N O T I C E 

THIS DOCUMENT HAS BEEN REPRODUCED FROM 

MICROFICHE. ALTHOUGH IT IS RECOGNIZED THAT 

CERTAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASED 

IN THE INTEREST OF MAKING AVAILABLE AS MUCH 

INFORMATION AS POSSIBLE 
RESE4RCH ON OUTPUT FEEDBACK CONTROL

INTERIM REPORT

January 1985 - September 1985

November 1985

Research Supported by NASA-Langley Research Center

NASA Grant No. NAG-1-243

Principal Investigator: Dr. Anthony J. Calise

Research Assistant: Mr. Friedrich S. Kramer

NASA Grant Monitor: Dr. Douglas B. Price

MECHANICAL ENGINEERING AND MECHANICS

DREXEL UNIVERSITY

PHILADELPHIA, PA 19104

(NASA-CF-174253) RESEARCB ON OUTPUT -

W86- 13028

PEED BACK CGHTHOL interiD report, Jas.

Sep. 1985 (Drexel Univ.) 20 p HC &0 2/11F A01

12A Cnclas

G3/64 04786

iL
-iSUMP' CRY

The research for the current year has dealt with the synthesis of fixed

order dynamic compensators for multivariable linear time invariant systems by

minimizing a linear quadratic performance cost functional. In particular,

attention has been given to robustness issues in terms of frequency domain

specifications, namely low frequency performance, high frequency robustness to

unmodeled dynamics and parasitics, and cross-over frequency sensitivity.

In designing fixed order compensators, an output feedback formulation has

been adopted by suitably augmenting the system description to include the

compensator states. However, the minimization of the performance index over

the range of possible compensator descriptions was impeded due to the

noruaiqueness of the compensator transfer function. A controller canonical

form of the compensator was chosen to reduce the number of free parameters to

its minimal number in the optimization. In the MIM0 case, the controller form

requires a prespecified set of ascending controllability indices. This

constraint on the compensator structure is rather :..:^ruous in relation to the

increase in convergence rate of the optimization. Moreover, the controller

form is easily relatable to a unique controller transfer function description.

This structure of the compensator does not require penalizing the compensator

states for a nonzero or coupled solution, a problem that occurs when following

a standard output feedback synthesis formulation.

Unlike full state LQR designs, output feedback designs do not have

guaranteed gain and phase margin properties. For that matter, neither does

an LQG design. Recent robustness recovery techniques try to asymptotically

approximate the full state LQR design by incorporating fictitious noise in the

performance index. A cheap control formulation results with high frequency
4 dynamics appearing in the compensator. The approach taken in this research is

to extend the concept of frequency -dependent cost functionals to the output

feedback problem. Lead - lag performance weightings have been used to

j

mvnipulate the performance index and provide a method to expand the

 compensator to enhance the systems closed loop robustness. Results on SISO

systems confirm our hypothesis that output feedback dynamic compensators can

t

f be designed to meet phase margin requirements of the system without

sacrificing control bandwidth constraints. Design examples for MIMO systems

are currently underway, and similar results are expected.

This report summarizes the results to date on the SISO case. The

contents of this report will be submitted for pr, ^ntation at the next AT.AA

Conference on Guidance, Navigation and Control.
z

1. Introduction

It is well known that Linear Quadratic Regulator (LQR) synthesis methods

have guaranteed stability margins. Unfortunately, these properties hold

only in case of full state feedback. Observer based compensator design

techniques exist to estimate un-tvailable plant states, and make the LQR

design viable. However, this combination of state estimation and regulation

may result in a compensator with poor stability margins, even though the

separate designs are robust. It has been shown that the robustness properties

of the LQR design for nonminimum phase plants can be recovered via an

asymptotic method called Loop Transmission Recovery (LTR) [1]. The LTR

method relies on a cheap control formulation with a subset of the compensator

dynamics becoming fast. In this post analog era, finite word length, sampling

rate or time delay may impede the use of high gain controllers when

implemented digitally. Several other shortcomings of the LTR method that are

related to plant inversion are discussed in [2]. Aside from the robustness

issue, the order of the compensator when designed for large scale systems may

prove unwarranted.

Optimal output Feedback design of fixed order compensators introduced in

the early seventies [3] has received limited attention throughout the years.

Part of the difficulty rests with the inability to characterize the stability

margin properties. Unlike the algebraic Ricatti equation that arises in the

case of full state feedback, the necessary conditions that result from the

optimal output feedback problem are no g conducive to analysis in the frequency

domain. The other difficulty in synthesizing optimal dynamic compensators is

that the standard approach of adjoining the compensator dynamics to the plant

dynamics (and reformulating as a static gain output feedback design) results

In an over parameterized formulation. This is a direct consequence of the

.A

-1-

7t
"f F {

i

fact that the compensator lacks a predefined structure, which invariably

results in difficulties with convergence to qn optimal solution. This is

avoided here by representing the compensator in controller canonical form.

This paper presents an LQ optimal design approach for designing fixed

order compensators that satisfy frequency domain performance and robustness

specifications. Only single-input single-output plants are considered;

however, the approach can be extended to the multivariable case. Implicit in

the system description is the pieaence of noise in the measurement, and to

avoid control activity gene -i , A by this noise direct output feedthrough in

the compensator has been eliminated.

The following section discusses the optimal outpu % feedback formulation

in controller canonical form. Next, it is shown how the concept of

frequency-dependent cost functionals can easily be handled in this canonical

setting. This is useful In improving the stability margins of the

compensator. Finally, examples are given to elucidate the design technique.

-2-
2. Fixed-Oreer Dynamic Compensation

Consider the linear time invariant single-input single-output system

x - Ax + bu x(0)-xo x e Rn

(la)

y - c tX y e R 1 (lb)

The system (1) is controlled by the fixed-order dynamic compensator without

direct feedthrough of the output

u - -ht z u E R l

(2a)

z - -Pz- ny z(0)=0 z e Rnc (2b)

where 0<nc4n is chosen a priori subject to the requirement that there exist a

stabilizing compensator for the closed loop system defined by

L

A -bh t C

A - (3)

t -P

The dynamic compensator (2) is defined in a controller canonical form:

u= -htz

z = Po z + nuc

uc Etz - y

where

0 1 . . . 0 0

0 0 0 0

Po =

0 0 0 1

0 0 0 0

ht - [hl . . . hn I

c

u e R 1(4a)

n

z -, R c (4b)

uc a R l(4c)

0

0

n = _ 0

1

Et = IP1 pn 1 (5)

c

This canonical form reduces the number of free parameters from nc 2 + 2nc to
2nc. Furthermore, the companion form of P . PO-Mt provides structure to the

compensator and is a natural form for eliminating direct feedthrough of the

output. This point will be further explained below.

In the spirit of an output feedback formulation, augment the plant states

to include the compensator states, that is, let x t - [ x t , z t ], yt [z t ) and

u t : [u,uc). The augmented system is then given by

x = Ax + Bu (6a)

Y . CX ( 6b)

u = -Gy u c R2 (6c)

where

-[ A 0

A =

-nc t Po

b 0

B C [ 0 In )

0 n c

^

G 

ht (7)

Pt

Note that the controller form reduces the number of compensator control

elements (uc) from nc to 1. The fixed order dynamic compensation problem has

now been converted to a static output feedback problem where the compensator

gains (h,E) are given by G. Notice that this formulation alleviates the need

for zeroing those elements associated with the static feedthrough which would

arise if the standard output feedback formulation is used.

Th: compensator transfer function

-4-
K(s) - h t (sIn +P) -1 n - h(s)/p(s)

' n -1 n -1 n

(hl + ... + hn s c ),( p l + + pn a c + s c ) (8)

c c

directly correlates with the elements of G in (7). Note that the controller

form is easily relatable to a transfer function description, and as such any

parametric optimization of individual compensator gains provides a direct

transfer function counterpart. In particular by zeroing the ascending

coefficients p1,p2.... of the monic polynomial p(s) definPd in (8), type

1,2,... feedback systems are formed. Likewise the degree of compensator

rolloff can be controlled by zeroing the appropriate elements in ht.

The resulting composite system can now be ast as a parametric optimization problem. The free parameters (h,p) are chosen to minimize the

quadratic performance index

J(h,p) - E{Jm [x t (; + pu 2 + p c u c 2 ] dt) (9)

0

where E denotes expectation over the distribution on x (0), Q - D t D such that

(A,D) is detectable, and p and pc are positive scalars. The input uc ptz-y

drives the compensator, and increasing or decreasing the penalty on u.c

correlates directly with decreasing or increasing the c:mpensator bandwidth.

The solution to the LQ optimal output feedback problem depends on the

Initial distribution of the states. Extending the etandard LQ c timal output

feedback design to include a compensator requires some xvowleage of the

compensator states' initial distribution. One approach is to assume that the

compensator states are known and at rest, and that E{[xo ,zo j[xo ,zo ] t ) - block

diag [Xo,Zo], Zo - On . In practice however, the initial distribution of the

c

compensator states may be needed for the existence of (CLC t ) -1 in the calculation of G. This fact is stated in the following thereom, which is proven in

the appendix. V

-Sa aacva vm

r'

1

A sufficient condition for the existence of (CLC t )- 1 with (P,n,ht)

in controller form is that Zo D nnt.

3. Frequency Shaped Compensation

Frequency-dependent cost functionals have been recently introduced as a

way of embedding classical design concepts within the context of LQ optimal

control [4,5]. These frequency-dependent cost functionals have been developed

for the design of a full state controller and its dual, the state estimator.

When applied to the design of observer based compensators, the resulting

controller is dimensionally larger than the system model due to the

introduction of the frequency shaping terms [4]. In this section we formulate

the use of frequency-dependent cost functionals in the context of LQ optimal

output feedback.

Modifying the performance index in (9) to include input and output

frequency shaping, the cost functional becomes

J(p,h) - E{ 1/2x J [x*(ju,)Qx(jw) + IQy(jw) y (ja) I 2 + p IR(jw)u(j4,) I 2 +

p cIttc(jw)I 2 1&} (,o)

where * denotes complex conjugate transpose, Q, p, and pc are as before, and

Qy(jw) and R(j 4,) are frequency shaping terms with the particular lead-lag

form:

Qy(jU') - (Y jw + 1)/(ej(. + 1) (11a)

R(ju,) - (Bj(. + 1)/(aju + 1) (11b)

'."he weighting functions Qy (j u, ) and R(ju,) were ch,asen for their effectiveness

in shaping the stability margins in classical designs; however, other rational

functions of frequency are possible [4]. Realistically one would select S > a

i

5 

A

-6-
> 0 to penalize higher frequency control activity and 9 > y 0 to weight low

frequency sensor information. More importantly, the frequency shaping

parameters are varied to adjust the loop phase and gain margins while still

optimizing the compensator.

The cost functional in (10) can be expressed in the time domain is

W

J(E,h) - r {x tQx + Y 1 2 + pu1 2 + pcuc 2 )dt (12)

'^ b

with the system ( 6,7) augmented to include the state realization of the

frequency shaping terms (11):

w l - wl/9 =- y w1 a R 1 (13a)

y l - wl(e—Y)/8 2 + Y/6 Y1 a R 1

(13b)

L2 - w2/S + u l w2 c R 1 ul E R 1 (14s)

2

u - w2(6-a ) /^ + ula/s (14b)

The dynamic compensator is expanded to include the states wl and w2:

u l -

gll wl 912w2 - htz (158)

z - Poz + nuc (15b)

uc - - 921w1 922w2 -2 tz - Y1 (15c)

Design via Optimal Output Feedback

The optimization problem can be reformulated using the augmented system

description (6) where x t - [x t , wl, w2, z t ), Y t - [wl, w2, z t ], ut - (ul, uc)

an.i

-7-
A 0 b(B-a?/62 On x nc

Y c t /e -1/9 0 Ot

A =

Ot 0 - 1 /0 Ot

-Y nct /e n(y -e)/ey 0 Po

b ta/B 0 1 Ot

Bt = —

O t0 0 nt

O t 1 0 Ot

C Ot 0 1 Ot

On x n r) 0 In cc

. 911 912 ht

G=

921 922 P t (16)

The frequency shaped fixed order compensator design is now in terms of a

static gain output feedback design, and the solution G does not require any

zeroing of elements. However, as .ill be shown in the next section, zeroing

the gll and 822 terms in G gives a classical interpretation of the Effect

of frequency shaping in the case of output feedback.

Transfer Function Description

The transfer function description of the fixed order dynamic compensator

with lead-lag frequency shaping ( 13-15) is given by

u(s)/y(s) - K(s) - (as + 1)K'c(s)/(es +1) (17)

where

K'c(s) - ((Y s+1+921Y) Kc( s ) - g11Y1 M 0s+1+g12e)-g726Kc(s)J 08a)

7

f

A

i

t

-a-
P

Kc(s) - h t (sI+P) -I n (18b)

By zeroing gll and 822, the frequency shaped compensator, K(s),

simplifies to

K(s) - Fu(s)Kc(s)Fy(s) (19)

where

Fu(s) - (as+l)/(0s+1+8128) (20s)

Fy(s) - (Ys+1+B21Y)/(6s+!) (20b)

which is just the nominal compensator (with its internal parameters

reoptimized) cascaded fore and aft with the lead-lag type first order filters,

Fu(s) and Fy(s), respectively.

Both the input and output frequency shaping cost functions (10) were

introduced to point out the benefits of each. Notice from (17) that by

shaping the output of the system with a lead-lag cost function Q010 in

(lla), an pole is placed in the compensator at -1/6. The location of the zero

in the compensator is influenced by the gains P21 and gl;. Conversel y , by

shaping the input of the system with a lead-lag cost function R(j(,) in (llb),

a zero is placed in the compensator at -1/a. The location of the pole in the

compensator is _nfluenced by the gains 912 and 922. Hence, because of the

commutative property of SISO sy stems, the only difference between shaping the

output or input of the system is that either a pole or zero of the resulting

compensator is fixed by the weighting parameters 9 or a.

4. Example

Consider the simple 2nd order SISO 9-stem with state description:

x l - x2

x2 - -3xl - 4x2 + u (21)

Y -2x1+x2

-. - ''lam 
-10-

and loop transfer. function:

G(s) - (s+2)/1(9+1)(s+3)1 (22)

The performance specifications call for 20 dB of gain for frequencies

less than 1 rad/s and ! -cies-over frequency 10 red/s.

A first order compensator (nc-1) was designed to meet the performance

• specifications with associated cost penalties

Q - block diag(Qx ,10 3 1 Qx - dd t , d t - 4151d35,1110-2

P - 0.0035 pc - 25.0 (21)

and

(Xo,Zo1 - diag(1,1,11 (24)

The example and plant weighting matrix, Qx, were taken from 161, where art 1.TR

design was illustrated. The algorithm described in [71 was used to solve the

optimal output feedback problem. This algorithm has guarantees convergence

properties. It also permits a constraint of the fore y(G) - ^ to 1^e

approximated using a penalty function approach. This can he used to zero

selected elements in G, or to p1Ace selected poles or zeros o` the compensator

as will be illustrated below. The above performance weightings resulted in

the following compensator design

K(s) - 124.9/(s+4.86) (25)

Next, a second order compensator was designed to exhibllt type 1 behavior

and have a zero at 1 rad/s. This design was carried out by zeroing the pl

element in G of (7) and by constraining hl to equal h2. The cost penalties

were the same as above with Q and Zo expanded to include the Extra compensator

pp.

s to to

k- Q - block diag (Qx,0,10 3 1 and (Xo,Zo1 - diag 11,1,0,11 (20

The performance index cost increased by 15% and resulted in the compensator

K(s) - 129.9(•+l)/s(s+5) (27)
The magnitude and phase responses of the loop transfer function G(s)K(s) for

the first order compensator (25) and the type 1 second order compensator (27)

are illustrated on a Bode diagram in figure 1.

Lead-Lag Augmentation

The preceeding first order compensator design (25) meets the performance

specification defined earlier in this example and compares favorably with the

full state feedback controller designed in [61. The first order compensator

results in 35 degrees of phase margin; while, the full state design gives 85

degrees. Manipulating the input and output performance weightings to improve

the phase margin of the first order compensator resulted in designs which

strayed from the original specifications with no real improvement in the phase

margin. The performance index was next augmented to include a lead-lag term

at the output

Qy(ju,) _ (0.2j (, +1)/(.02j(L + 1) (28)

which from classical compensation techniques should add about 50 degrees of

phase margin near 10 rad/s. fie compensator was then redesigned with the

associated cost penalties

Q - block diag[Qx,0,10 3 1, p - 0.30 pc - 40.0

and (29)

[Xo,Zo1 - diag [1,1,1,11

Frequency shaping resulted in the compensator transfer function

K(s) - 592.4(9+4.28)/(9+2.05)(s+50) (30)

The phase margin was increased to 75 degrees. Bode diagrams of the loop

transfer function G(s)K(s) for the first order compensator (25) and the second

order compensator with frequency shaping (30) are compared in Figure 2.

-11-

u 

I M
f

^ t

se -

---,---T--1--r-r,"rr ------1---T-Y-r-r1-rT ------- ---- r--r-r-r l-rr 1 1 1 1 1 1 1, 1 1 1 1 1 1 1 1 { 1 1 1 1 1 1 1

, 1 1 1 , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1

1 1 I 1 1 1, 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 40 `rl_-_Z_J..- _ _

► _aJ_al __----J___J. _.l_a_L J_La -___-_J___J,.__L_a _a.Lal A

, 1 1 1 1 1 1

1 ^` 1 1 1, 1 1 i 1 1 1 , 1 1 1

1 1 1 1 , 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1, 1 1 1 1 ^I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G 30 ------ 1 1 1^ 1 1 1 1 ^- 1 1 , 1 1 1 1 1

------^---,r--r-r-r^-rT 1 1 1 1 1 1 1 1

-----------r--r-r-r^r-rT 1 , 1 1 I Iti y 1 1 1 1 1 1 1 1 1 1 1 1 1 / 1 1

1 1 1 ^. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

I 20 ______.1 , 1 1 1 1 1 1 1 1 1 / f l l , 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1

T-

---- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ---- , 1 1 1 1 1 1

1 1 1 1 1 1 1/ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 U 1 1 1 1 1 1 1 1 t l 1 1 1 1 1 1 i 1 t!, 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 C 0 ------,--- --r-r-t ,-r* ------,---Y--1--f-r-rt. ----- ----r --I--f -f tf ♦ / 1 1 , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1/ 1

E

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 I I I I I t

1 I 1 1 1 1 1

I 1 I I 1 1 11

-10 ______J___1_J.._L_LJ_La ___L_L_LJ-La __--

-- -_-L__L_a-LJ_L1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 / 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 _^`^0

1

------ ------1___T_T_r-r l"ri _____-1---T r- T -rTrt

/^ 1 1 1 1 1 1 1 1, 1 1 1 1 1 1 1

1 1 1 1 1 1, 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 , 1 1 1 1

^I I , 1 t t 1 t I / I 1 t , 1 1 1 t 1 1 , 1 1

B

_30 __----^--- _._ ► J_aa -___--.a_---.l.--1__a_ &.4 Y 1 1 1 , 1 1 / 1 I I I t I 1 t/ 1 I 1 1 t 1 1

1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-41-0- 10 102

FREQUENCY (red)

-1e^e

^ 1 1 t 1 1 1 1 1 1 1 1 1 1 1 1 1

^^•^— 1 t 1 1 1 1 1 1 1 1 1 1 1 1 1

i 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 i t 1 e 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 / 1 1 1 1 1 1 1 l i e

----- Y 1 1 1 I 1 1 1 I 1 1 1 1 1 1

-r---r--r-r-rte 1 ------ t --- ^,--4-rn-ri-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 I 1 1 1 1 , 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 t 1 1( 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1, —___—= 1 1 1 1 1 1 1

---^--}^444-4 ------{ - 4 -44444-"

1 1 1 1 1 1, 1 1 1 1 1 1 1 1{ ^^'1^-^^ T TtM-li p 1 1 1 1 1 11 1 1 1 1 1 1 1 1 ^^ 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1^^1 1 1 1 11 1 1 1 t 1 1 1 1 1 1^ 1 1 1 1 1 1

1 1 1 { 1 1 1 1 1 111 1 1 1 1

------•IL---I_^-1 1 { 1 1 -----"-_- J__ ;^! -1 1-A

{ 1 1 1 1 1 1 1 { 1 1 A{ 1 1 1 1 1 t 1 1 1 1 , 1 1 1, ,

1 1 1 { 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 i 1 1 1 1! 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1

1 1 1 1 i/ l t 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 t 1 1 1 1

1

------r--r--r-r-r-rrT 1 1 1 1 1 1 1 1 1 1 1 1 1/ 1 1 1 1 I 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i , 1 1 1 1 1

------ 1 1 1 1 1 / 1 1 r---r--r-T-rrr^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

t 1 1 1 1 1 1{ 1 1 1 1 1 1 1,

------ ---}--F-+-+ l ti I 1 1 t 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 1 1 1 1 1 1 -_____f___f__r_1_f_Ff•

1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1

1 1 / 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 ^ 1 1 1 1 1 1 1 1 ^ 1 1 1 1 1 1 1 1

T 1 1 1 1 1 1 1

10 102

0

-20

P -40

H

A -60

E -80

-100

d -120

e -140

-160

FREQUENCY (red)

Figure 1. Frequency response of G(s)K(s) for the first order compensator

(solid) and the type 1 second order compensator (dashed).

—12—
/3@ --_---,---ti-----r—r,-rr ------,---- • -----,----r--r-r•r r 

1 1 1 1 1 1 1 1 1 1 1 t 1 1 6 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 M S S 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 I 1 1 l i M 1 i 1 , , ! { 1 1! 1 1 , 1 1 1 1 1,

A 20 ----------;---;--}-Fi-t --, - ---^ --s--t-t^-}t -------------^-+-t ,-+t

I 1 1 1 1 1 1 1 ^` 1 1 1 1 1 1 1 1 t 1 1 1 1 1 1

1 1 1 1 1, 1 5 1^ t I 111 ( 1 1 1! 1, 1

G 1 t 1 1 1 I t l 1 .f. 1 1 1 1 1 1 / / ! ! 1 1/

1 1 1 1 1 1 1 1 1 I. 1 1 1 1 1 1 1 1 1 1 1 1 1 N 1@ -----d----- .^_

1 1 1 ► — ► J—f L1 t 1 1 1

----_-J--- 1

J--.^.— LJ—b. 1 1 W 1 1 1 -----J-_ .^_.V

1 1 1 1 1 1 1 1

I 1 1 1 1 1 1 1 1 1 1 1 I t 1 1 1 i 1 I t l l l

i 1 1 t 1 1 1 1 f 1 1 1 1 / 1 1 1 1 , 1 1 1

1 , 1 1 1, 1 1 , 1 1 , 1 1 I 1 1 1 , 1 1 1

T @ ------ 

1 1 /

ti---^-•-r-r1 1 1 1 1

-rte—r• ----- 1

------

1 1 1 1 1 1

r-^--r—rte-r ♦

_ 1 1 1 1 1 1 1 1

-- -^_-•—r--r-r—r-rrt

1 1 1 1 1 1/ 1 1 1 1 1/ 1 1 1 ^` 1 1 1 i 1 1 1 1

1 ( 1 1 1 1 1 1 1 1 1 / 1 1 1 1 1 1 i 1 1 1 1 1

/ 1 / 1 1 1 1 1 1 1 1 1 1 1 1 1 1^ 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^1 A 1 1 1 1,

Q -10 ----------^- ;--E-Fi-F+ -----------;----F-f-F} ------ ---i4`

-^-+-ti-tt

1 1 1 1 1 i 1 1 i 1 1 1 1 1 1 1 ! 1 l ! S i / 1 E 1 1 1 1 1! 1 1 1 1 1 1 1 1 1 1 1 , ^) 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , T^ 1 1 S S

^ 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1!

II^^ ^f .20 ------J---'^—J'—►

._LJ—L ♦ --_—__J----^'.. —L_L—L A—L ♦ ------J----L L_L—L.11r < ♦

`i/ 1 1 1 1 1 1 1 1 t 1 1 ( ' 1 1 1 1 1 t 1 1( i

1 1 1 1 1 1 1 1 ! / 1 1 1 11/ 1 1 1 1 1 D l

1 1 1 1 1 1 1 1 1 1 1 ( 1 1 1 1 1 1 1 1 1 1 t

1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 ' 1 1 1

/ 1/ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — 30 ----------- -----------

1 1 1 1 I l t l 1 1 , 1 / 1 1 1 1 1 1 1 of

B

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t 1 1 1

1 1 1 1 1 1( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

FREQUENCY (red)

10 102

0

------- ----r--r-r—r,—rr

1 , 1 1 1 1 1 1

1 1 1 1 1 1 1

—^@ ---- 1 1 I 1 1 1 1

__j—_— V_L—LJ_LL

1 1 1 1 1 S 1

1 ! 1 1! 1 1

}]r —4@ 1 1 1 1 1 1 1

------ 4-4-44-1.,

 1 1 H

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

60

1 1 1 1 1 1 1 1

1 1 A 1 1 1 1 1

1 1 1 1 1 1 1 1 S 1 7 1 1 1 1 1,

E -80 ------ ---t---^-t-ti-tt

1 1 1 1 1 1 1 1

1 { 1 1 1 1 1 1

-100 ------^---:.-^-;-:, ^: 1 1 1 1 1 1 1 1

1 1 1 1 I f l l

1 1 { 1 1 1 1 1

d -120-------;----^--; -}-;-^

1 1 1 1 1 1 1 1

1 1 / 1 1( 1 1

-140------- ----Y-P-^;-^^ ('^ 1 1 1 . 1/ 1 1

7 1 1 1 A 1 1 1 1

-16 L

1 1 

.

1 1 1 1 1 1

__------L.► — ► _LJ-VL 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1

I 1 1 1 1 1 1 1

1 1 1 1 , 1 1 5

I 1 ! 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 ! 1 1 S 1

1 1 1 1 f ^ 1 1

1 S 1 1 1 1 1 1

1 1 1 1 1 5 1 1

i 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

i 1 i 1 1 1/ 1

1 1 t 1 1 1 1

------i ^3--i--^i-tiff

1 1` J -.1 ^ ^ ^ 1 1 T

1 1 i 1 ^'► j l l

1 1 ! 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 t 1

1 1 1 1 S 1 1

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 t 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

I 1 1 1 1 1 1 1

I 1 1 1 1 1 1 1

1 1 1 i 1 1 1 1

—_ ----r---r-- r — T - r -i-H

1 1 1 f 1 1 1 1

1 1 1 , 1 1 1 1

1 1 i i 1 1 1 1

1 S 1 I l l l t

1 t 1 1 1 1 1 1

1 1 1 / 1 1 1 1

------t---; -44-t-'; t+

i 1 1 1 1 1 1 1

/ , 1 , 1 1 1

7 , 1 1 1 1 1

^^ t 1 1 t 1 1 1 1

^1 1 1 1 1 1 1 1

1 ( 1 1 1 1 1

t 1 ^!, 1 t t l l

I 1 1^ I l l f

1 1 i 1 1 1 1 ------r---r--r-tJ1T r^

1 1 1 1 1? 1 1

1 1 1 I l e 1^1

t 1 1 1 1 1 1 1

1 1 1 1 1 1

'1e4`- 1 10 102

FREQUENCY (red)

Figure 2. Frequency response of G(s)K(s) for the first order compensator

(solid) and the second order compensator with frequency shaping

(dashed).

—13—
5. Conclusions

A design methodology has been introdt--ed for the LQ synthesis of fixed

order dynamic compensators without output feedthrough for SISO systems. A

controller canonical structure was imposed on the compensator description

which minimized the number of free parameters and correlated with a unique

transfer function realization. A frequency-dependent cost functional provided

the means to enhance the robustness of the fixed order compensator design to

improve the stability margins of the closed loop system. In particular,

first order lead-lag type cost functionals were introduced to shape the input

and output of the system and were shown to have a classical lead-lag

counterpart when used with the dynamic compensator.

-14-
Appendix

Proof of Theorem

The necessary conditions satisfying the LQ optimal output feedback

problem are given by

A tK+KA+Q +CA P C =0 (Al)

!L + LA + XXo 0 (A2)

RG(CLC t ) = B tKLC t(A3)

where A A -WC is asymptotically stable, the quadruple (A,%,G,G) is defined

In (6) and (7), Q > 0, R = diag [p,pc) > 0 and XXo = block diag [Xo,Zo)> 0.

In order to uniquely determine G in (0), it is necessary that (CLC t ) -1 exist.

For the form of C in (7), this inverse can be reduced by first partitioning; L

into

[L11 L12

L t

L 12 L22

and expanding CLC t . Thus

CLC t = L22

w ►

and the existence of (CLC) --1 is solely dependent

(L22) -1 -

Expanding equation (A2) results in

t

PL22 + L22P t + L12 cn t + nc tL12 + Zo =

(A4)

-15-
Referring to [81 and the references therein, the following remarks hold.

Equation (A6) can be written as

(P + nctL12L22)L22 + L22(P + nctL 12 L22 )t + Zo - 0 (A7)

where + denotes a pseudoinverse. Since (P,n) is controllable, then so is

(P + nc tL12L22,n). Let Zo - Mt . Furthermore, since L22 is at least

non-negative and (P + nctL12L22,1) is controllable, then (P + nctL12L22) is a

stability matrix; and the solution L22 to (A7) is positive definite.

We can broaden the class of Zo to that which makes (P + nctLl2L22,D)

controllable, where DD t - Zo ::) nnt.

-16-
References

1. Doyle, J.C. and G. Stein, "Multivariable Feedback Design: Concepts for a

Classical/Modern Synthesis," IEEE Trans. Automatic Control, Vol. AC-26,

No. 1, February 1981, pp. 607-611.

2. Ridgely, Capt. D.B. et. al., "An Application of LQG/LTR to an Unmanned

Aircraft," AIAA Guidance and Control Conference, Snowmass, CO, August

1985, paper no. AIAA-85-1927-CP.

3. Johnson, T.L. and M. Athans, "On the Design of Optimal Constrained

Dynamic Compensators for Linear Constant Systems," IEEE Trans. Automatic

Control, Vol. AC-15, December 1970, pp. 658-660.

4. Gupta, N.K., "Robust Control/Estimation Design by Frequency-Shaped

Cost Functionals," presented at the 20th IEEE Conference on Decision

and Control, San Diego, Calif., December 1981, pp. 1167-1172.

r; 5. Anderson, B.D.O. and D.L. Mingori, "Use of Frequency Dependence in

Linear Quadratic Control Problems to Frequency-Shape Robustness,"

AIAA Journal of Guidance and Control, Vol. 8, No. 3, May-June 1985,

pp. 97-401.

6. Doyle, J.C. and G. Stein, "Robustness with Observers," IEEE Trans.

Automatic Control, Vol. AC-24, No. 4, August 1979, pp. 607-611.

7. Moerder, D.D. and A.J. Calise, "Convergence of a Numerical Algoritrm

for Calculating Optimal Output Feedback Gains," IEEE Trans. Automatic

Control, Vol. AC-30, No. 9, September 1985, pp. 900-903.

8. Hyland, D.C. and D.S. Bernstein, "The Optimal Equations for Fixed-Order

Dynamic Compensation," IEEE Trans. Automatic Control, Vol. AC-29, No. 11,

November 1984, pp. 1034-1035.

-17-