ISA Transactions 53 (2014) 1216–1222
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Design and implementation of a 2-DOF PID compensation for magnetic levitation systems Arun Ghosh a,n, T. Rakesh Krishnan b, Pailla Tejaswy a, Abhisek Mandal a, Jatin K. Pradhan a, Subhakant Ranasingh a a b
School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha 751013, India Department of Computer Science and Communication, KTH (Royal Institute of Technology), Stockholm, Sweden
art ic l e i nf o
a b s t r a c t
Article history: Received 21 June 2013 Received in revised form 4 March 2014 Accepted 16 May 2014 Available online 16 June 2014 This paper was recommended for publication by Dr. Q.-G. Wang
This paper employs a 2-DOF (degree of freedom) PID controller for compensating a physical magnetic levitation system. It is shown that because of having a feedforward gain in the proposed 2-DOF PID control, the transient performance of the compensated system can be changed in a desired manner unlike the conventional 1-DOF PID control. It is also shown that for a choice of PID parameters, although the theoretical loop robustness is the same for both the compensated systems, in real-time, 2-DOF PID control may provide superior robustness if a suitable choice of the feedforward parameter is made. The results are verified through simulations and experiments. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: 1-DOF PID controller 2-DOF PID controller MAGLEV
1. Introduction The magnetic levitation (MAGLEV) is a non-contact technology; it reduces the cost of maintenance as there is no friction and so the energy efficiency is high. This technology is the future in transportation, non-contact actuators, precision engineering, noncontact structures, satellite launching, etc. However, the openloop MAGLEV system is highly nonlinear and highly unstable and therefore, designing a simple and effective controller for such a system is very challenging. (Of course, other than this instability, there are some important problems in a real MAGLEV system such as compensation against track-induced self-excited vibration, as considered in [15].) For compensation of MAGLEV system, many advanced control schemes have been proposed in the literature, such as variable structure grey control [1], adaptive robust output feedback control [2], robust dynamic sliding mode control [3–5], some advanced nonlinear controls [6–10], fuzzy compensation based adaptive PID [11], fuzzy PID [12], and references therein. All these compensations are complex in structure and therefore, need complex computations. Some conventional lead type controllers were also used in [13,14]. (A detailed comparison is presented in Section 4.4.) As is well known, conventional (1-DOF) PID control is widely
n
Corresponding author. E-mail address: [email protected] (A. Ghosh).
http://dx.doi.org/10.1016/j.isatra.2014.05.015 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
used in practice because of its simplicity of structure and past record of success. However, the problem associated with 1-DOF PID control when applied to a highly unstable system is that it usually exhibits large overshoots due to the occurrence of the controller zeros in the closed-loop input–output transfer function at inconvenient positions. These zeros also occur in closedloop reference input to controller output transfer function and therefore, the controller output becomes high as well. The same may also happen for general 1-DOF dynamic compensations. To remedy this matter, in [21, Chapters 7, 8], a feedforward type 2-DOF dynamic compensation is proposed and it is shown therein that the controller zeros then no longer occur in the input–output transfer functions, and also by choosing feedforward gains suitably a desired input–output transfer function can be obtained, while at the same time loop robustness and disturbance/noise rejection properties remain unaltered. Motivated by the general feedforward type 2-DOF dynamic compensation reported in [21, Chapters 7, 8], this paper proposes a 2-DOF PID controller for compensation of a physical MAGLEV system. The PID parameters are designed using pole-placement method to meet design objectives such as setting time and peak overshoot. The feedforward gain present in the 2-DOF structure is used to obtain superior plant output and control effort responses. The controller is tested through simulations as well as on the experimental setup. It is shown through simulations and experiments that the proposed 2-DOF PID controller is superior to the (conventional) 1-DOF one not only in terms of input–output
A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
response and control effort, but also in terms of robustness, although the theoretical robustness is the same for both the PID controls. (Note that while the pole-placement based PID design method adopted in this paper is very effective for second order systems, there are many general PID design methods available in the literature, e.g. [16–18] where the PID parameters are designed to obtain certain gain and phase margins specifications. When the system order is more than two or it has time delay or the process to be controlled is not known, the proposed 2-DOF PID parameters may be tuned using these general approaches. It may also be noted that there exists other type of 2-DOF PID structures in the literature, which are effective for some specific applications e.g. in [19] where a fraction of PD part is kept in the feedback path to obtain superior performance over 1-DOF PID.) The paper is organized as follows. Section 2 describes the MAGLEV system considered. The proposed 2-DOF PID control structure and its design are presented in Section 3. Finally, the simulation and experimental results are shown in Section 4.
2. The magnetic levitation system The schematic of the MAGLEV system used is shown in Fig. 1. The experimental setup is manufactured by Feedback Instruments Ltd and it works on MATLABs 7 and above. The system is computer controlled and a desired controller can be implemented in real-time SIMULINK environment of MATLAB. The feedback signal from the IR (Infrared) sensor to the computer and control signals from the computer to the physical system are sent through PCI 1711 A/D card manufactured by Advantech Technologies [20]. Here, the objective is to suspend the steel ball having mass m at a distance x from the electromagnet, as shown in Fig. 1. The simplest nonlinear model [20] of the system relating to the ball position x and the electromagnet coil current i is given by i2 mx€ ¼ mg k 2 x
ð1Þ
where k is a constant depending on the coil parameters, g is the acceleration due to gravity. As the system dynamics is nonlinear, for analysis and controller design purpose it is linearized around an equilibrium point x0, i0. At this point by letting x€ ¼ 0, one obtains mgx2 k¼ 2 0 i0
1217
Table 1 The physical MAGLEV system parameters [20]. Parameters
Value
m—Mass of the steel ball g—Acceleration due to gravity i0—Equilibrium value of current x0—Equilibrium value of position k1—Control voltage to coil current gain k2—Sensor gain, offset (η) Control input voltage level (u) Sensor output voltage level (xv)
0.02 kg 9.81 m/s2 0.8 A 0.009 m 1.05 A/V 143.48 V/m, 2.8 V 75 V þ 1.25 V to 3.75 V
2
where f ði; xÞ ¼ ki =x2 . On evaluating the partial derivatives and taking Laplace transform on both sides of (3) one obtains the transfer function
Δx Ki ¼ Δi s2 K x
ð4Þ
where K i ¼ 2g=i0 , K x ¼ 2g=x0 . (Note, such a model is available in [20].) As the physical system of Fig. 1 is equipped with an inner current control loop providing coil current i proportional to the control voltage u, i.e. i ¼ k1 u, k1 being the proportionality constant [20], one obtains the transfer function
Δx k1 K i ¼ Δu s 2 K x
ð5Þ
where Δu is the incremental control voltage from its equilibrium value. Finally, considering the (IR) sensor linear with gain k2, we obtain the overall plant transfer function as PðsÞ 9
Δxv k1 k2 K i ¼ 2 Δu s Kx
ð6Þ
where xv is the sensor output (in volt). Note from the expression of Ki and Kx that this linearized system is independent of levitationball mass. The parameters of the physical system considered are presented in Table 1. With these parameters, the plant transfer function (6) becomes PðsÞ 9
b 3518:85 ¼ 2 s2 p2 s 2180
ð7Þ
Note, the open-loop system is highly unstable; the poles are located at 746.69.
ð2Þ
For linearization purpose, let x ¼ x0 þ Δx, i ¼ i0 þ Δi where Δx is the small deviation from the equilibrium position x0 and Δi the small deviation from the equilibrium coil current i0. Then the nonlinear system (1) can be linearized to ! ∂f ði; xÞ ∂f ði; xÞ Δx€ ¼ Δiþ Δx ð3Þ ∂i i0 ;x0 ∂x i0 ;x0
Current Amplifier
i Electromagnetic Actuator
u
Air Gap (x)
xv Steel Ball
A/D –D/A Board
IR Detector
IR Source
Fig. 1. Schematic of the MAGLEV system.
Computer
3. The proposed 2-DOF PID controller 3.1. Structure and advantages The block diagram of the 2-DOF PID controller to be employed for the MAGLEV system is shown in Fig. 2. As noted in the Introduction, the idea of this structure came from the general feedforward type 2-DOF dynamic controller available in [21]. The advantages of this 2-DOF structure compared to its conventional 1-DOF counterpart (shown in Fig. 3) are discussed below. With the 2-DOF PID controller, the small-signal input to output transfer function becomes
Δxv ðq2 s þ q1 Þb ¼ Δr δðsÞ
ð8Þ
and the small-signal input to controller output transfer function becomes
Δu ðs2 p2 Þðq2 s þ q1 Þ ¼ Δr δðsÞ
ð9Þ
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A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
r
+_
q2 s q1
1 s
u
Then comparing (10) and (12), one obtains
MAGLEV System
xv
2ξωn þ α b ω2n þ 2ξωn α þ p2 kp ¼ b
kd ¼
kd s 2 k p s ki
ki ¼
2-DOF PID controller
ω2n α
ð13Þ
b
Next, for robust steady state tracking, by letting Δxv =Δr ¼ 1 at s¼ 0, one obtains
Fig. 2. 2-DOF PID controller.
q1 ¼ ki
kp
ð14Þ
In summary, the design of the 2-DOF PID controller is carried out following the steps described below. r
ki s
+_
+
u
MAGLEV System
xv
1. Set q2 ¼ 0. 2. Choose suitable ξ and ωn corresponding to q2 ¼ 0 (The choice of ωn may be made from desired settling time, t s ¼ 4=ξωn .). 3. Keep the pole s ¼ α of (12) far away from s ¼ ξωn so that both ‖S‖1 o 2; ‖T‖1 o 2 are satisfied. 4. Determine kp , ki , kd following (13), and set q1 following (14). 5. Tune q2 to obtain the desired speed of the response.
kd s 1-DOF PID controller
Fig. 3. 1-DOF PID controller.
where
δðsÞ ¼ s3 þ kd bs2 þðbkp p2 Þs þ bki
ð10Þ
is the closed-loop pole polynomial. An 1-DOF PID controller, shown in Fig. 3, on the other hand, yields Δxv =Δr ¼ bðkd s2 þ kp s þ ki Þ=δðsÞ, and Δu=Δr ¼ ðs2 p2 Þðkd s2 þ kp s þ ki Þ=δðsÞ. Note, both the controllers yield the same open loop transfer function (with the loop breaking at input or output side of the plant i.e. the MAGLEV system) LðsÞ ¼
bðkd s2 þ kp s þ ki Þ ðs2 p2 Þs
ð11Þ
Therefore, for a set of choices of kp , ki , kd , the (theoretical) loop robustness becomes the same for both 1-DOF and 2-DOF PID controls, but the responses for the former become poor and often have large overshoots due to the two zeros introduced in transfer functions Δxv =Δr and Δu=Δr by the PID parameters at inconvenient locations. (In fact, it will be shown in Section 4 that due to this large overshoot, the real-time system with 1-DOF PID control exhibits less robustness.) In case of feedforward type 2-DOF PID control, these zeros do not occur (see (8), (9)) and the presence of additional gain parameters q1 ; q2 helps to add a zero at desired location to achieve superior responses.
3.2. Design First, the design is to be carried out with the feedforward parameter q2 ¼ 0. Then a suitable choice of q2 is to be made to achieve desired speed of response. The pole-placement technique is used for design. For this purpose, the desired closed-loop pole polynomial is chosen in the form
δðsÞ ¼ ðs þ αÞðs2 þ 2ξωn s þ ω2n Þ
ð12Þ
where ξ is the desired damping ratio, ωn is the desired natural frequency of oscillations of closed loop system of (8) with q2 ¼ 0. The third pole at s ¼ α is to be chosen far away from s ¼ ξωn to ensure good robustness, disturbance/noise rejection property. This can be achieved by ensuring both ‖S‖1 o2; ‖T‖1 o 2 where S ¼ ð1 þ LÞ 1 , and T ¼ ð1 þLÞ 1 L are sensitivity and complementary sensitivity transfer functions, respectively [21, Chapters 6 and 7].
For the compensated system, we choose ξ ¼ 0:8; t s ¼ 2 s. Then following step 2 we get ωn ¼ 2:5 rad=s. Next we choose α following step 3. We check that with α ¼ 350ξωn the limits noted in step 3 are satisfied, but to additionally limit the initial (sensor) output due to sensor offset and the system's initial conditions well within sensor's limits (between 1.25 V and 3.75 V) we choose, through simulations, α ¼ 750ξωn . Consequently, following step 4, one obtains kp ¼ 2:3264, ki ¼ 2:6642, kd ¼ 0:4274, q1 ¼ 2:6642. Finally, the tuning of q2 may be carried out to increase the speed of the response as shown in the next section.
4. Simulation and experimental results The 2-DOF PID controller designed in the above section is implemented both in simulations and on the experimental setup. To compare its performance, the 1-DOF PID controller with the same PID parameters is also implemented. For both simulations and experimentation, the 2-DOF controller is (minimally) realized in the form as shown in Fig. 4. The simulation diagram is presented in Fig. 5. To interface this controller with the real-time system, first, it is drawn on MATLAB-SIMULINK, as shown in Fig. 6 (where the A/D and D/A blocks are provided by the manufacturer) and then is built using the Real-time workshop. Similarly, the 1-DOF controller is also interfaced. The sampling time of the PCI A/D card is set to 1 ms. As the physical system's equilibrium point is at xv0 ¼ 1:5 V, a square wave reference with mean value of 1.5 V is applied for both simulations and experiments (see figures below). Also, the reference input is kept fixed at 1.5 V for the first 15 s to settle the response due to initial disturbances. q2
r
q1
+_
ki
+
+
-
kp
+
kd
_
u
d . dt
xv Fig. 4. The minimal realization of the proposed 2-DOF PID controller.
A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
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0.5
r
u
k1
i
i
x(s) i(s)
x
x
x
k2
x0
i0
Reference Input Response of 1−DOF PID Response of 2−DOF PID
0
ball position (V)
2-DOF PID Controller
Plant Dynamics with Sensor
Fig. 5. Simulation diagram.
−0.5 −1 −1.5 −2 −2.5
Sensor Data
PCI - 1711
xv
A / D Channel
2 - DOF PID Controller
−3
u
To MAGLEV
PCI - 1711
0
5
10
15
20
25
30
35
40
45
50
time (sec)
D / A Channel
Fig. 8. Nominal response with q2 ¼ 0 (real-time). r 4
ball position (V)
0.5
control voltage (V)
Fig. 6. Real-time controller implementation diagram.
Response of 2-DOF PID Reference Input Response of 1-DOF PID
0 -0.5
2 0 −2 −4
-1
10
15
20
25
-1.5
35
40
45
50
Fig. 9. 2-DOF PID controller output with q2 ¼ 0 (real-time).
-2 -2.5
30
time (sec)
0
5
10
15
20
25
30
35
40
45
500
50
400
time (sec)
300
As the physical system's input voltage should lie within 7 5 V, a saturation block is added at the input of the linearized plant during simulations. The simulation of the nonlinear plant is also carried out.
control voltage (V)
Fig. 7. Nominal response with q2 ¼ 0 (simulation).
4.1. Nominal set point tracking responses
200 100 0 -100 -200 -300 -400 -500 10
15
20
25
30
35
40
45
50
45
50
time (sec) Fig. 10. 1-DOF PID controller output (real-time).
-0.6 -0.8 -1
ball position (V)
The nominal (i.e. under normal or unperturbed situation) tracking responses obtained through simulations of the linearized system for both 1-DOF PID control and 2-DOF PID control with q2 ¼ 0 are presented in Fig. 7. It shows that although the 2-DOF PID with q2 ¼ 0 has more initial overshoot due to sensor offset (η), it exhibits superior responses after the oscillations (due to sensor offset) die out. It is also verified through simulations that with an increase in q2, the speed of the response increases and with the choice of q2 ¼ q1 =1:2 the responses of 2-DOF and 1-DOF PIDs become exactly the same, including the initial overshoot due to disturbances. The designed controller is also applied to non-linear system for simulations. It is found through simulations that the non-linear and linearized systems responses are almost the same for q2 Zq1 =1:2. When q2 o q1 =1:2, the non-linear system, however, cannot be simulated, since the high initial overshoot (shown in Fig. 7) causes saturation. Fortunately, for the real-time system, as the steel ball is hold by hand during the initial phase (up to 6–7 s), it is not affected by the above high overshoot. The nominal responses obtained experimentally are presented in Fig. 8. The controller outputs obtained experimentally for 2-DOF and 1-DOF PID configurations are shown in Figs. 9 and 10, respectively. These plots show that, as expected, 1-DOF PID control exhibits large overshoot and large control outputs compared to 2-DOF PID control. As shown experimentally in Fig. 11, the speed of the 2-DOF PID compensated system's
q 2 =q1 /1.5 q 2 =q1 /4
-1.2 q 2 =0
-1.4 -1.6 -1.8 -2 -2.2 -2.4 -2.6 10
15
20
25
30
35
40
time (sec) Fig. 11. Nominal response of 2-DOF PID control for different values of q2 (realtime).
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A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
Table 2 Summary of loop robustness. Controller type
Experimental
Theoretical
Delay margin DM (in ms)
Gain margin, GM
Delay margin, DM (in ms)
Gain margin, GM
Delay margin, DM (in ms)
Gain margin, GM
2.1 2.0 1.7 1.0
3:9=0:36 ¼ 10:83 3:8=0:42 ¼ 9:04 3:8=0:5 ¼ 7:5 5=0:6 ¼ 8:33
3 3 3 3
1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1
1 1 1 1
1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1
0
-0.8
-0.5
-1
-1
-1.2
ball position (V)
ball position (V)
2-DOF PID q2 ¼ 0 q2 ¼ q1/4 q2 ¼ q1 =1:5 1-DOF PID
Simulation
-1.5 -2 -2.5
Reference Input Response of 2-DOF PID
-1.4 -1.6 -1.8
Response of 2-DOF PID Reference Input
-3 10
15
20
25
30
35
40
45
-2
50
time (sec)
-2.2 10
Fig. 12. Response of 2-DOF PID control for q2 ¼ 0, T d ¼ 0, k ¼ 0.36 (real-time).
response can be varied by changing q2. Note that the settling time remains at 2 s.
15
20
25
30
35
40
45
50
time (sec) Fig. 13. Response of 2-DOF PID control for q2 ¼ 0, T d ¼ 2:1 ms, k ¼ 1 (real-time).
0
4.2. Robustness
-0.5
ball position (V)
-1 -1.5 -2 -2.5
Reponse of 1-DOF PID
-3
Reference Input -3.5 10
15
20
25
30
35
40
45
50
time (sec) Fig. 14. Response of 1-DOF PID control for T d ¼ 0 ms, k ¼0.6 (real-time).
0 -0.5
ball position (V)
The loop robustness of both 1-DOF and 2-DOF PID compensated systems is studied through simulations and experiments for different values of q2. It is verified through MATLAB that the loop transfer function (11) corresponding to both the compensated systems yields gain margin ðGMÞ ¼ 1, phase margin (PM)¼ 89.81, and gain cross-over frequency (ωg)¼1500 rad/s leading to delay margin DM ¼ π PM=ð180ωg Þ ¼ 1 ms. Further, ‖S‖1 ; ‖T‖1 both being less than 2 signify good loop robustness. To verify the GM through simulations and experiments, a gain block, k, is added at the controller output and is varied from nominal value of unity. If the range of gain k for which the compensated system remains stable equals to ½k1 k2 then GM is given by k2 =k1 . Similarly, to compute the DM, an artificial delay block e sT d is inserted at the controller output and the value of delay Td is increased from its nominal value zero till the system becomes unstable. In every case, a square wave reference input with mean value of 1.5 V and peak to peak value of 1 V is applied. The results obtained are summarized in Table 2. It shows that although the robustness obtained theoretically and through simulations is the same for both the compensated systems, the robustness of 2-DOF PID compensated system, when checked experimentally, increases with a decrease in q2. In this regard, some real-time responses of the perturbed system are shown in Figs. 12–15. Fig. 15 shows that for 1-DOF PID control, the system becomes unstable for T d 4 1 ms due to high overshoot for which the sensor voltage exceeds its limit. Similarly, due to high overshoot, as shown in Fig. 14, the 1-DOF PID control also becomes unstable for gain k o 0:6. Figs. 12 and 13, on the other hand, show that due to less overshoot, the 2-DOF PID control can tolerate gain reduction up to k ¼0.36 and loop delay up to Td ¼2.1 ms, respectively. Thus, as the overshoot in 2-DOF PID control can be reduced by decreasing the feedforward parameter
-1 -1.5 -2 -2.5
Response of 1-DOF PID
-3 -3.5 10
Reference Input 15
20
25
30
35
40
45
time (sec) Fig. 15. Response of 1-DOF PID control for T d ¼ 1 ms, k ¼1 (real-time).
50
A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
q2, the robustness (against gain variations and loop delay) of the real-time MAGLEV system can be improved. As the system is highly nonlinear, its behavior at a point far away from the equilibrium one is also studied. For this purpose, a square wave reference input with mean value of 1.5 V and peak to peak value of A is applied and the value of A is increased till the compensated system becomes unstable. Experimentally, it is found that for 2-DOF PID control with q2 ¼ 0, the value of A becomes 3 V, and with an increase in q2, the value of A decreases. For 1-DOF PID control, however, we found A ¼2 V only. Clearly, 2-DOF PID control performance with q2 ¼ 0 in this respect is also superior and that also because of negligible overshoot in the input–output response. Finally, for a given equilibrium point as the linearized system dynamics given by (6) is independent of mass (m) of the steel ball, it is verified experimentally that both (1-DOF and 2-DOF) the compensated systems are capable to levitate two steel balls together and its behavior remains almost unaltered. In this regard, the response for 2-DOF PID control is shown in Fig. 16. 4.3. Disturbance rejection behavior During the set point variations, the behavior of the compensated systems, when an output disturbance is forced to the system, is also studied. As the output disturbance (occurring at the output of the plant) to the system output transfer function for both 1-DOF and 2-DOF PID controls is given by the sensitivity function S ¼ 1=ð1 þ LÞ, which is the same for both the controls, the disturbance rejection behavior of both the compensated systems with the same PID parameters should be identical. Now, with the PID parameters noted in Section 3, we have ‖S‖1 ¼ 1:33, which being -0.8
Reference Input Response of 2-DOF PID
-1
1221
o 2, signifies that both the compensated systems have good disturbance rejection behavior. The same is also verified through simulations and experiments. For simulation and experimentation purpose, a periodic output disturbance of magnitude 0.3 V (i.e. 20% of xv0 ) and period 10 s is applied. The responses obtained through simulation are shown in Fig. 17 and those obtained through experimentation in Fig. 18. These show that the disturbance rejection properties of both the compensated systems are almost identical. Moreover, the behavior of both the compensated systems, when the steel ball is disturbed by hand, is noticed and the responses are shown in Fig. 19, where the external disturbance is applied in between 35 s and 40 s. These also show that the disturbance attenuation capability of both the designs are almost similar. 4.4. Comparison with the existing controls Many compensated methods were reported in [1–14] and references therein. Most of these methods use nonlinear control and therefore, their structure and design both are complex. In [1], an integral variable structure grey control is used for MAGLEV system to reduce chattering or steady state errors. In [2], an adaptive robust output feedback control is used, where the controller is designed using backstepping approach and nonlinear observer is used to estimate the states. In [3], a robust dynamic sliding mode control is used, where a recurrent Elman neural network is employed to estimate uncertainties. In [4], a modified sliding mode control is employed to tackle non-affine MAGLEV systems. In [5], first the nonlinear plant is linearized using feedback linearization and then sliding mode control is used for the linearized system. In [8], a feedback linearizing nonlinear control is used. In [10], an adaptive controller based on fast, online parameter estimation, exact linearization, generalized PI control is proposed. In [11], an adaptive PID controller is used for tracking and a fuzzy controller is used to guarantee stabilization. Clearly, all the above control methods are nonlinear and complex, and additionally, the 1
Reference Input Response of 1−DOF PID Response of 2−DOF PID
-1.4
ball position (V)
ball position (V)
-1.2
-1.6 -1.8 -2
0 −1 −2 −3
-2.2 10
15
20
25
30
35
40
45
50
−4
0
5
10
15
20
time (sec) Fig. 16. Response of 2-DOF PID control with q2 ¼ 0 for two balls (real-time).
Reference Input Response of 1−DOF PID Response of 2−DOF PID
0
1
−1 −2 −3
0
5
10
15
20
25
30
35
40
45
50
Fig. 18. Response of 1-DOF and 2-DOF PID controls subjected to periodic output disturbance (real-time).
ball position (V)
ball position (V)
1
−4
25
time (sec)
30
35
40
45
50
time (sec) Fig. 17. Response of 1-DOF and 2-DOF PID controls subjected to periodic output disturbance (simulation).
Reference Input Response of 1−DOF PID Response of 2−DOF PID
0 −1 −2 −3 −4
0
5
10
15
20
25
30
35
40
45
50
time (sec) Fig. 19. Response of 1-DOF and 2-DOF PID controls subjected to external disturbance forced by hand (real-time).
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A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
current feedback is necessary to implement the control laws. The conventional lead-type controllers employed in [13,14] show large overshoot in the step responses, as expected for an 1-DOF control. The structure of the feedforward type 2-DOF PID control proposed, on the other hand, is much simpler and its design (as described in Section 3.2) is very easy. The achieved performance and robustness of this controller are also satisfactory. 5. Conclusion A 2-DOF PID controller has been proposed, designed, and implemented for a physical MAGLEV system. For comparison, the 1-DOF PID controller having the same PID parameters has also been implemented. It has been shown that because of the presence of a feedforward parameter in 2-DOF PID control and because of non-occurrence of controller zeros in the input–output and input–controller output transfer functions, one can obtain superior input–output response with much less control efforts. Moreover, it is shown that while the theoretical loop robustness is the same for both the PID compensated systems, in 2-DOF configuration superior robustness can be achieved in real-time implementation by properly setting the feedforward term. Acknowledgments The authors would like to thank the associate editor and reviewers for their valuable comments and suggestions to improve the presentation of this paper. The authors acknowledge SRIC, IIT Bhubaneswar for providing financial support to carry out this research work under the In-house project SP 0033. References [1] Chiang HK, Chen CA, Li MY. Integral variable-structure grey control for magnetic levitation system. Proc Inst Elect Eng, Electr Power Appl 2006;153(6): 809–14. [2] Yang ZJ, Kunitoshi K, Kanae S. Adaptive robust output-feedback control of a magnetic levitation system by K-filter approach. IEEE Trans Ind Electron 2008;55(January (1)):390–9.
[3] Lin FJ, Chen SY, Shyu KK. Robust dynamic sliding-mode control using adaptive RENN for magnetic levitation system. IEEE Trans Neural Netw 2009;20(6): 938–51. [4] Gutierrez HM, Ro PI. Magnetic servo levitation by sliding-mode control of nonaffine systems with algebraic input invertibility. IEEE Trans Ind Electron 2005;52(October (5)):1449–55. [5] Shieh HJ, Siao JH, Liu YC. A robust optimal sliding mode control approach for magnetic levitation systems. Asian J Control 2010;12(July (4)):480–7. [6] Bonivento C, Gentili L, Marconi L. Balanced robust regulation of a magnetic levitation system. IEEE Trans Control Syst Technol 2005;13(November (6)): 1036–9. [7] Lin F, Teng L, Shieh P. Intelligent adaptive backstepping control system for magnetic levitation apparatus. IEEE Trans Magn 2007;43(May (5)):2009–18. [8] Hajjaji AE, Ouladsine M. Modeling and nonlinear control of magnetic levitation systems. IEEE Trans Ind Electron 2001;48(August (4)):831–8. [9] Peterson KS, Grizzle JW, Stefanopoulou AG. Nonlinear control for magnetic levitation of automotive engine vales. IEEE Trans Control Syst Technol 2006; 14(March (2)):346–54. [10] Morales R, Feliu V, Ramirez H. Nonlinear control for magnetic levitation systems based on fast online algebraic identification of the input gain. IEEE Trans Control Syst Technol 2011;19(4):757–71. [11] Lin CM, Lin MH, Chen CW. SoPC-based adaptive PID control system design for magnetic levitation system. IEEE Syst J 2011;5(2):278–87. [12] Marjan G, Tovornik B. Modelling and control of the magnetic suspension system. ISA Trans 2003;42:89–100. [13] Banerjee S, Prasad D, Pal J. Large gap control in electromagnetic levitation. ISA Trans 2006;45:215–24. [14] Banerjee S, Prasad D, Pal J. Design, implementation, and testing of a single axis levitation system for the suspension of a platform. ISA Trans 2007;46:239–46. [15] D. Zhou, J. Li and K. Zhang, Amplitude control of the track-induced self-excited vibration for a maglev system, ISA Trans (2014), http://dx.doi.org/10.1016/j. isatra.2013.12.016 (in press). [16] Fung H-W, Wang Q-G, Lee TH. PI Tuning in terms of gain and phase margins. Automatica 1998;34:1145–9. [17] Wang Q-G, Fung H-W, Zhang Y. PID tuning with exact gain and phase margins. ISA Trans 1999;38:243–9. [18] Tan KK, Lee TH, Wang Q-G. Enhanced automatic tuning procedure for process control of PI/PID controllers. AIChE 1996;42:2555–62. [19] Cui J, Guo ZY, Yang ZC, Hao YL, Yan GZ. Electrical coupling suppression and transient response improvement for a microgyroscope using ascending frequency drive with a 2-DOF PID controller. J Micromech Microeng 2011;21:095020 (11 pp). [20] Magnetic Levitation: Control Experiments Feedback Instruments Limited, UK, 2011. [21] Wolovich WA. Automatic control systems: basic analysis and design. Rochester, NY: Saunders College Publishing; 1994.
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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Design and implementation of a 2-DOF PID compensation for magnetic levitation systems Arun Ghosh a,n, T. Rakesh Krishnan b, Pailla Tejaswy a, Abhisek Mandal a, Jatin K. Pradhan a, Subhakant Ranasingh a a b
School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha 751013, India Department of Computer Science and Communication, KTH (Royal Institute of Technology), Stockholm, Sweden
art ic l e i nf o
a b s t r a c t
Article history: Received 21 June 2013 Received in revised form 4 March 2014 Accepted 16 May 2014 Available online 16 June 2014 This paper was recommended for publication by Dr. Q.-G. Wang
This paper employs a 2-DOF (degree of freedom) PID controller for compensating a physical magnetic levitation system. It is shown that because of having a feedforward gain in the proposed 2-DOF PID control, the transient performance of the compensated system can be changed in a desired manner unlike the conventional 1-DOF PID control. It is also shown that for a choice of PID parameters, although the theoretical loop robustness is the same for both the compensated systems, in real-time, 2-DOF PID control may provide superior robustness if a suitable choice of the feedforward parameter is made. The results are verified through simulations and experiments. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: 1-DOF PID controller 2-DOF PID controller MAGLEV
1. Introduction The magnetic levitation (MAGLEV) is a non-contact technology; it reduces the cost of maintenance as there is no friction and so the energy efficiency is high. This technology is the future in transportation, non-contact actuators, precision engineering, noncontact structures, satellite launching, etc. However, the openloop MAGLEV system is highly nonlinear and highly unstable and therefore, designing a simple and effective controller for such a system is very challenging. (Of course, other than this instability, there are some important problems in a real MAGLEV system such as compensation against track-induced self-excited vibration, as considered in [15].) For compensation of MAGLEV system, many advanced control schemes have been proposed in the literature, such as variable structure grey control [1], adaptive robust output feedback control [2], robust dynamic sliding mode control [3–5], some advanced nonlinear controls [6–10], fuzzy compensation based adaptive PID [11], fuzzy PID [12], and references therein. All these compensations are complex in structure and therefore, need complex computations. Some conventional lead type controllers were also used in [13,14]. (A detailed comparison is presented in Section 4.4.) As is well known, conventional (1-DOF) PID control is widely
n
Corresponding author. E-mail address: [email protected] (A. Ghosh).
http://dx.doi.org/10.1016/j.isatra.2014.05.015 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
used in practice because of its simplicity of structure and past record of success. However, the problem associated with 1-DOF PID control when applied to a highly unstable system is that it usually exhibits large overshoots due to the occurrence of the controller zeros in the closed-loop input–output transfer function at inconvenient positions. These zeros also occur in closedloop reference input to controller output transfer function and therefore, the controller output becomes high as well. The same may also happen for general 1-DOF dynamic compensations. To remedy this matter, in [21, Chapters 7, 8], a feedforward type 2-DOF dynamic compensation is proposed and it is shown therein that the controller zeros then no longer occur in the input–output transfer functions, and also by choosing feedforward gains suitably a desired input–output transfer function can be obtained, while at the same time loop robustness and disturbance/noise rejection properties remain unaltered. Motivated by the general feedforward type 2-DOF dynamic compensation reported in [21, Chapters 7, 8], this paper proposes a 2-DOF PID controller for compensation of a physical MAGLEV system. The PID parameters are designed using pole-placement method to meet design objectives such as setting time and peak overshoot. The feedforward gain present in the 2-DOF structure is used to obtain superior plant output and control effort responses. The controller is tested through simulations as well as on the experimental setup. It is shown through simulations and experiments that the proposed 2-DOF PID controller is superior to the (conventional) 1-DOF one not only in terms of input–output
A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
response and control effort, but also in terms of robustness, although the theoretical robustness is the same for both the PID controls. (Note that while the pole-placement based PID design method adopted in this paper is very effective for second order systems, there are many general PID design methods available in the literature, e.g. [16–18] where the PID parameters are designed to obtain certain gain and phase margins specifications. When the system order is more than two or it has time delay or the process to be controlled is not known, the proposed 2-DOF PID parameters may be tuned using these general approaches. It may also be noted that there exists other type of 2-DOF PID structures in the literature, which are effective for some specific applications e.g. in [19] where a fraction of PD part is kept in the feedback path to obtain superior performance over 1-DOF PID.) The paper is organized as follows. Section 2 describes the MAGLEV system considered. The proposed 2-DOF PID control structure and its design are presented in Section 3. Finally, the simulation and experimental results are shown in Section 4.
2. The magnetic levitation system The schematic of the MAGLEV system used is shown in Fig. 1. The experimental setup is manufactured by Feedback Instruments Ltd and it works on MATLABs 7 and above. The system is computer controlled and a desired controller can be implemented in real-time SIMULINK environment of MATLAB. The feedback signal from the IR (Infrared) sensor to the computer and control signals from the computer to the physical system are sent through PCI 1711 A/D card manufactured by Advantech Technologies [20]. Here, the objective is to suspend the steel ball having mass m at a distance x from the electromagnet, as shown in Fig. 1. The simplest nonlinear model [20] of the system relating to the ball position x and the electromagnet coil current i is given by i2 mx€ ¼ mg k 2 x
ð1Þ
where k is a constant depending on the coil parameters, g is the acceleration due to gravity. As the system dynamics is nonlinear, for analysis and controller design purpose it is linearized around an equilibrium point x0, i0. At this point by letting x€ ¼ 0, one obtains mgx2 k¼ 2 0 i0
1217
Table 1 The physical MAGLEV system parameters [20]. Parameters
Value
m—Mass of the steel ball g—Acceleration due to gravity i0—Equilibrium value of current x0—Equilibrium value of position k1—Control voltage to coil current gain k2—Sensor gain, offset (η) Control input voltage level (u) Sensor output voltage level (xv)
0.02 kg 9.81 m/s2 0.8 A 0.009 m 1.05 A/V 143.48 V/m, 2.8 V 75 V þ 1.25 V to 3.75 V
2
where f ði; xÞ ¼ ki =x2 . On evaluating the partial derivatives and taking Laplace transform on both sides of (3) one obtains the transfer function
Δx Ki ¼ Δi s2 K x
ð4Þ
where K i ¼ 2g=i0 , K x ¼ 2g=x0 . (Note, such a model is available in [20].) As the physical system of Fig. 1 is equipped with an inner current control loop providing coil current i proportional to the control voltage u, i.e. i ¼ k1 u, k1 being the proportionality constant [20], one obtains the transfer function
Δx k1 K i ¼ Δu s 2 K x
ð5Þ
where Δu is the incremental control voltage from its equilibrium value. Finally, considering the (IR) sensor linear with gain k2, we obtain the overall plant transfer function as PðsÞ 9
Δxv k1 k2 K i ¼ 2 Δu s Kx
ð6Þ
where xv is the sensor output (in volt). Note from the expression of Ki and Kx that this linearized system is independent of levitationball mass. The parameters of the physical system considered are presented in Table 1. With these parameters, the plant transfer function (6) becomes PðsÞ 9
b 3518:85 ¼ 2 s2 p2 s 2180
ð7Þ
Note, the open-loop system is highly unstable; the poles are located at 746.69.
ð2Þ
For linearization purpose, let x ¼ x0 þ Δx, i ¼ i0 þ Δi where Δx is the small deviation from the equilibrium position x0 and Δi the small deviation from the equilibrium coil current i0. Then the nonlinear system (1) can be linearized to ! ∂f ði; xÞ ∂f ði; xÞ Δx€ ¼ Δiþ Δx ð3Þ ∂i i0 ;x0 ∂x i0 ;x0
Current Amplifier
i Electromagnetic Actuator
u
Air Gap (x)
xv Steel Ball
A/D –D/A Board
IR Detector
IR Source
Fig. 1. Schematic of the MAGLEV system.
Computer
3. The proposed 2-DOF PID controller 3.1. Structure and advantages The block diagram of the 2-DOF PID controller to be employed for the MAGLEV system is shown in Fig. 2. As noted in the Introduction, the idea of this structure came from the general feedforward type 2-DOF dynamic controller available in [21]. The advantages of this 2-DOF structure compared to its conventional 1-DOF counterpart (shown in Fig. 3) are discussed below. With the 2-DOF PID controller, the small-signal input to output transfer function becomes
Δxv ðq2 s þ q1 Þb ¼ Δr δðsÞ
ð8Þ
and the small-signal input to controller output transfer function becomes
Δu ðs2 p2 Þðq2 s þ q1 Þ ¼ Δr δðsÞ
ð9Þ
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A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
r
+_
q2 s q1
1 s
u
Then comparing (10) and (12), one obtains
MAGLEV System
xv
2ξωn þ α b ω2n þ 2ξωn α þ p2 kp ¼ b
kd ¼
kd s 2 k p s ki
ki ¼
2-DOF PID controller
ω2n α
ð13Þ
b
Next, for robust steady state tracking, by letting Δxv =Δr ¼ 1 at s¼ 0, one obtains
Fig. 2. 2-DOF PID controller.
q1 ¼ ki
kp
ð14Þ
In summary, the design of the 2-DOF PID controller is carried out following the steps described below. r
ki s
+_
+
u
MAGLEV System
xv
1. Set q2 ¼ 0. 2. Choose suitable ξ and ωn corresponding to q2 ¼ 0 (The choice of ωn may be made from desired settling time, t s ¼ 4=ξωn .). 3. Keep the pole s ¼ α of (12) far away from s ¼ ξωn so that both ‖S‖1 o 2; ‖T‖1 o 2 are satisfied. 4. Determine kp , ki , kd following (13), and set q1 following (14). 5. Tune q2 to obtain the desired speed of the response.
kd s 1-DOF PID controller
Fig. 3. 1-DOF PID controller.
where
δðsÞ ¼ s3 þ kd bs2 þðbkp p2 Þs þ bki
ð10Þ
is the closed-loop pole polynomial. An 1-DOF PID controller, shown in Fig. 3, on the other hand, yields Δxv =Δr ¼ bðkd s2 þ kp s þ ki Þ=δðsÞ, and Δu=Δr ¼ ðs2 p2 Þðkd s2 þ kp s þ ki Þ=δðsÞ. Note, both the controllers yield the same open loop transfer function (with the loop breaking at input or output side of the plant i.e. the MAGLEV system) LðsÞ ¼
bðkd s2 þ kp s þ ki Þ ðs2 p2 Þs
ð11Þ
Therefore, for a set of choices of kp , ki , kd , the (theoretical) loop robustness becomes the same for both 1-DOF and 2-DOF PID controls, but the responses for the former become poor and often have large overshoots due to the two zeros introduced in transfer functions Δxv =Δr and Δu=Δr by the PID parameters at inconvenient locations. (In fact, it will be shown in Section 4 that due to this large overshoot, the real-time system with 1-DOF PID control exhibits less robustness.) In case of feedforward type 2-DOF PID control, these zeros do not occur (see (8), (9)) and the presence of additional gain parameters q1 ; q2 helps to add a zero at desired location to achieve superior responses.
3.2. Design First, the design is to be carried out with the feedforward parameter q2 ¼ 0. Then a suitable choice of q2 is to be made to achieve desired speed of response. The pole-placement technique is used for design. For this purpose, the desired closed-loop pole polynomial is chosen in the form
δðsÞ ¼ ðs þ αÞðs2 þ 2ξωn s þ ω2n Þ
ð12Þ
where ξ is the desired damping ratio, ωn is the desired natural frequency of oscillations of closed loop system of (8) with q2 ¼ 0. The third pole at s ¼ α is to be chosen far away from s ¼ ξωn to ensure good robustness, disturbance/noise rejection property. This can be achieved by ensuring both ‖S‖1 o2; ‖T‖1 o 2 where S ¼ ð1 þ LÞ 1 , and T ¼ ð1 þLÞ 1 L are sensitivity and complementary sensitivity transfer functions, respectively [21, Chapters 6 and 7].
For the compensated system, we choose ξ ¼ 0:8; t s ¼ 2 s. Then following step 2 we get ωn ¼ 2:5 rad=s. Next we choose α following step 3. We check that with α ¼ 350ξωn the limits noted in step 3 are satisfied, but to additionally limit the initial (sensor) output due to sensor offset and the system's initial conditions well within sensor's limits (between 1.25 V and 3.75 V) we choose, through simulations, α ¼ 750ξωn . Consequently, following step 4, one obtains kp ¼ 2:3264, ki ¼ 2:6642, kd ¼ 0:4274, q1 ¼ 2:6642. Finally, the tuning of q2 may be carried out to increase the speed of the response as shown in the next section.
4. Simulation and experimental results The 2-DOF PID controller designed in the above section is implemented both in simulations and on the experimental setup. To compare its performance, the 1-DOF PID controller with the same PID parameters is also implemented. For both simulations and experimentation, the 2-DOF controller is (minimally) realized in the form as shown in Fig. 4. The simulation diagram is presented in Fig. 5. To interface this controller with the real-time system, first, it is drawn on MATLAB-SIMULINK, as shown in Fig. 6 (where the A/D and D/A blocks are provided by the manufacturer) and then is built using the Real-time workshop. Similarly, the 1-DOF controller is also interfaced. The sampling time of the PCI A/D card is set to 1 ms. As the physical system's equilibrium point is at xv0 ¼ 1:5 V, a square wave reference with mean value of 1.5 V is applied for both simulations and experiments (see figures below). Also, the reference input is kept fixed at 1.5 V for the first 15 s to settle the response due to initial disturbances. q2
r
q1
+_
ki
+
+
-
kp
+
kd
_
u
d . dt
xv Fig. 4. The minimal realization of the proposed 2-DOF PID controller.
A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
1219
0.5
r
u
k1
i
i
x(s) i(s)
x
x
x
k2
x0
i0
Reference Input Response of 1−DOF PID Response of 2−DOF PID
0
ball position (V)
2-DOF PID Controller
Plant Dynamics with Sensor
Fig. 5. Simulation diagram.
−0.5 −1 −1.5 −2 −2.5
Sensor Data
PCI - 1711
xv
A / D Channel
2 - DOF PID Controller
−3
u
To MAGLEV
PCI - 1711
0
5
10
15
20
25
30
35
40
45
50
time (sec)
D / A Channel
Fig. 8. Nominal response with q2 ¼ 0 (real-time). r 4
ball position (V)
0.5
control voltage (V)
Fig. 6. Real-time controller implementation diagram.
Response of 2-DOF PID Reference Input Response of 1-DOF PID
0 -0.5
2 0 −2 −4
-1
10
15
20
25
-1.5
35
40
45
50
Fig. 9. 2-DOF PID controller output with q2 ¼ 0 (real-time).
-2 -2.5
30
time (sec)
0
5
10
15
20
25
30
35
40
45
500
50
400
time (sec)
300
As the physical system's input voltage should lie within 7 5 V, a saturation block is added at the input of the linearized plant during simulations. The simulation of the nonlinear plant is also carried out.
control voltage (V)
Fig. 7. Nominal response with q2 ¼ 0 (simulation).
4.1. Nominal set point tracking responses
200 100 0 -100 -200 -300 -400 -500 10
15
20
25
30
35
40
45
50
45
50
time (sec) Fig. 10. 1-DOF PID controller output (real-time).
-0.6 -0.8 -1
ball position (V)
The nominal (i.e. under normal or unperturbed situation) tracking responses obtained through simulations of the linearized system for both 1-DOF PID control and 2-DOF PID control with q2 ¼ 0 are presented in Fig. 7. It shows that although the 2-DOF PID with q2 ¼ 0 has more initial overshoot due to sensor offset (η), it exhibits superior responses after the oscillations (due to sensor offset) die out. It is also verified through simulations that with an increase in q2, the speed of the response increases and with the choice of q2 ¼ q1 =1:2 the responses of 2-DOF and 1-DOF PIDs become exactly the same, including the initial overshoot due to disturbances. The designed controller is also applied to non-linear system for simulations. It is found through simulations that the non-linear and linearized systems responses are almost the same for q2 Zq1 =1:2. When q2 o q1 =1:2, the non-linear system, however, cannot be simulated, since the high initial overshoot (shown in Fig. 7) causes saturation. Fortunately, for the real-time system, as the steel ball is hold by hand during the initial phase (up to 6–7 s), it is not affected by the above high overshoot. The nominal responses obtained experimentally are presented in Fig. 8. The controller outputs obtained experimentally for 2-DOF and 1-DOF PID configurations are shown in Figs. 9 and 10, respectively. These plots show that, as expected, 1-DOF PID control exhibits large overshoot and large control outputs compared to 2-DOF PID control. As shown experimentally in Fig. 11, the speed of the 2-DOF PID compensated system's
q 2 =q1 /1.5 q 2 =q1 /4
-1.2 q 2 =0
-1.4 -1.6 -1.8 -2 -2.2 -2.4 -2.6 10
15
20
25
30
35
40
time (sec) Fig. 11. Nominal response of 2-DOF PID control for different values of q2 (realtime).
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A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
Table 2 Summary of loop robustness. Controller type
Experimental
Theoretical
Delay margin DM (in ms)
Gain margin, GM
Delay margin, DM (in ms)
Gain margin, GM
Delay margin, DM (in ms)
Gain margin, GM
2.1 2.0 1.7 1.0
3:9=0:36 ¼ 10:83 3:8=0:42 ¼ 9:04 3:8=0:5 ¼ 7:5 5=0:6 ¼ 8:33
3 3 3 3
1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1
1 1 1 1
1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1 1=0:26 ¼ 1
0
-0.8
-0.5
-1
-1
-1.2
ball position (V)
ball position (V)
2-DOF PID q2 ¼ 0 q2 ¼ q1/4 q2 ¼ q1 =1:5 1-DOF PID
Simulation
-1.5 -2 -2.5
Reference Input Response of 2-DOF PID
-1.4 -1.6 -1.8
Response of 2-DOF PID Reference Input
-3 10
15
20
25
30
35
40
45
-2
50
time (sec)
-2.2 10
Fig. 12. Response of 2-DOF PID control for q2 ¼ 0, T d ¼ 0, k ¼ 0.36 (real-time).
response can be varied by changing q2. Note that the settling time remains at 2 s.
15
20
25
30
35
40
45
50
time (sec) Fig. 13. Response of 2-DOF PID control for q2 ¼ 0, T d ¼ 2:1 ms, k ¼ 1 (real-time).
0
4.2. Robustness
-0.5
ball position (V)
-1 -1.5 -2 -2.5
Reponse of 1-DOF PID
-3
Reference Input -3.5 10
15
20
25
30
35
40
45
50
time (sec) Fig. 14. Response of 1-DOF PID control for T d ¼ 0 ms, k ¼0.6 (real-time).
0 -0.5
ball position (V)
The loop robustness of both 1-DOF and 2-DOF PID compensated systems is studied through simulations and experiments for different values of q2. It is verified through MATLAB that the loop transfer function (11) corresponding to both the compensated systems yields gain margin ðGMÞ ¼ 1, phase margin (PM)¼ 89.81, and gain cross-over frequency (ωg)¼1500 rad/s leading to delay margin DM ¼ π PM=ð180ωg Þ ¼ 1 ms. Further, ‖S‖1 ; ‖T‖1 both being less than 2 signify good loop robustness. To verify the GM through simulations and experiments, a gain block, k, is added at the controller output and is varied from nominal value of unity. If the range of gain k for which the compensated system remains stable equals to ½k1 k2 then GM is given by k2 =k1 . Similarly, to compute the DM, an artificial delay block e sT d is inserted at the controller output and the value of delay Td is increased from its nominal value zero till the system becomes unstable. In every case, a square wave reference input with mean value of 1.5 V and peak to peak value of 1 V is applied. The results obtained are summarized in Table 2. It shows that although the robustness obtained theoretically and through simulations is the same for both the compensated systems, the robustness of 2-DOF PID compensated system, when checked experimentally, increases with a decrease in q2. In this regard, some real-time responses of the perturbed system are shown in Figs. 12–15. Fig. 15 shows that for 1-DOF PID control, the system becomes unstable for T d 4 1 ms due to high overshoot for which the sensor voltage exceeds its limit. Similarly, due to high overshoot, as shown in Fig. 14, the 1-DOF PID control also becomes unstable for gain k o 0:6. Figs. 12 and 13, on the other hand, show that due to less overshoot, the 2-DOF PID control can tolerate gain reduction up to k ¼0.36 and loop delay up to Td ¼2.1 ms, respectively. Thus, as the overshoot in 2-DOF PID control can be reduced by decreasing the feedforward parameter
-1 -1.5 -2 -2.5
Response of 1-DOF PID
-3 -3.5 10
Reference Input 15
20
25
30
35
40
45
time (sec) Fig. 15. Response of 1-DOF PID control for T d ¼ 1 ms, k ¼1 (real-time).
50
A. Ghosh et al. / ISA Transactions 53 (2014) 1216–1222
q2, the robustness (against gain variations and loop delay) of the real-time MAGLEV system can be improved. As the system is highly nonlinear, its behavior at a point far away from the equilibrium one is also studied. For this purpose, a square wave reference input with mean value of 1.5 V and peak to peak value of A is applied and the value of A is increased till the compensated system becomes unstable. Experimentally, it is found that for 2-DOF PID control with q2 ¼ 0, the value of A becomes 3 V, and with an increase in q2, the value of A decreases. For 1-DOF PID control, however, we found A ¼2 V only. Clearly, 2-DOF PID control performance with q2 ¼ 0 in this respect is also superior and that also because of negligible overshoot in the input–output response. Finally, for a given equilibrium point as the linearized system dynamics given by (6) is independent of mass (m) of the steel ball, it is verified experimentally that both (1-DOF and 2-DOF) the compensated systems are capable to levitate two steel balls together and its behavior remains almost unaltered. In this regard, the response for 2-DOF PID control is shown in Fig. 16. 4.3. Disturbance rejection behavior During the set point variations, the behavior of the compensated systems, when an output disturbance is forced to the system, is also studied. As the output disturbance (occurring at the output of the plant) to the system output transfer function for both 1-DOF and 2-DOF PID controls is given by the sensitivity function S ¼ 1=ð1 þ LÞ, which is the same for both the controls, the disturbance rejection behavior of both the compensated systems with the same PID parameters should be identical. Now, with the PID parameters noted in Section 3, we have ‖S‖1 ¼ 1:33, which being -0.8
Reference Input Response of 2-DOF PID
-1
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o 2, signifies that both the compensated systems have good disturbance rejection behavior. The same is also verified through simulations and experiments. For simulation and experimentation purpose, a periodic output disturbance of magnitude 0.3 V (i.e. 20% of xv0 ) and period 10 s is applied. The responses obtained through simulation are shown in Fig. 17 and those obtained through experimentation in Fig. 18. These show that the disturbance rejection properties of both the compensated systems are almost identical. Moreover, the behavior of both the compensated systems, when the steel ball is disturbed by hand, is noticed and the responses are shown in Fig. 19, where the external disturbance is applied in between 35 s and 40 s. These also show that the disturbance attenuation capability of both the designs are almost similar. 4.4. Comparison with the existing controls Many compensated methods were reported in [1–14] and references therein. Most of these methods use nonlinear control and therefore, their structure and design both are complex. In [1], an integral variable structure grey control is used for MAGLEV system to reduce chattering or steady state errors. In [2], an adaptive robust output feedback control is used, where the controller is designed using backstepping approach and nonlinear observer is used to estimate the states. In [3], a robust dynamic sliding mode control is used, where a recurrent Elman neural network is employed to estimate uncertainties. In [4], a modified sliding mode control is employed to tackle non-affine MAGLEV systems. In [5], first the nonlinear plant is linearized using feedback linearization and then sliding mode control is used for the linearized system. In [8], a feedback linearizing nonlinear control is used. In [10], an adaptive controller based on fast, online parameter estimation, exact linearization, generalized PI control is proposed. In [11], an adaptive PID controller is used for tracking and a fuzzy controller is used to guarantee stabilization. Clearly, all the above control methods are nonlinear and complex, and additionally, the 1
Reference Input Response of 1−DOF PID Response of 2−DOF PID
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ball position (V)
ball position (V)
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-1.6 -1.8 -2
0 −1 −2 −3
-2.2 10
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−4
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time (sec) Fig. 16. Response of 2-DOF PID control with q2 ¼ 0 for two balls (real-time).
Reference Input Response of 1−DOF PID Response of 2−DOF PID
0
1
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5
10
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Fig. 18. Response of 1-DOF and 2-DOF PID controls subjected to periodic output disturbance (real-time).
ball position (V)
ball position (V)
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−4
25
time (sec)
30
35
40
45
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time (sec) Fig. 17. Response of 1-DOF and 2-DOF PID controls subjected to periodic output disturbance (simulation).
Reference Input Response of 1−DOF PID Response of 2−DOF PID
0 −1 −2 −3 −4
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5
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time (sec) Fig. 19. Response of 1-DOF and 2-DOF PID controls subjected to external disturbance forced by hand (real-time).
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current feedback is necessary to implement the control laws. The conventional lead-type controllers employed in [13,14] show large overshoot in the step responses, as expected for an 1-DOF control. The structure of the feedforward type 2-DOF PID control proposed, on the other hand, is much simpler and its design (as described in Section 3.2) is very easy. The achieved performance and robustness of this controller are also satisfactory. 5. Conclusion A 2-DOF PID controller has been proposed, designed, and implemented for a physical MAGLEV system. For comparison, the 1-DOF PID controller having the same PID parameters has also been implemented. It has been shown that because of the presence of a feedforward parameter in 2-DOF PID control and because of non-occurrence of controller zeros in the input–output and input–controller output transfer functions, one can obtain superior input–output response with much less control efforts. Moreover, it is shown that while the theoretical loop robustness is the same for both the PID compensated systems, in 2-DOF configuration superior robustness can be achieved in real-time implementation by properly setting the feedforward term. Acknowledgments The authors would like to thank the associate editor and reviewers for their valuable comments and suggestions to improve the presentation of this paper. The authors acknowledge SRIC, IIT Bhubaneswar for providing financial support to carry out this research work under the In-house project SP 0033. References [1] Chiang HK, Chen CA, Li MY. Integral variable-structure grey control for magnetic levitation system. Proc Inst Elect Eng, Electr Power Appl 2006;153(6): 809–14. [2] Yang ZJ, Kunitoshi K, Kanae S. Adaptive robust output-feedback control of a magnetic levitation system by K-filter approach. IEEE Trans Ind Electron 2008;55(January (1)):390–9.
[3] Lin FJ, Chen SY, Shyu KK. Robust dynamic sliding-mode control using adaptive RENN for magnetic levitation system. IEEE Trans Neural Netw 2009;20(6): 938–51. [4] Gutierrez HM, Ro PI. Magnetic servo levitation by sliding-mode control of nonaffine systems with algebraic input invertibility. IEEE Trans Ind Electron 2005;52(October (5)):1449–55. [5] Shieh HJ, Siao JH, Liu YC. A robust optimal sliding mode control approach for magnetic levitation systems. Asian J Control 2010;12(July (4)):480–7. [6] Bonivento C, Gentili L, Marconi L. Balanced robust regulation of a magnetic levitation system. IEEE Trans Control Syst Technol 2005;13(November (6)): 1036–9. [7] Lin F, Teng L, Shieh P. Intelligent adaptive backstepping control system for magnetic levitation apparatus. IEEE Trans Magn 2007;43(May (5)):2009–18. [8] Hajjaji AE, Ouladsine M. Modeling and nonlinear control of magnetic levitation systems. IEEE Trans Ind Electron 2001;48(August (4)):831–8. [9] Peterson KS, Grizzle JW, Stefanopoulou AG. Nonlinear control for magnetic levitation of automotive engine vales. IEEE Trans Control Syst Technol 2006; 14(March (2)):346–54. [10] Morales R, Feliu V, Ramirez H. Nonlinear control for magnetic levitation systems based on fast online algebraic identification of the input gain. IEEE Trans Control Syst Technol 2011;19(4):757–71. [11] Lin CM, Lin MH, Chen CW. SoPC-based adaptive PID control system design for magnetic levitation system. IEEE Syst J 2011;5(2):278–87. [12] Marjan G, Tovornik B. Modelling and control of the magnetic suspension system. ISA Trans 2003;42:89–100. [13] Banerjee S, Prasad D, Pal J. Large gap control in electromagnetic levitation. ISA Trans 2006;45:215–24. [14] Banerjee S, Prasad D, Pal J. Design, implementation, and testing of a single axis levitation system for the suspension of a platform. ISA Trans 2007;46:239–46. [15] D. Zhou, J. Li and K. Zhang, Amplitude control of the track-induced self-excited vibration for a maglev system, ISA Trans (2014), http://dx.doi.org/10.1016/j. isatra.2013.12.016 (in press). [16] Fung H-W, Wang Q-G, Lee TH. PI Tuning in terms of gain and phase margins. Automatica 1998;34:1145–9. [17] Wang Q-G, Fung H-W, Zhang Y. PID tuning with exact gain and phase margins. ISA Trans 1999;38:243–9. [18] Tan KK, Lee TH, Wang Q-G. Enhanced automatic tuning procedure for process control of PI/PID controllers. AIChE 1996;42:2555–62. [19] Cui J, Guo ZY, Yang ZC, Hao YL, Yan GZ. Electrical coupling suppression and transient response improvement for a microgyroscope using ascending frequency drive with a 2-DOF PID controller. J Micromech Microeng 2011;21:095020 (11 pp). [20] Magnetic Levitation: Control Experiments Feedback Instruments Limited, UK, 2011. [21] Wolovich WA. Automatic control systems: basic analysis and design. Rochester, NY: Saunders College Publishing; 1994.
