THE DESIGN OF REDUCED-ORDER LUENBERGER OBSERVERS WITH PRECISE LTR Moghen M. Monahemi" Jewel B. Barlow** Dianne P. O'Learyf Abstract This work concerns the design of reduced-order observers for controllable, observable, and regular systems in which the number of measurements is more than the number of controls. It uses eigenstructure assignment whereas other approaches use Kalman filter (LQG/LTR) methods. The advantages of this approach are the following: i) precise LTR rather than approximate LTR, ii) no restriction to minimum phase systems, iii) finite observer gain rather than asymptotic observer gain, iv) simpler and more efficient numerical calculation. Case studies are presented illustrating these features. I. Introduction The problem of designing an observer which can achieve loop transfer recovery (LTR) has been receiving continuous attention. It started when Doyle' presented a counter example with a Kalman filter based observer (LQG) design lacking robustness even though the full state feedback (LQR) controller had impressive robustness properties, namely gain margins of - 6db to + mdb and phase margins of rt 60 deg.2 To alleviate this problem, Doyle and Stein3 developed a robustness recovery procedure in which fictitious process noise is added to the input in the design model. The LQR robustness properties are recovered by recovering the loop transfer function with loop open at the input, asymptotically as the intensity of the fictitious process noise is increased. Stein and Athans4 call this procedure LQG/LTR. There have been further development and applications of LQG/LTR by a number of workers. A. N. Madiwale et al? extended the themy to reduced-order observer-based LQG designs for non-square, minimum phase, and left invertible plants. A. J. Calise et a1.6 developed an approach for designing a fixed-order compensator, and obtained an approximate LTR for non-square, minimum phase systems. It is similar to the full-order compensator design case of ordinary LQG/LTR. C. C. Tsui7 introduced a theoretical analysis of an entirely new approach to the problem of loop transfer recovery. His approach was to minimize the observer gain to the system input by observer pole selection. Furthermore, he claimed that this new approach aimed directly at achieving the necessary and sufficient condition of LTR. He also attempted to prove that this approach guaranteed observer stability for non-minimum phase systems. An approximate solution of the theoretical, analytical approach was presented including the derivation of finite observer gain. The purpose of this paper is to elaborate on the concept of Tsui7 for design of reduced order observers using filter eigenstructure assignment and to present a computational algorithm' for the solution of the constrained matrix Sylvester equation which arises. Precise LTR is achieved with finite observer gain and no restriction to minimum phase plants. Other approaches have been unable to attain these results. The method developed here provides freedom to select eigenvalues and eigenvectors for the observer. An outline of the paper is as follows. In Section 11, the problem formulation using the complete reduced-order Luenberger observer of Ref. 7 is reviewed, and the precise loop transfer recovery properties formulatkd. In Section 111, the procedure for designing such an observer is outlined algorithmically. A theorem is proved introducing the conditions under which such an observer produces the precise properties of the state-feedback. In Section IV, the above theory is illustrated by five case studies: a lightweight flexible arm, a helicopter flight control problem using a model for the fuselage and rotor dynamics, and three aircraft flight control syntheses. In Section V, concluding comments and directions for future research are noted. * Graduate Student, Aerospace Engineering Department, University of Maryland, College Park, Maryland 20742. Student Member, AIAA. Associate Professor, Aerospace Engineering Department, University of Maryland, College Park, Maryland 20742. Senior Member, AIAA. + Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742. The work of this author was supported by the Air Force Office of Scientific Research under Grant ** AFOSR-87-0158. 1145
Transcript
THE DESIGN OF REDUCED-ORDER LUENBERGER OBSERVERS WITH PRECISE LTR
Moghen M. Monahemi" Jewel B. Barlow**
Dianne P. O'Learyf
Abstract
This work concerns the design of reduced-order observers for controllable, observable, and regular systems in which the number of measurements is more than the number of controls. It uses eigenstructure assignment whereas other approaches use Kalman filter (LQG/LTR) methods. The advantages of this approach are the following: i) precise LTR rather than approximate LTR, ii) no restriction to minimum phase systems, iii) finite observer gain rather than asymptotic observer gain, iv) simpler and more efficient numerical calculation. Case studies are presented illustrating these features.
I. Introduction
The problem of designing an observer which can achieve loop transfer recovery (LTR) has been receiving continuous attention. It started when Doyle' presented a counter example with a Kalman filter based observer (LQG) design lacking robustness even though the full state feedback (LQR) controller had impressive robustness properties, namely gain margins of -6db to + mdb and phase margins of rt 60 deg.2 To alleviate this problem, Doyle and Stein3 developed a robustness recovery procedure in which fictitious process noise is added to the input in the design model. The LQR robustness properties are recovered by recovering the loop transfer function with loop open at the input, asymptotically as the intensity of the fictitious process noise is increased. Stein and Athans4 call this procedure LQG/LTR. There have been further development and applications of LQG/LTR by a number of workers. A. N. Madiwale et al? extended the themy to reduced-order observer-based LQG designs for non-square, minimum phase, and left invertible plants. A. J. Calise et a1.6 developed an approach for designing a fixed-order compensator, and obtained an approximate LTR for
non-square, minimum phase systems. It is similar to the full-order compensator design case of ordinary LQG/LTR. C. C. Tsui7 introduced a theoretical analysis of an entirely new approach to the problem of loop transfer recovery. His approach was to minimize the observer gain to the system input by observer pole selection. Furthermore, he claimed that this new approach aimed directly at achieving the necessary and sufficient condition of LTR. He also attempted to prove that this approach guaranteed observer stability for non-minimum phase systems. An approximate solution of the theoretical, analytical approach was presented including the derivation of finite observer gain.
The purpose of this paper is to elaborate on the concept of Tsui7 for design of reduced order observers using filter eigenstructure assignment and to present a computational algorithm' for the solution of the constrained matrix Sylvester equation which arises. Precise LTR is achieved with finite observer gain and no restriction to minimum phase plants. Other approaches have been unable to attain these results. The method developed here provides freedom to select eigenvalues and eigenvectors for the observer.
An outline of the paper is as follows. In Section 11, the problem formulation using the complete reduced-order Luenberger observer of Ref. 7 is reviewed, and the precise loop transfer recovery properties formulatkd. In Section 111, the procedure for designing such an observer is outlined algorithmically. A theorem is proved introducing the conditions under which such an observer produces the precise properties of the state-feedback. In Section IV, the above theory is illustrated by five case studies: a lightweight flexible arm, a helicopter flight control problem using a model for the fuselage and rotor dynamics, and three aircraft flight control syntheses. In Section V, concluding comments and directions for future research are noted.
* Graduate Student, Aerospace Engineering Department, University of Maryland, College Park, Maryland 20742. Student Member, AIAA.
Associate Professor, Aerospace Engineering Department, University of Maryland, College Park, Maryland 20742. Senior Member, AIAA.
+ Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742. The work of this author was supported by the Air Force Office of Scientific Research under Grant
**
AFOSR-87-0158.
1145
11. Reduced-Order Observer and Precise Loop Transfer The objective of precise LTR is to make (LTF), - = Recoverv Formulation RL TF.
Consider a controllable and observable (LTI) state space Recently, C. C. T ~ u i ~ , ~ has proven that a necessary and model of a system sufficient condition for LTR at all frequencies is
p = Cx, p c m < n , for every s, where N satisfies the equation
and its full-state feedback control law K , = NT + MC. (6)
- u = -Kc&. Condition (5) can be achieved if we find TER("-~)'~ satisfying
This K, can be separately designed to provide both stability robustness and performance robustness of the corresponding state feedback system using standard state space control techniques (e.g., LQR, pole placement, direct eigenspace assignment/full-state feedback).
(7) TB = 0.
T is further constrained to satisfy the Sylvester matrix equation
Because the state x is ordinarily not completely measurable, we usually need an estimator. We design a TA - FT = KfC reduced order observer Z ~ R @ ' - ~ ) ~ ' satisfying
= Fa + (TB)g + K f y , (2)
The observer-based feedback system (1) and (2) is shown in Figure 6.
Equation (2) is the general form of a Luenberger observer' that takes the system input g and output y as its inputs, and that estimates K c x . The familiar Kalman filter, in which the parameters (F,T,N,M) are fixed to be (A-Ks C,Z,K,,O) is one example of a Luenberger observer. The generalization to M # 0 has also been made,' but up until now, the generalization to T # Z and N # K, has not been successful. C. C. Tsui7p10 has demonstrated that the generalization to T # Z is essential for precise LTR.
The loop transfer function at the break point u in Figure 6 is
Tsui has shown that equations (7) and (8) are sufficient conditions for the observer (2) to estimate K c x .
c
It is clear that (7) cannot be achieved by the Kalman filtering algorithms that have the restriction T = I. These algorithms aim for approximate LTR by using conditions less restrictive than (7).3>" Commonly, the Doyle-Stein Identity is employed, an identity that can be only approximately achieved in the limit as system input noise is increased. Each Kalman filter pole is required to approach a system zero or negative infinity. Therefore, this approach produces a Kj with infinite gain. This approach breaks down when nonminimum-phase plants are present, since the Kalman filter poles cannot approach the unstable region of the s-plane, since that would result in an unstable controller. This is very undesirable from robustness point of view.
A more general observer with T # I, however, can acheve precise LTR, even if nonminimum-phase systems are present. In the next section we present an algorithm for designing such an observer, and prove that it satisfies (7) and (8).
The loop transfer function of the direct state feedback system, the regulator loop transfer function RLTF, is
111. Desiming an Observer
The observer dynamic matrix F may be chosen with wide latitude. This will be discussed later. Given A,B,C,K, for (1) and F, design the observer Kf using the following algorithm.
(4) RLTF = K&Z - A)'-'B.
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Algorithm Theorem
1) Perform a QR factorization of B: [W,q = qr(B)
W = n
P
Wl w2
P , s =
* -P
P
s,
,o ...
2) Let C, = CW, , A , = WlAW, , a d A, = W:AW,.
3 ) Perform QR factorization, [Q,R] = qr(C1)
P m -P
P 4)Let E = Q V W , =
m -P
P
8 -P El ...
E2
5) Solve the Sylvester Equation:
Z(A, - A&EJ - n = L,E,,
where the elements of L, are chosen at random.
and T = ZW:.
-1
Remark: A related approach has been developed by C. C. Tsui, but we have not been successful in using his algorithm. lo
The Luenberger observer Kf designed by this algorithm achieves precise loop transfer recovery, as the following theorem demonstrates.
If the system (1) is controllable and observable, if B has full rank (Le., p ) , if CB has full rank p (Le.,
if the eigenvalues of A, - A,Rl-'E, are distinct from
then the algorithm produces an observer matrix Kf that
the system is regular),
those of F, and if m > p ,
gives precise LTR.
The assumptions on the rank of B and of CB are sufficient to guarantee the existence of the QR factors in Steps 1 and 3 and the invertibility of R,. The distinct eigenvalue hypothesis guarantees that the Sylvester equation in Step 5 has a unique solution. Thus, all of the indicated computations in Steps 1 - 6 can be performed. We now verify that we have satisfied equations (7) and (8). In Step
6 , we set T = ZW:, so TB = (ZW:)(FVIS1) = 0 , since
W:Wl = 0 , so (7) is satisfied. Now, the matrix T satisfies the Sylvester equation (8) if and only if
ZWlA - FZW: = K j C ,
-or, multiplying by the nonsingular matrix W,
ZW:AW - FZW:W = KjCW,
and this last equation can be rewritten as
aAl i A,] - FZ[O i I ] = KJC, i CW2]
This is equivalent to the two conditions
24, = KjC,
and
(9)
We now verify that the matrices Z and Kf determined by the algorithm satisfy these relations. By Step 6 ,
KjC1 = [24,R1-' i L2]QTC1 = lRl-l i
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and by Step 5 and Step 4, The gain matrix K, is:
K, = [3.162 -28.507 4.434 -8.446]."6 - n = ZA,R;'E, + L ~ E ~
= [ZA,R,-' I L,] r1 ...
as desired. 0
The computation in Step 7 can be performed whenever
the matrix [TI is invertible. This depends on the observa-
bility of the reduced system (A = A, - A,R;'E, ,e = L2Ez) of Step 5.
The software modules required to 'implement the algorithm are readily available: matrix multiplication routines, QR factorization and triangular system solvers (see, for example, Linpack12), and a Sylvester Solver (see, for example, Barbels and S t e ~ a r t ' ~ ) . As illustrated in the case studies in the next section this algorithm overcomes several weaknesses of existing approaches: high gain in K the tradeoff in observer performance due to unselectab P e Kalman filter poles, the restriction to minimum-phase systems, and non-precise LTR.
IV. Case Studies
In this section, the use of the algorithm will be demonstrated via five case studies: lightweight flexible arm, a helicopter flight control problem and three aircraft flight dynamic problems taken from References 6, 14, 15, 17, and 18, respectively. The computational results were obtained using MAT LAB.^^
Case Study 1
The dynamic model of a flexible arm prototype developed by the Flexible Automation Laboratory at Georgia Institute of Technology is used. The dynamic model is described in Ref. 6 where A, B, and the sensor arrangements via C matrix are given.
"The full-state feedback design results in a damping ratio of 0.7 in the flexible mode and overdamped rigid-body modes with the following eigenvalues:
The observer pole has been selected as F = -9.
The necessary observer gain matrix K and associated f N and M matrices obtained from the algonthm are:
Figure (1) is a plot of C(s)P(s) singular values. The K&Z - A)-'B loop is precisely and perfectly recovered. The multivariable stability margins, gain margin, and phase margin when the observer is included in the control system are precisely the same as for the full-state feedback.
Comment: Note that in the above case study we obtained the precise LTR whereas Ref. 6 was restricted to approximate LTR. Furthermore in our design the order of the compensator is lower.
Case Study 2
"This is a case of a helicopter in hover. The system matrices A, B are given here as in Ref. 14.
A = Columns 1 through 5
1 0 0
-41.3 599 0 0 4.91 3.53 -15 21.9
0 0
-601. -56.7 0 0
-0.94 18.7 21.9 15
1 0
-30.2 42.6 0 0
-0.044 -0.013 1.03 -0.045
0 1
-42.6 -29.4 0 0
0.0034 -0.166 -0.045 -1.03
0 0 0 0 0 0 0 0
.32.2 0
Rigid body mode: - 1.089, -2.618 Flexible mode: -9.438 f 9.77 j
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Columns 6 through 10
-5 0 0 0 - 0 - 7 0 0 0 0 - 9 0 0 0 0 -11
0 0
0 0 0 0 0 0 0 32.2
where
0 0
-30.4 50.2 1 0
-0.0521 -0.105 4.37 1.44
B =
0 0
-50.1 -30.2 0 1.
0.0281 -0.196 1.44 -4.37
0 0
0.126 -0.283 0 0
0.0012 -0.0008 -0.0166 -0.0072
0 0 0 0
-601 -1.47 5.5200 -599 0 0 0 0
-0.938 1.32 -4.97 -3.52 21.8 -16.8 -16.8 -21.8
0 0
-0.284 -0.122 0 0
-0.0002 -0.0047 0.0072 -0.0166-
"In Ref. 14, it was found that in the absence of rotor state feedback, the rotor-made response would become unstable as the control loop was tightened, and it was concluded that tight autopilot design should include rotor state feedback to insure stability of rotor response. In Ref. 14, this was accomplished through a full-order Kalman filter design and the resulting system was shown to be stable even when the attitude loop was tightened. In this case study, we demonstrate the effectiveness of the new design procedure for an observer using two configurations of outputs.
For the given eigepvalues of closed-loop system as furnished in Ref. 14, we apply the robust pole placement technique" and derive a full state feedback matrix, K,,
" A reduced-order observer is carried out using the PLTR
technique. The resulting Kp Nand M matrices for the given (arbitrary) observer dynamics matrix F are calculated. Two configurations of measurements, each nonminimum phase are:
- u = [ec, &I' = control vector
The system dynamics, and related control system for that is discussed in Ref. 6 as follows:
"A tenth order model for the Sikorsky S-61 helicopter in hover flight condition is taken from Ref. 14, where a tight attitude control system is designed using full-state feedback. The model consists of fuselage longitudinal velocity, lateral velocity, pitch attitude, roll attitude, pitch rate, roll rate and two rigid degrees of freedom for the rotor, with body vertical and yawing motions decoupled from the rest of the dynamics. ...
Figure (2) is a plot of C(s)P(s) singular values. The regulator loop is recovered precisely by the observer-based control system.
Comment: With the configurations and arrangements of sensors in (I) and (11) we have been able to achieve precise loop transfer recovery, with arbitrary observer poles and finite observer gain. (See also the comments for the Case Study 4).
Case Study 315
The example aircraft is a linearized approximation to the AFTl/F-16 on landing approach with V = 139 knots. The objective will be to design feedback configurations using both an angle of attack sensor and a rate gyro. The small perturbation longitudinal-vertical equations of motion (1) are given by:
The plant is typical of a statically unstable aircraft and for the given arrangements of sensors, is minimum phase. Satisfactory and acceptable flying qualities for this aircraft would result if the airplane were augmented to produce the following short period and phugoid mode characteristics.
W,, = 2.5 radhec WpH = 0.1 radhec
S , = 0.5 s, = 0.1.
The full-state feedback gain matrix, using the Bass-Gura pole placement technique2' is
F = [,' The observer gain matrix Kp and the corresponding
N,M matrices are as follows:
-4.422Oe - 02
3e - 002 3.4572e - 002 5.3462e - 002 4 = [
N = [1.7019e + 003 -4.1912e + 0021.
M = [6.0014e + -5.9516e + 0001.
Figure (3) is a plot of target loop and the control system loop transfer function has been precisely recovered.
-6.0206 S; GM 5 V.H. db -60" S; PM 5 60".
Comment: The technique used in this case study is simple and straight-forward, as compared to the technique presented in Ref. 15 by E. G. Rynaski.
Case Study 4"
This case uses a model of a generic forward swept wing aircraft. The generic aircraft is roughly the same size as the X29. The wings are swept forward at a 30" angle. The operating point used in this study is level flight at a velocity of 1000 ft/sec at sea level, which is about Mach 0.9. This corresponds to a dynamic pressure of 1189 lb/ft2. Three models were given in Ref. 17, corresponding to three different center of gravity locations. The model with the center-of-mass location at 0.30 feet ahead of the wing root elastic axis was used in this study. The aircraft data, and the structural mode data developed in Ref. 17 are all the information needed to obtain the mathematical model of the FSW aircraft configuration under consideration.
The model is in linear state variable form (l), where the system matrices A and B are:
The stability margins of the observer-based system are the same as that of the regulator, and they exhibit excellent (optima1)'l gain and phase margins.
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1.2296e + OOO 4.925% - 001
-5.5524e + -1.5324e + 001.
0 0
-7 -4 0 0
4 - 7 0 0
0 0 -9 -5 F =
0 0 5 -9-
-2.3328e + 602 1.8399e + 003
0 0
F =
- 1.2985e + 003 1.4750~2 + 002
0 0 2.8023e - 001 3.4029e + 001
--9 -6 0
6 -9 0 .
0 0 -11
For the airplane configuration under consideration, the state vector .I is
The first four states are the usual rigid dynamic state variables, perturbations in
forward velocity, u(t) ft/sec angle of attack, a(t) rad. pitch attitude, O(t) rad pitch rate, q(t) radlsec.
The remaining four states represent flexible degrees of freedom. The first flexible mode represents the wing bending: nl(r) is the wing tip deflection in ft, ril(t) is its rate of deflection in ft/sec. The second flexible mode represents wing torsion, n,(r) is the wing rotation about the elastic axis in rad, A,@) is the rate of deflection in rad/sec.
The control vector &) is:
SF(r) is the perturbation deflection from trim of the full-span flaperon in rad, and S&) is the perturbation deflection from trim of the canard in rad. The aircraft under consideration has an unstable pole at 7.308 rad/sec corresponding to the short period mode, (A , = 7.308, -11.918 T&/SX).
A full-state feedback regulator is designed to stabilize the pitch rate and control the wing tip bending rate in the face of a wind gust.
Given this matrix, the multivariable phase and gaip margins are found to be,21
-34.26” 5 PM i 34.26”
when the full-state feedback is included in the flight control system.
We consider two configurations and arrangements of sensors, each nonminimum phase. The measurements are:
The corresponding observer dynamics matrices, observer gain matrices, associated N, M matrices, minimum return difference matrix of control system, loop transfer function (infn[Z + C(s)P(s)] = p) are:
Figure (4) shows the target loop and control system loop transfer function. Recovery has been achieved precisely in each situation.
Y
Comment: This case study reveals that smaller, more realistic and practical values for observer gain matrix Kfand associated N,M matrices can be achieved when more measurements of state variables are available. Also, the faster eigenvalues of observer dynamics would result in larger values of Ks, Nand M matrices.
The flexibility in selection of observer dynamics eigenvalues can be used to meet handling and flying quality requirements.
Case Studv 518
The vehicle considered is a STOL aircraft, with an airframe similar to an F-18 aircraft. The linearized dynamic model for the longitudinal axis includes four rigid-body degrees of freedom and three first order actuator lags (each at 15 rad/sec). The flight condition is the landing approach. The control inputs to be utilized are the horizontal tail (elevator), the thrust vector angle for a 2-D nozzle, and the trailing edge flap.
The statically unstable vehicle dynamics is written in the form (1) where A,B, the system matrices, are
When the direct eigenspace assignment (DEA) full-state feedback technique is used," the control objectives are stated in terms of a desired eigenstructure for the augmented systems. The full-state feedback is
The multivariable phase and gain margins were calculated as:
-2.65" < PM s 2.65"
It seems that the performance objective and flying quality requirements have been satisfied, l8 but the stability margins are very tight and present a highly non-robust control system w.r.t. multivariable stability margins.
In an attempt to improve the stability robustness of the system, we propose the control power being reduced to the form: u = 6, only. The B and C matrices are:
1.8382 2.0402 ,1089 0 573 2865 ,6302 c = [ 0 0 1 . o o o o o o 0 0
1.oooo 0 0 0 0 0 0
The corresponding matrices, full-state feedback Tc for this particular control, and reduced order observer matrices (F,KpN, and M) are:
Comment: We have been able to drastically improve the stability margins by reducing the control power to only control (SJ. The performance and flying quality requirements might be degraded because of reduction in control powers.
V. Conclusions
We have given an algorithm that achieves precise LTR and provides freedom of eigenstructure assignment when m > p and certain rank conditions are satisfied. Importantly, the algorithm yields finite observer gain, a critical requirement for pragmatic design. This approach is computationally simple, requiring on the order of n3 operations, and can be applied to non-minimal phase systems as well as minimal phase.
The flexibility in selection of observer eigenvalues can be used to meet other performance requirements, in particular, in the case of flight control problems this can be used to meet handling and flying quality requirements. l6 Much work can be done in the area of exploring the selection of F and L2 matrices needed to achieve certain desired good handling qualities and better performance in general.
The situation in which the number of actuators exceeds the number of sensors (p > m) is dual to this case, and corresponds to loop transfer recovery at the output point (xx) of Figure 6. From a robustness point of view, this means that the uncertainty is concentrated in (A,C) or y(t) instead of in (A,B) or ~ ( t ) , and we consider sensor noise attenuation and output response to output commands instead of actuator constraints and input response to input commands.
This dual version of LTR has been explained by a C. C. Tsui7 states that the number of worker^.^*^^^^
necessary and sufficient conditions for dual LTR are CT = 0 and AT - TF = BK,. In this case, the state feedback system does not have a reduced-order version.
There exists no full rank matrix T to exactly satisfy these dual conditions, but in forthcoming work we derive an algorithm for computing approximate solutions via an optimization problem. We also plan to investigate the situation in which m 5 n < p . This situation arises in aircraft in certain flight conditions such as super- maneuvering when the number of control actuators far exceeds the number of states (e.g., F/18 HARV a/c).
Acknowledgments
The authors would like to express their sincere appreciation to Dr. Chia-Chi Tsui of City University of New York, Staten Island College, for many discussions with tt. i first author. His interest is grateful1;acknowledged.
References
. J. C. Doyle, "Guaranteed margins for LQG regulators", IEEE Transactions on Automatic Control, Vol. AC-23, no. 4, August 1978, pp. 756-757.
2. M. G. Safono and M. Athans, "Gains and phase margin of multiloop LQG regulators", IEEE Transactions on Automatic Control, Vol. AC-22, April
3. J. C. Doyle and G. Stein, "Robustness with observers", IEEE Transactions on Automatic Control, Vol. AC-24, no. 4, August 1979, pp. 607-611.
4. G. Stein and M. Athans, "The LQSILTR procedure for multivariable feedback control design", IEEE Transactions on Automatic Control, Vol. AC-32, February 1987, pp. 105-114.
5. A. N. Madiwale and D. E. Williams, "Some extensions of loop transfer recovery", Proc. 1985 ACC, pp. 790-795, Boston.
6. A. J. Calise, J. V. R. Prossad, "An approximate loop transfer recovery method for designing fixed-order compensators", Journal of Guidance and Control and Dynamics, Vol. 13, Mar.-Apr. 1990, pp. 297-302.
7. C. C. Tsui, "On the loop transfer recovery", Proc. 1989 ACC, pp. 2184, Pittsburgh.
8. J. B. Barlow, M. M. Monahemi, D. P. O'Leary, "Constrained matrix Sylvester equations", Siam J on Matrix Analysis and Applications, to appear 1992.
9. D. G. Luenberger, "Introduction to observers", IEEE Transactions on Automatic Control, Vol. AC-16, pp.
10. C. C. Tsui, Personal communication, May-June, 1990. 11. J. C. Doyle and G. Stein, "Multivariable feedback
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12. J. J. Dongara, C. B. Moler, J. R. Bunch, G. W. Stewart, LINPACK User's Guide, SIAM, 1979.
13. R. H. Bartels, and G. H. Stewart, "Algorithm 432, solution of the matrix equation AX + X B = C", Commun. Ass. Comput. Mach., Vol. 15, pp. 820-826, 1972.
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17. M. Gilbert, "Dynamic modeling and active control of aeroelastic aircraft", MS Thesis, Purdue University, 1982.
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