Image Processing Reference
In-Depth Information
5
Closed-Loop System
Analysis and Design
5.1 INTRODUCTION
This chapter covers the design of closed-loop control systems in state space. The
most common technique is state feedback, where the control signal is made to be
proportional to the states of the system. The feedback gain is designed through pole
placement, where the poles of the closed-loop system are assigned for a speci
c time
response. This approach is valid when all the states are available. In practical
applications not all states are measurable and they have to be estimated from control
input and measured system output. The state estimation is another topic that will be
discussed later in this chapter. We conclude this chapter by introducing optimal
control for design of closed-loop control systems. The well-known linear quadratic
regulator (LQR) will be used as an example for design of optimal control loops.
5.2 STATE FEEDBACK
5.2.1 B
ASIC
C
ONCEPT
Consider a single-input single-output (SISO) open-loop LTI discrete-time system in
state-space form given by
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
)
(
5
:
1
)
where
A
2
R
N
N
B
2
R
N
u
2
R
1
x
2
R
N
The block diagram of the open-loop system is shown in Figure 5.1.
For an N-state SISO system, the state feedback control signal is generated by
using weighted sum of the states, assuming states are accessible. In other words,
u
(
k
) ¼
Kx
(
k
)
(
5
:
2
)
Then, the closed-loop control system will be described by the following equation:
x
(
k
þ
1
) ¼ (
A
BK
)
x
(
k
)
(
5
:
3
)
The block diagram of the closed-loop system with state feedback appears in Figure 5.2.
215
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