Nonlinear Control Sy.. - Free

292 CHAPTER 11. NONLINEAR OBSERVERS
11.1.1 Observability
Definition 11.1 The state space realization (11.1) is said to be observable if for any initial
state xo and fixed time tl > 0, the knowledge of the input u and output y over [0, t1] suffices
to uniquely determine the initial state xo.
Once xo is determined, the state x(ti) can be reconstructed using the well-known solution
of the state equation:
x( tl) = elxo + Lt1 eA(t-T)Bu(r) dr.
(11.2)
Also
t,
y(ti) = Cx(tl) = CeAtlxo + C r eA(t-T)Bu(rr) dr. (11.3)
0
Notice that, for fixed u = u`, equation (11.3) defines a linear transformation L : 1R" -+ ]R
that maps xo to y; that is, we can write
y(t) = (Lxo)(t).
Thus, we argue that the state space realization (11.1) is observable if and only if the mapping
Lx0 is one-to-one. Indeed, if this is the case, then the inversion map xo = L-ly uniquely
determines x0. Accordingly,
,b1 "observable" if and only if Lxl = Lx2 x1 = x2.
Now consider two initial conditions xl and x2. We have,
and
tl
y(ti) = (Lxl)(t) = CeAt'xl +C J eA(t-T)Bu*(r) d7 (11.4)
0
rt,
y(t2) = (Lx2)(t) = CeAtlx2 +CJ eA(t-T)Bu*(r) drr (11.5)
0
By definition, the mapping L is one-to-one if
yl - y2 = Lx1 - Lx2 = CeAtl (xl - x2).
y1 = Y2
For this to be the case, we must have:
CeAtx = 0 x=0