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Chapter 6 State-Space Design
Xmath Control Design Module 6-12 ni.com
pairs as poles( ). For each pole value in poles( ), poleplace( )
forms a vector by subtracting the pole’s value from each diagonal element
of S except for the last element (0). The resulting matrix is then divided by
the corresponding value in the random complex vector. The complex value
is padded with zeros to form a vector that is row compatible with the
matrix.
poleplace( ) then divides the last element of this quotient vector
by the negative of the first element of the quotient vector, and the result is
the gain required to move that pole value. This sequence of steps is
performed as a matrix operation so that the complete gain vector is
computed immediately.
rcond( ) is called to examine the condition of the
matrix formed by all the quotient vectors. If the condition number returned
is less than
eps × (the row size of A), poleplace( ) displays a warning
message indicating that the eigenvectors of the closed-loop system are
ill-conditioned.
Linear Quadratic Regulator
A regulator is a feedback controller designed to drive the states of a
controllable system using acceptable amounts of control and keeping the
states within acceptable levels (where the designer can mathematically
define what constitutes “acceptable” in both cases). Figure 6-3 shows a
continuous-time regulator where the design presumes availability of all
states, feeding them back through the optimal gain array K
r
to drive the
system so that the states return to zero as quickly as possible in the presence
of a disturbance or noise, represented by ω.
Figure 6-3. Continuous-Time Regulator
In designing a regulator, the goal is to find a controller that minimizes the
effects of disturbances on the states of the system. Xmath’s linear quadratic
regulator function,
regulator( ), uses a quadratic performance index to
establish the trade-off between the permissible state fluctuation and the
available energy, or amount of control, required to move the states.
Note In designing a regulator, assume that all the states of the system are available as
outputs.
ux
–K
r
x = Ax + Bu