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objective is then to drive the error between the actual and estimated outputs to zero
as rapidly as possible so that,in the limit, the estimated states approach the actual
states. The schematic diagram of such a full-state observer is shown in Fig. 5.43.
Suppose the dynamics of the plant and output are given by
x - Ax + Bu
y = Cx
Let the observer dynanucs and the output be given by
so that
x = Ax + Bu + L(y - y)
y -. Cx
y - y = C(x - x)
(5.285)
(5.286)
(5.287)
(5.288)
(5.289)
Fig. 5.43 Schematic diagram of fall-state obsener.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW 523
ex -x -x = A(x -x) - L(y - y)
- (A - LC)(x -- x)
ey = C(x - x)
(5.290)
(5.291)
(5.292)
P/ant given in dual phase-variable form. Suppose that the given plant A is
in dual p~ase-variable form
Then,let us assume that the matrix A - LC has the following form
A - LC -
The characteristic equation of the observer system is then given by
(5.293)
(5.294)
s" + (an -i +/i)s -1 + (a,, _2 +/2)Sn -2 + ...+ (t/0 + /n) = 0 (5.295)
Let the characteristic equation that gives the desired transient response of the
observer be given by
s I + Ltn - IS i-I + dr._ 2S r -2 + .. . + do = 0 (5.296)
Equating the coefficients oflike powers of s, we obtain
ll - dn -i - an -i, l2 -. d,, _ 2 - an -2,- - -, 1,, -. do - ao (5.297)
Here, /1,12,...,/n are the elements of the feedback gain matrix to achieve the
desired performance of the full-state observer.
Here, x is the estimated state vector, and y is the estimated output. Let e. - x - x
be the error between the actual states and the estimated states and ey = y - y be
the error between the measured and estimated outputs. Then,
I
i
The objective of the design is to choose the observer gain matrix / such that the
errors ex and ey approach zero as rapidly as possible. In other words, we choose
the observer gain matrix / so that the closed-loop eigenvalues produce the desired
transient response of the observer.
-an_i 1 O C
A- -an_3 O O 1
.N.
-ai - . .
-ao 0 0 0
-(an_i+/l) 1 O O . O
-(an_2+/2) 0 1 O . 0
-(an_3+/3) 0 O 1 . O
...
-(ai+/n) . . . 1
~(aO+ln) 0 O 0 . O
524 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
P/ant not given in dual phase-variable form. Suppose the given plant
x = Ax + BU . (5.298)
y = Cx (5.299)
is not in dual phase-variable form. Then we assume that there exists a matrix P
where x - Pz that transforms this plant into a dual phase-variable form as given
by
z = Azz + Bzu (5.300)
y = Czz (5.301)
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