Eq. 23 it is apparent that for q2 = 0 it minimizes at o,, -+ co. This corresponds to the wellknown optimal choice of the velocity error coefficient as infinite when the one-dimensional ISE criterion is used for optimization. For the variation of I with respect to 5, i321/a[2= S/I&,. Therefore, since (> 0 for stable systems, the extremal obtained by Eq. 25 is minimal for either positive or negative values of q2.The variation of I with respect to w0 gives m/ad = q2/2 0. A set of parameters 5 and w,, cannot be found which minimize the functional I for a finite q2. Practical considerations normally place restrictions or give guide-lines for the limits of wO. Figures 1 and 2 give the information that is necessary to provide complete correlation between the weighting factor qa and time or frequency response performance measures.

106

Journal of The Franklin

Institute

Weighting lkctors

of the Quadratic Performance

Index

4-

3-

2qz

I-

FIG. 2. Second-order variation of q, with qz for a fixed 5. Third-order,

Type-one

System

The third-order system to be considered first is the type-one, zeroposition-error system. Again the normalized representation is used, i.e.

G(s) =

4!l

E(s) =

s(s2+a,w,s+a,w$’

s2+a,w,s+a2w2, s3+a,w,s2+a2w~s+w~’

(27)

i3 = AE,

A=

i

0

1

0

0

0

1

-CO!

cl(O)

= 1,

-a2wi

-a1wo

ez(0) = 0,

The cost functional is I = ST E* QE dt; [PA + A* P] to - [Q] determines

7 i

e,(O) = 0.

assuming

I = E* PE and

forcing

a2-a,a~--a~--a,w~q2--w~q, Pll

Evaluating

=

2w,(l

-a,a,)

*

at t = 0 and t = CO; 1=

Vol. 2Pi, No. 2. February

1969

[ei(~)l*P[ei(~)l= Pll. 107

W. R. Wakehd The resulting Eq. 20 is

(;;;

Considering follows :

:ij

(zl)

= (l-~I~~~~--a$j

u+, as a normalizing

parameter,

(28)

Eq. 28 can be rewritten

as

Wa) Solving Eq. 2Sa simultaneously gives a relation for q2 and q3 in terms of the system coefficients ai and the normalizing parameter w,,:

4q2 = 443

=

alai--a,-2at

(29)

2

a1 af+l-aa,a,

(30)

a2

I-

I

I I

FIG. 3. Third-order,

I

at

I 2

I

I 3

type-one weighting factor-parameter second-order minimization.

relationships for

From Eqs. 29 and 30 values of wtq2 and wtq3 can be produce an extremal in the cost functional for any given a, and a2. These values must be tested to ascertain if minimum. For this system and for the values of interest they do indeed produce minimums.

108

determined which set of parameters the extremal is a shown on Fig. 3,

Journal of The FranklinInstitute

Weighting Factors of the Quadratic Performance Index Figure 3 shows the relationship between the normalized weighting factors and the system parameters ai which result from an optimization process using the ISE cost functional. As shown in Fig. 3, except for areas in the parameter plane close to the stability line a, a2 = 1 and the line alas = l-5 which represents a gain margin of O-33, a particular combination of weighting factors determines a unique pair of parameters. As will be shown in the example problem, once the designer is satisfied with a particular set of parameters ai, the constants of the system can be readily determined by straightforward relationships between the parameters ai and the system constants. Towill (8,9) has presented on a parameter plane curves of overshoot, response time and other design figures of merit for the third-order system. For design convenience overshoot and settling time loci are plotted on the weighting factor plane in Fig. 4. Using Fig. 4, the designer can tell at a glance what the overshoot and settling time will be for a particular pair of normalized weighting factors. If he is satisfied, Fig. 3 can give him the coefficients a, and a2. The system constants can then be determined. 4G%30%

percent 15%

Overshooi 10% 5%

7

2%

95%

6

USponse Tlf?W

5

9.0 sec.

4

&I,

3

8.4sec.

2

FIG.

4.

I

7.5sec.

0

6.9 sec. 6.3 sec.

Third-order, type-one weighting factor relationship with overshoot 95 per cent response time. Heavy lines are response time loci.

Third-order,

Type-two

For the type-two,

and

System

zero-velocity-error

system which has been subjected

to

a unit step input,

E(s) = Equation /

S2+a,U,S s3+a,w,s2+a2w;s+w;

(31)

20 becomes l+ag

vol. 287,No.Z. February 1969

109

W. R. Wakehnd The two equations solved simultaneously result in Figs. 5 and 6.As for the type-one system, the solution guarantees extremals only and must be tested for minimums.

ai

La,=

I

FIG. 5. Third-order, type-two weighting factor-parameter minimization. Heavy lines are

-.6 -.4 -.2 FIG. 6. Third-order,

Negative

Weighting

type-two

0

.2 .4

.6

3

relationship for second-order

qs loci.

I.0 1.2 1.4

weighting factor relationship with overshoot.

Factors

Optimal control law determination by Kalman (lo), and Athans and Falb (ll), require that the weighting factor matrix Q in Eq. 2 be positive definite. However, the results of Eqs. 20 and 21 lead to some negative weighting factors. This is shown in Figs. 1 through 6. Optimization does occur to the

110

Journal

of The Franklin

Institute

Weighting Factors of the Quadratic Performance

Index

parameter values indicated. This has been repeatedly demonstrated for negative weighting factors utilizing a digital computer for the optimization. Of course, the analytical development previously presented makes no requirement that Q be positive definite. Example

Problem

Consider the attitude stabilization of a satellite in orbit. The combination of horizon sensors and a strapped-down inertial reference system can provide attitude reference about all three axes. There is no viscous friction present, so the attitude thruster and satellite structure transfer function is represented by G(s) =

&*

Position feedback is necessary to maintain the desired orientation, and rate feedback is necessary for stability. In addition, the gyro output has second-order oscillations which should be filtered. The system for one set of axes can be represented by the block diagram in Fig. 7. Then

c where

K/J

3 = s3+ps2+(KK,/J)s+K/J wo =

4 s3+a,w,s2+a,w~s+o_$’

(34)

W/JF, (35)

Now, the actual system is only piecewise linear, but the thrust controller logic within the saturation limits produces average thrust proportional to E, (Fig. 7). A linear analysis is, therefore, reasonably accurate. Filter

K = 180 ft.lbs. radian J = 990 slug-ft* FIG. 7. Satellite attitude control.

The nature of the mission of the satellite determines the design requirements and also provides the basis for the structure of the cost functional. In this problem it is decided that the expenditure of propellant in recovering from displacement disturbances is of primary importance. The ISE, I=

sw

[e2(t) + q2C2(t)+ q3 i;“(t)] dt

0

Vol. 287, No. 0, February

1969

(36)

111

W. R. Wakeland is used as the cost functional. Any signal at E, (outside of the dead band) produces a torque which in turn produces an acceleration in direct proportion. Consequently, @a*(t)dt is representative of the amount of fuel expended during a recovery. A designer wouId certainly desire to eliminate the error eventually, but there is no particular concern about error rate. Therefore, the cost functional weighting factors are chosen as q2 = 0, q3$1.0. Referring to Fig. 4, it can be seen that for 0: q2 = 0 and ~0”q3 = 7, the dynamic response of the linear system, Eq. 34, would give about a 2.5 per cent overshoot and require about 9.5 set to settle to 5 per cent of the initial error. If the designer is satisfied with this response, he proceeds. If not, he then realizes he must compromise between cost functional weighting factors and system response. Figure 4 is a guide in this compromise omitting the necessity of any trial calculations. The moment of inertia J must normally be accepted as given and the thrust controller gain is set by the geometry, the thruster size, and saturation limits. Considerations other than dynamic response determine these values. Then for W: q3 = 7, the following values result : w0 = 0.565, q3 = 68.5. From Fig. 3, a1 = 3,

a2 = 2.7,

p = a,0+ = l-7, K, = 5

= 4.8.

WO

The break-point of the - BdB/octave filter occurs at 1.7 rad/sec, and the rate feedback gain is 4.8 sec. Actual system simuIation gives responses to disturbances within the saturation limit of 0.5 rad, which are very close to the linear system dynamic response. By the use of Figs. 3 and 4 the designer is able to perform an optimal design of his system, continually aware of the relationships between the cost functional and the dynamic response. In addition, the process of optimization is reduced to selecting the optimally chosen coefficients ai from Fig. 3 and determining the system constants from the system, Eqs. 34 and 35.

Conclusions

Exact relationships are obtained between weighting factors for the generalized quadratic performance index and the coefficients of the secondorder, type-one system and third-order, type-one and -two systems. For these problems, correlation between weighting factors and classical time response figures of merit is easily made, as in Figs. 1, 2, 4, and 6 by reference to

112

Journal of The Frauklin

Institute

Weighting Factors of the Quadratic Performance Index previous works. A straightforward method of deriving similar relationships for other classes of problems is demonstrated. It is shown that negative weighting factors not only are possible, but are useful. In some cases they are necessary in order to achieve a desired response characteristic. Although this paper covers only the second-order and two basic thirdorder systems, the system applicability is widened considerably by simplification techniques as in Chen (12) and Marshall (13). Correlation between weighting factors and time response criteria is useful either in the initial selection of weighting factors or their adjustment, if first effort gives a system characteristic which is undesirable. There seems to be no reason why a sufficient body of correlations cannot be established so that moving back and forth between optimization cost functionals and time response criteria is an easy routine.

References The C. Hall, “The Analysis and Synthesis of Linear Servomechanisms”, Technology Press, Massachusetts Institute of Technology, Cambridge, Mass., p. 19, 1943. Trans. AIEE, (2) P. T. Nims, “Some Design Criteria for Automatic Controls”, Vol. 70, Part I, pp. 606-611, 1951. (3) D. Graham and R. C. Lathrop, “The Synthesis of Optimum Transient Response: Criteria and Standard Forms”, Trans. AIEE, Vol. 72, Part II, pp. 278-288, Nov. 1953. (4) Tien C. Hsia, “On the Optimal Control of Plants with Saturation Non-linearity”, IEEE Trans. on Automatic Control, June 1967. (5) W. G. Tuel, “An Improved Algorithm for the Solution of Discrete Regulation Problems”, IEEE Trans. on Automatic Control, Oct. 1967. (6) M. A. Aizerman, “Theory of Automatic Control”, Oxford, England, Pergamon Press, Ltd., Chap. 4, 1963. (7) J. T. Tou, “Modern Control Theory”, New York, McGraw-Hill Book Co., p. 46, 1964. (8) D. R. Towill, “Analysis and Synthesis of Feedback Compensated Third Order Control Systems via the Coefficient Plane”, The Radio and Electronic Engineer, Aug. 1966. (9) D. R. Towill, “Coefficient Plane Synthesis of Zero Velocity Lag Servomechanisms”, The Radio and Electronic Engineer, Dec. 1967. (10) R. E. Kalman, “When is a Linear Control System Optimal?” Journal of Basic Engineering (Trans. ASME, Part D), March, 1964. (11) M. Athans and P. L. Falb, “Optimal Control”, New York, McGraw-Hill Book Co., Chap. 9, Sec. 5, 1966. (12) Kan Chen, “A Quick Method for Estimating Closed Loop Poles of Central Systems”, AIEE Trans., Vol. 76, Part II, May 1957. (13) S. A. Marshall, “An Approximate Method for Reducin the Order of Linear Systems”, Control, Dec. 1966. (1) A.

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