Robust Nonlinear Control of a Voltage-Controlled Magnetic Levitation ...
16th IEEE International Conference on Control Applications Part of IEEE Multi-conference on Systems and Control Singapore, 1-3 October 2007
TuB04.1
Robust Nonlinear Control of a Voltage-Controlled Magnetic Levitation System with Disturbance Observer Zi-Jiang Yang, Hiroshi Tsubakihara, Shunshoku Kanae, Kiyoshi Wada and Chun-Yi Su
II. STATEMENT OF THE PROBLEM
Abstract— This paper considers the control problem of a popular magnetic levitation system, which is open-loop unstable and strongly nonlinear associated with the electromechanical dynamics. The system dynamics is governed by a third-order nonlinear differential equation. The overall controller is designed through a backstepping manner by combining both the robust control and disturbance observer techniques. With the help of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proved. Rigorous analysis of the ISS property is given, and experimental results are included to show the excellent position tracking performance of the designed control system.
I. INTRODUCTION Due to strong open-loop instablility and inherent nonlinearities associated with the electromechanical dynamics, the control problem of a magnetic levitation system is usually quite challenging to the control engineers. In our previous works, we designed adaptive robust controller for a magnetic levitation system and verified excellent position tracking performance through experimental studies [6], [7]. However, a major drawback is that the controller is very complicated. In this paper, we propose a robust nonlinear controller of a voltage-controlled magnetic levitation system using disturbance observer(DOB). The DOB based motion controllers have been widely accepted in the industrial side, due to their simplicity and transparency of design, and excellent disturbance compensation ability. However, the DOB based motion controllers are usually designed according to the linear control theory [1], [4], even if the actual controlled plant may be strongly nonlinear. Unfortunately, the rigorous stability of these controllers for nonlinear systems has not been well studied in the literature. The overall controller is designed through a backstepping manner by combining both the robust control and DOB techniques. With the help of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proved. Rigorous analysis of the ISS property is given, and experimental results are included to show the excellent position tracking performance of the designed control system. Z. J. Yang, H. Tsubakihara, S. Kanae and K. Wada are with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, 744 Motooka,Nishi-ku, Fukuoka, 819-0395 Japan. TEL:+81-92-802-3708, FAX:+81-92-802-3705, E-mail: [email protected] C. Y. Su is with the Department of Mechanical Engineering, Concordia University, 1455 de Maisonneuve Blvd. W., H549, Montreal, Quebec H3G 1M8 Canada.
1-4244-0443-6/07/$20.00 ©2007 IEEE.
Fig. 1. Diagram of the magnetic levitation system.
Consider the magnetic levitation system shown in Fig. 1, whose dynamics are described in the following equations [3]. x˙ 1 x2 0 0 x˙ 2 = α(x) + 0 u + g β(x) x˙ 3 γ(x) 0 Qx23 α(x) = − 2M (X∞ + x1 )2 x3 {Qx2 − R(X∞ + x1 )2 } β(x) = Q(X∞ + x1 ) + L∞ (X∞ + x1 )2 X∞ + x1 γ(x) = Q + L∞ (X∞ + x1 )
(1)
where x = [x1 , x2 , x3 ]T = [x, x, ˙ i]T is state variable vector. And, x: air gap (vertical position) of the steel ball; i: coil current; g: gravity acceleration; M : mass of the steel ball; R: electrical resistance; u: voltage control input; L∞ , Q and X∞ : positive constants determined by the characteristics of the coil, magnetic core and steel ball. Denote the nominal physical parameters as g0 , M0 , R0 , L∞0 , Q0 and X∞0 . Then we have the nominal nonlinear functions and the modelling errors respectively as the fol-
747
TuB04.1 Step 1: Define the position error signal and velocity error signal respectively as
lowing. Q0 x23
α0 (x) = −
2M0 (X∞0 + x1 )2 x3 {Q0 x2 − R0 (X∞0 + x1 )2 } β0 (x) = Q0 (X∞0 + x1 ) + L∞0 (X∞0 + x1 )2 X∞0 + x1 γ0 (x) = Q0 + L∞0 (X∞0 + x1 )
z1 = ξ1 − yr , (2)
III. COORDINATE TRANSFORMATION
(3)
Notice that ξ = T (x) is only locally defined in a compact feasible region Ωx = {x0 ≤ x1 ≤ x1M , x3 > 0} ⊂ R3 , no matter what the control strategy is. The restriction x3 > 0 is in order to avoid the singular point of the control input u, see equation (26). Hence the system model (1) is transformed into ξ˙1 = ξ2
where α1 is the virtual input to stabilize z1 . Then we have subsystem S1 as the following. z˙1 = α1 + ξ2 − y˙ r
(11)
0
To convert the original nonlinear system into a system that is “simpler” in the sense that controller synthesis is more straightforward, we adopt the following nonlinear coordinate transformation. T
∆α (x) ξ˙2 = g0 + ∆g + ξ3 1 + α0 (x) ξ˙3 = F1 (x) + F0 (x) + ∆F (x) + u (G0 (x) + ∆G (x)) (4)
where c1p > 0, c1i > 0. Notice that α˙ 1 = −c1p (x2 − y˙ r ) − c1i z1 + y¨r
z3 = ξ3 − α2
(14)
where α2 is a virtual input to stabilize z2 . Then we have the subsystem S2 as z˙2 = −α˙ 1 + α2 + g0 + α2
∆α (x) α(x) + ∆g + z3 α0 (x) α0 (x)
M0 (X∞0 + x1 )3
x2
Q0 x23 Q0 x2 − R0 (X∞0 + x1 )2 F0 (x) = − (X∞0 + x1 )3 {Q0 + L∞0 (X∞0 + x1 )} G0 (x) = −
Q0 x3 M0 (X∞0 + x1 ) {Q0 + L∞0 (X∞0 + x1 )}
(5)
(16c)
where, w2∆ is due to the modeling error, w2z is due to the control error z3 of the next (electrical) subsystem: w2∆ = α2
(6)
(7)
(15)
Lump all the uncertain terms in this (mechanical) subsystem into w2 :
(16a) w2 = z˙2 − g0 + α2 − α˙ 1 ∆α (x) α(x) = α2 + ∆g + z3 (16b) α0 (x) α0 (x) = w2∆ + w2z
F1 (x) =
(13)
Step 2: Define the error signal of the nominal acceleration exerted by the electromagnet as
where Q0 x23
(10)
The virtual input α1 is designed based on the common PI control technique. t (12) z1 dt + y˙ r α1 = −c1p z1 − c1i
∆α (x) = α(x) − α0 (x) ∆β (x) = β(x) − β0 (x) ∆γ (x) = γ(x) − γ0 (x)
ξ = [ξ1 , ξ2 , ξ3 ]T = [x1 , x2 , α0 (x)]
z2 = ξ2 − α1
∆α (x) + ∆g , α0 (x)
w2z = z3
α(x) α0 (x)
(17)
Since z˙2 in (16a) is usually noisy, we pass w2 through a low-pass filter to obtain its estimate w: 2∆ + w w 2 = Q(s)w2 = Q(s)w2∆ + Q(s)w2z = w 2z (18)
(9)
This is the so called DOB studied extensively in the literature [1], [4]. In the low-frequency domain, we can expect w2 ≈ w 2 . In is paper, we adopt a simple second-order filter: 1 Q(s) = (19) (1 + τ2 s)2
In this section, we show the design procedure of the robust nonlinear controller. It is assumed here that the reference position yr of the steel ball and its first, second and third (3) derivatives, i.e., y˙ r , y¨r and yr are uniformly bounded, and available. The concrete design procedure is given as follows.
Since at the next step, we have to calculate w ˙2 = sw 2 , it is necessary to let the relative degree of Q(s) higher than 1. 2 , a simple By compensating the virtual input α2 by w controller can be designed for the approximated nominal model. Replacing α2 by α2 = v2 + α˙ 1 − g0 , and assuming w 2 ≈ w2 , we have
Q0 x3 ∆F (x) = − ∆β (x) M0 (X∞0 + x1 )2 ∆G (x) = −
Q0 x3 ∆γ (x) M0 (X∞0 + x1 )2
(8)
IV. C ONTROLLER D ESIGN
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z˙2 ≈ v2
(20)
TuB04.1 α30 = −c3 z3 − Ψ0 α31 = κ31 1 − 0.5e−λ1 |z3 | F0d z3 α32 = κ32 1 − 0.5e−λ2 |z3 | |F3 |(|α0 (x)| + g0 )z3 α33 = κ33 1 − 0.5e−λ3 |z3 | |α30 |z3
where v2 is a nominal linear input. This paves the way to design a simple controller. The simplest design is to let v2 = −c2 z2 . However, it should be pointed out that we can only expect w 2 ≈ w2 at low-frequencies. If the disturbance and model mismatch are fast changing, the estimation error w2 − w 2 can not be neglected and even can destroy the stability of the closed-loop in the case of large model mismatch [8]. To stabilize subsystem S2, we design the robust virtual input α2 as follows.
x23 {Q0 |x2 | + R0 (X∞0 + x1 )2 } (X∞0 + x1 ){Q0 + L∞0 (X∞0 + x1 )} Q0 Fc = − M0 (X∞0 + x1 )2
F0d = |Fc |
α2 = α20 − α2w − α21 − α22 − α23 α20 = −c2 z2 + α˙ 1 − g0 α2w = w 2 α21 = κ21 g0 z2 α22 = κ22 α220 + νz2 α23 = κ23 w 22 + νz2
(21)
where, c2 , κ21 , κ22 , κ23 > 0; ν is a small positive number, and is given as ν = 0.01; α20 is a nominal feedback controller with nominal model compensation; α2w is a compensating term by the DOB’s output; α21 is a linear damping term to counteract ∆g ; α22 is a nonlinear damping term to counteract ∆α (x)/α0 (x); α23 is a nonlinear damping term to ensure boundedness of z2 when w 2 is used. Notice that the nonlinear damping terms employ time-varying control gains so that they grow at least as the same order as the corresponding uncertain terms grow. Step 3: The derivative of α2 is calculated as follows. α˙ 2 = F2 − F3 (α0 (x) + g0 ) − F3 (∆α (x) + ∆g )
(22)
where F2 = 1 − κ22 (α220 + ν)−0.5 α20 z2 × (3) c1p y¨r − c1i z˙1 + yr + c2 α˙ 1 ˙2 − 1 + κ23 (w 22 + ν)−0.5 w 2 z2 w
+ κ21 g0 + κ22 (α220 + ν)0.5 + κ23 (w 22 + ν)0.5 α˙1
where, c3 , κ31 , κ32 , κ33 > 0; α20 is a nominal feedback controller with nominal model compensation; α31 , α32 and α33 are nonlinear damping terms employed to counteract the modelling errors. Also, notice that (1 − 0.5e−λi|z3 | ), i = 1, 2, 3 are introduced to reduce control efforts due to the nonlinear damping terms, when |z3 | is relatively small. In this study, we choose λ1 = λ2 = λ3 = 0.1. Remark 1: Since the signals of the electrical subsystem change much faster than those of the mechanical subsystem, the DOB which uses a low-pass filter is not so effective. Therefore, we do not employ a DOB for the electrical subsystem. However, as can be seen in (16b), since the error signal z3 of the electrical subsystem is included in the lumped disturbance term w2 , the low-frequency components of z3 can be compensated by the DOB employed at step 2, so that the influences by z3 to the mechanical subsystem can be reduced at low-frequencies. V. S TABILITY ANALYSIS Step 1: Applying α1 to subsystem S1, we have t z˙1 = z2 − c1p z1 − c1i z1 dt
where z 1a
(24)
where (25)
α30 − α31 − α32 − α33 G0 (x)
z˙ 1a = A z 1a + B z2 t = [ 0 z1 dt, z1 ]T , 0 1 A= , B = [0 1]T −c1i −c1p
(30)
(31)
The ISS property of subsystem S1 can be described in the following lemma [5] : Lemma 1: If the virtual input α1 is applied to subsystem S1, and if z2 is made uniformly bounded at the next step, then S1 is ISS, i.e., for ∃ λ0 > 0, ∃ ρ0 > 0, λ0 sup |z2 (τ )| |z 1a (t)| ≤ λ0 e−ρ0 t |z 1a (0)| + ρ0 0≤τ ≤t Step 2: Applying α2 to subsystem S2, we have
Then we design the control voltage as follows. u=
(29)
Rewrite it into the state-space form:
Therefore, we have the electrical subsystem S3 as
Ψ0 = F0 (x) + F1 (x) − F2 + F3 (α0 (x) + g0 ) ∆Ψ = ∆F (x) + F3 (∆α (x) + ∆g )
(28)
0
F3 = c2 + c1p + κ21 g0 + κ22 (α220 + ν)0.5 +κ23 (w 22 + ν)0.5 − (c2 + c1p )κ22 (α220 + ν)−0.5 α20 z2 (23) z˙3 = ξ˙3 − α˙ 2 = Ψ0 + ∆Ψ + G0 (x)u + ∆G (x)u
(27)
(26)
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z˙2 = −c2 z2 − (α21 + α22 + α23 ) − w 2 + w2 (32a) α(x) = −c2 z2 − (α21 + α22 + α23 ) α0 (x) α(x) α(x) ∆α (x) + ∆g + z3 − w 2 (32b) +α20 α0 (x) α0 (x) α0 (x)
TuB04.1 And then, c d z22 c2 2 + D2 |z2 | |z2 | − µ2 (t) ≤ − z22 − dt 2 2 2 ∆α (x) α(x) α(x) 2 α0 (x) α20 + ∆g + α0 (x) z3 − α0 (x) w µ2 (t) = c2 + D2 2 α(x) κ21 g0 + κ22 α220 + ν + κ23 w 22 + ν D2 = α0 (x) (33) It can be verified that the the nonlinear damping terms in the denominator grow at least as the same order as the uncertain terms in the numerator grow. Furthermore, if z3 is made uniformly bounded at step 3, we can conclude that µ2 (t) is uniformly bounded. More precisely, we have |z2 (t)| ≥ µ2 (t) ⇒
d 2 z ≤ −c2 z22 dt 2
|z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2 (τ ) 0≤τ ≤t
(34)
(35)
In the above analysis, the main attention is to show the boundedness of the internal signals. No analysis yet be done for the attenuation effects of w2 − w 2 . Without such an analysis, we cannot clearly see how the DOB’s output w 2 can bring improvement. We now attempt to make such an effort. Rewrite (16)∼(18): α(x) ∆α (x) + z3 + ∆g − w 2 (36a) α0 (x) α0 (x)
∆α (x) − D2w z2 + α20 − w 2 + ∆g = α0 (x) α(x) −w 2 (36b) +z3 α0 (x) = w2∆ + w2z − w 2 (36c) 2∆ + w2z − w 2z (36d) = w2∆ − w
w2 − w 2 = α2
where
D2w = κ21 g0 + κ22 α220 + ν + κ23 w 22 + ν
And from (32a) and (36), we have c c2 d z22 2 + D2w |z2 | [|z2 | − µ2wa ] ≤ − z22 − dt 2 2 c2 c2 2 + D2w |z2 | [|z2 | − µ2wb ] ≤ − z22 − 2 2
= µ2wb (t)
|w 2∆ − w2∆ | µw 2wb (t) = c2 + D2w 2
(1 − Q2 (s)) α(x)/α0 (x) z3 z µ2wb (t) = c2 + D2w 2 Therefore, we have |z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2wa (τ )
(41a)
≤ |z2 (0)|e−c2 t/2 + sup µ2wb (τ )
(41b)
0≤τ ≤t
Remark 2: So far, we have obtained (35), (41a) and (41b) for z2 , which seem confusing. These results are explained here. The inequality (35) is to show the boundedness of the internal signals of the first two subsystems. It can be seen in (36b), (37) and (39a) that in the numerator of µ2wa (t), there is a term D2w z2 . However, in the denominator, the correspoding term is D2w . Therefore, we should at first ensure the boundedness of z2 as shown in (35), then we can discuss the boundedness of µ2wa (t) in (41a). Notice that µ2wa (t) has very transparent physical meaning. At lowfrequencies, we can expect µ2w ≈ 0. And any nonzero 2 at high-frequencies is counteracted by c2 /2 + D2w w2 − w so that z2 is quite robust against w2 − w 2 . However, for the sake of proving the ISS property of the overall error system, we have to express the overall error system as a cascade of the three subsystems [2]. Therefore, as shown in (39c), we separate the effects of the next control error z3 from the modeling uncertainty in µ2wb (t). Then we have Lemma 2: If the virtual input α2 is applied to subsystem S2, and if z3 is made uniformly bounded at the next step, then S2 is ISS: |z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2wb (τ ) 0≤τ ≤t
Step 3: Applying the control voltage u to the subsystem S3, we have z˙3 = −c3 z3 + ∆Ψ − α31 − α32 − α33 + ∆G (x)u = −c3 z3 + ∆F (x) + F3 ∆α + F3 ∆g α31 + α32 + α33 α30 − G(x) +∆G (x) G0 (x) G0 (x)
(38)
(39a)
(39b) (39c) (39d)
(40)
0≤τ ≤t
(37)
where |w 2 − w2 | µ2wa (t) = c2 + D2w 2 |w 2∆ − w2∆ | |w 2z − w2z | ≤ c2 + c2 + D2w + D2w 2 2 z = µw 2wb (t) + µ2wb (t)
and
(42)
and d z32 = −c3 z32 + ∆F (x)z3 + F3 (∆α (x) + ∆g ) z3 dt 2 ∆G (x) α31 + α32 + α33 α30 z3 − G(x) z3 + G0 (x) G0 (x) c3 c3 ≤ − z32 − z32 − D3 z32 2 2 ∆G (x)α30 z3 + ∆F (x)z3 + F3 (∆α (x) + ∆g ) z3 + G0 (x) c c3 3 + D3 |z3 | [|z3 | − µ3 (t)] ≤ − z32 − 2 2 (43)
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TuB04.1 where
∆F (x) + F3 (∆α (x) + ∆g ) + ∆G (x)α30 G0 (x) µ3 (t) = c3 + D3 2 D3 = G(x) (κ31 F0d + κ32 |F3 |(|α0 (x)| + g0 ) + κ33 |α30 |) 0.5 G0 (x) (44)
Just as the case of step 2, it can be verified that the nonlinear damping terms in the denominator of µ3 (t) grow at least as the same order as the uncertain terms in the numerator grow. Therefore we can conclude that µ3 (t) Lemma 3: If the control input u is applied to subsystem S3, then S3 is ISS: |z3 (t)| ≤ |z3 (0)|e−c3 t/2 + sup µ3 (τ ) 0≤τ ≤t
Stability of the overall error system: Since the overall error system is a cascade of the three subsystems characterized by Lemmas 1∼3, along the same line of the proof of Lemma C.4 in [2], we can prove the cascaded system is also ISS. Define the error signal vector T z(t) = z T1a (t), z2 (t), z3 (t) (45) Then based on Lemmas 1∼3, we can prove the following results. √ |z(t)| ≤ 2λ2 |z(0)|e−ρ2 t (46) +γ3 sup µw 2wb (τ ) + β3 sup µ3 (τ ) 0≤τ ≤t
where
0≤τ ≤t
|z(t)| = |z 1a (t)|2 + |z 2 (t)|2 + |z3 (t)|2 γ3 = (λ1 + 1)γ2 β3 = (λ1 + 1)γ2 β2 + 1 λ2 = λ21 + (λ1 + 1)γ2 β2 + 1 ρ2 = min(ρ0 /4, c2 /8, c3 /4)
VI. E XPERIMENTAL RESULTS To verify the performance of the proposed robust nonlinear controller with DOB, experimental studies have been carried out on the magnetic levitation system shown in Fig. 1. The physically allowable operating region of the steel ball shown in Fig. 1 is limited to 0[m] < x1 ≤ 0.013[m]. The output of the controllable voltage source is limited to −60.0[V ] ≤ u ≤ 60.0[V ]. The velocity x2 is measured by pseudo-differentiation of the measured position x1 as sx1 /(0.004s + 1). The resolution of the laser distance sensor is ±0.00018[m], which is considered to be relatively noisy. The physical parameters are identified as follows. M = 0.54[kg], g = 9.8[m/s2 ] X∞ = 0.008114[m], Q = 0.001624[Hm] L∞ = 0.8052[H], R = 11.88[Ω] The following nominal system parameters with considerable errors are used for experimental studies. X∞0 = 0.0020[m], Q0 = 0.0003[Hm]
(50)
L∞0 = 0.50[H], R0 = 10.0[Ω] (47)
The following two controllers are implemented: (1)Robust nonlinear controller without DOB: c1p = 40, c1i = 202 , c2 = 40, c3 = 20
λ20
where, h(t) is the impulse response of the transfer function H(s) = 1 − Q(s). The results are summarized in the following theorem. Theorem 1: If the proposed robust nonlinear controller with DOB is applied to the magnetic levitation system under study, then the following results hold. 1) The overall error system is ISS such that √ |z(t)| ≤ 2λ2 |z(0)|e−ρ2 t +γ3 sup µw 2wb (τ ) + β3 sup µ3 (τ ) 0≤τ ≤t
(49)
M0 = 0.30[kg], g0 = 9.0[m/s2]
λ0 + +1 ρ0 ρ0 λ0 λ2 γ2 = 0 + +1 ρ0 ρ0 β2 = 1/(c2 /2 + D2w ) ∞ α(x)/α0 (x) ∞ h(t) 1 (48)
λ1 = λ20 +
Remark 3: The result 1) of Theorem 1 characterizes the ISS property of the overall error system, which ensures the boundedness of the error signals. According to (40), µw 2wb (τ ) is small at low-frequencies, owing to the DOB. On the other hand, µ3 (τ ) of the electrical subsystem may not be so small. Therefore, |z(t)| itself may not be so small. However, our final purpose is to make the position error z1 small. According to Lemma 1 and (41a), we can conclude that z1a (t), z2 (t) can be made small at low-frequencies. The comments will be confirmed by experimental results.
0≤τ ≤t
2) The steady offset (zero-frequency component) of z1 approaches zero.
κ21 = 1, κ22 = 3
(51)
κ31 = 0.3, κ32 = 0.3, κ33 = 0.3 (2)Robust nonlinear controller with DOB: c1p = 40, c1i = 202 , c2 = 40, c3 = 20 τ2 = 0.02 κ21 = 1, κ22 = 3, κ23 = 1
(52)
κ31 = 0.3, κ32 = 0.3, κ33 = 0.3 The results are shown Fig. 2 and Fig. 3. In each figure, from the top to the bottom are the position x1 , velocity x2 , coil current x3 , control volatge u, error signals z1 , z2 , z3 . It can be found in Fig. 2 that owing to the nonlinear damping terms, the error signals are made bounded. However, empirically, we found it is difficult to further reduce the position tracking error z1 by simply increasing the control gains. A comparison between Fig. 2 and Fig. 3 can be made here. For the electrical subsystem, since the DOB is not used, the
751
x [m/s]
1
x [mm]
TuB04.1 For the sake of physical transparency, the controller was designed directly on the model of the magnetic levitation system. However, we should emphasize here that the basic idea and theory can be extended to a general electro-mechanical system governed by the following third-order strict feedback form [2]:
15 10 5 0
3 2 1 0 50 0 −50
z1[mm]
u[V]
x3[A]
2
0.1 0 −0.1
2
z [m/s]
z3[m/s ]
0
1
2
3
4 5 Time[sec]
6
7
(53)
where , x1 : position; x2 : velocity; x3 : current or equivalent driving force, torque, acceleration; u: voltage control input; f2 (x1 , x2 ), g2 (x1 , x2 ), f3 (x1 , x2 , x3 ), g3 (x1 , x2 , x3 ) nonlinear functions which may include uncertainties. In contrast to most DOB based controllers reported in the literature, our major academic contribution is to propose a theoretically guaranteed robust nonlinear controller with DOB for a strongly nonlinear and unstable system.
1 0 −1
0.1 0 −0.1 5 0 −5
2
x˙ 1 = x2 x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )x3 x˙ 3 = f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )u
8
R EFERENCES Experimental results of the robust nonlinear controller without
15 10 5 0
0.1 0 −0.1 3 2 1 0 50 0 −50
2
1 0 −1
0.1 0 −0.1 5 0 −5
z3[m/s ]
2
z [m/s]
z1[mm]
u[V]
x3[A]
2
x [m/s]
1
x [mm]
Fig. 1. DOB.
Fig. 2.
0
1
2
3
4 5 Time[sec]
6
7
8
Experimental results of the robust nonlinear controller with DOB.
error signal z3 is not suppressed sufficiently. However, for the mechanical subsystem, owing to the DOB, in Fig. 3, the low-frequency components of z2 are significantly reduced and hence the position error z1 is largely reduced. The results reflect the comments in Remark 3 quite well. VII. C ONCLUSIONS In this paper, a robust nonlinear controller with DOB was proposed for a voltage-controlled magnetic levitation system. We have found that owing to the DOB, the position tracking error can be significantly reduced.
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[1] S. Komada, N. Machii and T. Fukuda, Control of redundant manipulators considering order of disturbance observer, IEEE Trans. Industrial Electronics, Vol. 47, 413/420, 2000. [2] M. L. Krstic, Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995. [3] A. J. Joo and J. H. Seo, Design and analysis of the nonlinear feedback linearizing control for an electromagnetic suspension system, IEEE Transactions on Control Systems Technology, Vol. 5, 135/144, 1997. [4] T. Mita, M. Hirata, K. Murata and H. Zhang, H∞ control versus disturbance-observer-based control, IEEE Trans. Industrial Electronics, Vol. 45, 488/495, 1998. [5] J. H. Park and F. T. Park, Adaptive fuzzy observer with minimal dynamic order for uncertain nonlinear systems, IEE Proceeding– Control Theory and Applications, Vol. 150, 188/197, 2003. [6] Z. J. Yang and D. Miyazaki: Adaptive robust nonlinear control of a magnetic levitation system, 5th IFAC Symposium on Nonlinear Control Systems, 2001. [7] Z. J. Yang, K. Miyazaki, S. Kanae and K. Wada: Adaptive robust dynamic surface control for a magnetic levitation system, 42nd IEEE Conference on Decision and Control, 2003. [8] B. Yao, M. Almajed and M. Tomizuka, High performance robust motion control of machine tools: an adaptive robust control approach and comparative experiments, IEEE/ASME Trans. on Mechatronics, Vol.2, 62/76, 1997.
TuB04.1
Robust Nonlinear Control of a Voltage-Controlled Magnetic Levitation System with Disturbance Observer Zi-Jiang Yang, Hiroshi Tsubakihara, Shunshoku Kanae, Kiyoshi Wada and Chun-Yi Su
II. STATEMENT OF THE PROBLEM
Abstract— This paper considers the control problem of a popular magnetic levitation system, which is open-loop unstable and strongly nonlinear associated with the electromechanical dynamics. The system dynamics is governed by a third-order nonlinear differential equation. The overall controller is designed through a backstepping manner by combining both the robust control and disturbance observer techniques. With the help of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proved. Rigorous analysis of the ISS property is given, and experimental results are included to show the excellent position tracking performance of the designed control system.
I. INTRODUCTION Due to strong open-loop instablility and inherent nonlinearities associated with the electromechanical dynamics, the control problem of a magnetic levitation system is usually quite challenging to the control engineers. In our previous works, we designed adaptive robust controller for a magnetic levitation system and verified excellent position tracking performance through experimental studies [6], [7]. However, a major drawback is that the controller is very complicated. In this paper, we propose a robust nonlinear controller of a voltage-controlled magnetic levitation system using disturbance observer(DOB). The DOB based motion controllers have been widely accepted in the industrial side, due to their simplicity and transparency of design, and excellent disturbance compensation ability. However, the DOB based motion controllers are usually designed according to the linear control theory [1], [4], even if the actual controlled plant may be strongly nonlinear. Unfortunately, the rigorous stability of these controllers for nonlinear systems has not been well studied in the literature. The overall controller is designed through a backstepping manner by combining both the robust control and DOB techniques. With the help of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proved. Rigorous analysis of the ISS property is given, and experimental results are included to show the excellent position tracking performance of the designed control system. Z. J. Yang, H. Tsubakihara, S. Kanae and K. Wada are with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, 744 Motooka,Nishi-ku, Fukuoka, 819-0395 Japan. TEL:+81-92-802-3708, FAX:+81-92-802-3705, E-mail: [email protected] C. Y. Su is with the Department of Mechanical Engineering, Concordia University, 1455 de Maisonneuve Blvd. W., H549, Montreal, Quebec H3G 1M8 Canada.
1-4244-0443-6/07/$20.00 ©2007 IEEE.
Fig. 1. Diagram of the magnetic levitation system.
Consider the magnetic levitation system shown in Fig. 1, whose dynamics are described in the following equations [3]. x˙ 1 x2 0 0 x˙ 2 = α(x) + 0 u + g β(x) x˙ 3 γ(x) 0 Qx23 α(x) = − 2M (X∞ + x1 )2 x3 {Qx2 − R(X∞ + x1 )2 } β(x) = Q(X∞ + x1 ) + L∞ (X∞ + x1 )2 X∞ + x1 γ(x) = Q + L∞ (X∞ + x1 )
(1)
where x = [x1 , x2 , x3 ]T = [x, x, ˙ i]T is state variable vector. And, x: air gap (vertical position) of the steel ball; i: coil current; g: gravity acceleration; M : mass of the steel ball; R: electrical resistance; u: voltage control input; L∞ , Q and X∞ : positive constants determined by the characteristics of the coil, magnetic core and steel ball. Denote the nominal physical parameters as g0 , M0 , R0 , L∞0 , Q0 and X∞0 . Then we have the nominal nonlinear functions and the modelling errors respectively as the fol-
747
TuB04.1 Step 1: Define the position error signal and velocity error signal respectively as
lowing. Q0 x23
α0 (x) = −
2M0 (X∞0 + x1 )2 x3 {Q0 x2 − R0 (X∞0 + x1 )2 } β0 (x) = Q0 (X∞0 + x1 ) + L∞0 (X∞0 + x1 )2 X∞0 + x1 γ0 (x) = Q0 + L∞0 (X∞0 + x1 )
z1 = ξ1 − yr , (2)
III. COORDINATE TRANSFORMATION
(3)
Notice that ξ = T (x) is only locally defined in a compact feasible region Ωx = {x0 ≤ x1 ≤ x1M , x3 > 0} ⊂ R3 , no matter what the control strategy is. The restriction x3 > 0 is in order to avoid the singular point of the control input u, see equation (26). Hence the system model (1) is transformed into ξ˙1 = ξ2
where α1 is the virtual input to stabilize z1 . Then we have subsystem S1 as the following. z˙1 = α1 + ξ2 − y˙ r
(11)
0
To convert the original nonlinear system into a system that is “simpler” in the sense that controller synthesis is more straightforward, we adopt the following nonlinear coordinate transformation. T
∆α (x) ξ˙2 = g0 + ∆g + ξ3 1 + α0 (x) ξ˙3 = F1 (x) + F0 (x) + ∆F (x) + u (G0 (x) + ∆G (x)) (4)
where c1p > 0, c1i > 0. Notice that α˙ 1 = −c1p (x2 − y˙ r ) − c1i z1 + y¨r
z3 = ξ3 − α2
(14)
where α2 is a virtual input to stabilize z2 . Then we have the subsystem S2 as z˙2 = −α˙ 1 + α2 + g0 + α2
∆α (x) α(x) + ∆g + z3 α0 (x) α0 (x)
M0 (X∞0 + x1 )3
x2
Q0 x23 Q0 x2 − R0 (X∞0 + x1 )2 F0 (x) = − (X∞0 + x1 )3 {Q0 + L∞0 (X∞0 + x1 )} G0 (x) = −
Q0 x3 M0 (X∞0 + x1 ) {Q0 + L∞0 (X∞0 + x1 )}
(5)
(16c)
where, w2∆ is due to the modeling error, w2z is due to the control error z3 of the next (electrical) subsystem: w2∆ = α2
(6)
(7)
(15)
Lump all the uncertain terms in this (mechanical) subsystem into w2 :
(16a) w2 = z˙2 − g0 + α2 − α˙ 1 ∆α (x) α(x) = α2 + ∆g + z3 (16b) α0 (x) α0 (x) = w2∆ + w2z
F1 (x) =
(13)
Step 2: Define the error signal of the nominal acceleration exerted by the electromagnet as
where Q0 x23
(10)
The virtual input α1 is designed based on the common PI control technique. t (12) z1 dt + y˙ r α1 = −c1p z1 − c1i
∆α (x) = α(x) − α0 (x) ∆β (x) = β(x) − β0 (x) ∆γ (x) = γ(x) − γ0 (x)
ξ = [ξ1 , ξ2 , ξ3 ]T = [x1 , x2 , α0 (x)]
z2 = ξ2 − α1
∆α (x) + ∆g , α0 (x)
w2z = z3
α(x) α0 (x)
(17)
Since z˙2 in (16a) is usually noisy, we pass w2 through a low-pass filter to obtain its estimate w: 2∆ + w w 2 = Q(s)w2 = Q(s)w2∆ + Q(s)w2z = w 2z (18)
(9)
This is the so called DOB studied extensively in the literature [1], [4]. In the low-frequency domain, we can expect w2 ≈ w 2 . In is paper, we adopt a simple second-order filter: 1 Q(s) = (19) (1 + τ2 s)2
In this section, we show the design procedure of the robust nonlinear controller. It is assumed here that the reference position yr of the steel ball and its first, second and third (3) derivatives, i.e., y˙ r , y¨r and yr are uniformly bounded, and available. The concrete design procedure is given as follows.
Since at the next step, we have to calculate w ˙2 = sw 2 , it is necessary to let the relative degree of Q(s) higher than 1. 2 , a simple By compensating the virtual input α2 by w controller can be designed for the approximated nominal model. Replacing α2 by α2 = v2 + α˙ 1 − g0 , and assuming w 2 ≈ w2 , we have
Q0 x3 ∆F (x) = − ∆β (x) M0 (X∞0 + x1 )2 ∆G (x) = −
Q0 x3 ∆γ (x) M0 (X∞0 + x1 )2
(8)
IV. C ONTROLLER D ESIGN
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z˙2 ≈ v2
(20)
TuB04.1 α30 = −c3 z3 − Ψ0 α31 = κ31 1 − 0.5e−λ1 |z3 | F0d z3 α32 = κ32 1 − 0.5e−λ2 |z3 | |F3 |(|α0 (x)| + g0 )z3 α33 = κ33 1 − 0.5e−λ3 |z3 | |α30 |z3
where v2 is a nominal linear input. This paves the way to design a simple controller. The simplest design is to let v2 = −c2 z2 . However, it should be pointed out that we can only expect w 2 ≈ w2 at low-frequencies. If the disturbance and model mismatch are fast changing, the estimation error w2 − w 2 can not be neglected and even can destroy the stability of the closed-loop in the case of large model mismatch [8]. To stabilize subsystem S2, we design the robust virtual input α2 as follows.
x23 {Q0 |x2 | + R0 (X∞0 + x1 )2 } (X∞0 + x1 ){Q0 + L∞0 (X∞0 + x1 )} Q0 Fc = − M0 (X∞0 + x1 )2
F0d = |Fc |
α2 = α20 − α2w − α21 − α22 − α23 α20 = −c2 z2 + α˙ 1 − g0 α2w = w 2 α21 = κ21 g0 z2 α22 = κ22 α220 + νz2 α23 = κ23 w 22 + νz2
(21)
where, c2 , κ21 , κ22 , κ23 > 0; ν is a small positive number, and is given as ν = 0.01; α20 is a nominal feedback controller with nominal model compensation; α2w is a compensating term by the DOB’s output; α21 is a linear damping term to counteract ∆g ; α22 is a nonlinear damping term to counteract ∆α (x)/α0 (x); α23 is a nonlinear damping term to ensure boundedness of z2 when w 2 is used. Notice that the nonlinear damping terms employ time-varying control gains so that they grow at least as the same order as the corresponding uncertain terms grow. Step 3: The derivative of α2 is calculated as follows. α˙ 2 = F2 − F3 (α0 (x) + g0 ) − F3 (∆α (x) + ∆g )
(22)
where F2 = 1 − κ22 (α220 + ν)−0.5 α20 z2 × (3) c1p y¨r − c1i z˙1 + yr + c2 α˙ 1 ˙2 − 1 + κ23 (w 22 + ν)−0.5 w 2 z2 w
+ κ21 g0 + κ22 (α220 + ν)0.5 + κ23 (w 22 + ν)0.5 α˙1
where, c3 , κ31 , κ32 , κ33 > 0; α20 is a nominal feedback controller with nominal model compensation; α31 , α32 and α33 are nonlinear damping terms employed to counteract the modelling errors. Also, notice that (1 − 0.5e−λi|z3 | ), i = 1, 2, 3 are introduced to reduce control efforts due to the nonlinear damping terms, when |z3 | is relatively small. In this study, we choose λ1 = λ2 = λ3 = 0.1. Remark 1: Since the signals of the electrical subsystem change much faster than those of the mechanical subsystem, the DOB which uses a low-pass filter is not so effective. Therefore, we do not employ a DOB for the electrical subsystem. However, as can be seen in (16b), since the error signal z3 of the electrical subsystem is included in the lumped disturbance term w2 , the low-frequency components of z3 can be compensated by the DOB employed at step 2, so that the influences by z3 to the mechanical subsystem can be reduced at low-frequencies. V. S TABILITY ANALYSIS Step 1: Applying α1 to subsystem S1, we have t z˙1 = z2 − c1p z1 − c1i z1 dt
where z 1a
(24)
where (25)
α30 − α31 − α32 − α33 G0 (x)
z˙ 1a = A z 1a + B z2 t = [ 0 z1 dt, z1 ]T , 0 1 A= , B = [0 1]T −c1i −c1p
(30)
(31)
The ISS property of subsystem S1 can be described in the following lemma [5] : Lemma 1: If the virtual input α1 is applied to subsystem S1, and if z2 is made uniformly bounded at the next step, then S1 is ISS, i.e., for ∃ λ0 > 0, ∃ ρ0 > 0, λ0 sup |z2 (τ )| |z 1a (t)| ≤ λ0 e−ρ0 t |z 1a (0)| + ρ0 0≤τ ≤t Step 2: Applying α2 to subsystem S2, we have
Then we design the control voltage as follows. u=
(29)
Rewrite it into the state-space form:
Therefore, we have the electrical subsystem S3 as
Ψ0 = F0 (x) + F1 (x) − F2 + F3 (α0 (x) + g0 ) ∆Ψ = ∆F (x) + F3 (∆α (x) + ∆g )
(28)
0
F3 = c2 + c1p + κ21 g0 + κ22 (α220 + ν)0.5 +κ23 (w 22 + ν)0.5 − (c2 + c1p )κ22 (α220 + ν)−0.5 α20 z2 (23) z˙3 = ξ˙3 − α˙ 2 = Ψ0 + ∆Ψ + G0 (x)u + ∆G (x)u
(27)
(26)
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z˙2 = −c2 z2 − (α21 + α22 + α23 ) − w 2 + w2 (32a) α(x) = −c2 z2 − (α21 + α22 + α23 ) α0 (x) α(x) α(x) ∆α (x) + ∆g + z3 − w 2 (32b) +α20 α0 (x) α0 (x) α0 (x)
TuB04.1 And then, c d z22 c2 2 + D2 |z2 | |z2 | − µ2 (t) ≤ − z22 − dt 2 2 2 ∆α (x) α(x) α(x) 2 α0 (x) α20 + ∆g + α0 (x) z3 − α0 (x) w µ2 (t) = c2 + D2 2 α(x) κ21 g0 + κ22 α220 + ν + κ23 w 22 + ν D2 = α0 (x) (33) It can be verified that the the nonlinear damping terms in the denominator grow at least as the same order as the uncertain terms in the numerator grow. Furthermore, if z3 is made uniformly bounded at step 3, we can conclude that µ2 (t) is uniformly bounded. More precisely, we have |z2 (t)| ≥ µ2 (t) ⇒
d 2 z ≤ −c2 z22 dt 2
|z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2 (τ ) 0≤τ ≤t
(34)
(35)
In the above analysis, the main attention is to show the boundedness of the internal signals. No analysis yet be done for the attenuation effects of w2 − w 2 . Without such an analysis, we cannot clearly see how the DOB’s output w 2 can bring improvement. We now attempt to make such an effort. Rewrite (16)∼(18): α(x) ∆α (x) + z3 + ∆g − w 2 (36a) α0 (x) α0 (x)
∆α (x) − D2w z2 + α20 − w 2 + ∆g = α0 (x) α(x) −w 2 (36b) +z3 α0 (x) = w2∆ + w2z − w 2 (36c) 2∆ + w2z − w 2z (36d) = w2∆ − w
w2 − w 2 = α2
where
D2w = κ21 g0 + κ22 α220 + ν + κ23 w 22 + ν
And from (32a) and (36), we have c c2 d z22 2 + D2w |z2 | [|z2 | − µ2wa ] ≤ − z22 − dt 2 2 c2 c2 2 + D2w |z2 | [|z2 | − µ2wb ] ≤ − z22 − 2 2
= µ2wb (t)
|w 2∆ − w2∆ | µw 2wb (t) = c2 + D2w 2
(1 − Q2 (s)) α(x)/α0 (x) z3 z µ2wb (t) = c2 + D2w 2 Therefore, we have |z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2wa (τ )
(41a)
≤ |z2 (0)|e−c2 t/2 + sup µ2wb (τ )
(41b)
0≤τ ≤t
Remark 2: So far, we have obtained (35), (41a) and (41b) for z2 , which seem confusing. These results are explained here. The inequality (35) is to show the boundedness of the internal signals of the first two subsystems. It can be seen in (36b), (37) and (39a) that in the numerator of µ2wa (t), there is a term D2w z2 . However, in the denominator, the correspoding term is D2w . Therefore, we should at first ensure the boundedness of z2 as shown in (35), then we can discuss the boundedness of µ2wa (t) in (41a). Notice that µ2wa (t) has very transparent physical meaning. At lowfrequencies, we can expect µ2w ≈ 0. And any nonzero 2 at high-frequencies is counteracted by c2 /2 + D2w w2 − w so that z2 is quite robust against w2 − w 2 . However, for the sake of proving the ISS property of the overall error system, we have to express the overall error system as a cascade of the three subsystems [2]. Therefore, as shown in (39c), we separate the effects of the next control error z3 from the modeling uncertainty in µ2wb (t). Then we have Lemma 2: If the virtual input α2 is applied to subsystem S2, and if z3 is made uniformly bounded at the next step, then S2 is ISS: |z2 (t)| ≤ |z2 (0)|e−c2 t/2 + sup µ2wb (τ ) 0≤τ ≤t
Step 3: Applying the control voltage u to the subsystem S3, we have z˙3 = −c3 z3 + ∆Ψ − α31 − α32 − α33 + ∆G (x)u = −c3 z3 + ∆F (x) + F3 ∆α + F3 ∆g α31 + α32 + α33 α30 − G(x) +∆G (x) G0 (x) G0 (x)
(38)
(39a)
(39b) (39c) (39d)
(40)
0≤τ ≤t
(37)
where |w 2 − w2 | µ2wa (t) = c2 + D2w 2 |w 2∆ − w2∆ | |w 2z − w2z | ≤ c2 + c2 + D2w + D2w 2 2 z = µw 2wb (t) + µ2wb (t)
and
(42)
and d z32 = −c3 z32 + ∆F (x)z3 + F3 (∆α (x) + ∆g ) z3 dt 2 ∆G (x) α31 + α32 + α33 α30 z3 − G(x) z3 + G0 (x) G0 (x) c3 c3 ≤ − z32 − z32 − D3 z32 2 2 ∆G (x)α30 z3 + ∆F (x)z3 + F3 (∆α (x) + ∆g ) z3 + G0 (x) c c3 3 + D3 |z3 | [|z3 | − µ3 (t)] ≤ − z32 − 2 2 (43)
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TuB04.1 where
∆F (x) + F3 (∆α (x) + ∆g ) + ∆G (x)α30 G0 (x) µ3 (t) = c3 + D3 2 D3 = G(x) (κ31 F0d + κ32 |F3 |(|α0 (x)| + g0 ) + κ33 |α30 |) 0.5 G0 (x) (44)
Just as the case of step 2, it can be verified that the nonlinear damping terms in the denominator of µ3 (t) grow at least as the same order as the uncertain terms in the numerator grow. Therefore we can conclude that µ3 (t) Lemma 3: If the control input u is applied to subsystem S3, then S3 is ISS: |z3 (t)| ≤ |z3 (0)|e−c3 t/2 + sup µ3 (τ ) 0≤τ ≤t
Stability of the overall error system: Since the overall error system is a cascade of the three subsystems characterized by Lemmas 1∼3, along the same line of the proof of Lemma C.4 in [2], we can prove the cascaded system is also ISS. Define the error signal vector T z(t) = z T1a (t), z2 (t), z3 (t) (45) Then based on Lemmas 1∼3, we can prove the following results. √ |z(t)| ≤ 2λ2 |z(0)|e−ρ2 t (46) +γ3 sup µw 2wb (τ ) + β3 sup µ3 (τ ) 0≤τ ≤t
where
0≤τ ≤t
|z(t)| = |z 1a (t)|2 + |z 2 (t)|2 + |z3 (t)|2 γ3 = (λ1 + 1)γ2 β3 = (λ1 + 1)γ2 β2 + 1 λ2 = λ21 + (λ1 + 1)γ2 β2 + 1 ρ2 = min(ρ0 /4, c2 /8, c3 /4)
VI. E XPERIMENTAL RESULTS To verify the performance of the proposed robust nonlinear controller with DOB, experimental studies have been carried out on the magnetic levitation system shown in Fig. 1. The physically allowable operating region of the steel ball shown in Fig. 1 is limited to 0[m] < x1 ≤ 0.013[m]. The output of the controllable voltage source is limited to −60.0[V ] ≤ u ≤ 60.0[V ]. The velocity x2 is measured by pseudo-differentiation of the measured position x1 as sx1 /(0.004s + 1). The resolution of the laser distance sensor is ±0.00018[m], which is considered to be relatively noisy. The physical parameters are identified as follows. M = 0.54[kg], g = 9.8[m/s2 ] X∞ = 0.008114[m], Q = 0.001624[Hm] L∞ = 0.8052[H], R = 11.88[Ω] The following nominal system parameters with considerable errors are used for experimental studies. X∞0 = 0.0020[m], Q0 = 0.0003[Hm]
(50)
L∞0 = 0.50[H], R0 = 10.0[Ω] (47)
The following two controllers are implemented: (1)Robust nonlinear controller without DOB: c1p = 40, c1i = 202 , c2 = 40, c3 = 20
λ20
where, h(t) is the impulse response of the transfer function H(s) = 1 − Q(s). The results are summarized in the following theorem. Theorem 1: If the proposed robust nonlinear controller with DOB is applied to the magnetic levitation system under study, then the following results hold. 1) The overall error system is ISS such that √ |z(t)| ≤ 2λ2 |z(0)|e−ρ2 t +γ3 sup µw 2wb (τ ) + β3 sup µ3 (τ ) 0≤τ ≤t
(49)
M0 = 0.30[kg], g0 = 9.0[m/s2]
λ0 + +1 ρ0 ρ0 λ0 λ2 γ2 = 0 + +1 ρ0 ρ0 β2 = 1/(c2 /2 + D2w ) ∞ α(x)/α0 (x) ∞ h(t) 1 (48)
λ1 = λ20 +
Remark 3: The result 1) of Theorem 1 characterizes the ISS property of the overall error system, which ensures the boundedness of the error signals. According to (40), µw 2wb (τ ) is small at low-frequencies, owing to the DOB. On the other hand, µ3 (τ ) of the electrical subsystem may not be so small. Therefore, |z(t)| itself may not be so small. However, our final purpose is to make the position error z1 small. According to Lemma 1 and (41a), we can conclude that z1a (t), z2 (t) can be made small at low-frequencies. The comments will be confirmed by experimental results.
0≤τ ≤t
2) The steady offset (zero-frequency component) of z1 approaches zero.
κ21 = 1, κ22 = 3
(51)
κ31 = 0.3, κ32 = 0.3, κ33 = 0.3 (2)Robust nonlinear controller with DOB: c1p = 40, c1i = 202 , c2 = 40, c3 = 20 τ2 = 0.02 κ21 = 1, κ22 = 3, κ23 = 1
(52)
κ31 = 0.3, κ32 = 0.3, κ33 = 0.3 The results are shown Fig. 2 and Fig. 3. In each figure, from the top to the bottom are the position x1 , velocity x2 , coil current x3 , control volatge u, error signals z1 , z2 , z3 . It can be found in Fig. 2 that owing to the nonlinear damping terms, the error signals are made bounded. However, empirically, we found it is difficult to further reduce the position tracking error z1 by simply increasing the control gains. A comparison between Fig. 2 and Fig. 3 can be made here. For the electrical subsystem, since the DOB is not used, the
751
x [m/s]
1
x [mm]
TuB04.1 For the sake of physical transparency, the controller was designed directly on the model of the magnetic levitation system. However, we should emphasize here that the basic idea and theory can be extended to a general electro-mechanical system governed by the following third-order strict feedback form [2]:
15 10 5 0
3 2 1 0 50 0 −50
z1[mm]
u[V]
x3[A]
2
0.1 0 −0.1
2
z [m/s]
z3[m/s ]
0
1
2
3
4 5 Time[sec]
6
7
(53)
where , x1 : position; x2 : velocity; x3 : current or equivalent driving force, torque, acceleration; u: voltage control input; f2 (x1 , x2 ), g2 (x1 , x2 ), f3 (x1 , x2 , x3 ), g3 (x1 , x2 , x3 ) nonlinear functions which may include uncertainties. In contrast to most DOB based controllers reported in the literature, our major academic contribution is to propose a theoretically guaranteed robust nonlinear controller with DOB for a strongly nonlinear and unstable system.
1 0 −1
0.1 0 −0.1 5 0 −5
2
x˙ 1 = x2 x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )x3 x˙ 3 = f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )u
8
R EFERENCES Experimental results of the robust nonlinear controller without
15 10 5 0
0.1 0 −0.1 3 2 1 0 50 0 −50
2
1 0 −1
0.1 0 −0.1 5 0 −5
z3[m/s ]
2
z [m/s]
z1[mm]
u[V]
x3[A]
2
x [m/s]
1
x [mm]
Fig. 1. DOB.
Fig. 2.
0
1
2
3
4 5 Time[sec]
6
7
8
Experimental results of the robust nonlinear controller with DOB.
error signal z3 is not suppressed sufficiently. However, for the mechanical subsystem, owing to the DOB, in Fig. 3, the low-frequency components of z2 are significantly reduced and hence the position error z1 is largely reduced. The results reflect the comments in Remark 3 quite well. VII. C ONCLUSIONS In this paper, a robust nonlinear controller with DOB was proposed for a voltage-controlled magnetic levitation system. We have found that owing to the DOB, the position tracking error can be significantly reduced.
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