multi-agent pid and fractional pid control of the three...
TRANSCRIPT
Multi-agent PID and fractional PID control
of the three-tank benchmark system ?
Gonçalo Vinagre ∗ Duarte Valério ∗ José Sá da Costa ∗
∗ IDMEC/IST, Technical University of Lisbon, Portugal (e-mail:[email protected], [email protected],
Abstract: This paper addresses the development of a multi-agent system to control the three-tank benchmark plant with PID and fractional PID controllers. Each tank was controlled usingan agent implemented in a different computer; under normal circumstances, agents are expectedto communicate to exchange information and cooperate to achieve their goals; if there shouldbe a communication failure, each agent is able to pursue control on its own, attempting to keepperformance as good as possible by using simulations to supply unavailable data. Furthermoreagents are able to identify the plant and change the control laws accordingly. Simulation andexperimental results are given showing the control system’s robustness properties.
Keywords: multi-agent system, three-tank system, distributed control, fault tolerance,fractional control
1. INTRODUCTION
This paper addresses the development of a multi-agentsystem to control the three-tank benchmark plant withPID and fractional PID controllers. Each tank was con-trolled using an agent implemented in a different com-puter; under normal circumstances, agents are expectedto communicate to exchange information and cooperateto achieve their goals; if there should be a communicationfailure, each agent is able to pursue control on its own,attempting to keep performance as good as possible byusing simulations to supply unavailable data. Furthermoreagents are able to identify the plant and change the controllaws accordingly, which means that it is possible to beginworking without knowing values for plant specificationssuch as tank cross-section or valve characteristics, and thatit is possible to cope with changing parameters (such ashappens when valves get clogged).
Section 2 presents details on the plant and its model,section 3 introduces the multi-agent control system, andsection 4 gives simulation and results showing the robust-ness achieved.
2. THE PLANT
This section follows Amira (2002) in its general lines;results about the control of this plant (without usinga multi-agent system) have been published in (Vinagreet al., 2009). The plant shown in figure 1 consists inthree equal constant section cylindrical tanks, connectedby pipes with valves (see data in table 1). Water flows in
? Research for this paper was partially supported by grantPTDC/EME-CRO/70341/2006 of FCT, funded by POCI 2010,POS C, FSE and MCTES; and by the Portuguese Government andFEDER under program “Programa de Financiamento Plurianual dasUnidades de I&D da FCT para as atividades de investigação dolaboratório associado LAETA” (POCTI-SFA-10-46-IDMEC).
a closed circuit, fed by two pumps, as seen in the diagramof figure 6. Each of the six valves (identified in figure 6)may be open or closed; the case of partially open valveswill not be addressed, save as to simulate a malfunction;the basic configuration used in this paper has valves VL1,VL2 and VD2 open (see figure 6) and the others closed.This system, produced by Amira (model DTS200), is usedin the Control, Automation and Robotics Laboratory ofIST for didactic purposes, and conveniently reproducesthe behaviour of industrial plants where some fluid flowsin pipes connecting tanks because of gravity. The systemis controlled by a PC using Matlab/Simulink and withHumusoft MF614 and AD622 input/output boards; thisPC is connected to the Internet and thereby to other fourPCs used to implement the plant’s controllers.
Fig. 1. The three tank plant
2.1 Model of the plant
Let hn be the height of water in tank n = 1, 2, 3, Qnm thevolume flow from tank n = 1, 2, 3 to tank m = 0, 1, 2, 3(see figure 6), Qn the volume flow of water provided by
Table 1. Three-tank system data
Tank cross-section area A = 0.0154 m2
Valve cross-section area Sv = 5 × 10−5 m2
Maximum height of water in a tank hmax = 62 cm ± 1 cmMaximum pump flow Qmax = 7 L/min
pump n = 1, 2, g = 9.8 m/s2. The controlled variables areh1 and h2 and the control actions are Q1 and Q2; theseare normalised: hn = 1 means that tank Tn is full, andQn = 1 means that pump n is delivering its maximumflow. Conservation of mass means that
Ah1 = Q1 − Q13 − Q10 (1)
Ah2 = Q2 − Q23 − Q20 − Q200 (2)
Ah3 = Q13 + Q23 − Q30 (3)
(the superimposed dot denotes differentiation with respectto time t). The flows can be found using the generalisedTorricelli rule:
Qnm = anmSvsgn(hn − hm)√
2g|hn − hm| (4)
where anm is a flux coefficient for the valve connectingtanks n and m, and h0 = 0. Defining
h = [h1 h2 h3]T (5)
Q = [Q1 Q2]T (6)
A(h) =
[−Q13 − Q10
−Q23 − Q20 − Q200
Q13 + Q32 − Q30
]
A(7)
B =1
A
[1 00 10 0
]
(8)
we can sum up (1)–(3) as
h = A(h) + BQ (9)
y =
[1 0 00 1 0
]
h (10)
where y is the plant output vector. Of course, if a valveis closed its flow is equal to 0 (because so is its coeffi-cient anm). The values of coefficients anm for each openvalve were experimentally found emptying a filled-up tankthrough the valve under consideration, all the other onesbeing closed, and with the pumps stopped; under suchconditions equation (4) solved in order to anm providesthe desired value. Results obtained show that differencescaused by the direction of the flow are minimal, and thatcoefficients depend on the difference of water heights oneither side of the valve, anm(|hn − hm|) . Their evolutionis shown in figure 2.
2.2 Decoupling
The plant is a non-linear MIMO (multiple-input, multiple-output) system. Since the plant is square (the number ofinputs being equal to that of outputs), it is possible toapply a control strategy called uncoupling, whereby eachof the inputs is made to affect one output only. The MIMOplant can then be controlled as a set of several (in thiscase, two) SISO (single-input, single-output) plants, each
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
1.2
|h1−h
3|
a nm
a
13a
23a
10a
30a
20a
200
Fig. 2. Non-linear behaviour of the valves
dealt with independently of the others. While a perfectdecoupling is seldom (if ever) possible, and each inputalways has some effect in all outputs, it is in practicepossible to reduce cross-effects to minimal values, withsuch small magnitudes that can be overcome by SISOcontrollers, as disturbances are expected to be.
Consider the following non-linear plant:
x(t) = A(x, t) + B(x, t)u(t) (11)
y(t) = C(x, t) + D(x, t)u(t) (12)
where t is the time, x is the state vector, y is the outputvector, and A, B, C and D are the matrixes of the state-space model. To decouple the plant, a new input w isintroduced by means of feedback
u(t) = F(x, t) + G(x, t)w(t) (13)
To determine these new matrixes F and G it is necessaryto know, for each output yi, the order di such that the
input vector u directly affectsddiyi
dtdi. If the i-th line of D
has at least one element different from 0, then di = 0. Inour case it is clear that d1 = d2 = 1, since the inputs Qaffect directly the derivatives of y. In any case it is possibleto write
y∗
i =ddiyi
dtdi(14)
y∗ = C∗(x, t) + D∗(x, t)u(t) (15)
which, in our case, is
[h1 h2]T
︸ ︷︷ ︸
y∗
=1
A[−Q13 −Q23 ]T
︸ ︷︷ ︸
C∗
+1
A
[1 00 1
]
︸ ︷︷ ︸
D∗
Q (16)
By making
u = F + Gw ≡ −D∗−1C∗ −D∗−1
M︸ ︷︷ ︸
F
+D∗−1L
︸ ︷︷ ︸
G
w (17)
we will gety∗ = −M + Lw (18)
If L is a diagonal matrix, this means the outputs will bedecoupled. For our system in its basic configuration, a goodchoice of M and L is
M = [a1h1 a2h2]T (19)
L =
[l1 00 l2
]T
(20)
where coefficients l1, l2, a1 and a2 must be chosen toensure a response which is fast enough for our purposes,where variables are indeed decoupled, and where controlactions do not saturate the actuators. By trial and errorthe values l1 = l2 = 100 and a1 = a2 = 300 were chosen.The resulting control loop is represented in figure 3. Themajor obstacle to a complete decoupling of variables inthis system is the saturation of actuators: pumps not onlydeliver a limited maximum flow, as are unable to providenegative control actions (which would be negative flows;this means, for instance, that it is impossible to keep atank empty if another is to have some water inside).
Fig. 3. Decoupling control loop plus SISO controllersemployed with the decoupled plant
2.3 Plant simulator
A plant simulator with a graphical interface was imple-mented in Matlab/Simulink, implementing the controlloop of figure 3, as defined by equations (4)–(10), (17)and (19)–(20), together with the non-linear behavioursregistered in figure 2. Figure 4 shows experimental andsimulation responses to unit steps, for the basic configu-ration. Steady state errors between simulated and exper-imental responses to steps are always inferior to 3.1% forall configurations tested.
0 50 100 150 200 250 300 3500
0.05
0.1
0.15
0.2
0.25
t / s
wat
er h
eigh
t / n
on−d
imen
sion
al
experimentalsimulation
Fig. 4. Decoupling results for tank T1 in the basic config-uration, with a unit step input from 0 s to 200 s
3. MULTI-AGENT CONTROL
For the purpose of a multi-agent system, agents are entitieswith some degree of autonomy (capable of taking actionsby themselves) and some capacity of interaction betweenthem (capable of taking actions accounting for the oth-ers) (Wooldridge, 2002; Vidal, 2010). Such systems are fitfor achieving fault-tolerance in control systems (Mendes,
2008). A multi-agent system was used with this plantto achieve the objectives mentioned in the Introduction.The architecture is shown in figure 5 and is collaborative(agents achieve their goals by working together, not bycompetition). Since there are three tanks it was decidedto include three autonomous agents, one for each tank.Each agent is responsible for one tank only, controlled witha controller C(s), either a PID controller or a fractionalPID controller. For this control strategy to be possible,the system should be already decoupled. But since thedecoupling strategy needs, among other things, values forwater heights and flow rates, it is necessary that eachagent should know how the system is evolving: thereforeeach agent will have a complete model of the system.Each agent has the information corresponding to its tank,and communicates it to the other agents, receiving theirsin return. Agents 1 and 2 also send a control action tothe three-tank-system; agent 3 only receives and sendsinformation. There is a fourth agent (“Main” in figure 5)responsible for the exchange of the data between the three-tank-system and the agents; data fluxes are schematisedin figure 5 and detailed in table 2. (Time delays δ aretime lags between information sent and received, from themain agent to other agents, to verify the existence of acommunication failure. Other variables have been definedor have a clear meaning.)
Notice that this “Main” agent preserves the characteristicsof a multi-agent system: all agents are autonomous (atleast partially), have a local view of the system (theydepend on one another for a full global view of the system,and if agent communications fail they will lack this globalview), and are decentralised, since there is no controllingagent (agent “Main” has no such role) (Panait and Luke,2005).
The multi-agent system was implemented in Simulink us-ing the FTNCS-MAS toolbox (Mendes et al., 2009). Thistoolbox plus Matlab’s Distributed Computing Engine en-sure the services of a multi-agent platform compliant withspecifications from FIPA (The Foundation for IntelligentPhysical Agents, an IEEE Computer Society standards or-ganisation). Each agent was implemented in one computer,and the three-tank system was handled by a fifth computerwith input/output boards (thus charged of controlling thehardware or, in the case of simulation runs, of handlingthe simulator).
3.1 Controllers
PID controllers are given by C(s) = P + Is + Ds and were
obtained for this plant using one of two different methods:
• Ziegler-Nichols tuning rules (Ziegler and Nichols,1942);
• tuning rules aiming a behaviour similar to a fractionalPID (Valério and Sá da Costa, 2006).
The former method proved to lead to the best resultswith this plant. Fractional PID controllers are given byC(s) = P + I
sλ + Dsα and were obtained for this plantusing one of two different methods:
• tuning rules similar to Ziegler-Nichols rules (Valérioand Sá da Costa, 2007);
• minimisation of a performance function (Monje et al.,2004).
The latter method proved to lead to the best resultswith this plant. Notice that of all four methods only thelast one requires knowing a model of the plant. Eachtechnique resulted in two SISO feedback controllers thatallow finding Q1 and Q2 to ensure that h1 and h2 followr1 and r2. Fractional PIDs were implemented using twoCrone approximations (Oustaloup, 1991), given by
sα ≈ k
N∏
m=1
1 + sωz,m
1 + sωp,m
(21)
ωz,m = ωl
(ωh
ωl
) 2m−1−α2N
(22)
ωp,m = ωl
(ωh
ωl
) 2m−1+α2N
(23)
where the number of poles and zeros was N = 5, therange of their placement was [ωl, ωh] = [0.01, 100] rad/s,and the gain factor k is not needed since its role is takenby parameters I and D. All controllers were implementeddigitalised, using Tustin’s method and a sampling time ofTs = 0.1 s.
Water height readings are corrupted with noise (whichcan be seen in figures 4–6 and 7–10, and are causedmainly by water surface oscillations), and cause somechattering in control actions, which are thus filtered byF = 1
1+s/0.3 , to protect the hardware; the position of the
pole was chosen so as not to affect the system’s dynamics.
Fig. 6. Diagram of the plant (taken from the simulator’sgraphical user interface)
3.2 Online identification
The multi-agent system is capable of periodically reassess-ing the experimental results to update its model of theplant. The calculations are performed every 40 seconds;this value was found to be reasonable in face of thedynamics of the system, but may be modified throughthe graphical interface (much smaller values are unaccept-able, since this time constant for updating the controllermust be significantly slower than the characteristic time-constant of the system). Experimental data shows thatthe two decoupled systems may be modelled by first ordertransfer functions with delay. This would lead, however,to a closed-loop transfer function which is not rational.For this reason, a second order transfer function model
b0n
s2+a1ns+a0n, n = 1, 2 is assumed for the decoupled sys-
tems; the corresponding closed-loop models are employedin their observable canonical form state-space represen-tations, which are the most convenient ones to take intoaccount initial conditions (which are water heights in tanksat the beginning of the last period of 40 seconds). Fromthe evolution of h1 and h2, the knowledge of the controllerparameters employed, and the reference tracked, the sim-plex search minimisation algorithm is employed to find theplant’s parameters, minimising∫
t
t−40
{
L−1
[Dnb0ns2+Pnb0ns+Inb1n
s3+(a1n+Dnb0n)s2+(a0n+Pnb0n)s+Inb0nL (rn)
]
−hn
}2
dt
n=1,2 (24)
where the time response is given by Matlab’s functionlsim.
Sometimes the procedure above leads (because of numeri-cal reasons, such as the possibility that the simplex searchalgorithm gets trapped in local minima) to parameterswith values which are clearly outside their possible rangeof variation; if this situation is detected, the existingcontroller is kept. Otherwise, apparent delay and charac-teristic time-constant values are found from the second-order models, and the tuning rules mentioned above insubsection 3.1 are employed to update the PID controller.(No such adaptive algorithm was provided for fractionalPIDs, since they were tuned by minimising a performancefunction, not by tuning rules.)
This procedure makes it possible to begin working witha poorly designed controller (that however must be goodenough to keep the system working during at least 40 sec-onds) and then improve it iteratively until an optimisedbehaviour is achieved. It also allows to cope with changingparameters, such as those resulting from clogged valves orfrom leaks in valves that should be closed.
4. RESULTS
The control system described is robust because: 1) PIDand fractional PID controllers show robustness in faceof parameter variations; 2) online identification providesrobustness by coping with different configurations (of openand closed valves); 3) online adjustment of controller pa-rameters allows reflecting the effects of identification incontrol actions; and 4) if communication fails, each agentuses a copy of the plant simulator to provide data for un-available variables, until communication is reestablished.
Figure 7 shows the evolution of h1 obtained in four cases.In two of them the best performing PID is used, in theother two it is the best performing fractional PID. Bothcontrollers are employed in two manners: implemented inthe same computer that has the input/output boards, andimplemented as part of the multiagent control system.The plot illustrates that the use of agents in severalcomputers, with consequent communication delays, doesnot significantly change the performance of the PID, butsomewhat deteriorates the performance of the fractionalPID.
Figure 8 shows the performance of the online plant iden-tification and controller update. The performance of theZiegler-Nichols PID from figure 7, kept constant duringthe experiment, is compared to what happens when three
Fig. 5. Multi-agent architecture
Table 2. Data exchanged by agents
1 2 3
Package AGi
Water height reference r1 r2 —Water height measured h1 h2 h3
Position of valves (open/closed) VD1 VD3, VD4 VD2, VL1, VL2Delays δ1 δ2 δ3FeedBack AGi
Control action Q1 Q2 —Valves coefficients az10 az20, az200 az30, az13, az23
Flow rate Q10 Q20, Q200 Q30, Q13, Q23
Delays δ1 δ2 δ3Info AGi
Water height reference r1 r2 —Water height measured h1 h2 h3
Position of valves (open/closed) VD1 VD3, VD4 VD2, VL1, VL2Control parameters P1, I1, D1 P2, I2, D2 —Parameters PIDi
Control parameters P1, I1, D1 P2, I2, D2 —Notice that if a fractional PID is used there will be additional λ and µ control parameters.
poorly guessed PIDs are employed together with onlineidentification. It is seen that the latter present a clearlyinferior performance, but are iteratively improved so thatafter 350 seconds performance becomes indistinguishable.PID parameters obtained are of the same order of magni-tude of those obtained with Ziegler-Nichols rules.
Figure 9 shows the performance of the multiagent systemwhen there are communication failures. All controllers,including the one iteratively improved from a poor initialestimate, are ultimately able to keep their performancewhen there is communication failure and thus experimen-tal data from other agents becomes unavailable, by usingthe simulator to provide estimates of necessary values. Butnotice that when the reference is sinusoidal the agents keepthe last known value for the reference and thus an error
accumulates until communications are restored. For thissituation, Figure 10 shows the quality of the simulationby comparing experimental results with those calculatedand used instead of experimental ones in case of commu-nication failure, when the controller is the Ziegler-NicholsPID.
ACKNOWLEDGEMENTS
The authors wish to appreciate the support of Prof. MárioMendes and Bruno Santos, who provided constant help onthe MA toolbox.
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0 200 400 600 800 1000 1200 1400 16000
0.2
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0.2
0.4
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