AbstractâThis paper addresses a practical networked based control application. A Profibus DP is considered in order to coordinate a triangular maglev platform.
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Maglev Platform Networked Control: A Profibus DP Application ´ Ricardo Piz´a, Juli´an Salt, Member, IEEE, Antonio Sala, Member, IEEE, and Angel Cuenca, Member, IEEE,
Abstract—This paper addresses a practical networked based control application. A Profibus DP is considered in order to coordinate a triangular maglev platform. The control of this magnetic levitation process is a complicated system from an control engineering point of view due to its unstable, nonminimum phase and non-linear nature; the difficulty is growing up because the bandwidth limitation and some other problems introduced by the network which consideration is neccesary in order to prepare control, supervision and coordination of local loops tasks. A rude local classical control and a refinated based LMI solution in remote side allow to solve this problem where the packets have a variable time delays compromising performance or even stability. The results shows a good stabilization and reference position accuracy even when the operating point is changing with an external load. Index Terms—Networked control system, network delay, multirate control systems, stability analysis, LMI, Maglev. Fig. 1.
I. I NTRODUCTION
I
N recent years, the magnetic levitation has been used in a wide range of applications where the non-contact actuators leads to better results. It seems that the high-speed transportation systems [1] is the most important contribution of this technology, but other valuables possibilities are in semiconductor manufacturing by photolithography [2], haptic applications (teleoperation, surgeon training) [3], or microrobots [4]. When a control application is projected on a network based environment, in which different devices (sensor, actuator, controller) are connected by means of a shared communication medium [5], [6], [7] typical problems as data packet losses, lack of synchronization among devices, bandwidth limitations, and variable time delays occur. This problem is dramatic when the process control is very sensitive to these problems. That is the case of a maglev based platform [8], [9] networked based control [10]. This magnetic levitation process control problem is challenging because each of the systems that configure the platform are unstable and non-minimum phase and the linear model is valid for a specific operating point. In this case a Profibus DP with asynchronous operation has been selected [11], because the long experience of our research group in this specific field-bus control applications [12], [13]. In this work with difference to another similar like [14], [15], an original ´ Cuenca are with Departamento de Ingenieria R. Piz´a , J. Salt, A. Sala and A. de Sistemas y Automatica, Instituto Universitario de Automatica e Informatica Industrial, Universidad Polit´ecnica de Valencia, Camino de Vera s/n, 46022 Valencia (Spain) (e-mail: {acuenca,julian,asala,rpiza}@isa.upv.es). The authors Salt, Piza, Cuenca are grateful to project DPI 2009-14744C03-03.A. Sala is grateful to the financial support of Spanish Ministry of Education research grant DPI 2008-06731- C 02-01, and Generalitat Valenciana grant PROMETEO/2008/088.
978-1-4244-7300-7/10/$26.00 ©2010 IEEE
160
Experimental Setup
hierchical control system is proposed. Due to the packets delay magnitudes it is not viable to consider just a remote control. So an approximated local stabilizing controller is assumed and in the following step a more sophisticated remote control will be designed. This part next to host with intelligence capacity will be able to implement supervision, monitoring and platform coordination tasks as well. That is a hierarchical structure is planned. The scheme of the paper is as follows: next section describes the system to be controlled. At third section it is analyzed the delays introduced in the Profibus-DP installation and the sampling periods that permits a maglev’s feasible control. The fourth section is dedicated to introduce both the control objectives and structure considered. Finally, the experimental results and conclusions are introduced.
II. S YSTEM DESCRIPTION The experimental setup is a levitated platform that is shown in Fig. 1. The levitated platform is an equilateral triangle shape with permanent magnets located at the ends of the platform. The vertical position of each magnet is controlled performed with a maglev system operated through an electromagnet. Thus, there are three maglevs located at terminals of platform. Each maglev takes a voltage input signal to generate the magnetic field and takes the vertical position measure using a set of infrared sensors array. First of all, a single Maglev continuous model was obtained, giving the set of equations:
2
M · z¨(t) = F (t) 3 F (t) = S · I(t) + T · z(t) ˙ L · I(t) + Q · z(t) ˙ + R · I(t) = V (t)
(2) (3)
y(t) = K · z(t)
(4)
(1)
In this group of equations the system physics are reflected. This is a electromechanical system, where M is the mass of the whole levitated platform, F the force that acts over it, and S, T, and Q are physical constants. With respect to the electric circuit, I is the current, V the voltage and R and L the circuit’s resistance and inductance. Finally z and z˙ are respectively the position and speed of the load measured along Z axis. The variable y is the measure of the position taken with the infrared sensor system, being K a calibration constant. These equations can be expressed in state-space form as: ⎛
⎞ ⎛ −R ˙ I(t) L ⎝ z(t) ⎠=⎝ 0 ˙ 3S z¨(t) M
0 0
3S M
−Q L
1 0
⎞ ⎛
⎞ I(T ) ⎠ · ⎝ z(t) ⎠ + z(t) ˙ ⎛ 1 ⎞ L
+ ⎝ 0 ⎠ · V (t) 0 ⎛ ⎞ I(t) y(t) = 0 K 0 · ⎝ z(t) ⎠ z(t) ˙
(5)
III. H IERARCHICAL STRUCTURE : L OCAL AND R EMOTE C ONTROLLERS D ESCRIPTION
Fig. 2.
As previously discussed, the use of Profibus-DP network introduces certain bandwidth and time delay in communications. So for this reason, due to platform dynamics characteristics, is not possible to perform the overall control in remote assuring the stability. For this reason, it is necessary to perform certain local control directly. Thus, in order to deal with these issues, a hierarchical structure composed of a local controller plus a networked higher level was chosen and experimentally demonstrated in this paper. For each maglev a continuous local controller is designed and implemented based on classical control using root locus controller design techniques. Afterwards assuming a typical discretization technique a discrete local controller is achieved. Inside the master node a remote controller is designed using LMI gridding techniques [16] in order to implement a state feedback controller designed for controlling the overall platform model, including the remote controllers as part of the local plant. Therefore, the control architecture is a hierarchical structure where the slave nodes of the network perform a low level control layer, the network acts as a shared media and the master node with the LMI controller acts as a high level control layer. This control structure is presented in Fig. 2. One low smart with easy implementation control is proposed at local nodes although for this field there are a large number of applications based on predictive control [17], fuzzy control [18], [19] or some non-linear control methods based as for instance [20] in the scientific background. 161
Control Structure
Let us discuss now the different elements of the abovedescribed structure. A. Lower control level For the plant model described before a continuous controller (one for each maglev) was designed with the simple aim of roughly stabilizing the single-maglev model whithout taking into account the interactions. The controller was a low-gain phase lead controller. The gains were intentionally low, leaving for the remote layer the coordination with additional control action commands. For example, one of the three controller expressed in statespace becomes: ˙ X(t) = Ac · X(t) + Bc · U (t)
(6)
V (t) = Cc · X(t) + Dc · U (t)
(7)
All the controllers are very similar, so the the numerical values of just one of them are presented: ˙ X(t) = −2000 · X(t) + U (t) V (t) = −59400 · X(t) + 300 · U (t)
(8) (9)
This kind of local controller is implemented in a local node with T0 = 5 ms as sampling time. In order to implement
3
20
Sampling Delays
15
M · z¨(t) = F1 (t) + F2 (t) + F3 (t)
(10)
Jx · α ¨ (t) = T1α (t) − T2α (t) − T3α (t) ¨ = T2 (t) − T3 (t) Jy · β(t) β β
(11) (12)
T1α (t) = F1 (t) · L T2α (t) = F2 (t) · L · sin 30
(13) (14)
T2β (t) = F2 (t) · L · sin 60
(15)
T3α (t) = F3 (t) · L · sin 30 T3β (t) = F3 (t) · L · sin 60
(16) (17)
F1 (t) = S1 · I1 (t) + T1 · z1 (t) F2 (t) = S2 · I2 (t) + T2 · z2 (t)
(18) (19)
F3 (t) = S3 · I3 (t) + T3 · z3 (t) ˙ L1 · I1 (t) + Q1 · z˙1 (t) + R1 · I˙1 (t) = V1 (t) L2 · I˙2 (t) + Q2 · z˙2 (t) + R2 · I˙2 (t) = V2 (t) L3 · I˙3 (t) + Q3 · z˙3 (t) + R3 · I˙3 (t) = V3 (t)
(20) (21)
z1 (t) = z(t) + L · α(t) z2 (t) = z(t) + L · (sin 60 · β(t) − sin 30 · α(t))
(24) (25)
z3 (t) = z(t) + L · (− sin 60 · β(t) − sin 30 · α(t)) ˙ + L · α(t) ˙ z˙1 (t) = z(t) ˙ z˙2 (t) = z(t) ˙ + L · (sin 60 · β − sin 30 · α) ˙
(26) (27)
10
5
0
Fig. 3.
0
0.5
1
1.5 Time (ms)
2
2.5
3 4
x 10
Round Trip Delays Analysis
it, Tustin discretization was used. Five milliseconds was the slowest sampling frequency which stabilised one maglev simulation. This controller can perform simple tasks around a reference but can not synchronize the different platforms because is a simple controller with decoupled plant models and the possible interaction might lead to unacceptable performance or even unstability when used in the 3-maglev real system.
B. Network Delays and permissible control sampling periods As discussed above, local controllers lost stability for a sampling time greater than 5 ms. However, the disposed Profibus-DP bandwidth allows us to use a remote coordinating control action just every 20 miliseconds. Indeed, the chosen Profibus configuration parameters were bus rate 96000 KBps, with asynchronous operation way. The round trip delay appears in Fig. 3. This delay was determined to approximately lie between 3 and 14 miliseconds, which amount to one to three local sampling periods (multiples of 5 ms). Apart from experimental measurements, it is possible to consider results about the delays with a particular ProfibusDP configuration via a simulation model developed by our group [21].
C. Higher control level At higher level, in remote, a plant model is developed considering as the process the set composed by the three maglev systems and the corresponding local control subsystems. All the set of elements is considered to extract the set of equations that describes the overall platform. 162
˙ + L · (− sin 60 · β˙ − sin 30 · α) ˙ z˙3 (t) = z(t) X˙ 1 (t) = AC 1(t) · X1 (t) + BC 1 · U1 (t) X˙ 2 (t) = AC 2(t) · X2 (t) + BC 2 · U2 (t) X˙ 3 (t) = AC 3(t) · X3 (t) + BC 3 · U3 (t)
(22) (23)
(28) (29) (30) (31) (32)
V1 (t) = CC 1 · X1 (t) + DC 1 · U1 (t) V2 (t) = CC 2 · X2 (t) + DC 2 · U3 (t)
(33) (34)
V3 (t) = CC 3 · X3 (t) + DC 3 · U3 (t)
(35)
Where the subindexes 1, 2, 3 reference the corresponding maglev subsystem and the subindexes α and β are referred to the rotation angles yaw (around X axis) and pitch (aroung Y axis). This set (each maglev with its local controller) must be considered discrete with sampling period of T0 = 5ms, although as we will discuss later the remote controller will get the perspective of local side every N T0 = 20ms. Therefore, the discrete model for T0 becomes: ⎛ ⎞ ⎛ ⎞ I1 I1 ⎜ I2 ⎟ ⎜ I2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ I3 ⎟ ⎜ I3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ z1 ⎟ ⎜ z1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ z2 ⎟ ⎜ z2 ⎟ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ U1 ⎜ z3 ⎟ ⎜ z3 ⎟ ⎜ ⎟ ⎜ ⎟ + Blocal ⎝ U2 ⎠ (36) ⎜ vz1 ⎟ = Alocal ⎜ vz1 ⎟ ⎜ ⎟ ⎜ ⎟ U 3 ⎜ vz2 ⎟ ⎜ vz2 ⎟ kT0 ⎜ ⎟ ⎜ ⎟ ⎜ vz3 ⎟ ⎜ vz3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X1 ⎟ ⎜ X1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ X2 ⎠ ⎝ X2 ⎠ X3 X3 (k+1)T kT 0
0
being Alocal = A0 and Blocal = B0 :
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At this level, a controller based in an LMI gridding procedure [16] is designed. In order to develop this control system structure, the time delay introduced by Profibus network and the time delay introduced by remote controller (computation time) must be considered. The basic idea is designing a higher level controller with a sampling rate of 20 ms and computing the LMI gridding procedure allowing for a variation in input delays from 5 to 15 milliseconds. It is assumed that a time-driven policy is used in sensor and an event-driven one in the actuator part; however, considering the discrete (T0 ) nature in this side, that is the arrived control action samples will be applied at the next local sampling instant. From the remote controller -that operates each Tc = N ∗ T0 = 4 ∗ 5 = 20 ms- point of view the state equation will be:
Fig. 4.
Control and State Update Chronogram
delay will be: N −dk χ(K+1)N T0 = AN 0 χKN T0 + A0
dK
Aj−1 B0 ul−1 0
j=1
χ(K+1)N T0 = AN 0 χKN T0 +
N
+
Ah−1 B0 u((K+1)N −h)T0 0
(37) Fig. 4 shows how the actuator works. The control action is updated after a delay of δK = dK T0 . So it is possible to express:
= AN 0 χKN T0 + B(N, dk )l−1 ul−1 + B(N, dk )l ul
+
(38)
j=1
−dk χ(K+1)N T0 = AN χKN T0 +dk T0 0 N
−dK
Ai−1 0 B0 u((K+1)N −dk −i)T0 =
i=1
Ψk+1 =
AN 0 0
B(N, dk )l B(N, dk )l−1 Ψk + ul (41) 0 I Ψk+1 = A∗ (N, dk )Ψk + B ∗ (N, dk )ul (42)
where notations A∗ and B ∗ have been introduced. The design of the high level control layer can be split in two stages, state-feedback controller and observer. In this way, the first control systhesis problem can be cast as a state-feedback one for this augmented model, leading to:
and for this reason:
+
(40)
It must be noted that always Tc will be strictly greater than dk ∀k. So if an a augmented state ψk = (χk , ul−1 ) referred to the period Tc is considered, it is possible to express:
χKN T0 +dk T0 = Ad0k χKN T0 Aj−1 0 B0 u(KN +dk −j)T0
Ai−1 0 B0 ul =
i=1
h=1
dK
N
−dK
uk = −F ∗ Ψk = −Fx χKN T0 − Fu ul−1
−dk dk = AN A0 χKN T0 0
(43)
(39)
Note that subindex k is referred to Tc . So, the design problem is to obtain F ∗ . It must be observed that this computation are function of two variables N T0 and δk = dk T0 . An LMI gridding procedure has been considered to ensure the resulting controller is robust to variations in dk . A detailed explanation can be found in [16]. Basically, the global LMI gridding problem assumes the variation of Tc designed by Tk and δk . This question would be involve setting LMIs such as:
∗∗∗ e−2βT X >0 (44) X A∗ (T, δk )X − B ∗ (T, δ)M (T, δ))
Actually a ZOH operating each T0 at local side allow us to consider as it is shown in Fig. 4 the control actions before ul−1 and after ul the control update in time (KN + dk )T0 . So the model equation at the slow rate with a varying input
for all possible values of sampling period T and delay δ. In this case, T = 0.02 seconds and δ ∈ {0.005, 0.01, 0.015} were used, as discussed above. As sampling period is constant and the future input delay is unknown in the current application being explored, the feedback parameter M (T, δ) is actually a constant decision variable M .
−dk +AN 0
dK
Aj−1 0 B0 u(KN +dk −j)T0
j=1
+
N
−dK
Ai−1 0 B0 u((K+1)N −dk −i)T0 =
i=1 N −dk = AN 0 χKN T0 + A0
dK
Aj−1 0 B0 u(KN +dk −j)T0
j=1
+
N
−dK
Ai−1 0 B0 u((K+1)N −dk −i)T0
i=1
163
5
Regarding the observer case, observer gains depending on sampling period must be carried out. Time-varying gains are known in the context of Kalman filtering. The discretization of a linear CT stochastic process is
0.5
0.4
xk+1 = Ak xk + Bk uk + υk
(45)
yk = Cxk + ωk
(46)
white noise with variance where ωk and υk are uncorrelated 0 T matrices Wk and Vk = Tk eAt ΓeA t dt, and Γ is a CT variance parameter. In that case, the optimal observer is the Kalman filter with Lk obtained by
Levs position
0.3
0.2
0.1
0
−0.1
−0.2
0
1000
2000
3000
4000
5000
Time
Pk = Aˆk Λk AˆTk + Vk
mk = CPk C T + Wk
(47)
Lk = Pk C m
Λk+1 = (I − Lk C)Pk
(48)
T
−1
When packet arrives at remote side in time k, the Tk and yk are known, and also δk−1 . So in the set of equations before ˆ k , δk−1 ) would be considered. A(T Hence, the model equation to be adjointed to the variance estimation ones (47) is: x ˆk = (I−Lk C)(A(Tk−1 δk−1) x ˆk−1 +B(Tk−1 δk−1 )uk )+Lk yk (49) The separation principle is proved in [16]. In order to implement the above equation, the packet received by the remote (master) node should contain the vector of measures, the time of those ones, and the delay δ in the preceding cycle.
D. Experimental Results Once the above design is carried out, it has been tested on the experimental apparatus. The system configured in the described way keeps the platform stable despite disturbances. Fig. 5 shows the system response when, starting from a stable operation point, a load of 8.5 grams is added to the center of the levitating platform with the consequent change of operating point. In the experiment a 5% increment of mass is added in certain instant. Note that the control works well although the conditions are quite different. The Y axis represents the position in mm.
IV. C ONCLUSIONS In this paper a practical networked based control application has been developed. A Profibus DP has been considered in order to coordinate a triangular maglev platform. A simple stabilization local control and a refined based LMI solution in remote side allows to solve the problem caused by limited network bandwidth even with time-variable delays. Due to these factors and the process (unstable and non-minimum phase) the global problem had a great complexity. 164
Fig. 5.
Platform Control with perturbations
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