Chapter 6 State-Space Design
Xmath Control Design Module 6-2 ni.com
matrix B, then the mode of the system associated with the corresponding
eigenvalue cannot be controlled with any input. You can think of this in
the SISO transfer function case as a cancellation between a numerator and
denominator root—where you cannot control the system in the direction
of that root (mode).
It can be shown (refer to [Kai80]) that for a continuous-time system with
the state update equation:
(6-1)
you can define the controllability matrix for both continuous and discrete
systems as:
(6-2)
For all modes of the system to be controllable, the controllability matrix C
must contain a linearly independent column vectors for each system mode.
Thus, with A an n×n matrix, C must have rank n for the system to be
controllable.
In the context of gain-state feedback, a system’s controllability determines
whether you may be able to change the effective dynamics of the system to
ones that yield a more desirable response.
Using full-state feedback, as shown in Figure 6-1, so that u=v–Kx.
Working through the system equations, you obtain
(6-3)
for the new state-update equation. If the system is controllable, you can
relocate the eigenvalues of the closed-loop system to any value by choosing
the vector of state gains K appropriately. Conversely, the eigenvalues
associated with uncontrollable modes remain unchanged, no matter what
value you choose for K.
x
·Ax Bu+=
CBABA2B…An1–B[]=
x
·ABK–()xBv+=