Transition processes in the SISO system incorporating an observer in the control loop are discussed. The symmetry of the transition matrix with respect to the eigenvalues assigned for the controller and the observer is proved using state space methods. The interrelation with the peaking phenomenon is discussed, asymptotic behavior of the system with a fast observer is analyzed, and illustrative examples are presented. [S0022-0434(00)02102-X]

1.
Kwakernaak, H., and Sivan, R., 1972, Linear Optimal Control Systems, Wiley, New York.
2.
Polotsky
,
V.
,
1980
, “
Estimation of the state of single-output linear systems by means of observers
,”
Autom. Remote Control (Engl. Transl.)
,
12
, pp.
1640
1648
.
3.
Kimura
,
H.
,
1981
, “
A New Approach to the Perfect Regulation and the Bounded Peaking in Linear Multivariable Systems
,”
IEEE Trans. Autom. Control.
,
AC-26
, pp.
253
270
.
4.
Sussmann
,
H. J.
, and
Kokotovic
,
P.
,
1991
, “
The Peaking Phenomenon and the Global Stabilization of Nonlinear Systems
,”
IEEE Trans. Autom. Control.
,
AC-36
, pp.
424
440
.
5.
Sada
,
N.
, and
Fujii
,
T.
,
1981
, “
The Optimality Property of an Optimal Regulator Incorporating an Observer
,”
Int. J. Control
,
33
, pp.
617
647
.
6.
Andry
,
A. N.
,
Chung
,
J. C.
, and
Shapiro
,
E. T.
,
1984
, “
Modalized Observers
,”
IEEE Trans. Autom. Control.
,
AC-29
, pp.
669
672
.
7.
Jabberi
,
F.
, and
Shmitendorf
,
W.
,
1991
, “
Robust Linear Controllers Using Observers.
IEEE Trans. Autom. Control.
,
AC-36
, pp.
1509
1514
.
8.
Esfandari
,
F.
, and
Khalil
,
H. K.
,
1992
, “
Output Feedback Stabilization of Fully Linearizable Systems
,”
Int. J. Control
,
56
, pp.
1007
1037
.
9.
Friedland
,
B.
,
Meutzlopoulou
,
S.
, and
Antoniou
,
G.
,
1993
, “
On a Property of Compensators Designed by the Separation Principle
,”
IEEE Trans. Autom. Control.
,
AC-38
, pp.
370
373
.
10.
Bender
,
D. J.
,
1985
, “
Some Consideration for Estimator-Based Compensator Design
,”
Int. J. Control
,
41
, pp.
1577
1588
.
11.
Fowel
,
D. J.
, and
Benderand
,
R. A.
,
1985
, “
Computing the Estimator-Controller Form of a Compensator
,”
Int. J. Control
,
41
, pp.
1565
1576
.
12.
Polotski, V., 1997, “The Response of the System Incorporating an Observer in the Control Loop,” ACC97, New Mexico, pp. 3465–3469.
13.
Polotsky
,
V.
,
1978
, “
On the Maximal Errors of an Asymptotic State Identifier
,”
Autom. Remote Control (Engl. Transl.)
,
8
, pp.
1116
1121
.
14.
Atassi
,
A. N.
, and
Khalil
,
H. K.
,
1999
, “
A Separation Principle for the Stabilization of a Class of Nonlinear Systems
,”
IEEE Trans. Autom. Control.
,
AC-44
, pp.
1672
1687
.
15.
Gantmacher, F. R., 1969, The Theory of Matrices, Chelsea, New York.
16.
Kokotovic, P. V., and Khalil, H. K., 1986, Singular Perturbation in Systems and Control, IEEE Press, New York.
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