User`s guide

Pole Placement
7-5
Pole Placement
The closed-loop pole locations have a direct impact on time response
characteristics such as rise time, settling time, and transient oscillations. This
suggests the following method for tuning the closed-loop behavior:
1 Based on the time response specifications, select desirable locations for the
closed -loop poles.
2 Compute feedback gains that achieve these locations.
This design technique is known as pole placement.
Pole placement requires a state-space model of the system (use
ss to convert
other LTI models to state space). In continuous time, this model should be of
the form
where i s the vector of control inputs and is the vector of measurements.
Designing a dynamic compensator for this system involves two steps:
state-feedback gain selection, and state estimator design.
State-Feedback Gain Selection
Under state feedback , the closed-loop dynamics are given by
and the closed-loop poles are the eigenvalues of . Using pole placement
algorithms, you can compute a gain matrix that assigns these poles to any
desired locations in the complex p lane (provided that is controllable).
State Estimator Design
You cannot implement the state-feed back law unless the full st at e
is mea su r e d. Howev er, you can cons tru ct a state estimate such that the
law retains the same pole assignmentproperties. This is a chieved by
designing a state estimator (or observer) of the form
x
·
Ax Bu+=
yCxDu+=
u
y
uKx
–=
x
·
ABK–()x=
ABK
–
K
AB
,()
uKx
–=
x
ξ
uK
ξ–=