Page 64 - Servo Motors and Industrial Control Theory -

56 3 State Variable Feedback Control Theory

v := eigenvecA( ,−0 .102 +1 .192 i)
−
− . 0 228 0 .414 i
−
.
v = 0 .516 023 i
. 0 22110 639+ . i
(
= : v eigenvecA, − .
2 796)
012
− .
.
v = 0 334
0 935
− .
For complex conjugates eigenvalues, there are also two conjugate eigenvectors.
For real root, there is a real eigenvector. The eigenvectors will be used to assess the
controllability and observability of the control system.

3.4 State Variable Feedback Control Theory

It was shown in the previous chapter that control systems might have one or several
outputs. The output is fed back in the controller and various control strategies may
be used to control the system. The problem with complex system with higher order
transfer function than third order is that it is not possible to control the location of
the roots of characteristic equation on the s-plane. A compromise has to be made
between the steady state error and the transient response.
In the state variable form, if all the state variables are measurable then state vari-
able feedback control theory may be used. The condition is that the system must
be controllable. The condition of controllability will be discussed later. If the state
variables are not measurable for direct feedback, observer may be used to predict
the state variables. The idea is that if the system model and input are known, with
measurement taken on one or several state variables all state variables can be pre-
dicted. The condition of observability will be discussed later.
Consider the state equation,
d X := AXBu+
dt
In the above equation, it is assumed that there is only one input variable. The pro-
cedure for controllable system with more than one input is the same and it can be
shown that by a single input it is possible to control the location of all roots of the
characteristic equation. The control strategy is that a summation junction will be
added to the input u which represents the command signal and all state variables
i
with appropriate gain are used as negative feedback. The state equation becomes