NASA Contractor Report 182302 

Stability Robustness Improvement 

of Direct Eigenspace Assignment 

Based Feedback Systems Using 

Singular Value Sensitivities 

Sanjay Garg 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

Cleveland, Ohio 

August 1989 

Prepared for 

Lewis Research Center 

Under Contract NAS3-25266 

National Aeronautics and 

Space Administration 

[a AS A-CR- 18230 2) 

IMPROVEMEOT OP DIRECT EIGEMSPACE US16MMmJT 

BASED FEEDBACK SYSTEMS USING SIHGULAR VALUE 

SlSElSfTIBITXES Final Beport (Sverdrnp Unclas 

Technology) 32 p CSCL OtC G3/08 0225000 

STABILITY ROB USTbiE S S M09-2767 2 
Stability Robustness Imnrovernent of Direct Eierensnace Assiment 

Based Feedback Svstems Using Sindax - Value Sensitivities 

Sanjay Garg] 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

Cleveland, Ohio 44135 

Abstract 

A methodology to improve the stability robustness of feedback control systems 

designed using direct eigenspace assignment techniques is presented. The method consists 

of considering the sensitivity of the minimum singular value of the return difference 

transfer matrix at the plant input to small changes in the desired closed-loop eigenvalues 

and the specified elements of the desired closed-loop eigenvectors. Closed-form expressions 

for the gradient of the minimum return difference singular value with respect to desired 

closed-loop eigenvalue and eigenvector parameters are derived. Closed-form expressions 

for the gradients of the control feedback gains with respect to the specified eigenspace 

parameters are obtained as an intermediate step. The use of the gradient information to 

improve the guaranteed gain and phase margins in eigenspace assignment based designs is 

demonstrated by application to an advanced fighter aircraft. 

Introduction 

A fundamental objective in the design of flight control systems is to change the 

transient response of the flight vehicle to a desirable one using feedback control. As 

discussed in Ref. [l], the so called direct eigenstructure (or eigenspace) assignment 

techniques are really well suited to designing feedback control systems to meet this 

objective. Various applications of the eigenspace assignment techniques have appeared in 

Controls Engineer, Member AIAA. 

1 
the literature in the recent past. Some of these are - design of pitch pointing flight control 

systems [2], design of flutter and gust load alleviation systems [3], and transient response 

shaping for flexible vehicles [4]. All these applications were multivariable in nature and 

demonstrated that direct eigenstructure assignment is a viable multi-input multi-output 

(MIMO) control system design technique. 

One of the drawbacks of direct eigenspace techniques, as compared to some other 

multivariable techniques - specially the Linear Quadratic Gaussian/Loop Transfer 

Recovery (LQG/LTR) approach [5], is that the synthesis procedure does not guarantee 

stability robustness with respect to variations in plant dynamics. Even using the Linear 

Quadratic Regulator (LQR) based methodology to asymptotically approach the desired 

eigenspace [6] does not guarantee the well known stability margins of LQ Regulators [7] as 

the procedure results in non-diagonal control weighting matrices which violates the 

conditions under which the LQ Regulator stability margins are guaranteed. Also, as 

discussed in Ref. [4], the direct eigenspace assignment techniques are preferable to the LQR 

based approach because the LQR approach requires very high actuator bandwidths for the 

desired eigenstructure to be achieved. 

In direct eigenspace assignment techniques the design parameters are the desired 

closed-loop eigenvalues and specified elements of the closed-loop eigenvectors. Once the 

design parameters are specified, the feedback control gains are uniquely determined 

(provided enough parameters are specified - see Ref. [8] for discussion of limits on 

achievable eigenspace using direct eigenspace assignment). So, given a set of specificat ions, 

the feedback control gains will provide the desired closed-loop transient response (or come 

as close to it as possible within the system constraints), but they might result in a system 

with poor stability robustness, i.e. a small change in the plant dynamics may cause the 

closed-loop system to go unstable. The designer is then faced with the dilemma of how to 

change the design specifications such that the resulting feedback system will also provide 

adequate stability robustness. Note that in general the designer does have a certain amount 

2 
of freedom in choosing the design specifications - rarely does he want an exact value for a 

closed-loop eigenvalue or exact shape for a corresponding eigenvector, the specifications 

are rather in terms of desired regions for the closed-loop eigenvalues and acceptable sets of 

eigenvector shapes. The objective of this paper then is to develop a methodology which will 

provide adequate information to the designer to change the design specifications in a 

systematic step-wise manner such that at each step the guaranteed stability robustness of 

the feedback system is improved while the eigenstructure is within the desirable regions. 

.. 

In multivariable feedback systems, a reliable (but sometimes conservative ['i]) 

measure of stability robustness is the minimum singular value of the return difference 

matrix at the plant input evaluated as a function of frequency. The methodology presented 

in this paper is based on sensitivities of the minimum singular value of the return difference 

matrix to the design parameters, which in this case are the desired closed-loop eigenvalues 

and eigenvectors. Note that the notion of using return difference singular value sensitivities 

to design robust controllers is not new. Singular value sensitivities to compensator 

parameters were used in Ref. [9] to directly design robust reduced-order compensators, and 

singular value sensitivities to plant parameters (elements of the plant system matrices) 

were used in Ref. [lo] to determine which elements need to be modeled more accurately for 

the feedback system to guarantee stability. 

In the following a technique for solving the feedback gains for direct eigenspace 

assignment is first briefly described, and the mathematical problem formulation for deriving 

analytical expressions for the return difference singular value sensitivities is presented. 

Closed-form expressions for the singular value gradients with respect to closed-loop 

eigenvalues and elements of the closed-loop eigenvectors are then developed by first 

considering the singular value gradients w.r.t. the feedback gains and then deriving the 

closed-form expressions for the gradients of the feedback gains to the closed-loop 

eigenspace parameters. Finally, the use of the gradient information to improve guaranteed 

stability robustness is demonstrated by application to a modern fighter aircraft. 

3 
After the completion of the present manuscript, it was brought to the author's 

notice that somewhat similar research has been reported earlier in Ref. [ll]. However, as 

pointed out in a later section, the research reported herein differs from that reported in 

Ref. [ll] in the important detail of the choice of independent parameters with respect to 

which the singular value sensitivities are calculated. 

Problem Formulation 

Direct Eigenspace Assignment Gain Synthesis 

In the direct eigenspace assignment technique, the control objectives are stated in 

terms of a desired eigenstructure for the augmented system. For the full-state feedback 

case, the synthesis problem is as follows : 

Given a linear, time-invariant dynamical system with the state-space 

represent ation 

- 

x=AF+Bu 

with xdRn and KtlRm, find a control law of the form 

- 

u = -I<? 

to achieve some desired eigenspace for the augmented system 

X = (A-BK)F (3) 

To determine the feedback gains K, note that the augmented (closed-loop) system 

eigenvalues and eigenvectors are related by 

(4) - (A-BK)Vc. = X.v , i = 1 ,... n 

where Xi is the ith closed-loop eigenvalue and Tc. is the corresponding closed-loop 

eigenvector. 

1 ci 1 

1- 

For full-state feedback, the limitation on the achievable eigenspace is that all the 

desired closed-loop eigenvalues can be exactly placed while only llm" elements of their 

associated eigenvectors can be exactly achieved [8] (here, m = dimension of and it is 

4 
assumed that the system given by (1) is controllable). Since in general m<n, we cannot 

exactly obtain all elements of the desired eigenvector for each closed-loop mode. One 

approach for determining the feedback gains is to obtain the "best" achievable 

eigenvectors, for each of the closed-loop modes, so as to minimize the mode's cost function 

Ji given by 

* 1- - 

211 

J. = - (v -Vd.) Qi (Va.-V ) , i = 1 ,... n 

1 di 1 a. 

where 

- 

v = ith achievable eigenvector associated with eigenvalue Xi 

'd.= ith desired eigenvector 

Q. = ith n-by-n symmetric positive semi-definite weighting matris 011 

a. 1 

- 

1 

1 

eigenvector error elements, 

* 

and [ -1 denotes complex-conjugate transpose of [ .]. 

Equation (4) can be rewritten as 

(XiI-A)v = -BKT 

a. a. 1 1 

-a Defining the vector wi = -KV 

a. 

eqn. (5) is obtained as (see Ref. [4] for a complete derivation) : 

and using eqn. (6), the solution to minimizing the cost in 

1 

where Li = (XiI-A)-lB. 

Once Wi are obtained, the achievable eigenvectors are given by 

- - 

v = L.w. i = 1, ..a 

a. 11' 

1 

and the feedback gains are obtained as 

K = -WV-l (9) 

- - - where W = W2 .... wn] and V = v .... v 1. 

"1 "2 an 

Note that this algorithm requires that the specified closed-loop eigenvalues Xi be 

distinct and different from the open-loop eigenvalues (eigenvalues of plant system matrix 

5 
A). Also, we are considering full-state feedback rather than reduced-order output feedback 

because reduced-order output feedback further restricts the achievable eigenspace [SI, and 

full-state feedback can always be implemented with state estimation without any 

significant loss in stability robustness by using either the loop recovery procedure of the 

LQG/LTR approach or an eigenspace assignment based loop recovery procedure discussed 

in Ref. [3]. 

Problem Statement 

Given a multivariable system as in eqn. (1) with a control law of the form (2) and 

the state feedback gains given by eqn. (9) such that the closed-loop eigenvalues are 

A = i = 1, ... k , j = a and bi>O -yi+jbi x 2i-l,2i , 

and A. = -ql . , i = 21~+1, ... n 

1 

and the desired eigenvectors are 

- 

and Vdi= col.[u..] for X. = -7. 1' 1J 1 

we wish to derive analytical espressions for the sensitivities of the 

of the return difference matrix at the plant input to the specified 

i.e. closed-form expressions for the partial derivatives 

aa[I+I<G(s)] - 

minimum singular value 

eigens tructure elements, 

where G(s)=(sI-A)-lB with s being the Laplace operator, - a[ -1 denotes minimum singular 

value of [e], and t represents the eigenspace parameters yi, bi, vi, p.., p.. and u.. defined in 

(10) and (11). 

1J 1J 1J 

In Ref. [Ill, the parameters with respect to which the singular value sensitivities are 

calculated are the desired closed-loop eigenvalues Xi (and hence -yi, bi and qi as above), and 

the elements of the vectors vi given by Eqn. 7. (Wi corresponds to the ti of Ref. [ll]). As 

seen from Eqn. 7, Wi depend on the choice of Xi, and hence it is inappropriate to choose wi 

and Xi as independent parameters for calculating the return difference singular value 

6 
sensitivities. It is important to note that the design specifications available to the control 

designer are in terms of the desired eigenvalues and eigenvectors, Xi and V respectively, 

and not in terms of Ai and Wi (ti). Therefore, it is more meaningful to choose Xi and as 

the independent parameters with respect to which the stability robustness singular value 

sensitivities are obtained. 

di 

di 

The discussion in the present paper is limited to singular value sensitivity analysis 

for the return difference matrix at the plant input. In general, it is important to guarantee 

the stability robustness at the plant output also. As will be apparent from later discussions, 

the ideas presented in this paper can be extended to deriving sensitivities of the minimum 

singular value of the return difference matrix at the output simply by replacing I+I<G(s) 

with I+G(s)K. Also note that the singular value sensitivities to the natural frequency and 

damping, wn and < respectively, of a complex closed-loop mode can be obtained by using 

dw n. ari awn 

1 

and 

with 

aa - 

+- - asi awn 

where we have used the equalities yi = Ciwn., and 6. = wn.j I--<;. 

1 

1 1 

Lehtomaki et al. [7] have shown that if 

- u[I+KG(jw)] >_ a. , 0 <w <o;, (13) 

for some constant a. 5 1, then simultaneously in each loop of the feedback system there is 

guaranteed gain margin (GM) given by 

7 
1 

l*ao 

GM = - 

and also a guaranteed phase margin (PM) given by 

Therefore 

move the 

PM = ic0s-l il 

I 21 L 1 

the gradient information provided by expressions of the form (12) can be used to 

eigenspace parameters within the desirable regions in a systematic, step-wise 

manner such that the lowest value of the minimum singular value of the return difference 

matrix, and hence the guaranteed gain and phase margins: are improved at each step. 

Singular Value Sensitivity Derivation 

In Ref. [12] it has been shown that for a general complex matrix H of rank m which 

has distinct singular values ai, i = 1, ... rn? the sensitivity of the singular value a. 1 with 

respect to a real parameter p is given by 

where vi and Ui are the right and left singular vectors [13], respectively, corresponding to 

the singular value ai. 

The eigenspace parameters defined in (10) and (11) are all real and the open-loop 

state frequency respoiise matrix (G(jw)) is independent of these closed-loop eigenspace 

parameters. Therefore, using (16), we get 

ag[ I + K G ( ju)] 

a< 

where 

corresponding to the minimum singular value - a. With I< given by (9), we have 

and U are the right and left singular vectors, respectively, of [I+I<G(jw)] 

The expressions for the feedback gain sensitivities to eigenspace parameters are fully 

expanded in the Appendix by first considering the eigenvalue parameters (vi, yi and fii) and 

8 
then the eigenvector parameters (v.., p.. and p..). An algorithm which uses the sensitivity 

information for stability robustness improvement is briefly discussed in the following 

section and finally an example application is presented. 

1J 1J 'J 

Stability Robustness Improvement Algorithm 

Based on the derivation of the return difference singular value sensitivities to 

eigenspace parameters in the previous section and the Appendix, a step-by-step algorithm 

for improving the guaranteed stability robustness of direct eigenspace assignment feedback 

control designs is as follows : 

Step 1 : 

Sten 2 : 

Sten 3 : 

Sten 4 : 

Sten 5 : 

Formulate the eigenspace requirements; the desired closed-loop eigenvalues 

and closed-loop eigenvectors, and identify the design freedom available in 

the closed-loop eigenspace specification. 

Solve for the control feedback gains K, and check for the guaranteed stability 

robustness using the minimum singular value of the return difference matrix 

at the input, - a[I+MG(jw)]. If the guaranteed stability margins are 

acceptable, then stop, otherwise go to step 3. 

aw i3v 

a< 

Calculate - and - for the eigenspace parameters <, identified in Step 1, 

for which some design freedom is available. The calculation procedure is as 

discussed in the Appendix for each case of eigenspace parameter. 

da - [ I+ I< G ( jw) ] dK Using the results of Step 3, calculate - from (18) and then 

a< 

from (17). 

Using the information from Step 4, make small changes in the eigenspace 

parameters such that - a[I+KG(jw)] will increase in the desired frequency 

region while making sure that the changed eigenspace parameters are within 

the bounds specified in Step 1. Go to Step 2. 

The above algorithm can be implemented in the form of a constrained optimization 

9 
technique with the objective of minimizing the area under a specified value for the 

minimum return difference singular value frequency response within the constraint that the 

design parameters stay within certain specified bounds. Such an optimization approach is 

discussed in Ref. [9]. However, the choice of changing the eigenspace parameters in step 5 

requires a considerable amount of "engineering intuition" which cannot be easily put in the 

form of a computer program. Therefore, for the present study, a computer program was 

developed just to implement steps 2 to 4 and the selection in step 5 was based on 

knowledge about the particular dynamics being controlled. 

Example 

The flight vehicle model considered is the short period approximation for the 

AFTI/F-16 aircraft as discussed in Ref. [a]. The model is for a flight condition 

corresponding to an altitude h = 3000 ft and Mach number M = 0.6. The equations of 

motion are given in the form of (1) with 

T - 

x = [Y, q, Q, 'e, 'd 

where 

y = flight path angle 

q = pitch rate 

Q = angle of attack 

be = elevator deflection 

bf = flaperon deflection, 

and 

where 

- 

u=[6 s IT ec' fc 

SeC = elevator deflection command 

bfC = flaperon deflection command. 

The system matrix A aid the control distribution matrix B have the following numerical 

10 
values : 

A= 

0 0.00665 1.3411 0.16897 0.25183- 

0 4.56939 43.223 -17.251 -1.5766 

0 0.99335 -1.3411 -0.16897 -0.25183 

0 0 0 -20 0 

0 0 0 0 -20 

; B = 

- 00 

00 

00 

20 0 

0 20 

The eigenvalues of the open-loop system are given by 

unstable short period mode I X --7.662 1- 

X2 = 5.452 

X3 = 0.0 

X - -20 elevator actuator mode 

X - -20 flaperon actuator mode 

pitch attitude mode 

4- 

5- 

The control design objective is to provide decoupled tracking of flight path and 

pitch attitude commands with a well damped response and zero steady-state error to step 

commands. The control law is of the form 

(19) - 

u = Fy - KX 

C 

with = [y , 0 IT. F is the feed-forward gain matrix and K is the feedback gain matrix. 

The feedback gains are to be obtained using direct eigenspace assignment techniques with 

the objective of providing decoupled flight path and pitch response modes. The feedback 

gains should also be such as to guarantee gain margins of at least i3.5 dB for simultaneous 

gain changes in each control loop at the plant input and phase margins of at least h30 deg 

for simultaneous phase changes in each control loop. These stability margin specifications 

translate into the requirement that - a[I+KG(jw)] 2 0.31. Once a set of feedback gains that 

satisfy the transient response and stability robustness requirements are obtained, t,he 

feedforward gains (F) will then be obtained using a special case of Broussard's command 

generator tracker [14]. As discussed in Ref. [2], a set of feedforward gains that provide 

command following with zero steady-state error is given by 

C cc 

= 022 + KO12 (20) 

with 0.. given by 

1J 

11 
.=[ O11 .12 I=[ AB 

.21 O22 HO 

where H is defined by 

y = HF (22) 

T with 7 being = [y, 03 . the controlled variables of the plant. For the present example, 

A set of feedback gains was obtained initially to achieve the desired eigenspace 

listed in Table 1. In Table 1, the short period frequency and damping (corresponding to 

were chosen to be wn = 7 rad/s and < = O.S, respectively, so as to meet the 

MIL-F-8785C [15] specifications for Category A, Level I flight, and the flight path mode 

(A,) was chosen to provide adequate bandwidth for flight path control. The eigenvectors 

corresponding to the short period mode and the flight path mode were chosen to minimize 

the coupling between the pitch rate and the flight path angle. The actuator mode 

eigenvalues were chosen to be close to their open-loop values and the corresponding 

eigenvectors were chosen to minimize actuator cross-feed. Note that we are only specifying 

two elements for each desired eigenvector ('XI in Table 1 denotes arbitrary), and since we 

have two control inputs the specified eigenspace can be exactly achieved as seen from the 

achieved eigenvectors listed in Table 1. The control feedback gains for this initial design 

are also listed in Table 1. The minimum singular value of the return difference frequency 

response at the plant input is shown in Fig. 1. We note from Fig. 1 that the minimum 

singular value at low frequency is much lower than that required to guarantee the desired 

stability margins. Therefore, although this initial design will meet the performance 

requirements, a control law redesign is necessary for the stability robustness specifications 

to be met. 

A1,d 

The singular value sensitivity procedure was then applied to the initial feedback 

design. The eigenspace parameters for which there is design freedom are : 

(a) Short period frequency and damping (and hence X ). Based on R/LIL-F-87S5C, the 

allowable regions for Level I flight response, at the given flight condition, are 0.35 5 

172 

12 
5 5 1.35 and 2.5 5 wn 5 S.5 radslsec. 

Flight path mode (A3). Any value 2 1 rad/s will provide adequate flight path 

control, however there will be an upper limit due to maximum allowable flap 

deflect ion. 

The desired eigenvector elements Vd (and corresponding element v ) and vd , 11 d2 1 32 

i.e. the short period contribution to the flight path and the flight path mode 

contribution to pitch rate. Although these elements were chosen to be 0 (zero) in 

the initial design, the only requirement is that they be "small" (<< 1) in order to 

keep the coupling of the modes to a low level. 

There is no design freedom available in placing the actuator modes as it is desirable to keep 

these close to their open-loop values. 

The predicted change in the minimum singular value of the return difference matris 

for a 10 % (percent) increase in the short period frequency and damping, obtained using 

8U - dU 

- and 1 respectively, is shown in Fig. 2 and that for a 10 % increase in the flight path 

'*n 

mode (/A,[) is shown in Fig. 3. The return difference singular value sensitivities to the real 

and imaginary parts of the desired eigenvector element Vd , pll and pll respectively, are 

shown in Fig. 4 and the sensitivity to eigenvector element Vd (v~~) is shown in Fig. 5. 

From these figures we note that changing the eigenvector element v , decreasing pll and 

increasina will be most effective in increasing the minimum singular value of the 

return difference matrix. 

11 

32 

dl 1 

41' 

Based on the results in Fig. 5, letting vd = -0.01+~.01, and keeping the rest of 

the desired eigenspace parameters the same as in Table 1, should result in an increase of 

20.115 in the lowest value of the minimum singular value of the return difference matrix 

while still keeping the coupling between flight path and pitch rate, for the short period 

mode, to be very low. The minimum return difference singular value with the feedback 

11 

13 
gains corresponding to this choice of eigenspace parameters, referred to as trial 1 from here 

onwards, is shown in Fig.6. Also shown in Fig. 6 is the minimum singular value predicted 

using the singular value sensitivities, i.e. a[I+K G(ju)] - 0.01 - + 0.01 - where ICo 

are the feedback gains corresponding to the initial design. Note that although there is not, 

very good agreement between the actual and predicted values at low frequencies. the 

sensitivities did accurately predict the direction of the change (increase) in the sinsular 

value. Also note that the actual feedback gains for trial 1 and those predicted using 

and - , both listed in Table 2, are virtually identical. From Fig. 6 we get 8K 

- a[I+KG(jw)] 2 0.4 for trial 1, which is much improved over the initial design but still not 

high enough to meet the stated stability robustness requirement. Therefore we need to 

further change the eigenspace design parameters in order to improve the guaranteed 

stability margins. 

aa aa - - 

1 

- 0 

1 

- dI< 

%l %l 

For trial 1 the singular value sensitivity calculations showed that the most effective 

way to increase the minimum singular value of the return difference matrix was to further 

decrease pll and increase pll. However, doing so will result in increased coupling in the 

flight path and pitch rate response for the short period mode which will be undesirable. 

Next to the desired eigenvector element vd , the return difference singular value was most 

sensitive to changes in the short period mode. The changes in - a[I+I<G(jw)] from that for 

trial 1, using singular value sensitivities for a 10 % increase in short period frequency and a 

10 % increase in damping are shown in Fig. 7. From Fig. 7 we note that increasing both the 

short period frequency and aamping by 10 % each over that for trial 1 will result in an 

increase of 0.114 in the minimum singular value of the return difference matrix. We will 

then have - a[I+KG(jw)] 2 0.51 which will satisfy the design requirement. 

11 

The short period frequency and damping for a 10 % increase over that for trial 1 are 

wn = 7.7 rad/s and C = 0.85 which are within the region for Level I flight requirements. 

14 
The desired and the achieved eigenspace for this change are summarized in Table 3 and the 

corresponding control feedback gains are also listed there. The resulting minimum singular 

value of the return difference matrix is shown in Fig. 8. From Fig. 8 we note good 

agreement between the actual singular value response and that predicted using singular 

value sensitivities. For this case also there was excellent agreement between the actual 

feedback gains and those predicted using feedback gain sensitivities for trial 1. The plot in 

Fig. 8 shows that this set of feedback gains satisfy the stability robustness design 

requirement, therefore this case will be referred to as the final design. The closed-loop 

performance of the initial and the final designs is compared in the following. 

The feed-forward gains for the initial and final designs, calculated using eqns. (37) 

and (38), are listed in Table 4. The response of the initial design to a unit step gamma 

command (Tc(t) = 1 deg) is shown in Fig. 9. Shown in Fig. 9 are the time histories of the 

controlled variables y and 8, and also the control input deflections be and bf. The response 

of the final design to a unit step gamma command was identical to that of the initial 

design, so those time histories are not shown here. From Fig. 9, then, we note that both the 

designs provide well-damped tracking of flight path commands with a reasonably fast rise 

time and without any perturbations in the pitch attitude. The responses of the initial and 

final designs to a unit step pitch attitude command (Oc(t) = 1 deg) are shown in Figs. 10 

and 11 respectively. Note that both the designs provide fast tracking of pitch attitude 

commands, however the initial design does so with no perturbation in the flight path angle 

while the final design does result in a small initial perturbation in the flight path ansle.. 

This coupliiig of the flight path with the pitch attitude is due to small contribution of the 

short period mode to the flight path that was allowed in order to improve the guaranteed 

stability robustness. Therefore, in just two iterations the judicious use of the singular value 

gradient information led to a feedback control law design with much improved stability 

robustness while still maintaining acceptable performance. 

15 
Summary and Conclusion 

A methodology to improve the stability robustness of feedback control systems 

designed using direct eigenspace assignment techniques was presented. A full-state 

feedback gain synthesis technique for direct eigenspace assignment was briefly discussed 

and closed-form expressions for the sensitivity of the minimum singular value of the return 

difference matrix, at plant input , to changes in desired closed-loop eigenvalues and 

specified elements of the desired closed-loop eigenvectors were derived. The closed-form 

expressions for the sensitivity of the control feedback gains were obtained as an 

intermediate step. An algorithm discussing the steps involved in calculating the return 

difference singular value gradients was presented and the use of the gradient information to 

improve the guaranteed gain and phase margins was demonstrated by application to an 

advanced fighter aircraft. The aircraft example consisted of the short period approsimatioll 

of the longitudinal dynamics with the design objective of providing decoupled tracking of 

flight path and pitch attitude commands using the elevator and flaperon as control 

effectors. The nominal feedback control design which provides ideal decoupling between 

flight path and pitch rate was found to have very low guaranteed stability robustness. 

Using the singular value gradient information it was shown that by allowing the short 

period mode contribution to the flight path angle to be non-zero, but still small (<< 

contribution to pitch rate), and by increasing the short period mode frequency and 

damping by 10 percent from that for the nominal design, the guaranteed stability margins 

could be increased significantly while still maintaining acceptable performance. 

Appendix 

Feedback Gain Sensitivity to Eigenvalue Parameters 

aK - : We first consider the n-2k real closed-loop eigenvalues, -vi, i = 2k+l, ... n. Then 

Vi 
ith col. 

because, as seen from (7), w., j#i, does not depend on qi. Using ('7) we further have 1 

which can be expanded to give 

1 

-1 T 

aLi] T 1 1 

-i & = [L. T Q.L.] -' f [gQ.L. + L. T Q. - [L. Q.L.] Li + 

111 11 11 

d'li 'V i alii 

Furthermore, using the definition of Li7 we have 

1 8Li 

- = (-ViI-.4)- Li 

'Vi 

ay. 

arli 

Substituting (A.4) into (A.3) we will get an expression for -1 in terms of known 

aw 

8Vi 

quantities, and substituting the result in (A.1) we will get - . 

aV Nest we determine - where V is the matrix of achievable eigenvectors as defined 

a Vi 

earlier. Noting that Va., j#i, (as defined in (S)), does not depend on vi, we have 

J 

(A.5) 

and 

% 

'Vi hi 'vi 

I%. 

dLi 

Knowing - from (A.4) and -1 from (A.3), we can determine Ai from (A.6) and then 

dv &V dV dI< substitute in (A.5) to get - . Once - and - are known, - are determined from eqn. 

d'lj 'Vi 'Vi Vi 

17 
(18). 

- aK : Next we consider the k complex modes, -3ijJi, i=l, ... k, by first deriving the 

Ti 

feedback gain sensitivities to the real part of the complex eigenvalues. If we arrange the 

columns of W such that 

- - - - -- w . ( j = 2 k+ 1 ). . . n)] w = F17 9, e.. W2i-1, W2i' w21-1' W21' J 

where F2i-l corresponds to X2i-1= -yi+jYi and G2i corresponds to X2i= -?-jfii, then 

w2i- 

- 

- conj(E2i-l) where conj( a) denotes complex conjugate of ( -) and we have 

(.4.7) 

Proceeding just as in the case of - above, we get 

'Vi 

* 

and 

(AS) 

- a[conj (3i-l)l aw , we can get -by making 

8Yi Using the relationship *2i = 

aY i '7 i 

use of (A.9), (A.8) and (A.7). 

Next, just as in (A.7), we have 

E = [o I 52i-l 

I I ayi I ayi I 

1 ' Yi 

Also V = conj(V ) and from the definition of? in (8) we have 

"2i "2i-1 'i 

(A.lO) 

18 
(A. 11) 

0- 

i 

0- 

av Using (A.9), (A.8) and (A.ll), we can determine - and hence determine an expression 

+fi 

- jth element 

dI< for - . a Yi 

- aK : The procedure for determining the control feedback gain sensitivities to the 

a si 

imaginary part of the complex closed-loop eigenvalues is similar to that for determining 

Using (A.12) ,instead of (A.9), in (AS) and (All), we can get analytical 

tG dj dIi expressions for -2i-1 and 2%-1; respectively, and then we can determine - using dSi asi d hi 

(A.7) and (A.lO) with yi replaced by Si. 

Feedback Gain Sensitivity to Eigenvector Parameters 

: We first consider the feedback gain sensitivities to the eigenvector elements u.. 

1J 

- 

ay;; 

n and - 

dW corresponding to the real eigenvalues -qi for i = 2k+l, ... n. The sensitivities - 

av.. 

1J 

a v .. 

'3 

'J 

are as given by eqns. (A.l) and (A.5), respectively, with 3 replaced by u... Furthermore, 

noting that Li does not depend on elements of the desired closed-loop eigenvectors, we 

have 

1J 

fi T -1 T -i = [L. Q.L.] 

dU.. 

1J 

L. Q. 111 11 

and 

(p1.13) 

19 
a. dw. (A.14) 

% 

-1 = L. -1 

'vi 1 avij 

aw dv aI1; 

av.. av.. au.. 

The expressions (A.13) and (A.14) can be used to obtain - and - , and then - can 

be obtained using (18). 

1J 1.l 'J 

- ilK : Next we consider the eigenvector elements corresponding to the k comples modes, 

'Pij 

-Ti*jbi, i=l,..k, by first deriving the feedback gain sensitivities to the real part of the -- 

and - a' are obtained from (A.7) and (AlO)! 'W eigenvectors (p.). The forms for - 

1J 'P ij 'Pij 

respectively, with ri replaced by p .. Furthermore, we have 

1J 

-1 * * - 

k2i-1 = [L.Q.L.] 111 LiQi 

apij 

and 

01 

+- jth element 

& -21-1 

(A.15) 

(A. 

hi 

'Pij 

Using (A.15) and (A.16) along with the equality -2i = conj (and similarly for 

%a 'W av 'I< 

'pi j 'P ij apij 'Pij 

-2i), we can obtain - and - , and then determine - . 

- : The procedure for determining the feedback gain sensitivities to the imaginary part 

'P ij 

of the desired complex closed-loop eigenvectors is similar to that for determining the 

sensitivities to the real part with p.. replaced by p.. . The only difference is in calculating 1.l 1J - 

B2i-1, which is now 

'Pij 

fi -1 * * 

-2i-1 = [L. 111 Q.L.] 

'Pij L. ii Q (A.17) 

20 
i3w Using (A.17), instead of (A.151, and proceeding as for the previous case, we can get - 

"Pij 

dV and - and hence obtain - . 

'Pij 8Pij 

Acknowledgement 

A major portion of this research was done while the author was a research assistaiit 

in the School of Aeronautics and Astronautics at Purdue University, \\rest Lafayette: IY. 

The author is grateful to Professor David I<. Schmidt for that support. The author would 

also like to thank Mr. Carl F. Lorenzo, Chief: Advanced Controls Technology Branch. 

NASA Lewis Research Center, for having provided the opportunity to complete and 

document this work. 

References 

[l] Andry, A.N., Shspiro, E.Y., and Chung, J.C., "On Eigenstructure Assignment for 

Linear Systems," IEEE Transactions on Aerospace and Electronic Sgstems, Vol. 

AES-19, Sept. 1983, pp. 711-729. 

[2] Sobel, K.M., and Shapiro, E.Y., "A design hlethodology for Pitch Pointins Flight 

Control Systems," Journal of Guidance, Control and Dynamics, Vol. S, No. 2, 

March-April 19SS, pp. 1S1-187. 

[3] Leibst, B.S., Garrard, W.L., and Farm, Jerome A., "Design of Multivariable Flutter 

Suppression / Gust Load Alleviation System," Journal of Guidance, ControZ and 

Dynamics, Vol. 11, NO. 3, May-June 1988, pp. 220-229. 

Schmidt, D.K., and Davidson, J.B., "Synthesis and Analysis of a Modal Control 

Law for an Aeroelastic Vehicle," AIAA Paper 85-1898, presented at the AIAA 

Guidance, Navigation and Control Conference, Snowmass, CO, August 19%. 

Stein, G., and Athans, M., "The LQG-LTR Procedure for Multivariable Feedback 

Control Design," Report LIDS-R-13S4, Massachusetts Institute of Technology, 

Cambridge, MA, May 19S4. 

Harvey, C.A., and Stein, G., "Quadratic Weights for Asymptotic Regulator 

Properties," IEEE Transactions on Automatic Control, Vol. AC-23, No. 3 , June 

1978. 

[4] 

[5] 

[6] 

[7] Lehtomaki, N.A., Sandell, N.S., Jr., and Athans, M., "Robustness results in Linear 

Quadratic Ga,ussia.n based Multivariable Control Designs," IEEE Transactions on 

Automatic Control, Vol. AC-26, Feb. 1981, pp. 75-92. 

21 
Srinathkumar, S., "Eigenvalue/Eigenvector Assignment Using Output Feedback," 

IEEE Transactions on Automatic Control, Vol. AC-23, No. 1, February 1975. 

Mukhopadhyay, V., "Stability Robustness Improvement using Constrained 

Optimization Techniques," Journal of Guidance, Control and Dynamics, Vol. 10, 

No. 2, March-April 1987, pp. 172-177. 

Herrera-Vaillard, A., Paduans J., and Downing D., "Sensitivity Analysis of 

Automatic Flight Control Systems using Singular-Value Concepts," Journal of 

Guidance, Control and Dynamics, Vol. 9, No. 6, Nov.-Dec. 19S6, pp. 621-626. 

Andersen, Per Sogaard, Trostmann Erik, and Conrad, Finn, "A Singular Value 

Sensitivity Approach to Robust Eigenstructure Assignment," Proceedings of 25th 

Conference on Decision and Control, Athens, Greece, Dec. 19SG. 

Mukhopadhyay, V., and Newsom, J.R., "A klultiloop System Stability Margin 

Study Using Matrix Singular Values," Journal of Guidance, Control and Dynamics. 

Vol. 7, No. 5, Sept.-Oct. 19S4, pp. 5S2-557. 

Noble, B., and Daniel, J.W., "Applied Linear Algebra," Prentice-Hall Inc., New 

Jersey, 1977. 

O'Brien, M.J., and Broussard, J.R., "Feedforward Control to Track the Output of a 

Forced Model," Proceedings of the 17th IEEE Conference on Decision and Control, 

"Military Specification - Flying Qualities of Piloted Airplanes," blIL-F-87S5C, 

USAF, Wright Patterson AFB, Ohio, 1980. 

1975, pp. 1149-1155. 

22 
Table 1 Injtial Desien Eicenspace Summxrv 

-4.014 

0.060 

0.011 

0 

1 

- 

i 

T State : X = [y, q, Q, de: 6d 

- 0 1 - 

1 0 

-0.114~fl.086 -1 

-0.0707j0.533 -2. SO1 

0.629hfl.Sl4 3. ‘231 

Eigenvalues : X1,2 = -5.6*j4.2, X. - -1.0, X - -19.0, ,\r = -13.5 3- 4- 3 

-4.006 

1.072 

-0.051 

1 

0 

Eigenvectors : 

Desired 

1 -3.250 -0.S91 -7.112 0.526 O.OS4 

6.101 0.S9S 10.02 -0.4’30 -0.102 Feedback Gains : I< = 

Table 2 Feedback Gains for Tria.1 1 

1 

1 

-2.865 -0.837 -6.624 0.45 O.OS1 

1.533 0.196 4.612 0.035 -0.061 

-2.S65 -0.527 -6.624 0.435 O.OS1 

1.S30 0.195 4.609 0.035 -0.061 

Actual : I< = 

Predicted : K = 

23 
Table 3 Final Desien Eie;ensna.ce Summarv 

, T 

State : X = [y, q, a, Se, Sd 

Eigenvalues : X1,2 = -6.7Sktj3.66, X3 = -1.0; X 4- - -19.0, X 5 = -19.5 

I 

-4.0 1 oq. 0 10 

0.023TjO.414 

0.637*$.245 

1 

-0.1 04 rjo . 0 72 

Eigenvectors : 

Desired 

0.060 

0.011 

1 

0 1.072 

-1 -0.051 

-2. SO1 1 

3.234 0 

--0.01+jO.o1- -1- -x1 oxx 

X xxx 

X x10 

X xo1 

-x1 -3.2SO -0.954 -7.239 0.5S3 0.090 

0.13s -0.101 2.423 0.232 -0.043 Feedback Gains : K = 

Table 4 Feedforward Ga.ins 

y.373 -2.s77J 

4.124 1.976 Initial Design : F = 

Final Design : F = [4.373 -2.9061 

4.122 -3.954 

24 
W 

n 

2 

.1 1 10 100 .I 

FREQUENCY, RAD/SEC 

(OCI+KG(jw)l) - FOR INITIAL DESIGN. 

FIGURE 1. - MINIMUM RETURN DIFFERENCE SINGULAR VALUE 

- Ag FOR Awn 

AO FOR A{ 0-w-o .035 c - 

.030 c / 

.020 F !\ ‘. 

0 

t - .005 

-.010 

25 
.006 - W 

.003 - 

.002 

,001 - 

- 

.1 1 10 100 

-6 

-8 

-10 

-12 

FREQUENCY, RAD/SEC 

CREASE IN I h3 I (FLIGHT PATH MODE). 

FIGURE 3. - CHANGE IN I+KG( j 0) I. FOR 10 PERCENT IN- 

- 

- 

- 

- 

I I Ii11111 

6 

4 

2 

0 

a -2 L 

-4 

W 

c) 

L 

3 

t %l 

FREQUENCY, RAD/SEC 

FIGURE 4. - MINIMUM RETURN DIFFERENCE SINGULAR VALUE 

SENSITIVITY TO REAL AND IMAGINARY PARTS OF THE DESIRED FLIGHT PATH ELEMENT OF THE SHORT PERIOD MODE 

do 

EIGENVECTOR ( AND !% ) . % 1 dp 11 

26 
,011 n 

-.01 

- I L 3 -.03f 

-.04 - 

/ 

-.05 - 

-.06 - 

I I I 1 I llll I an 

1 

W 

=) 

n 

c 

LLdd 

.1 I IW 100 

FREQUENCY, RAD/SEC 

SENSITIVITY TO DESIRED PITCH RATE ELEMENT OF THE 

FIGURE 5. - MINIMUM RETURN DIFFERENCE SINGULAR VALUE 

FLIGHT PATH MODE EIGNEVECTOR (E). 

,. 

----- GRADIENT BASED PREDICTION 

1: F - ACTUAL cCI+KGl 

0- I I I1111 I I I I11111 

.1 -&+++srtd 1 10 100 

FREQUENCY, RAD/SEC 

FIGURE 6. -gCI+KG(jo)l FOR TRAIL 1 - ACTUAL AND PREDICTED (USING GRADIENT INFORMATION). 

27 
# 

3 

-09 

.08 

.07 

.06 

.OS c I 

W 

s 

c 

0 L-JlLul 

A2 FOR Awn 

AO FOR AT - 

.1 1 

- 

10 100 

FREQUENCY. RAD/SEC 

FIGURE 7. - CHANGE INCI+KG(jU)I. FROM TRAIL 1, FOR 

10 PERCENT INCREASE IN SHORT PERIOD 0, AND T. 

ACTUAL a[ I+KGI 

-.I--- GRADIENT BASED PREDICTION 

.90 

.85 

.80 

m 75 

.70 

.65 

.60 

m 55 

.50 

.1 1 10 100 

FREQUENCY, RAD/SEC 

FIGURE 8. - oC1+KG(j0)1 FOR FINAL DESIGN - ACTUAL AND 

PREDICTED 

28 
1.0 

.8 

ubo 

bo" . -5 

.2 

0 

I 

0-- 1 be I I I I I I I I I I I 

re 

/ 

I / 

/ I I I I L I 

.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

TIME, SEC 

FIGURE 9. - INITIAL DESIGN RESPONSE TO STEP yc = 1 DEG. 

a 

rY .3 

I I I ,'I I I I I I I / 

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

bo 

- \ 

. -5-\ W 

e 

-10 

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

TIME, SEC 

FIGURE io. - INITIAL DESIGN RESPONSE TO STEP e, = 1 DEG. 

29 
rY 

I I I 1 

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

10 

. -5 4 bo= 

-10 

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

TINE, SEC 

FIGURE 11. - FINAL DESIGN RESPONSE TO STEP 0, = 1 DEG. 

30 
National Aeronautics and 

1. Report No. 

NASA CR- 182302 

Report Documentation Page 

2. Government Accession No. 3. Recipient's Catalog No. 

Sanjay Garg 

I I 

4. Title and Subtitle 5. Report Date 

None (E-4853) 

Stability Robustness Improvement of Direct Eigenspace Assignment 

Based Feedback Systems Using Singular Value Sensitivities 

August 1989 

L 

10. Work Unit No. 

505-62-01 

9. Performing Organization Name and Address 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

Cleveland, Ohio 44135 

i2. Sponsoring Agency Name and Address 

- 

11. Contract or Grant No. 

NAS3-25266 

13. Type of Report and Period Covered 

Contractor Report 

Final 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

14. Sponsoring Agency Code t------ 

,5. Supplementary Notes 

Project Manager, Carl F. Lorenzo, Instrumentation and Control Technology Division, NASA Lewis Research 

Center. Prepared for the American Control Conference sponsored by the American Automatic Control Council, 

Pittsburgh, Pennsylvania, June 21-23, 1989. 

6. Abstract 

A methodology to improve the stability robustness of feedback control systems designed using direct eigenspace 

assignment techniques is presented. The method consists of considering the sensitivity of the minimum singular 

value of the return difference transfer matrix at the plant input to small changes in the desired closed-loop 

eigenvalues and the specified elements of the desired closed-loop eigenvectors. Closed-form expressions for the 

gradient of the minimum return difference singular value with respect to desired closed-loop eigenvalue and 

eigenvector parameters are derived. Closed-form expressions for the gradients of the control feedback gains with 

respect to the specified eigenspace parameters are obtained as an intermediate step. The use of the gradient information to improve the guaranteed gain and phase margins in eigenspace assignment based designs is demonstrated 

by application to an advanced fighter aircraft. 

7. Key Words (Suggested by Author(s)) 1 18. Distribution Statement 

Direct eigenspace assignment 

Stability robustness 

Singular value sensitivities 

Unclassified - Unlimited 

Subject Category 08 

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