In this paper, we derive an expression for the loss of optimal performance (compared to the corresponding linear-quadratic optimal performance with the instantaneous full-state feedback) when the continuous-time finite-horizon linear-quadratic optimal controller uses the estimates of the state variables obtained via a reduced-order observer. It was shown that the loss of optimal performance value can be found by solving the differential Lyapunov equation whose dimensions are equal to dimensions of the reduced-order observer. A proton exchange membrane fuel cell example is included to demonstrate the loss of optimal performance as a function of the final time. It can be seen from the simulation results that the loss of optimal performance value can be very large. The loss of optimal performance value can be drastically reduced by using the proposed least-square formulas for the choice of the reduced-order observer initial conditions.

References

References
1.
Aly
,
M.
,
Roman
,
M.
,
Rabie
,
M.
, and
Shaaban
,
S.
,
2017
, “
Observer-Based Optimal Position Control for Electrohydraulic Steer-by-Wire System Using Gary-Box System Identification Model
,”
ASME J. Dyn. Syst. Meas. Control
,
139
(12), p.
121002
.
2.
Qu
,
P.
,
Zhu
,
D.
, and
Sun
,
S.
,
2017
, “
Observer-Based Guidance Law With Impact Angle Constraint
,”
ASME J. Dyn. Syst. Meas. Control
,
139
(11), p.
114504
.
3.
Luenberger
,
D.
,
1964
, “
Observing the State of a Linear System
,”
IEEE Trans. Mil. Electron.
,
8
(2), pp.
74
80
.
4.
Luenberger
,
D.
,
1966
, “
Observers for Multivariable Systems
,”
IEEE Trans. Autom. Control
,
11
(2), pp.
190
197
.
5.
Luenberger
,
D.
,
1971
, “
An Introduction to Observers
,”
IEEE Trans. Autom. Control
,
16
(6), pp.
596
602
.
6.
O'Reilly
,
J.
,
1983
,
Observers for Linear Systems
,
Academic Press
,
London
.
7.
Sinha
,
A.
,
2007
,
Linear Systems: Optimal and Robust Control
,
CRC Press
,
Boca Raton, FL
.
8.
Chen
,
T.
,
2013
,
Linear System Theory and Design
,
4th ed.
,
Oxford University Press
,
Oxford, UK
.
9.
Antsaklis
,
J.
, and
Michel
,
N.
,
2005
,
Linear Systems
,
Bikhauser
,
Boston, MA
.
10.
Kwakernaak
,
H.
, and
Sivan
,
R.
,
1972
,
Linear Optimal Control Systems
,
Wiley-Interscience
,
New York
.
11.
Stefani
,
R.
,
Savant
,
C.
,
Shahian
,
B.
, and
Hostetter
,
G.
,
1994
,
Design of Feedback Control Systems
,
Saunders College Publishing
,
Orlando, FL
.
12.
Franklin
,
G.
,
Powel
,
J.
, and
Emami-Naeini
,
A.
,
2002
,
Feedback Control of Dynamic Systems
,
Prentice Hall
,
Upper Saddle River, NJ
.
13.
Ogata
,
K.
,
2002
,
Modern Control Engineering
,
Prentice Hall
,
Upper Saddle River, NJ
.
14.
Friedland
,
B.
,
1996
,
Control Systems Design
,
Prentice Hall
,
Englewood Hills, NJ
.
15.
Radisavljevic-Gajic
,
V.
,
2015
, “
Full-Order and Reduced-Order Observer Implementations in MATLAB/SIMULINK
,”
IEEE Control Systems Magazine: Lecture Notes
, pp.
91
101
.
16.
Anderson
,
B.
, and
Moore
,
J.
,
2005
,
Optimal Control: Linear-Quadratic Methods
,
Dover Publications
,
Mineola, NY
.
17.
Lewis
,
F.
,
Vrabie
,
D.
, and
Syrmos
,
V.
,
2012
,
Optimal Control
,
Wiley
,
Hoboken, NJ
.
18.
Kilicasian
,
S.
, and
Banks
,
S.
,
2012
, “
Existence of Solutions of Riccati Differential Equation
,”
ASME J. Dyn. Syst. Meas. Control
,
134
(3), p.
031001
.
19.
Strang
,
G.
,
2009
,
Introduction to Linear Algebra
,
Wellesley Cambridge Press
,
Wellesley, MA
.
20.
Johnson
,
C. D.
,
1988
, “
Optimal Initial Condition for Full-Order Observers
,”
Int. J. Control
,
48
(3), pp.
857
864
.
21.
Pukrushpan
,
J.
,
Peng
,
H.
, and
Stefanopoulou
,
A.
,
2004
, “
Control Oriented Modeling and Analysis for Automotive Fuel Cell Systems
,”
ASME J. Dyn. Syst. Meas. Control
,
126
(1), pp.
14
25
.
22.
Radisavljevic-Gajic
,
V.
,
Milanovic
,
M.
, and
Clayton
,
G.
,
2017
, “
Three-Stage Feedback Controller Design With Applications to Three Time-Scale Linear Control Systems
,”
ASME J. Dyn. Syst. Meas. Control
,
139
(10), p.
104502
.
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