This section briefly discusses the properties of the PID algorithm in control of NCS and points out
some modifications that have been proposed to better tackle the network-induced delays and
packet losses. When applying the ideal PID algorithm (28) in a control loop, where varying time-
delays are present, one can show by using the concept of jitter margin (26) that with a FOTD
process the stability of the system cannot be guaranteed, at least in the sense of the jitter margin
condition, for any additional delays. In such case the jitter margin is zero. It should be noted,
though, that the jitter margin is a conservative criterion and only sufficient, so the zero jitter
margin does not necessarily imply instability for additional delays. The reason for having zero jitter
margin is that the complementary sensitivity function of the system does not have roll-off at high
frequencies and in the frequency response plot, the closed-loop magnitude crosses the bounding
curve that the jitter margin defines, see Figure 13. This can also be shown analytically, see [P7].
The problem can be easily solved by using a measurement filter, such as (34) or (59), which will
saturate the gain of the controller derivative term at high frequencies.
Figure 13 shows the advantage of using measurement filters in the PID controller from the jitter
margin point of view. It is seen in the figure that without the filter the complementary sensitivity
function of a PID controlled FOTD process hits the jitter margin bound and obviously the jitter
margin is zero. When filtering is applied the jitter margin is positive, because the complementary
sensitivity function remains below the jitter margin bound for all frequencies. Nevertheless, if the
controller is not well tuned, the lines may intersect and the required jitter margin is not achieved.
Frequency rad/s
Complementary sensitivity functions of PID controlled FOTD process,
with and without the measurement filter, and the jitter margin bound for ?max = 0.5 [P7].
Packet losses cause undesired behavior in the signals of a PID-controlled system, if the losses are
not considered in the controller design procedure. Under packet losses in communications, the D-
term of the PID controller is likely to cause spikes in the control signal [76]. This would happen
when the communications are reestablished after a period of disconnectivity. The amplitude of the
spike is determined based on how much the measured value of the process variable has evolved
between the times of successful reception of packets (before and after the packet losses). It is
pointed out in [76] that during the time of packet losses the control signal is a linearly changing
function of error, since there is no new information available of the process variable. Thus the
error is constant and the I-term integrates the error over time. The set-point and measurement
signals are static functions of error, and as a result, a linearly increasing or decreasing control signal
is obtained. This can endanger the stability of the process or cause undesired actions. It should also
be noted that during packet losses there will be no information regarding actuator saturation and
the possible anti-windup functions may fail. In addition, on controller output losses the actuator
may experience bumps in the control signal.
In [76], an enhanced PID algorithm for wireless control systems is proposed to overcome the
problems of discontinuous communications. The integral and derivative parts of the controller are
remodeled such that they are updated only on the arrival of new packets. The integrator is replaced
by a filter that receives information about the actuator position at the same time as new
measurement packets arrive. The filter output is calculated
where O(k) is the new filter output, Ok(1)- its previous value, uk(1)- the controller output for
the last execution (based on the actuator position feedback), ?T the elapsed time since a new value
was communicated, and TReset a tuning parameter (integrator resetting time). The last
communicated actuator position is used in the filter output calculation and hence this structure
should be able to automatically compensate for any loss in the output of the controller.
The derivative part of the proposed algorithm is basically an event-based version of the regular
where d(k) is the controller derivative term, kd the gain, and e(k) – ek(1)- the difference of current
and previous error. The difference with respect to the normal PID algorithm is that the sample
time is updated based on reception of measurement packets and thus the derivative action is
smaller the longer the time interval between the two last packets. Note that the control law is
updated only upon receiving new information.
Event-based PID control has also been considered in [101], but from a different perspective. The
basic idea of event-based control is to transmit sensor and command data only when needed.
Especially in bandwidth limited systems, such as networked control systems, event-based control is
desirable, because the communication medium is only used if something has happened since the
last event [26]. Although the main motivation for developing the event-based PID controller in
[101] arises from the field of embedded control systems and the problems faced with scheduling
and control co-design, a similar controller structure is very usable in networked control systems (or
even networked embedded control systems). In event-based control the execution of the control
algorithm is not time-based, but the control law is updated upon certain variables exceeding
predefined thresholds, that is, when events occur. Different criteria for event detection may be
defined based on the process and the scenario. An event could occur, for example, if a measured
variable would exceed a certain value, or relative changes of variables could be tracked. Sampling
should also be performed at reference changes. The tracking of error signal rather than the
measurement signal might be more effective, because on the sudden change of the reference
signal the error changes immediately, whereas the measured variable reacts much slower. An event
could be triggered if the absolute value of current error would deviate from the error at the
previous sample more than a predefined threshold value. It might also be useful to set a maximum
sample time after which sampling would occur even though none of the triggering events were
activated. Event-based control may be very efficient from the CPU utilization and networking
points of view, since presumably the control law is updated more rarely than in the case of time-
based control. The drawback of event-based control, however, is that the control system becomes
more difficult to analyze [101].
In the literature, the structure of the PID controller is sometimes discussed in the light of NCS and
modifications such as presented above have been proposed to the algorithm. The tuning of the
PID controller for NCS has not been comprehensively dealt with, even though it has a significant
impact on robustness properties especially with respect to varying time-delays and packet loss. The
discrete-time PID controller tuning problem is discussed in [70] in systems with random delays.
The NCS is assumed to be fully-distributed and the tuning is based on simulation based
optimization of the controller parameters for a specific process model. In [41], a dual PID
structure is proposed to overcome the effects of load disturbances and to improve the dynamic
performance and disturbance rejection in NCS. In addition, a relay auto-tuning method is applied
for obtaining the process parameters, which are then used for determining the controller gains.
The framework is based on certain simplifications of the NCS model, for example, the network
sc ca
delays (?k and ?k ) are lumped together as a single delay.
In [17], fuzzy logic is used to set the integral and derivative terms of the PID controller. For the
proportional term, the paper suggests using a fuzzy immune algorithm that is based on the ideas of
the biological immune system. The proportional term is modified so that the adaptive gain kp1 is
calculated by
kKp1 =-??1(),()?fukuk()? ??, (80)
where K is gain, ? is an adaption speed parameter, f is a selected nonlinear function, which is
approximated by a simple fuzzy logic scheme, and ?u(k) is the change of control signal u at k. The
PID controller algorithm is implemented in the incremental form.
Genetic algorithms (GA, see e.g. [12]) have been proposed to tune the PID controller, also in the
NCS setup. The GA in [40] updates the given initial PID gains based on evaluating a fitness
function that is a weighted sum of several performance criteria, including settling time, overshoot
and normalized integral of square error (ISE) cost criterion (see Section 3.5.1). The experiments
reported show that the delay in the Profibus-DP fieldbus setup that is used in the study varies
between one and four sample times. The performance of a modified Ziegler-Nichols tuning and
the proposed GA based PID tuning method is compared in the Profibus-DP fieldbus testbed. This
GA based tuning procedure resembles those proposed in [P2] and [P4], but the optimization
method is different. The idea is the same in the sense that the performance of the control system is
evaluated in the presence of varying time-delays and the performance is optimized. If the
optimization criteria are carefully chosen, these methods provide good performance and
robustness against delays.
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