Page 781 - The Mechatronics Handbook

0066_Frame_C24 Page 29 Thursday, January 10, 2002 3:45 PM

The controllability matrix is then given by

1 ( – R + R )
1
1
------------------ --------------------------
R R C 1 ( R R C ) 2
2
1
Γ c A, B] = [ B AB] = 1 2 1 (24.167)
[
1 ( – R C + R C )
3
3
2
1
------------ ---------------------------------------
R C 3 ( R C ) R C 1
2
3
3
3
2
and
(
R 2
[
det Γ c A, B]) = ------------------------------------- −R 1 C 1 +( R 3 C 3 ) (24.168)
( R 1 R 2 R 3 C 1 C 2 ) 2
where we can observe that the system is completely controllable if, and only if, R 1 C 1 ≠ R 3 C 3 .
This issue has a very important interpretation if we analyze it from the transfer function point of view.
Applying Laplace transform to Eqs. (24.161)–(24.164), the transfer function from v i (t) to v o (t) (recall
that V + (s) = V − (s)) is given by
1  s + ------------- 
1
V o s() V o s()V + s() -------------  R C 
R C
⋅
------------ = -------------------------- = ---------------------- ------------------------- (24.169)
1 1
3
3
V i s() V − s() V i s()  s + ------------   s + R + R 2 
1
1
 R C   -------------------
R R C 
3 3
1 2 1
where we can observe that the loss of complete controllability, when R 1 C 1 = R 3 C 3 obtained from (24.168),
means that there is a zero-pole cancellation in the transfer function, i.e., the zero from the left half of
the circuit in Fig. 24.13 is cancelled by the pole from the other part of the circuit. This issue will be
discussed in more detail in section “Canonical Decomposition.”
Controllability Gramian
The test of controllability gives us a yes or no answer about the controllability of a system model. However,
to conclude that a system is completely controllable says nothing about the degree of controllability. For
stable systems, we can quantify the effort to control the system state through the energy involved in the
input signal u(t) applied from t = −∞ to reach the state x(0) = x 0 at t = 0:
J u() = ∫ 0 ||u t()|| td = ∫ 0 u t() u t() t (24.170)
T
2
d
– ∞ – ∞
It can be shown that the minimal control energy is
(
1
T
–
J u opt ) = x o P x o (24.171)
where
∞ T
P = ∫ e BB e t d (24.172)
At
T A t
0
The matrix P is called the controllability gramian, and it measures the controllability of the state
vector x(0). If this matrix is small, it means that we need a lot of energy in the control input u(t) to steer
the state vector to x 0 . Indeed, we can appreciate the necessary effort for each one of the state variables,
T
making, for example x 0 = [0,…, 0, 1, 0,…, 0] .
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