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A retunable PID multi-rate controller for a networked control system

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A Retunable PID Multi-rate Controller for aNetworked Control System
Antonio Sala, ´Angel Cuenca, Juli´an Salt
System Engineering and Control Department Univ. Politecnica de Valencia, Camino de Vera s/n, 46022 Valencia (SPAIN)
{
asala,julian,acuenca
}
@isa.upv.es
Abstract
This paper introduces a control strategy based on retuning a multi-rate PID con-troller in accordance with the variable delays detected in a networked control system,in order to avoid a decreased control performance. The basic idea is minimising theﬁrst-order Taylor terms of a performance measure via gain scheduling, i.e., makingthe controller gains delay dependent. As network delay is time-variant, the stabilityof this control approach will be proved by means of Linear Matrix Inequalities.Index terms: Communication systems, multi-rate control systems, delay eﬀects,ﬁeld buses
1 Introduction
Networked Control Systems (NCS) [26,12,10,25] are control systems in whichdiﬀerent devices (sensor, actuator, controller) are connected by means of ashared communication medium. NCS can be found in several kinds of controlapplications: teleoperation, supervisory control, avionics, chemical plants, etc.Their main advantages are: wiring reduction, easier and cheaper maintenance,and cost optimization [30], which must be confronted with the possibility of network outages and the appearance of time-varying delays and sampling pe-riods. There are diﬀerent topologies for NCS. For instance, two perspectives(Direct Control and Hierarchical Control) are considered in [26]: the formerinvolves only a remote controller, the latter involves a local controller in cas-cade with a remote one. There are also topologies involving middleware orsmart actuators [27].Basically, the fundamental problem in NCS is that signal transmission (usu-ally, sensor-to-controller and controller-to-actuator)through the shared mediumcan introduce delays, degrading the control performance with respect to the
Preprint submitted to Elsevier Science 9 January 2009
nominal no-delay case. Such delays may be time varying. Also, in some cases,networks introduce signiﬁcant bandwidth limitations, missing data, and forceinformation from diﬀerent time instants to travel as a single packet. Thereare also event-driven tasks and time-driven ones, and some procedures requiresynchronization [11].As discussed above, the analysis and design of NCS is a complex problem and,usually, some simplifying assumptions are made. There are many approachesin literature on control of NCS: for instance, inference or prediction capabil-ities appear in [16]; fuzzy methodologies are discussed in [15]; missing-dataapproaches have been discussed in [2]; if time-stamped data are interchanged,state-feedback and time-varying observers are discussed in [21]; gain schedul-ing in networked control is discussed in [27]; quantization and bandwidth-limited stabilization are discussed in [29,7]; packet-based transmission of sev-eral control signals on a network is also an option [22,25], related to the dual-rate ones to be later discussed. Further issues are considered in [26,30] andreferences therein. The convenience of each of the cited approaches dependson the network characteristics, communication protocols, existence of time-stamped data, controller parametrization, controller software and hardwarecomplexity, etc.In this work, the network setup under consideration will involve a remotesensor which sends its information through a network to a controller which islocated close to the plant actuator. The sensor will follow a time-driven policywhereas the controller (together with the actuator) follows an event-drivenpolicy (triggered when sensor data become available from the network). Also,the sensor-controller delay is assumed to be measurable, perhaps requiring asuitable initial synchronisation procedure or time-stamping.In situations where the sampling period is long (say, in the same order asthe settling time), dual-rate setups (fast control) may also be advantageous interms of achievable performance [4,16]; this could be the case if the networkconﬁguration imposes a limitation in the frequency at which measurements aretransmitted (for instance, because of an excessive number of devices sharinga communication link). In this context, the consideration of a dual-rate con-troller might be useful: as it is directly connected to the actuator, it can workat a faster rate than the network which provides the measurements. Dual-ratecontrollers in linear time-invariant setups have been considered, for instance,in an internal-model inferential structure in [16], in an algebraic transfer func-tion setting in [8], or in a state-space representation in [14].In this paper, a multi-rate networked control system is addressed: there is aneed to both consider the multiple sampling rates as well as the variation onthose sampling rates and the communication delays induced by the network,giving rise to a time-varying problem.2
The objective of this paper is to set up a delay-dependent gain-schedulingapproach for PID controllers, in dual-rate implementations. Indeed, in orderto reduce the inﬂuence of network delays and to improve the control systemperformance, an approach based on re-tuning the PID parameters based on themeasured delay will be set up. Dual-rate control techniques will be adoptedas they will prove a better performance in the particular chosen exampleunder long sampling period values. Of course, there are other more powerfulapproaches for networked control if using a non-PID controller [20,21], but thePID control structure is adopted due to its practical interest [1,5].In [27] a gain-scheduling procedure for PIDs in networked control is also pre-sented. However, they adjust only the overall regulator gain (by simulationtests) and they do not include a closed-loop LMI stability analysis step, sostability is also checked by simulation. In this work, all the PID controllergains (proportional, integral, derivative) are adjusted, and stability for arbi-trary time-varying network delays is proved.The basic idea to be used will be cancellation of the ﬁrst-order Taylor termsof a performance measure
p
(
θ,τ
S
−
C
), where
θ
are the adjustable controllerparameters and
τ
S
−
C
is the sensor-to-controller delay: gain scheduling will setup a suitable scheduling policy
θ
(
τ
S
−
C
). Then, in order to account for fastvariations of the network delay from sample to sample, a set of linear matrixinequalities [6] will be used to analyse the actually achieved worst-case LTVdecay-rate.The structure of the paper is as follows. In sections 2 and 3 the problem sce-nario and the process realisation are respectively exposed. Section 4 proposesa generic retuning approach for the regulator and section 5 presents the state-space representation for a particular choice of dual-rate PID controller. Finally,section 6 presents simulation results by means of an example and shows theexpected improvements, and section 7 enumerates the main conclusions.
2 Problem scenario
The network setup to be considered will be a time-driven sensor, which pe-riodically sends a measurement to the controller. The measurement reachesthe controller after a certain time has elapsed. Then, an event-driven controlaction is produced at that instant and subsequent ones are produced period-ically, with nominal controller sampling period
T
. Measurements are carriedout with period
T
sens
, which is an integer multiple of the control period
T
,i.e.,
T
sens
=
NT
. From non-conventional sampling literature,
NT
will be alsodenoted as global period or metaperiod.3
Fig. 1. Chronogram of the proposed NCS; controller sampling period
T
, sensorsampling period
T
sens
=
NT
.
Figure 1 shows graphically the overall behavior of the proposed dual-rate NCS.The meaning of the encircled numbers is now detailed:1
The sensor samples at period
T
sens
=
NT
the process output This deviceworks in a time-driven operation mode,
i.e.
, measurements are periodicallytriggered by a clock signal.2
The controller is going to receive the sampled process output via the net-work. From the moment the sensor sends the sample until it is received bythe controller, a processing and propagation time ellapses,
τ
S
−
C
, which is as-sumed measurable. Depending on diﬀerent reasons (network load, distanceamong devices, etc.), this delay time can be variable, usually randomly.3
When the sample arrives, an event is triggered so the controller generates
N
faster-rate control actions. Such actions are nominally scheduled to beapplied every
T
time units but network delays will force the actual time of application to be diﬀerent. In particular, the ﬁrst of the computed actions willbe applied at the time of arrival of the measurement, i.e.,
τ
S
−
C
time units afterthe measurement was taken. The remaining control actions, as they are notinﬂuenced by the network delay, will be applied every
T
time units, triggeredby a fast-rate clock signal. If the delay is greater than one control samplingperiod
T
, some of the
N
control actions (the ﬁrst ones) will never be actuallyapplied to the plant. To avoid causality and packet order issues, it will beassumed that the propagation time
τ
S
−
C
is always less than
T
sens
.Figure 2 summarizes the structure of the described NCS by means of a block-diagram representation. On the following
τ
S
−
C
will be denoted with shorthand
τ
if no confusion arises from the context.4
C( )
Y
NT
delayed
S-C
Y(s)
Y
NT
[ ]
S-C
,
T, 2T, ..., (N-1)T
U
T
delayed
S-C
the first sample
[ ]
S-C
[0]
network delay
S-C
referenceController+Actuator
Sensor
Process
ZOH
A
,
B
,
C
,
D
Fig. 2. Structure of the proposed NCS
The faster the measurement sampling period
T
sens
=
NT
is, the better theachievable performance will be possible; however, such measurement periodis constrained by network characteristics. With a suitable control structure,such as the internal-model inferential one [16], the smaller the control period
T
is (the larger
N
is), the better tracking performance will be possible evenif the measurement period is kept constant (
NT
). So, there is the possibilityof improving performance by a faster actuation even if the measurement rateis constrained.The controller structure to be chosen in this work is a PID one, adapted todual rate. Details of it will be given in Section 5. In the example section, itwill be shown how improved performance can be achieved in a dual-rate PIDsetting with respect to single (slow) rate controllers, at least for the plant inconsideration.
3 Preliminaries and notation
Consider a continuous linear time-invariant plant
P
, which admits a state-space realisation given by the quadruple
{A
,
B
,
C
,
D}
with suitable dimensionsfulﬁlling:˙
x
(
t
) =
A
x
(
t
) +
B
u
(
t
) (1)
y
(
t
) =
C
x
(
t
) +
D
u
(
t
) (2)Being
ξ
an arbitrary real number, denote as
B
(
ξ
) =
ξ
0
e
A
γ
B
dγ ξ >
00
ξ
≤
0(3)It is well known [28,1] that, in the case the input changes every
T
time units,being constant in the inter-sample period (zero-order hold, ZOH) and the5

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