7 Intelligent and Connected Cyber-Physical Systems: A Perspective. . .
367
The control input taking discrete values is denoted as u[k], which is passed through
a zero-order hold (ZOH) and applied to the plant. The output of the ZOH is given
by
u(t) = u [k] , t k ≤ t < t k+1 .
Now the discretized dynamics can be derived:
x [k + 1] = A d x [k] + B d u [k] ,
y [k] = Cx [k] ,
where
A d = e
Ah , B d =
h
0
e
Aτ dτ
B.
Settling time is a widely used metric to quantify the control performance,
especially for real-time control applications [6, 7]. The time it takes for the system
output y[k] to reach and stay in a closed region around the reference value r (e.g.,
0.98r to 1.02r) is the settling time of a control loop and denoted as t s . Shorter settling
time implies better control performance. In order to ensure safety of the CPS, there
is often a requirement on the settling time. That is, t s must be shorter than or equal
to certain bound t 0
s .
Besides the control performance, there are system constraints related to the
CPS safety. For instance, in almost every real-world system, due to the physical
constraint of the actuator, there is some maximum available control input signal,
and the controller needs to be designed such that the maximum value of u[k] does
not exceed this limit U max , i.e., u[k] ≤ U max . This is the constraint of the input
saturation. Another constraint is on the peak overshoot, which is defined as
y max − r ≤ φ 0 r,
where y max is the maximum system output and φ 0 is the overshoot threshold. The
constraint on the steady-state error has been discussed when defining the settling
time. The system output y[k] has to reach and stay in a closed region around r, i.e.,
the system has to settle. If the region is [0.98r, 1.02r], then the steady-state error
tolerance is φ e = 2%.
In the state-feedback control algorithm, the control input u[k] is computed
based on the system state x[k]. There can be both linear and nonlinear controllers,
depending on the relationship between u[k] and x[k]. The general structure of a
linear controller is as follows:
u [k] = Kx [k] + F r,
367
The control input taking discrete values is denoted as u[k], which is passed through
a zero-order hold (ZOH) and applied to the plant. The output of the ZOH is given
by
u(t) = u [k] , t k ≤ t < t k+1 .
Now the discretized dynamics can be derived:
x [k + 1] = A d x [k] + B d u [k] ,
y [k] = Cx [k] ,
where
A d = e
Ah , B d =
h
0
e
Aτ dτ
B.
Settling time is a widely used metric to quantify the control performance,
especially for real-time control applications [6, 7]. The time it takes for the system
output y[k] to reach and stay in a closed region around the reference value r (e.g.,
0.98r to 1.02r) is the settling time of a control loop and denoted as t s . Shorter settling
time implies better control performance. In order to ensure safety of the CPS, there
is often a requirement on the settling time. That is, t s must be shorter than or equal
to certain bound t 0
s .
Besides the control performance, there are system constraints related to the
CPS safety. For instance, in almost every real-world system, due to the physical
constraint of the actuator, there is some maximum available control input signal,
and the controller needs to be designed such that the maximum value of u[k] does
not exceed this limit U max , i.e., u[k] ≤ U max . This is the constraint of the input
saturation. Another constraint is on the peak overshoot, which is defined as
y max − r ≤ φ 0 r,
where y max is the maximum system output and φ 0 is the overshoot threshold. The
constraint on the steady-state error has been discussed when defining the settling
time. The system output y[k] has to reach and stay in a closed region around r, i.e.,
the system has to settle. If the region is [0.98r, 1.02r], then the steady-state error
tolerance is φ e = 2%.
In the state-feedback control algorithm, the control input u[k] is computed
based on the system state x[k]. There can be both linear and nonlinear controllers,
depending on the relationship between u[k] and x[k]. The general structure of a
linear controller is as follows:
u [k] = Kx [k] + F r,